Organization: Pearson Product Name: Connected Mathematics 3 Grade 8 Product Version: 1.0 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-e6bbf756-d1de-3160-8bb4-9601cd776c8d Timestamp: Friday, November 30, 2018 10:44 AM EST Status: VALID! Conformant: true ----- VALID! ----- Resource Validation Results The document is valid. ----- VALID! ----- Schema Location Results Schema locations are valid. ----- VALID! ----- Schema Validation Results The document is valid. ----- VALID! ----- 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 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 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 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 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 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 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 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 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 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 List of all Files Validated: imsmanifest.xml I_00851c16-a47e-30b5-b750-ee173c33b787_1_R/BasicLTI.xml I_0272de21-b0c2-326c-baba-3d35c3ed687d_R/BasicLTI.xml I_03526f6f-5df4-39bf-8252-1e3e104ccb61_1_R/BasicLTI.xml I_0546ec7e-192d-3401-900f-bee9e21bb551_R/BasicLTI.xml I_05a4881e-aa84-3912-ab5f-a78437f3b675_R/BasicLTI.xml I_05c9f6a0-603b-3b9e-afdb-44cf79aedd2a_1_R/BasicLTI.xml I_05f7361e-4b00-35ec-9d01-ca3ee788231e_1_R/BasicLTI.xml I_067a25b3-92eb-3c7a-915e-95fbbc9bdaa6_1_R/BasicLTI.xml I_079bd1ff-b026-3404-9538-ca81785ea5ee_R/BasicLTI.xml I_08278e63-6788-3d2a-b66f-ee6ca6f55dee_R/BasicLTI.xml I_08d55ef7-e4d9-3059-b52c-96b6555b3c3a_1_R/BasicLTI.xml I_09d1aa85-216d-34fd-abc5-37dad513e8bb_R/BasicLTI.xml 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I_fccfad32-292c-3db1-a093-d70174671015_R/BasicLTI.xml I_fcf59440-a21d-3117-af17-7accdc48074e_1_R/BasicLTI.xml I_fdac786f-9f37-3096-b405-5c78fa721c86_1_R/BasicLTI.xml I_fe33267d-d5a9-3507-9b63-86eb3fdeac0f_R/BasicLTI.xml I_feb01ca1-2a26-369e-b5ce-d8af7e507fac_R/BasicLTI.xml I_fee292f7-e0ea-3092-a798-8c2e1a2d4455_1_R/BasicLTI.xml I_feff1aa7-6d93-3930-9984-41343bf380b6_R/BasicLTI.xml I_ff11e9cb-2ecc-3576-b829-829fb06db2f4_R/BasicLTI.xml I_ff67baf5-04a6-3852-816e-b13fe7cd8981_R/BasicLTI.xml I_ffbd0d33-eb09-3807-904b-6f240b41179a_1_R/BasicLTI.xml Title: Connected Mathematics 3 Grade 8 2018 Tools Math Tools Glossary Student Activities HTMLBook: Thinking With Mathematical Models HTMLBook: Looking for Pythagoras HTMLBook: Growing, Growing, Growing HTMLBook: Butterflies, Pinwheels, and Wallpaper HTMLBook: Say It With Symbols HTMLBook: It's In The System 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: 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. 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. 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 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 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. 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. 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. 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. 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. 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 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. Bridge Length and Strength Student Edition - Problem 1.2 - Thinking With Mathematical Models 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 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. 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. 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. 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 ?? = ??(??). 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. 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 ??. 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. 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. 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. 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. 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. Custom Construction Parts: Finding Patterns Student Edition- Problem 1.3 - Thinking With Mathematical Models 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). 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. 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. 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. 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. 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. 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. 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). 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. 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. 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 - 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. 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). 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. 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. 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. 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 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. 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 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. 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 ??. 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). 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. 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. 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. Informally assess the fit of a function by plotting and analyzing residuals. Informally assess the fit of a function by plotting and analyzing residuals. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. 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. 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. 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 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 ?? = ??(??). 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. 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. 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. 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. 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). 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. 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). 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. 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. 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. 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. 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. 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. 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 ?? = ??(??). 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. 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. 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). 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. 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. 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. 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. 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. Boat Rental Business: Solving Linear Equations Student Edition - Problem 2.4 - Thinking With Mathematical Models 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. 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. 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 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. 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. 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. 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. 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 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. 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. 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 c. 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: 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. 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). 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. 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. 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. 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 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. 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 ??. 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). 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. 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). 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. Distance, Speed, and Time Student Edition - Problem 3.2 - Thinking With Mathematical Models 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. 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). 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. 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. 