Organization: Pearson
Product Name: Connected Mathematics 3 Grade 8 Algebra
Product Version: 1.0


Source:     IMS Online Validator
Profile:    1.2.0
Identifier: realize-577e4fe1-c2b0-3df4-a9ce-17a6e159a84e
Timestamp:  Monday, December 3, 2018 01:21 PM EST
Status:     VALID!
Conformant: true

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Resource Validation Results
The document is valid.

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Schema Location Results
Schema locations are valid.

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Schema Validation Results
The document is valid.

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Schematron Validation Results
The document is valid.

Curriculum Standards:

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - 1F4E8D1A-7053-11DF-8EBF-BE719DFF4B22
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). - 201A8BE0-7053-11DF-8EBF-BE719DFF4B22
Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. - 20491316-7053-11DF-8EBF-BE719DFF4B22
Compute (using technology) and interpret the correlation coefficient of a linear fit. - 212668B0-7053-11DF-8EBF-BE719DFF4B22
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. - 2066E814-7053-11DF-8EBF-BE719DFF4B22
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - 20098A98-7053-11DF-8EBF-BE719DFF4B22
Factor a quadratic expression to reveal the zeros of the function it defines. - 1FCF76A0-7053-11DF-8EBF-BE719DFF4B22
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. - 1F6B252E-7053-11DF-8EBF-BE719DFF4B22
Fit a linear function for a scatter plot that suggests a linear association. - 2121F942-7053-11DF-8EBF-BE719DFF4B22
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - 2107150A-7053-11DF-8EBF-BE719DFF4B22
Solve an equation of the form _???(_??_) = _??_ for a simple function _??? that has an inverse and write an expression for the inverse. Example: For example, _???(_??_) =2 _??_?_ or _???(_??_) = (_??_+1)/(_??_???1) for _??_ ??? 1. - 20552DCC-7053-11DF-8EBF-BE719DFF4B22
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. - 200598A2-7053-11DF-8EBF-BE719DFF4B22
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. - 202DFCA2-7053-11DF-8EBF-BE719DFF4B22
Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. - 1F41FC3A-7053-11DF-8EBF-BE719DFF4B22
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. - 1F5BA5FE-7053-11DF-8EBF-BE719DFF4B22
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. - 1F3F09DA-7053-11DF-8EBF-BE719DFF4B22
Distinguish between correlation and causation. - 21280AA8-7053-11DF-8EBF-BE719DFF4B22
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. - 2029EA4A-7053-11DF-8EBF-BE719DFF4B22
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. - 2061D40A-7053-11DF-8EBF-BE719DFF4B22
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm??s law _??? = _??__??? to highlight resistance _???. - 1FFDCE7E-7053-11DF-8EBF-BE719DFF4B22
Use the properties of exponents to interpret expressions for exponential functions. Example: For example, identify percent rate of change in functions such as y = (1.02) to the _??? power, _??? = (0.97) to the _??? power, _??? = (1.01) to the 12_??? power, _??? = (1.2) to the _???/10 power, and classify them as representing exponential growth or decay. - 203BE466-7053-11DF-8EBF-BE719DFF4B22
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - 2023EA78-7053-11DF-8EBF-BE719DFF4B22
Interpret parts of an expression, such as terms, factors, and coefficients. - 1FC1804A-7053-11DF-8EBF-BE719DFF4B22
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. - 2034CEF6-7053-11DF-8EBF-BE719DFF4B22
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 1F665990-7053-11DF-8EBF-BE719DFF4B22
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180?; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. - 20C0B0A6-7053-11DF-8EBF-BE719DFF4B22
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 1F319AA2-7053-11DF-8EBF-BE719DFF4B22
Identify the effect on the graph of replacing _???(_??_) by _???(_??_) + _???, _??? _???(_??_), _???(_???_??_), and _???(_??_ + _???) for specific values of _??? (both positive and negative); find the value of _??? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. - 2051CA1A-7053-11DF-8EBF-BE719DFF4B22
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). - 2022443E-7053-11DF-8EBF-BE719DFF4B22
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. - 211C3D22-7053-11DF-8EBF-BE719DFF4B22
Explain why the _??_-coordinates of the points where the graphs of the equations _??? = _???(_??_) and _??? = _???(_??_) intersect are the solutions of the equation _???(_??_) = _???(_??_); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where _???(_??_) and/or _???(_??_) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. - 201C0FBA-7053-11DF-8EBF-BE719DFF4B22
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - 203F6C30-7053-11DF-8EBF-BE719DFF4B22
Use the properties of exponents to transform expressions for exponential functions. Example: For example the expression 1.15 to the _??? power can be rewritten as ((1.15 to the 1/12 power) to the 12_??? power) is approximately equal to (1.012 to the 12_??? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. - 1FD28A34-7053-11DF-8EBF-BE719DFF4B22
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. - 1FF53976-7053-11DF-8EBF-BE719DFF4B22
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - 20334950-7053-11DF-8EBF-BE719DFF4B22
Represent data with plots on the real number line (dot plots, histograms, and box plots). - 21148EA6-7053-11DF-8EBF-BE719DFF4B22
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. - 1F6E148C-7053-11DF-8EBF-BE719DFF4B22
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation _??? = _???_??_ for a line through the origin and the equation _??? = _???_??_ + _??? for a line intercepting the vertical axis at _???. - 1F34E2FC-7053-11DF-8EBF-BE719DFF4B22
Informally assess the fit of a function by plotting and analyzing residuals. - 2120C0EA-7053-11DF-8EBF-BE719DFF4B22
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. - 1F4FD706-7053-11DF-8EBF-BE719DFF4B22
Use the structure of an expression to identify ways to rewrite it. Example: For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). - 1FC900FE-7053-11DF-8EBF-BE719DFF4B22
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. - 1F48652A-7053-11DF-8EBF-BE719DFF4B22
Graph linear and quadratic functions and show intercepts, maxima, and minima. - 2031EE3E-7053-11DF-8EBF-BE719DFF4B22
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. - 213F8494-7053-11DF-8EBF-BE719DFF4B22
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function _???(_???) gives the number of person-hours it takes to assemble _??? engines in a factory, then the positive integers would be an appropriate domain for the function. - 202B0812-7053-11DF-8EBF-BE719DFF4B22
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. - 211F39C8-7053-11DF-8EBF-BE719DFF4B22
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). - 2065846A-7053-11DF-8EBF-BE719DFF4B22
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - 203752B6-7053-11DF-8EBF-BE719DFF4B22
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. - 20256704-7053-11DF-8EBF-BE719DFF4B22
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. - 1F1FA9D2-7053-11DF-8EBF-BE719DFF4B22
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. - 1FF95682-7053-11DF-8EBF-BE719DFF4B22
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 1F3AE526-7053-11DF-8EBF-BE719DFF4B22
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - 1FF7725E-7053-11DF-8EBF-BE719DFF4B22
Interpret the parameters in a linear or exponential function in terms of a context. - 206AD910-7053-11DF-8EBF-BE719DFF4B22
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? - 1F715E62-7053-11DF-8EBF-BE719DFF4B22
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. - 2062C6F8-7053-11DF-8EBF-BE719DFF4B22
Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. - 20361D92-7053-11DF-8EBF-BE719DFF4B22
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). - 211788D6-7053-11DF-8EBF-BE719DFF4B22
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. - 1F3DA5D6-7053-11DF-8EBF-BE719DFF4B22
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. - 203A6D70-7053-11DF-8EBF-BE719DFF4B22
Determine an explicit expression, a recursive process, or steps for calculation from a context. - 20476DA4-7053-11DF-8EBF-BE719DFF4B22
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. - 1F6CB2F4-7053-11DF-8EBF-BE719DFF4B22
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - 1FD11118-7053-11DF-8EBF-BE719DFF4B22
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. - 20BF272C-7053-11DF-8EBF-BE719DFF4B22
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. - 211634CC-7053-11DF-8EBF-BE719DFF4B22
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. - 1F4742BC-7053-11DF-8EBF-BE719DFF4B22
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. - 1F819E58-7053-11DF-8EBF-BE719DFF4B22
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - 2124F836-7053-11DF-8EBF-BE719DFF4B22
Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. - 1F4AA754-7053-11DF-8EBF-BE719DFF4B22


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Title: Connected Mathematics 3 Grade 8 Algebra 1 2018


         Tools

            Math Tools

            Glossary

            Student Activities

            HTMLBook: Thinking With Mathematical Models

            HTMLBook: Looking for Pythagoras

            HTMLBook: Growing, Growing, Growing

            HTMLBook: Frogs, Fleas, and Painted Cubes

            HTMLBook: Butterflies, Pinwheels, and Wallpaper

            HTMLBook: Say It With Symbols

            HTMLBook: It's In The System

            HTMLBook: Function Junction

         Thinking With Mathematical Models: Linear and Inverse Variation

            Thinking With Mathematical Models - Student Edition

            Exploring Data Patterns

               Student Edition - Investigation 1 - Thinking With Mathematical Models

               Bridge Thickness and Strength

                  Student Edition - Problem 1.1 - Thinking With Mathematical Models
                    Curriculum Standards: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 

               Bridge Length and Strength

                  Teacher Edition - Problem 1.2 - Thinking With Mathematical Models
                    Curriculum Standards: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 

                  Student Edition - Problem 1.2 - Thinking With Mathematical Models
                    Curriculum Standards: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 

               Custom Construction Parts: Finding Patterns

                  Student Edition- Problem 1.3 - Thinking With Mathematical Models
                    Curriculum Standards: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation _  = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

               ACE - Investigation 1 - Thinking With Mathematical Models

               Mathematical Reflections - Investigation 1 - Thinking With Mathematical Models

            Linear Models and Equations

               Student Edition - Investigation 2 - Thinking With Mathematical Models

               Modeling Linear Data Patterns

                  Student Edition - Problem 2.1 - Thinking With Mathematical Models
                    Curriculum Standards: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Informally assess the fit of a function by plotting and analyzing residuals. Informally assess the fit of a function by plotting and analyzing residuals. 

               Up and Down the Staircase: Exploring Slope

                  Student Edition - Problem 2.2 - Thinking With Mathematical Models
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 

               Tree Top Fun: Equations for Linear Functions

                  Student Edition - Problem 2.3 - Thinking With Mathematical Models
                    Curriculum Standards: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

               Boat Rental Business: Solving Linear Equations

                  Student Edition - Problem 2.4 - Thinking With Mathematical Models
                    Curriculum Standards: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Amusement Park or Movies: Intersecting Linear Models

                  Teacher Edition - Problem 2.5 - Thinking With Mathematical Models
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

                  Student Edition - Problem 2.5 - Thinking With Mathematical Models
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

               ACE - Investigation 2 - Thinking With Mathematical Models

               Mathematical Reflections - Investigation 2 - Thinking With Mathematical Models

            Inverse Variation

               Student Edition - Investigation 3 - Thinking With Mathematical Models

               Rectangles With Fixed Area

                  Student Edition - Problem 3.1 - Thinking With Mathematical Models
                    Curriculum Standards: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm??s law _??? = _??__??? to highlight resistance _???. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 

               Distance, Speed, and Time

                  Student Edition - Problem 3.2 - Thinking With Mathematical Models
                    Curriculum Standards: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm??s law _??? = _??__??? to highlight resistance _???. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 

               Planning a Field Trip: Finding Individual Cost

                  Student Edition - Problem 3.3 - Thinking With Mathematical Models
                    Curriculum Standards: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 

               Modeling Data Patterns

                  Student Edition - Problem 3.4 - Thinking With Mathematical Models
                    Curriculum Standards: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm??s law _??? = _??__??? to highlight resistance _???. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. 

               ACE - Investigation 3 - Thinking With Mathematical Models

               Mathematical Reflections - Investigation 3 - Thinking With Mathematical Models

            Variability and Associations in Numeric Data

               Student Edition - Investigation 4 - Thinking With Mathematical Models

               Vitruvian Man: Relating Body Measurements

                  Student Edition - Problem 4.1 - Thinking With Mathematical Models
                    Curriculum Standards: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Informally assess the fit of a function by plotting and analyzing residuals. Informally assess the fit of a function by plotting and analyzing residuals. 

               Older and Faster: Negative Correlations

                  Student Edition - Problem 4.2 - Thinking With Mathematical Models
                    Curriculum Standards: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Fit a linear function for a scatter plot that suggests a linear association. 

               Correlation Coefficients and Outliers

                  Student Edition - Problem 4.3 - Thinking With Mathematical Models
                    Curriculum Standards: Compute (using technology) and interpret the correlation coefficient of a linear fit. Compute (using technology) and interpret the correlation coefficient of a linear fit. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Distinguish between correlation and causation. Distinguish between correlation and causation. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 

               Measuring Variability: Standard Deviation

                  Student Edition - Problem 4.4 - Thinking With Mathematical Models
                    Curriculum Standards: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Represent data with plots on the real number line (dot plots, histograms, and box plots). Represent data with plots on the real number line (dot plots, histograms, and box plots). 

               ACE - Investigation 4 - Thinking With Mathematical Models

               Mathematical Reflections - Investigation 4 - Thinking With Mathematical Models

            Variability and Associations in Categorical Data

               Student Edition - Investigation 5 - Thinking With Mathematical Models

               Wood or Steel: That's the Question

                  Student Edition - Problem 5.1 - Thinking With Mathematical Models
                    Curriculum Standards: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: 
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: 
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. 

               Politics of Girls and Boys: Analyzing Data in Two-Way Tables

                  Student Edition - Problem 5.2 - Thinking With Mathematical Models
                    Curriculum Standards: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: 
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: 
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. 

               After-School Jobs and Homework: Working Backward: Setting Up a Two-Way Table

                  Student Edition - Problem 5.3 - Thinking With Mathematical Models
                    Curriculum Standards: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: 
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: 
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Distinguish between correlation and causation. Distinguish between correlation and causation. Combine standard function types using arithmetic operations. Example: 
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine standard function types using arithmetic operations. Example: 
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 