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. 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. 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 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. 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. 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. 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 ??. 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). 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. 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). 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. Planning a Field Trip: Finding Individual Cost Student Edition - Problem 3.3 - Thinking With Mathematical Models 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. Modeling Data Patterns Student Edition - Problem 3.4 - Thinking With Mathematical Models 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. 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. 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 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. 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 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 ??. 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. 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. 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. 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. 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. 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. 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. 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: 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 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). 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. 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. 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. 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 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. 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. 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). 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. 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. 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 ??. 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. 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. 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. 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. 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. Informally assess the fit of a function by plotting and analyzing residuals. Informally assess the fit of a function by plotting and analyzing residuals. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Older and Faster: Negative Correlations Student Edition - Problem 4.2 - Thinking With Mathematical Models 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. 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 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). 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. 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, 3th + 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Correlation Coefficients and Outliers Student Edition - Problem 4.3 - Thinking With Mathematical Models 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. 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. 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). Compute (using technology) and interpret the correlation coefficient of a linear fit. Compute (using technology) and interpret the correlation coefficient of a linear fit. 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. 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. 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. 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 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. 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. 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. 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 ??. Distinguish between correlation and causation. Distinguish between correlation and causation. 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. 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. 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. 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. 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. 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. 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. Measuring Variability: Standard Deviation Student Edition - Problem 4.4 - Thinking With Mathematical Models 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 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. 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. 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. 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. 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 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. 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). 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. 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 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. 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. 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. 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 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. 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. 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. 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. 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. 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. 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. 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. 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 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 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. 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. 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. 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. 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). 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. 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. 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 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. 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. 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. 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. 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). Distinguish between correlation and causation. Distinguish between correlation and causation. 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). 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. 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. 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. 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. 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 Representing Decimals as Fractions Student Edition - Problem 4.3 - Looking for Pythagoras Getting Real: Irrational Numbers Student Edition - Problem 4.4 - Looking for Pythagoras ACE - Investigation 4 - Looking for Pythagoras Mathematical Reflections - Investigation 4 - Looking for Pythagoras 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 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 Requesting a Reward: Representing Exponential Functions Student Edition - Problem 1.2 - Growing, Growing, Growing Making a New Offer: Growth Factors Student Edition- Problem 1.3 - Growing, Growing, Growing 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 Growing Mold: Interpreting Equations for Exponential Functions Student Edition - Problem 2.2 - Growing, Growing, Growing Studying Snake Populations: Interpreting Graphs of Exponential Functions Student Edition - Problem 2.3 - Growing, Growing, Growing 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 Investing for the Future: Growth Rates Student Edition - Problem 3.2 - Growing, Growing, Growing Teacher Connection: Summarize Problem 3.2 - Growing, Growing, Growing Making a Difference: Connecting Growth Rate and Growth Factor Student Edition - Problem 3.3 - Growing, Growing, Growing 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 Fighting Fleas: Representing Exponential Decay Student Edition - Problem 4.2 - Growing, Growing, Growing Cooling Water: Modeling Exponential Decay Student Edition - Problem 4.3 - 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 Analyzing Triangles Student Edition - Problem 5.2 - Growing, Growing, Growing Analyzing Circles Student Edition - Problem 5.3 - Growing, Growing, Growing 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 Mathematical Reflections - Investigation 5 Growing, Growing, Growing - Looking Back Growing, Growing, Growing - Unit Test 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 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. 