               ACE - Investigation 5 - Thinking With Mathematical Models

               Mathematical Reflections - Investigation 5 - Thinking With Mathematical Models

            Thinking With Mathematical Models - Looking Back

            Thinking With Mathematical Models - Unit Test

            Student Activities

            Math Tools

         Looking for Pythagoras: The Pythagorean Theorem

            Looking for Pythagoras - Student Edition

            Coordinate Grids

               Student Edition - Investigation 1 - Looking for Pythagoras

               Driving Around Euclid: Locating Points and Finding Distances

                  Student Edition - Problem 1.1 - Looking for Pythagoras

               Planning Parks: Shapes on a Coordinate Grid

                  Student Edition - Problem 1.2 - Looking for Pythagoras

               Finding Areas

                  Student Edition- Problem 1.3 - Looking for Pythagoras

               ACE - Investigation 1 - Looking for Pythagoras

               Mathematical Reflections - Investigation 1 - Looking for Pythagoras

            Squaring Off

               Student Edition - Investigation 2 - Looking for Pythagoras

               Looking for Squares

                  Student Edition - Problem 2.1 - Looking for Pythagoras

               Square Roots

                  Student Edition - Problem 2.2 - Looking for Pythagoras

               Using Squares to Find Lengths

                  Student Edition - Problem 2.3 - Looking for Pythagoras

               Cube Roots

                  Student Edition - Problem 2.4 - Looking for Pythagoras

               ACE - Investigation 2 - Looking for Pythagoras

               Mathematical Reflections - Investigation 2 - Looking for Pythagoras

            The Pythagorean Theorem

               Student Edition - Investigation 3 - Looking for Pythagoras

               Discovering the Pythagorean Theorem

                  Student Edition - Problem 3.1 - Looking for Pythagoras

               A Proof of the Pythagorean Theorem

                  Student Edition - Problem 3.2 - Looking for Pythagoras

               Finding Distances

                  Student Edition - Problem 3.3 - Looking for Pythagoras

               Measuring the Egyptian Way: Lengths That Form a Right Triangle

                  Student Edition - Problem 3.4 - Looking for Pythagoras

               ACE - Investigation 3 - Looking for Pythagoras

               Mathematical Reflections - Investigation 3 - Looking for Pythagoras

            Using the Pythagorean Theorem: Understanding Real Numbers

               Student Edition - Investigation 4 - Looking for Pythagoras

               Analyzing the Wheel of Theodorus: Square Roots on a Number Line

                  Student Edition - Problem 4.1 - Looking for Pythagoras

               Representing Fractions as Decimals

                  Student Edition - Problem 4.2 - Looking for Pythagoras
                    Curriculum Standards: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 

               Representing Decimals as Fractions

                  Student Edition - Problem 4.3 - Looking for Pythagoras
                    Curriculum Standards: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               Getting Real: Irrational Numbers

                  Student Edition - Problem 4.4 - Looking for Pythagoras
                    Curriculum Standards: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 

               ACE - Investigation 4

               Mathematical Reflections - Investigation 4

            Using the Pythagorean Theorem: Analyzing Triangles and Circles

               Student Edition - Investigation 5 - Looking for Pythagoras

               Stopping Sneaky Sally: Finding Unknown Side Lengths

                  Student Edition - Problem 5.1 - Looking for Pythagoras

               Analyzing Triangles

                  Student Edition - Problem 5.2 - Looking for Pythagoras

               Analyzing Circles

                  Student Edition - Problem 5.3 - Looking for Pythagoras
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 

               ACE - Investigation 5 - Looking for Pythagoras

               Mathematical Reflections - Investigation 5 - Looking for Pythagoras

            Looking for Pythagoras - Looking Back

            Looking for Pythagoras - Unit Test

            Student Activities

            Math Tools

         Growing, Growing, Growing: Exponential Functions

            Growing, Growing, Growing - Student Edition

            Exponential Growth

               Student Edition - Investigation 1 - Growing, Growing, Growing

               Making Ballots: Introducing Exponential Functions

                  Student Edition - Problem 1.1 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. 

               Requesting a Reward: Representing Exponential Functions

                  Student Edition - Problem 1.2 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 

               Making a New Offer: Growth Factors

                  Student Edition- Problem 1.3 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               ACE - Investigation 1 - Growing, Growing, Growing

               Mathematical Reflections - Investigation 1 - Growing, Growing, Growing

            Examining Growth Patterns

               Student Edition - Investigation 2 - Growing, Growing, Growing

               Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1

                  Student Edition - Problem 2.1 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 

               Growing Mold: Interpreting Equations for Exponential Functions

                  Student Edition - Problem 2.2 - Growing, Growing, Growing
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Studying Snake Populations: Interpreting Graphs of Exponential Functions

                  Student Edition - Problem 2.3 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 

               ACE - Investigation 2 - Growing, Growing, Growing

               Mathematical Reflections - Investigation 2 - Growing, Growing, Growing

            Growth Factors and Growth Rates

               Student Edition - Investigation 3 - Growing, Growing, Growing

               Reproducing Rabbits: Fractional Growth Patterns

                  Student Edition - Problem 3.1 - Growing, Growing, Growing
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Investing for the Future: Growth Rates

                  Student Edition - Problem 3.2 - Growing, Growing, Growing
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Use the properties of exponents to interpret expressions for exponential functions. Example: 
For example, identify percent rate of change in functions such as y = (1.02) to the _??? power, _??? = (0.97) to the _??? power, _??? = (1.01) to the 12_??? power, _??? = (1.2) to the _???/10 power, and classify them as representing exponential growth or decay. Use the properties of exponents to interpret expressions for exponential functions. Example: 
For example, identify percent rate of change in functions such as y = (1.02) to the ?? power, ?? = (0.97) to the ?? power, ?? = (1.01) to the 12?? power, ?? = (1.2) to the ??/10 power, and classify them as representing exponential growth or decay. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 

               Making a Difference: Connecting Growth Rate and Growth Factor

                  Student Edition - Problem 3.3 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 

               ACE - Investigation 3 - Growing, Growing, Growing

               Mathematical Reflections - Investigation 3 - Growing, Growing, Growing

            Exponential Decay

               Student Edition - Investigation 4 - Growing, Growing, Growing

               Making Smaller Ballots: Introducing Exponential Decay

                  Student Edition - Problem 4.1 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Fighting Fleas: Representing Exponential Decay

                  Student Edition - Problem 4.2 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Cooling Water: Modeling Exponential Decay

                  Student Edition - Problem 4.3 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Combine standard function types using arithmetic operations. Example: 
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine standard function types using arithmetic operations. Example: 
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 

               ACE - Investigation 4  - Growing, Growing, Growing
               ACE - Investigation 4 - Growing, Growing, Growing

               Mathematical Reflections - Investigation 4 - Growing, Growing, Growing

            Patterns With Exponents

               Student Edition - Investigation 5 - Growing, Growing, Growing

               Stopping Sneaky Sally: Finding Unknown Side Lengths

                  Student Edition - Problem 5.1 - Growing, Growing, Growing
                    Curriculum Standards: Use the properties of exponents to interpret expressions for exponential functions. Example: 
For example, identify percent rate of change in functions such as y = (1.02) to the _??? power, _??? = (0.97) to the _??? power, _??? = (1.01) to the 12_??? power, _??? = (1.2) to the _???/10 power, and classify them as representing exponential growth or decay. Use the properties of exponents to interpret expressions for exponential functions. Example: 
For example, identify percent rate of change in functions such as y = (1.02) to the ?? power, ?? = (0.97) to the ?? power, ?? = (1.01) to the 12?? power, ?? = (1.2) to the ??/10 power, and classify them as representing exponential growth or decay. Use the properties of exponents to transform expressions for exponential functions. Example: 
For example the expression 1.15 to the _??? power can be rewritten as ((1.15 to the 1/12 power) to the 12_??? power) is approximately equal to (1.012 to the 12_??? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Use the properties of exponents to transform expressions for exponential functions. Example: 
For example the expression 1.15 to the ?? power can be rewritten as ((1.15 to the 1/12 power) to the 12?? power) is approximately equal to (1.012 to the 12?? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Analyzing Triangles

                  Student Edition - Problem 5.2 - Growing, Growing, Growing
                    Curriculum Standards: Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Use the properties of exponents to transform expressions for exponential functions. Example: 
For example the expression 1.15 to the _??? power can be rewritten as ((1.15 to the 1/12 power) to the 12_??? power) is approximately equal to (1.012 to the 12_??? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Use the properties of exponents to transform expressions for exponential functions. Example: 
For example the expression 1.15 to the ?? power can be rewritten as ((1.15 to the 1/12 power) to the 12?? power) is approximately equal to (1.012 to the 12?? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Extending the Rules of Exponents

                  Student Edition - Problem 5.3 - Growing, Growing, Growing
                    Curriculum Standards: Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Use the properties of exponents to transform expressions for exponential functions. Example: 
For example the expression 1.15 to the _??? power can be rewritten as ((1.15 to the 1/12 power) to the 12_??? power) is approximately equal to (1.012 to the 12_??? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Use the properties of exponents to transform expressions for exponential functions. Example: 
For example the expression 1.15 to the ?? power can be rewritten as ((1.15 to the 1/12 power) to the 12?? power) is approximately equal to (1.012 to the 12?? power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Operation With Scientific Notation

                  Student Edition - Problem 5.4 - Growing, Growing, Growing
                    Curriculum Standards: Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. 

               Revisiting Exponential Functions

                  Student Edition - Problem 5.5 - Growing, Growing, Growing
                    Curriculum Standards: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. 

               ACE - Investigation 5 - Growing, Growing, Growing

               Mathematical Reflections - Investigation 5 - Growing, Growing, Growing

            Growing, Growing, Growing - Looking Back

            Growing, Growing, Growing - Unit Test

            Student Activities

            Math Tools

         Frogs, Fleas, and Painted Cubes: Quadratic Functions

            Frogs, Fleas, and Painted Cubes - Student Edition

            Introduction to Quadratic Functions

               Student Edition - Investigation 1 - Frogs, Fleas, and Painted Cubes

               Staking a Claim: Maximizing Area

                  Student Edition - Problem 1.1 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: 
For example, if the function _???(_???) gives the number of person-hours it takes to assemble _??? engines in a factory, then the positive integers would be an appropriate domain for the function. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: 
For example, if the function ??(??) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. 

               Reading Graphs and Tables

                  Student Edition - Problem 1.2 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Writing an Equation

                  Student Edition- Problem 1.3 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               ACE - Investigation 1 - Frogs, Fleas, and Painted Cubes

               Mathematical Reflections - Investigation 1 - Frogs, Fleas, and Painted Cubes

            Quadratic Expressions

               Student Edition - Investigation 2 - Frogs, Fleas, and Painted Cubes

               Trading Land: Representing Areas of Rectangles

                  Student Edition - Problem 2.1 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Changing Dimensions: The Distributive Property

                  Student Edition - Problem 2.2 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Factoring Quadratic Expressions

                  Student Edition - Problem 2.3 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Quadratic Functions and Their Graphs

                  Student Edition - Problem 2.4 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 

               ACE - Investigation 2 - Frogs, Fleas, and Painted Cubes

               Mathematical Reflections - Investigation 2 - Frogs, Fleas, and Painted Cubes

            Quadratic Patterns of Change

               Student Edition - Investigation 3 - Frogs, Fleas, and Painted Cubes

               Exploring Triangular Numbers

                  Student Edition - Problem 3.1 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Counting Handshakes: Another Quadratic Function

                  Student Edition - Problem 3.2 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. 

               Examining Patterns of Change

                  Student Edition - Problem 3.3 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Quadratic Functions and Patterns of Change

                  Student Edition - Problem 3.4 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

               ACE - Investigation 3 - Frogs, Fleas, and Painted Cubes

               Mathematical Reflections - Investigation 3 - Frogs, Fleas, and Painted Cubes

            Frogs Meet Fleas on a Cube: More Applications of Quadratic Functions

               Student Edition - Investigation 4 - Frogs, Fleas, and Painted Cubes

               Tracking a Ball: Interpreting a Table and an Equation

                  Student Edition - Problem 4.1 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Measuring Jumps: Comparing Quadratic Functions

                  Student Edition - Problem 4.2 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Painted Cubes: Looking at Several Functions

                  Student Edition - Problem 4.3 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

               Putting It All Together: Comparing Functions

                  Student Edition - Problem 4.4 - Frogs, Fleas, and Painted Cubes
                    Curriculum Standards: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

               ACE - Investigation 4 - Frogs, Fleas, and Painted Cubes

               Mathematical Reflections - Investigation 4 - Frogs, Fleas, and Painted Cubes

            Frogs, Fleas, and Painted Cubes - Looking Back

            Unit Test: Frogs, Fleas and Painted Cubes

            Student Activities

            Math Tools

         Butterflies, Pinwheels, and Wallpaper: Symmetry and Transformation

            Butterflies, Pinwheels, and Wallpaper - Student Edition

            Symmetry and  Transformations

               Student Edition - Investigation 1 - Butterflies, Pinwheels, and Wallpaper

               Butterfly Symmetry: Line Reflections

                  Student Edition - Problem 1.1 - Butterflies, Pinwheels, and Wallpaper

               In a Spin: Rotations

                  Student Edition - Problem 1.2 - Butterflies, Pinwheels, and Wallpaper

               Sliding Around: Translations

                  Student Edition- Problem 1.3 - Butterflies, Pinwheels, and Wallpaper

               Properties of Transformations

                  Student Edition - Problem 1.4 - Butterflies, Pinwheels, and Wallpaper

               ACE - Investigation 1 - Butterflies, Pinwheels, and Wallpaper

               Mathematical Reflections - Investigation 1 - Butterflies, Pinwheels, and Wallpaper

            Transformations and Congruence

               Student Edition - Investigation 2 - Butterflies, Pinwheels, and Wallpaper

               Connecting Congruent Polygons

                  Student Edition - Problem 2.1 - Butterflies, Pinwheels, and Wallpaper

               Supporting the World: Congruent Triangles I

                  Student Edition - Problem 2.2 - Butterflies, Pinwheels, and Wallpaper

               Minimum Measurement: Congruent Triangles II

                  Student Edition - Problem 2.3 - Butterflies, Pinwheels, and Wallpaper

               ACE - Investigation 2 - Butterflies, Pinwheels, and Wallpaper

               Mathematical Reflections - Investigation 2 - Butterflies, Pinwheels, and Wallpaper

            Transforming Coordinates

               Student Edition - Investigation 3 - Butterflies, Pinwheels, and Wallpaper

               Flipping on a Grid: Coordinate Rules for Reflections

                  Student Edition - Problem 3.1 - Butterflies, Pinwheels, and Wallpaper

               Sliding on a Grid: Coordinate Rules for Translations

                  Student Edition - Problem 3.2 - Butterflies, Pinwheels, and Wallpaper

               Spinning on a Grid: Coordinate Rules for Rotations

                  Student Edition - Problem 3.3 - Butterflies, Pinwheels, and Wallpaper

               A Special Property of Translations and Half-Turns

                  Student Edition - Problem 3.4 - Butterflies, Pinwheels, and Wallpaper

               Parallel Lines, Transversals, and Angle Sums

                  Student Edition - Problem 3.5 - Butterflies, Pinwheels, and Wallpaper
                    Curriculum Standards: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180?; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 

               ACE - Investigation 3 - Butterflies, Pinwheels, and Wallpaper

               Mathematical Reflections - Investigation 3 - Butterflies, Pinwheels, and Wallpaper

            Dilations and Similar Figures

               Student Edition - Investigation 4 - Butterflies, Pinwheels, and Wallpaper

               Focus on Dilations

                  Student Edition - Problem 4.1 - Butterflies, Pinwheels, and Wallpaper
                    Curriculum Standards: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation _??? = _???_??_ for a line through the origin and the equation _??? = _???_??_ + _??? for a line intercepting the vertical axis at _???. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation ?? = ???? for a line through the origin and the equation ?? = ???? + ?? for a line intercepting the vertical axis at ??. 