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. 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. 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 - 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 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 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 Thinking in Different Ways: Determining Equivalence Student Edition - Problem 1.2 - Say It With Symbols The Community Pool Problem: Interpreting Expressions Student Edition- Problem 1.3 - Say It With Symbols Diving In: Revisiting the Distributive Property Student Edition - Problem 1.4 - Say It With Symbols 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 Predicting Profit: Substituting Expressions Student Edition - Problem 2.2 - Say It With Symbols Making Candles:Volumes of Cylinders, Cones, and Spheres Student Edition - Problem 2.3 - Say It With Symbols Selling Ice Cream: Solving Volume Problems Student Edition - Problem 2.4 - Say It With Symbols 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 Comparing Costs: Solving More Linear Equations Student Edition - Problem 3.2 - Say It With Symbols Factoring Quadratic Equations Student Edition - Problem 3.3 - Say It With Symbols Solving Quadratic Equations Student Edition - Problem 3.4 - Say It With Symbols 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 Area and Profit - What's the Connection? Using Equations Student Edition - Problem 4.2 - Say It With Symbols Generating Patterns: Linear, Exponential, Quadratic Student Edition - Problem 4.3 - Say It With Symbols What's the Function? Modeling With Functions Student Edition - Problem 4.4 - Say It With Symbols 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 Odd and Even Revisited Student Edition - Problem 5.2 - Say It With Symbols Squaring Odd Numbers Student Edition - Problem 5.3 - Say It With Symbols 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 Connecting Ax + By = C and y = mx + b Student Edition - Problem 1.2 - It's In The System Booster Club Members: Intersecting Lines Student Edition- Problem 1.3 - It's In The System 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 Taco Truck Lunch: Solving Systems by Combining Equations I Student Edition - Problem 2.2 - It's In The System Solving Systems by Combining Equations II Student Edition - Problem 2.3 - It's In The System 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 Solving Linear Inequalities Symbolically Student Edition - Problem 3.2 - It's In The System Operating at a Profit: Systems of Lines and Curves Student Edition - Problem 3.3 - It's In The System 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 What Makes a Car Green: Solving Inequalities by Graphing I Student Edition - Problem 4.2 - It's In The System Feasible Points: Solving Inequalities by Graphing II Student Edition - Problem 4.3 - It's In The System Miles of Emissions: Systems of Linear Inequalities Student Edition - Problem 4.4 - It's In The System 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 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 - 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 Benchmark Assessment 2 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 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 Benchmark Assessment 3 Teacher Resources Container Teacher Resources: Grade 8 Intended Role: Instructor MATHDashboard Intended Role: Instructor Next Generation Assessments Intended Role: Instructor ExamView Intended Role: Instructor Unit 1 - Teacher Resources Intended Role: Instructor Thinking With Mathematical Models - Teacher Edition Intended Role: Instructor Teacher Connection: Supporting ELL and Struggling Students Intended Role: Instructor Teacher Edition - Investigation 1 - Thinking With Mathematical Models Intended Role: Instructor Problem 1.1 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 1.1 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 1.1 - Thinking With Mathematical Models Intended Role: Instructor Problem 1.2 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 1.2 - Thinking With Mathematical Models Intended Role: Instructor Teacher Connection: Explore Problem 1.2 - Thinking With Mathematical Models Intended Role: Instructor Teacher Connection: Summarize Problem 1.2 - Thinking With Mathematical Models Intended Role: Instructor Problem 1.3 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 1.3 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 1.3 - Thinking With Mathematical Models Intended Role: Instructor Teacher Resources Intended Role: Instructor Teacher Edition - Investigation 2 - Thinking With Mathematical Models Intended Role: Instructor Problem 2.1 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 2.1 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 2.1 - Thinking With Mathematical Models Intended Role: Instructor Problem 2.2 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 2.2 - Thinking With Mathematical Models Intended Role: Instructor Problem 2.3 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 2.3 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 2.3 - Thinking With Mathematical Models Intended Role: Instructor Classroom Connection: Explore Problem 2.3 - Thinking With Mathematical Models Intended Role: Instructor Problem 2.4 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 2.4 - Thinking With Mathematical Models Intended Role: Instructor Problem 2.5 - Teacher Resources Intended Role: Instructor Classroom Connection: Summarize Problem 2.5 - Thinking With Mathematical Models Intended Role: Instructor Teacher Resources Intended Role: Instructor Teacher Edition - Investigation 3 - Thinking With Mathematical Models Intended Role: Instructor Problem 3.1 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 3.1 - Thinking With Mathematical Models Intended Role: Instructor Problem 3.2 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 3.2 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 3.2 - Thinking With Mathematical Models Intended Role: Instructor Problem 3.3 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 3.3 - Thinking With Mathematical Models Intended Role: Instructor Problem 3.4 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 3.4 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 3.4 - Thinking With Mathematical Models Intended Role: Instructor Teacher Resources Intended Role: Instructor Teacher Edition - Investigation 4 - Thinking With Mathematical Models Intended Role: Instructor Problem 4.1 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 4.1 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 4.1 - Thinking With Mathematical Models Intended Role: Instructor Problem 4.2 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 4.2 - Thinking With Mathematical Models Intended Role: Instructor Problem 4.3 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 4.3 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 4.3 - Thinking With Mathematical Models Intended Role: Instructor Problem 4.4 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 4.4 - Thinking With Mathematical Models Intended Role: Instructor Teacher Resources Intended Role: Instructor Teacher Edition - Investigation 5 - Thinking With Mathematical Models Intended Role: Instructor Problem 5.1 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 5.1 - Thinking With Mathematical Models Intended Role: Instructor Launch Video - Problem 5.1 - Thinking With Mathematical Models Intended Role: Instructor Problem 5.2 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 5.2 - Thinking With Mathematical Models Intended Role: Instructor Teacher Connection: Summarize Problem 5.2 - Thinking With Mathematical Models Intended Role: Instructor Problem 5.3 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 5.3 - Thinking With Mathematical Models Intended Role: Instructor Teacher Resources Intended Role: Instructor Unit 2 - Teacher Resources Intended Role: Instructor Looking for Pythagoras - Teacher Edition Intended Role: Instructor Teacher Connection: Supporting ELL and Struggling Students Intended Role: Instructor Teacher Edition - Investigation 1 - Looking for Pythagoras Intended Role: Instructor Problem 1.1 - Teacher Resources Intended Role: Instructor Teacher Edition - Problem 1.1 - Looking for Pythagoras 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 Teacher Resources 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 Classroom Connection: Launch 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 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 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 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 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 eText Container Grade 8 - Teacher Edition Grade 8 - Spanish Student Edition Grade 8 - Student Edition