               Return of Super Sleuth: Similarity Transformations

                  Student Edition - Problem 4.2 - Butterflies, Pinwheels, and Wallpaper

               Checking Similarity Without Transformations

                  Student Edition - Problem 4.3 - Butterflies, Pinwheels, and Wallpaper
                    Curriculum Standards: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation _??? = _???_??_ for a line through the origin and the equation _??? = _???_??_ + _??? for a line intercepting the vertical axis at _???. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation ?? = ???? for a line through the origin and the equation ?? = ???? + ?? for a line intercepting the vertical axis at ??. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180?; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 

               Using Similar Triangles

                  Student Edition - Problem 4.4 - Butterflies, Pinwheels, and Wallpaper

               ACE - Investigation 4 - Butterflies, Pinwheels, and Wallpaper

               Mathematical Reflections - Investigation 4 - Butterflies, Pinwheels, and Wallpaper

            Butterflies, Pinwheels, and Wallpaper - Looking Back

            Butterflies, Pinwheels, and Wallpaper - Unit Test

            Student Activities

            Math Tools

         Say It With Symbols: Making Sense of Symbols

            Say It With Symbols - Student Edition

            Making Sense of Symbols: Equivalent Expressions

               Student Edition - Investigation 1 - Say It With Symbols

               Tiling Pools: Writing Equivalent Expressions

                  Student Edition - Problem 1.1 - Say It With Symbols
                    Curriculum Standards: Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               Thinking in Different Ways: Determining Equivalence

                  Student Edition - Problem 1.2 - Say It With Symbols
                    Curriculum Standards: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               The Community Pool Problem: Interpreting Expressions

                  Student Edition- Problem 1.3 - Say It With Symbols
                    Curriculum Standards: Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Diving In: Revisiting the Distributive Property

                  Student Edition - Problem 1.4 - Say It With Symbols
                    Curriculum Standards: Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               ACE - Investigation 1 - Say It With Symbols

               Mathematical Reflections - Investigation 1 - Say It With Symbols

            Combining Expressions

               Student Edition - Investigation 2 - Say It With Symbols

               Walking Together: Adding Expressions

                  Student Edition - Problem 2.1 - Say It With Symbols
                    Curriculum Standards: Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Predicting Profit: Substituting Expressions

                  Student Edition - Problem 2.2 - Say It With Symbols
                    Curriculum Standards: Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Making Candles:Volumes of Cylinders, Cones, and Spheres

                  Student Edition - Problem 2.3 - Say It With Symbols
                    Curriculum Standards: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Selling Ice Cream: Solving Volume Problems

                  Student Edition - Problem 2.4 - Say It With Symbols
                    Curriculum Standards: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               ACE - Investigation 2 - Say It With Symbols

               Mathematical Reflections - Investigation 2 - Say It With Symbols

            Solving Equations

               Student Edition - Investigation 3 - Say It With Symbols

               Selling Greeting Cards: Solving Linear Equations

                  Student Edition - Problem 3.1 - Say It With Symbols
                    Curriculum Standards: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Comparing Costs: Solving More Linear Equations

                  Student Edition - Problem 3.2 - Say It With Symbols
                    Curriculum Standards: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2 + = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Explain why the _??_-coordinates of the points where the graphs of the equations _??? = _???(_??_) and _??? = _???(_??_) intersect are the solutions of the equation _???(_??_) = _???(_??_); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where _???(_??_) and/or _???(_??_) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Explain why the ??-coordinates of the points where the graphs of the equations ?? = ??(??) and ?? = ??(??) intersect are the solutions of the equation ??(??) = ??(??); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ??(??) and/or ??(??) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Factoring Quadratic Equations

                  Student Edition - Problem 3.3 - Say It With Symbols
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Explain why the _??_-coordinates of the points where the graphs of the equations _??? = _???(_??_) and _??? = _???(_??_) intersect are the solutions of the equation _???(_??_) = _???(_??_); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where _???(_??_) and/or _???(_??_) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Explain why the ??-coordinates of the points where the graphs of the equations ?? = ??(??) and ?? = ??(??) intersect are the solutions of the equation ??(??) = ??(??); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ??(??) and/or ??(??) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Solving Quadratic Equations

                  Student Edition - Problem 3.4 - Say It With Symbols
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               ACE - Investigation 3 - Say It With Symbols

               Mathematical Reflections - Investigation 3 - Say It With Symbols

            Looking Back at Functions

               Student Edition - Investigation 4 - Say It With Symbols

               Pumping Water: Looking at Patterns of Change

                  Student Edition - Problem 4.1 - Say It With Symbols
                    Curriculum Standards: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Area and Profit - What's the Connection? Using Equations

                  Student Edition - Problem 4.2 - Say It With Symbols
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Generating Patterns: Linear, Exponential, Quadratic

                  Student Edition - Problem 4.3 - Say It With Symbols
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for -1 greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               What's the Function? Modeling With Functions

                  Student Edition - Problem 4.4 - Say It With Symbols
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Interpret the parameters in a linear or exponential function in terms of a context. Interpret the parameters in a linear or exponential function in terms of a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 

               ACE - Investigation 4 - Say It With Symbols

               Mathematical Reflections - Investigation 4 - Say It With Symbols

            Reasoning With Symbols

               Student Edition - Investigation 5 - Say It With Symbols

               Using Algebra to Solve a Puzzle

                  Student Edition - Problem 5.1 - Say It With Symbols
                    Curriculum Standards: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 

               Odd and Even Revisited

                  Student Edition - Problem 5.2 - Say It With Symbols
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               Squaring Odd Numbers

                  Student Edition - Problem 5.3 - Say It With Symbols
                    Curriculum Standards: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see _??_??? ??? _?????? as (_??_?_)?_ ??? (_????_)?_, thus recognizing it as a difference of squares that can be factored as (_??_?_ ??? _????_)(_??_?_ + _????_). Use the structure of an expression to identify ways to rewrite it. Example: 
For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 

               ACE - Investigation 5 - Say It With Symbols

               Mathematical Reflections - Investigation 5 - Say It With Symbols

            Say It With Symbols - Looking Back

            Say It With Symbols - Unit Test

            Student Activities

            Math Tools

         It's In The System: Systems of Linear Equations and Inequalities

            It's In The System - Student Edition

            Linear Equations With Two Variables

               Student Edition - Investigation 1 - It's In The System

               Shirts and Caps: Solving Equations With Two Variables

                  Student Edition - Problem 1.1 - It's In The System
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               Connecting Ax + By = C and y = mx + b

                  Student Edition - Problem 1.2 - It's In The System
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpret the equation _??? = _???_??_ + _??? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function _??? = _????_ giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Interpret the equation ?? = ???? + ?? as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function ?? = ??² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm??s law _??? = _??__??? to highlight resistance _???. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               Booster Club Members: Intersecting Lines

                  Student Edition- Problem 1.3 - It's In The System
                    Curriculum Standards: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               ACE - Investigation 1 - It's In The System

               Mathematical Reflections - Investigation 1 - It's In The System

            Solving Linear Systems Symbolically

               Student Edition - Investigation 2 - It's In The System

               Shirts and Caps Again: Solving Systems with y = mx + b

                  Student Edition - Problem 2.1 - It's In The System
                    Curriculum Standards: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm??s law _??? = _??__??? to highlight resistance _???. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: 
For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               Taco Truck Lunch: Solving Systems by Combining Equations I

                  Student Edition - Problem 2.2 - It's In The System
                    Curriculum Standards: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               Solving Systems by Combining Equations II

                  Student Edition - Problem 2.3 - It's In The System
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               ACE - Investigation 2 - It's In The System

               Mathematical Reflections - Investigation 2 - It's In The System

            Systems of Functions and Inequalities

               Student Edition - Investigation 3 - It's In The System

               Comparing Security Services: Linear Inequalities

                  Student Edition - Problem 3.1 - It's In The System
                    Curriculum Standards: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               Solving Linear Inequalities Symbolically

                  Student Edition - Problem 3.2 - It's In The System
                    Curriculum Standards: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

               Operating at a Profit: Systems of Lines and Curves

                  Student Edition - Problem 3.3 - It's In The System
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               ACE - Investigation 3 - It's In The System

               Mathematical Reflections - Investigation 3 - It's In The System

            Systems of Linear Inequalities

               Student Edition - Investigation 4 - It's In The System

               Limiting Driving Miles: Inequalities With Two Variables

                  Student Edition - Problem 4.1 - It's In The System
                    Curriculum Standards: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               What Makes a Car Green: Solving Inequalities by Graphing I

                  Student Edition - Problem 4.2 - It's In The System
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               Feasible Points: Solving Inequalities by Graphing II

                  Student Edition - Problem 4.3 - It's In The System
                    Curriculum Standards: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               Miles of Emissions: Systems of Linear Inequalities

                  Student Edition - Problem 4.4 - It's In The System
                    Curriculum Standards: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3_??_ + 2_??? = 5 and 3_??_ + 2_??? = 6 have no solution because 3_??_ + 2_??? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3?? + 2?? = 5 and 3?? + 2?? = 6 have no solution because 3?? + 2?? cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: 
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 

               ACE - Investigation 4 - It's In The System

               Mathematical Reflections - Investigation 4 - It's In The System

            It's In The System - Looking Back

            It's In the System - Unit Test

            Student Activities

            Math Tools

         Function Junction: The Families of Functions

            Function Junction - Student Edition

            The Families of Functions

               Student Edition - Investigation 1 - Function Junction

               Filling Functions

                  Student Edition - Problem 1.1 - Function Junction
                    Curriculum Standards: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). 

               Domain, Range, and Function Notation

                  Student Edition - Problem 1.2 - Function Junction
                    Curriculum Standards: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If _??? is a function and _??_ is an element of its domain, then _???(_??_) denotes the output of _??? corresponding to the input _??_. The graph of _??? is the graph of the equation _??? = _???(_??_). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: 
For example, if the function _???(_???) gives the number of person-hours it takes to assemble _??? engines in a factory, then the positive integers would be an appropriate domain for the function. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: 
For example, if the function ??(??) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. 

               Taxi Fares, Time Payments, and Step Functions

                  Student Edition- Problem 1.3 - Function Junction
                    Curriculum Standards: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Piecewise-Defined Functions

                  Student Edition - Problem 1.4 - Function Junction
                    Curriculum Standards: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: 
For example, if the function _???(_???) gives the number of person-hours it takes to assemble _??? engines in a factory, then the positive integers would be an appropriate domain for the function. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: 
For example, if the function ??((?) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. 

               Inverse Functions

                  Student Edition - Problem 1.5 - Function Junction
                    Curriculum Standards: Solve an equation of the form _???(_??_) = _??_ for a simple function _??? that has an inverse and write an expression for the inverse. Example: 
For example, _???(_??_) =2 _??_?_ or _???(_??_) = (_??_+1)/(_??_???1) for _??_ ??? 1. Solve an equation of the form ??(??) = ?? for a simple function ?? that has an inverse and write an expression for the inverse. Example: 
For example, ??(??) =2 ??³ or ??(??) = (??+1)/(??–1) for ?? ? 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 

               ACE - Investigation 1 - Function Junction

               Mathematical Reflections - Investigation 1 - Function Junction

            Arithmetic and Geometric Sequences

               Student Edition - Investigation 2 - Function Junction

               Arithmetic Sequences

                  Student Edition - Problem 2.1 - Function Junction
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 

               Geometric Sequences

                  Student Edition - Problem 2.2 - Function Junction
                    Curriculum Standards: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by _???(0) = _???(1) = 1, _???(_???+1) = _???(_???) + _???(_???-1) for _??? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: 
For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Determine an explicit expression, a recursive process, or steps for calculation from a context. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (_??_, _???) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (??, ??) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 

               ACE - Investigation 2 - Function Junction

               Mathematical Reflections - Investigation 2 - Function Junction

            Transforming Graphs, Equations, and Functions

               Student Edition - Investigation 3 - Function Junction

               Sliding Up and Down: Vertical Translations of Functions

                  Student Edition - Problem 3.1 - Function Junction
                    Curriculum Standards: Identify the effect on the graph of replacing _???(_??_) by _???(_??_) + _???, _??? _???(_??_), _???(_???_??_), and _???(_??_ + _???) for specific values of _??? (both positive and negative); find the value of _??? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ??  ?(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 

               Stretching and Flipping Up and Down: Multiplicative Transformations of Functions

                  Student Edition - Problem 3.2 - Function Junction
                    Curriculum Standards: Identify the effect on the graph of replacing _???(_??_) by _???(_??_) + _???, _??? _???(_??_), _???(_???_??_), and _???(_??_ + _???) for specific values of _??? (both positive and negative); find the value of _??? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? +  +) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 

               Sliding Left and Right: Horizontal Translations of Functions

                  Student Edition - Problem 3.3 - Function Junction
                    Curriculum Standards: Identify the effect on the graph of replacing _???(_??_) by _???(_??_) + _???, _??? _???(_??_), _???(_???_??_), and _???(_??_ + _???) for specific values of _??? (both positive and negative); find the value of _??? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 

               How can you use algebraic expression for any quadratic function with rule in the form f(x) = a(x ?? b)?? ?? c to predict the shape and location of the graph?

                  Student Edition - Problem 3.4 - Function Junction
                    Curriculum Standards: Identify the effect on the graph of replacing _???(_??_) by _???(_??_) + _???, _??? _???(_??_), _???(_???_??_), and _???(_??_ + _???) for specific values of _??? (both positive and negative); find the value of _??? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 

               ACE - Investigation 3 - Function Junction

               Mathematical Reflections - Investigation 3 - Function Junction

            Solving Quadratic Equations Algebraically

               Student Edition - Investigation 4 - Function Junction

               Applying Square Roots

                  Student Edition - Problem 4.1 - Function Junction
                    Curriculum Standards: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 

               Completing the Square

                  Student Edition - Problem 4.2 - Function Junction
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 

               The Quadratic Formula

                  Student Edition - Problem 4.3 - Function Junction
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 

               Complex Numbers

                  Student Edition - Problem 4.4 - Function Junction

               ACE - Investigation 4 - Function Junction

               Mathematical Reflections - Investigation 4 - Function Junction

            Polynomial Expressions, Functions, and Equations

               Student Edition - Investigation 5 - Function Junction

               Properties Expressions, Functions, and Equations

                  Student Edition - Problem 5.1 - Function Junction
                    Curriculum Standards: Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: 
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 

               Combining Profit Functions: Operating With Polynomials I

                  Student Edition - Problem 5.2 - Function Junction
                    Curriculum Standards: Combine standard function types using arithmetic operations. Example: 
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine standard function types using arithmetic operations. Example: 
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 

               Product Time: Operating With Polynomials II

                  Student Edition - Problem 5.3 - Function Junction
                    Curriculum Standards: Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Factor a quadratic expression to reveal the zeros of the function it defines. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 

               The Factor Game Revisited

                  Student Edition - Problem 5.4 - Function Junction

               ACE - Investigation 5 - Function Junction

               Mathematical Reflections - Investigation 5 - Function Junction

            Function Junction - Looking Back

            Function Junction - Unit Test

            Student Activities

            Math Tools

         Pearson-Created Practice and Assessments

            Practice Powered by MathXL - Thinking With Mathematical Models

               Practice Powered by MathXL - Investigation 1 - Thinking With Mathematical Models

               Practice Powered by MathXL - Investigation 2 - Thinking With Mathematical Models

               Practice Powered by MathXL - Investigation 3 - Thinking With Mathematical Models

               Practice Powered by MathXL - Investigation 4 - Thinking With Mathematical Models

               Practice Powered by MathXL - Investigation 5 - Thinking With Mathematical Models

            Practice Powered by MathXL - Looking for Pythagoras

               Practice Powered by MathXL - Investigation 1 - Looking for Pythagoras

               Practice Powered by MathXL - Investigation 2 - Looking for Pythagoras

               Practice Powered by MathXL - Investigation 3 - Looking for Pythagoras

               Practice Powered by MathXL - Investigation 4 - Looking for Pythagoras

               Practice Powered by MathXL - Investigation 5 - Looking for Pythagoras

            Benchmark Assessment 1

            Practice Powered by MathXL - Growing, Growing, Growing

               Practice Powered by MathXL - Investigation 1 - Growing, Growing, Growing

               Practice Powered by MathXL - Investigation 2 - Growing, Growing, Growing

               Practice Powered by MathXL - Investigation 3 - Growing, Growing, Growing

               Practice Powered by MathXL - Investigation 4 - Growing, Growing, Growing

               Practice Powered by MathXL - Investigation 5 - Growing, Growing, Growing

            Practice Powered by MathXL - Frogs, Fleas, and Painted Cubes

               Practice Powered by MathXL - Investigation 1 - Frogs, Fleas, and Painted Cubes

               Practice Powered by MathXL - Investigation 2 - Frogs, Fleas, and Painted Cubes

               Practice Powered by MathXL - Investigation 3 - Frogs, Fleas, and Painted Cubes

               Practice Powered by MathXL - Investigation 4 - Frogs, Fleas, and Painted Cubes

            Benchmark Assessment 2

            Practice Powered by MathXL - Butterflies, Pinwheels, and Wallpaper

               Practice Powered by MathXL - Investigation 1 - Butterflies, Pinwheels, and Wallpaper

               Practice Powered by MathXL - Investigation 2 - Butterflies, Pinwheels, and Wallpaper

               Practice Powered by MathXL - Investigation 3 - Butterflies, Pinwheels, and Wallpaper

               Practice Powered by MathXL - Investigation 4 - Butterflies, Pinwheels, and Wallpaper

            Practice Powered by MathXL - Say It With Symbols

               Practice Powered by MathXL - Investigation 1 - Say It With Symbols

               Practice Powered by MathXL - Investigation 2 - Say It With Symbols

               Practice Powered by MathXL - Investigation 3 - Say It With Symbols

               Practice Powered by MathXL - Investigation 4 - Say It With Symbols

               Practice Powered by MathXL - Investigation 5 - Say It With Symbols

            Benchmark Assessment 3

            Practice Powered by MathXL - It's In The System

               Practice Powered by MathXL - Investigation 1 - It's In The System

               Practice Powered by MathXL - Investigation 2 - It's In The System

               Practice Powered by MathXL - Investigation 3 - It's In The System

               Practice Powered by MathXL - Investigation 4 - It's In The System

            Practice Powered by MathXL - Function Junction

               Practice Powered by MathXL - Investigation 1 - Function Junction

               Practice Powered by MathXL - Investigation 2 - Function Junction

               Practice Powered by MathXL - Investigation 3 - Function Junction

               Practice Powered by MathXL - Investigation 4 - Function Junction

               Practice Powered by MathXL - Investigation 5 - Function Junction

            Benchmark Assessment 4

         Teacher Resources Container

            Teacher Resources: Grade 8 Algebra 1
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            MATHDashboard
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            Next Generation Assessments
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            Thinking With Mathematical Models - Teacher Edition
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            Teacher Edition - Investigation 1 - Looking for Pythagoras
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              Intended Role: Instructor

            Launch Video - Problem 1.1 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - Looking for Pythagoras
              Intended Role: Instructor

            Launch Video - Problem 2.2 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 2.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.3 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 2.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.4 - Looking for Pythagoras
              Intended Role: Instructor

            Launch Video - Problem 2.4 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Connection: Launch Problem 2.4 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - Looking for Pythagoras
              Intended Role: Instructor

            Classroom Connection: Summarize Problem 3.2 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - Looking for Pythagoras
              Intended Role: Instructor

            Classroom Connection: Launch Problem 3.3 - Looking for Pythagoras
              Intended Role: Instructor

            Classroom Connection: Explore Problem 3.3 - Looking for Pythagoras
              Intended Role: Instructor

            Classroom Connection: Summarize Problem 3.3 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 3.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.4 - Looking for Pythagoras
              Intended Role: Instructor

            Launch Video - Problem 3.4 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 4.1
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - Looking for Pythagoras
              Intended Role: Instructor

            Launch Video - Problem 4.1 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 4.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.4 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 5 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 5.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.1 - Looking for Pythagoras
              Intended Role: Instructor

            Launch Video - Problem 5.1 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 5.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.2 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Connection: Summarize Problem 5.2 - Looking for Pythagoras
              Intended Role: Instructor

            Problem 5.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.3 - Looking for Pythagoras
              Intended Role: Instructor

            Launch Video - Problem 5.3 - Looking for Pythagoras
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Unit 3 - Teacher Resources
              Intended Role: Instructor

            Growing, Growing, Growing - Teacher Edition
              Intended Role: Instructor

            Teacher Connection: Supporting ELL and Struggling Students
              Intended Role: Instructor

            Teacher Edition - Investigation 1 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 1.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 1.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Connection: Explore Problem 1.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 1.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 2.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 2.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 3.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Connection: Summarize Problem 3.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 3.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 4.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Connection: Launch Problem 4.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 4.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 5 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 5.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.1 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 5.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.2 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 5.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 5.3 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 5.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.4 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 5.4 - Growing, Growing, Growing
              Intended Role: Instructor

            Problem 5.5 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.5 - Growing, Growing, Growing
              Intended Role: Instructor

            Launch Video - Problem 5.5 - Growing, Growing, Growing
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Growing, Growing, Growing - Unit Project
              Intended Role: Instructor

            Unit 4 - Teacher Resources
              Intended Role: Instructor

            Frogs, Fleas, and Painted Cubes - Teacher Edition
              Intended Role: Instructor

            Teacher Connection: Supporting ELL and Struggling Students
              Intended Role: Instructor

            Teacher Edition - Investigation 1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 1.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Launch Video - Problem 1.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Connection: Explore Problem 2.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 2.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Launch Video - Problem 2.3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 2.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.4 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Launch Video - Problem 3.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Launch Video - Problem 3.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Connection: Explore Problem 3.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 3.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.4 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 4.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Connection: Launch Problem 4.1 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Launch Video - Problem 4.2 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Launch Video - Problem 4.3 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Problem 4.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.4 - Frogs, Fleas, and Painted Cubes
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Unit 5 - Teacher Resources
              Intended Role: Instructor

            Butterflies, Pinwheels, and Wallpaper - Teacher Edition
              Intended Role: Instructor

            Teacher Connection: Supporting ELL and Struggling Students
              Intended Role: Instructor

            Teacher Edition - Investigation 1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 1.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 1.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 1.3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 1.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Connection: Summarize Problem 1.4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 2.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 2.2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 2.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 2.3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 3.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 3.4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 3.5 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.5 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Connection: Launch Problem 3.5 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 4.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 4.1 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Connection: Explore Problem 4.2 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Problem 4.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Launch Video - Problem 4.4 - Butterflies, Pinwheels, and Wallpaper
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Butterflies, Pinwheels, and Wallpaper - Unit Project
              Intended Role: Instructor

            Unit 6 - Teacher Resources
              Intended Role: Instructor

            Say It With Symbols - Teacher Edition
              Intended Role: Instructor

            Teacher Connection: Supporting ELL and Struggling Students
              Intended Role: Instructor

            Teacher Edition - Investigation 1 - Say It With Symbols
              Intended Role: Instructor

            Problem 1.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.1 - Say It With Symbols
              Intended Role: Instructor

            Launch Video - Problem 1.1 - Say It With Symbols
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - Say It With Symbols
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - Say It With Symbols
              Intended Role: Instructor

            Launch Video - Problem 1.3 - Say It With Symbols
              Intended Role: Instructor

            Teacher Connection: Explore Problem 1.3 - Say It With Symbols
              Intended Role: Instructor

            Problem 1.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.4 - Say It With Symbols
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - Say It With Symbols
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - Say It With Symbols
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - Say It With Symbols
              Intended Role: Instructor

            Problem 2.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.3 - Say It With Symbols
              Intended Role: Instructor

            Launch Video - Problem 2.3 - Say It With Symbols
              Intended Role: Instructor

            Problem 2.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.4 - Say It With Symbols
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - Say It With Symbols
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - Say It With Symbols
              Intended Role: Instructor

            Launch Video - Problem 3.1 - Say It With Symbols
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - Say It With Symbols
              Intended Role: Instructor

            Launch Video - Problem 3.2 - Say It With Symbols
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - Say It With Symbols
              Intended Role: Instructor

            Teacher Connection: Launch Problem 3.3 - Say It With Symbols
              Intended Role: Instructor

            Problem 3.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.4 - Say It With Symbols
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - Say It With Symbols
              Intended Role: Instructor

            Problem 4.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - Say It With Symbols
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - Say It With Symbols
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - Say It With Symbols
              Intended Role: Instructor

            Teacher Connection: Explore Problem 4.3 - Say It With Symbols
              Intended Role: Instructor

            Problem 4.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.4 - Say It With Symbols
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 5 - Say It With Symbols
              Intended Role: Instructor

            Problem 5.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.1 - Say It With Symbols
              Intended Role: Instructor

            Launch Video - Problem 5.1 - Say It With Symbols
              Intended Role: Instructor

            Problem 5.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.2 - Say It With Symbols
              Intended Role: Instructor

            Problem 5.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.3 - Say It With Symbols
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Say It With Symbols - Unit Project
              Intended Role: Instructor

            Unit 7 - Teacher Resources
              Intended Role: Instructor

            It's In The System - Teacher Edition
              Intended Role: Instructor

            Teacher Connection: Supporting ELL and Struggling Students
              Intended Role: Instructor

            Teacher Edition - Investigation 1 - It's In The System
              Intended Role: Instructor

            Problem 1.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.1 - It's In The System
              Intended Role: Instructor

            Launch Video - Problem 1.1 - It's In The System
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - It's In The System
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - It's In The System
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - It's In The System
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - It's In The System
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - It's In The System
              Intended Role: Instructor

            Launch Video - Problem 2.2 - It's In The System
              Intended Role: Instructor

            Teacher Connection: Launch Problem 2.2 - It's In The System
              Intended Role: Instructor

            Problem 2.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.3 - It's In The System
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - It's In The System
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - It's In The System
              Intended Role: Instructor

            Launch Video - Problem 3.1 - It's In The System
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - It's In The System
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - It's In The System
              Intended Role: Instructor

            Teacher Connection: Summarize Problem 3.3 - It's In The System
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - It's In The System
              Intended Role: Instructor

            Problem 4.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - It's In The System
              Intended Role: Instructor

            Launch Video - Problem 4.1 - It's In The System
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - It's In The System
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - It's In The System
              Intended Role: Instructor

            Problem 4.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.4 - It's In The System
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Unit 8 - Teacher Resources
              Intended Role: Instructor

            Function Junction - Teacher Edition
              Intended Role: Instructor

            Teacher Connection: Supporting ELL and Struggling Students
              Intended Role: Instructor

            Teacher Edition - Investigation 1 - Function Junction
              Intended Role: Instructor

            Problem 1.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.1 - Function Junction
              Intended Role: Instructor

            Problem 1.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.2 - Function Junction
              Intended Role: Instructor

            Problem 1.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.3 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 1.3 - Function Junction
              Intended Role: Instructor

            Problem 1.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.4 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 1.4 - Function Junction
              Intended Role: Instructor

            Problem 1.5 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 1.5 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 1.5 - Function Junction
              Intended Role: Instructor

            Teacher Connection: Summarize Problem 1.5 - Function Junction
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 2 - Function Junction
              Intended Role: Instructor

            Problem 2.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.1 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 2.1 - Function Junction
              Intended Role: Instructor

            Problem 2.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 2.2 - Function Junction
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 3 - Function Junction
              Intended Role: Instructor

            Problem 3.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.1 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 3.1 - Function Junction
              Intended Role: Instructor

            Problem 3.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.2 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 3.2 - Function Junction
              Intended Role: Instructor

            Problem 3.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.3 - Function Junction
              Intended Role: Instructor

            Problem 3.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 3.4 - Function Junction
              Intended Role: Instructor

            Teacher Connection: Explore Problem 3.4 - Function Junction
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 4 - Function Junction
              Intended Role: Instructor

            Problem 4.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.1 - Function Junction
              Intended Role: Instructor

            Problem 4.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.2 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 4.2 - Function Junction
              Intended Role: Instructor

            Problem 4.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.3 - Function Junction
              Intended Role: Instructor

            Problem 4.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 4.4 - Function Junction
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Investigation 5 - Function Junction
              Intended Role: Instructor

            Problem 5.1 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.1 - Function Junction
              Intended Role: Instructor

            Problem 5.2 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.2 - Function Junction
              Intended Role: Instructor

            Launch Video - Problem 5.2 - Function Junction
              Intended Role: Instructor

            Problem 5.3 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.3 - Function Junction
              Intended Role: Instructor

            Problem 5.4 - Teacher Resources
              Intended Role: Instructor

            Teacher Edition - Problem 5.4 - Function Junction
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Classroom Connection: Assessments in CMP
              Intended Role: Instructor

            Classroom Connection: Three-Phase Instructional Model
              Intended Role: Instructor

            Teacher Connection: Meeting Students' Needs
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

            Teacher Resources
              Intended Role: Instructor

         eText Container

            Grade 8 - Student Edition

            Grade 8 - Teacher Edition

            Grade 8 - Spanish Student Edition