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Curriculum Standards:


Know that real numbers are either rational or irrational. Understand informally that every number has a decimal expansion which is repeating, terminating, or is non-repeating and non-terminating. - 8.NS.1
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, e.g., π². For example, by truncating the decimal expansion of √2, , show that √2, is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. - 8.NS.2
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? - 7.SP.C.8c
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - 7.SP.C.8a
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. - 7.SP.C.8b
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest, generating multiple samples to gauge variation and making predictions or conclusions about the population. - 7A.26d
Use informal arguments to establish that the sum of the interior angles of a triangle is 180 degrees. - 7A.38a
Differentiate between a sample and a population. - 7A.26a
Compare sampling techniques to determine whether a sample is random and thus representative of a population, explaining that random sampling tends to produce representative samples and support valid inferences. - 7A.26b
Determine whether conclusions and generalizations can be made about a population based on a sample. - 7A.26c
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. - 7.RP.A.2c
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - 7.RP.A.2b
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. - 7.RP.A.2a
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? - 7.SP.C.7b
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. - 7.RP.A.2d
Represent proportional relationships within and between similar figures. - 7.G.1b
Compute actual lengths and areas from a scale drawing and reproduce a scale drawing at a different scale. - 7.G.1a
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. - 7.SP.C.7a
Evaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000). - 7A.15a
Explain that the square root of a non-perfect square is irrational. - 7A.15b
Know and apply the properties of integer exponents to generate equivalent numerical expressions. - NY-8.EE.1
Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology. - NY-8.EE.4
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. - NY-8.EE.5
Focus on constructing quadrilaterals with given conditions noticing types and properties of resulting quadrilaterals and whether it is possible to construct different quadrilaterals using the same conditions. - 7.G.2b
Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 7.G.2a
Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational. - NY-8.EE.2
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. - NY-8.EE.3
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions. - NY-8.NS.2
Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational. - NY-8.NS.1
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. - 7A.16a
Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. - 7A.16b
Interpret scientific notation that has been generated by technology. - 7A.16c
Understand, explain, and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27. - 8.EE.1
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - 8.EE.2
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108; and the population of the world as 7 × 109; and determine that the world population is more than 20 times larger. - 8.EE.3
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are useUse scientific notation and choose units of appropriate size for measurements of very large or very small quantities, e.g., use millimeters per year for seafloor spreading. Interpret scientific notation that has been generated by technology. - 8.EE.4
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.5
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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EE.6
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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - NY-8.EE.6
Solve linear equations in one variable. - NY-8.EE.7
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 7A.12
Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers. - 7A.11
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. - 7.RP.A.1
Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions. - 7A.14
Generate expressions in equivalent forms based on context and explain how the quantities are related. - 7A.13
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. - 7.RP.A.3
Recognize and represent proportional relationships between quantities. - 7.RP.A.2
Define the real number system as composed of rational and irrational numbers. - 7A.10
Apply properties of operations as strategies to multiply and divide rational numbers. - 7.NS.A.2c
Create equations in two variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear functions. - 7A.19
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. - 7.NS.A.2d
Estimate and compare very large or very small numbers in scientific notation. - 7A.16
Use square root and cube root symbols to represent solutions to equations. - 7A.15
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - NY-7.SP.8a
Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities. - 7A.18
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - 7.EE.B.4
Represent sample spaces for compound events using methods such as organized lists, sample space tables, and tree diagrams.For an event described in everyday language, identify the outcomes in the sample space which compose the event. - NY-7.SP.8b
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. - 7.EE.B.3
Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies. - 7A.17
Design and use a simulation to generate frequencies for compound events. - NY-7.SP.8c
Find all possible outcomes of a compound event. - A7.DP.9.5
Find the probability of a compound event. - A7.DP.9.6
Determine the experimental probability of an event. - A7.DP.9.3
Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications. - 8A.50
Use probability models to find probabilities of events. - A7.DP.9.4
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. - 7.SP.A.1
Describe the likelihood that an event will occur. - A7.DP.9.1
Determine the theoretical probability of an event. - A7.DP.9.2
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. - 7.SP.A.2
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - 7.NS.A.2a
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real- world contexts. - 7.NS.A.2b
Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates. - 7A.10a
Convert a decimal expansion that repeats into a rational number. - 7A.10b
Simulate a compound event to approximate its probability. - A7.DP.9.7
Mathematical Modeling: Probability - A7.DP.9MM
Describe and analyze distributions. - 7.SP.3
Broaden statistical reasoning by using the GAISE model. - 7.SP.2
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihooA probability near 0 indicates an unlikely event; a probability around 1/2 indicates an event that is neither unlikely nor likely; and a probability near 1 indicates a likely event. - 7.SP.5
Parallel lines are taken to parallel lines. - 8.G.A.1c
Angles are taken to angles of the same measure. - 8.G.A.1b
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. - 7.SP.7
Lines are taken to lines, and line segments to line segments of the same length. - 8.G.A.1a
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. - 7.SP.6
Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship or from two points in a table or graph. - 7A.23a
Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. - 7.1.1.5
Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane. - 8A.49
Informally justify the Pythagorean Theorem and its converse. - 8A.48
Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator. - 7.1.1.2
Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is not rational, but that it can be approximated by rational numbers such as 22/7 and 3.14. - 7.1.1.1
Understand that statistics can be used to gain information about a population by examining a sample of the population. - 7.SP.1
Locate positive and negative rational numbers on the number line, understand the concept of opposites, and plot pairs of positive and negative rational numbers on a coordinate grid. - 7.1.1.3
Construct geometric shapes (freehand, using a ruler and a protractor, and using technology), given a written description or measurement constraints with an emphasis on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 7A.34
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. - NY-7.NS.2
Solve real-world and mathematical problems involving the four operations with rational numbers.Note: Computations with rational numbers extend the rules for manipulating fractions to complex fractions limited to (a/b)/(c/d) where a, b, c, and d are integers and b, c, and d ≠ 0. - NY-7.NS.3
Solve problems involving scale drawings of geometric figures, including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale. - 7A.33
Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle. - 7A.36
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. Represent addition and subtraction on a horizontal or vertical number line. - NY-7.NS.1
Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections. - 7A.35
Define and develop a probability model, including models that may or may not be uniform, where uniform models assign equal probability to all outcomes and non-uniform models involve events that are not equally likely. - 7A.30
Find probabilities of simple and compound events through experimentation or simulation and by analyzing the sample space, representing the probabilities as percents, decimals, or fractions. - 7A.32
Approximate the probability of an event using data generated by a simulation (experimental probability) and compare it to the theoretical probability. - 7A.31
Analyze and apply properties of parallel lines cut by a transversal to determine missing angle measures. - 7A.38
Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure. - 7A.37
Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms. - 7A.39
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. - 7.SP.B.4
Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. - 7A.2c
Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - 7A.2b
Compare populations using the mean, median, mode, range, interquartile range, and mean absolute deviation. - A7.DP.8.4
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. - 7.G.A.1
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. - 7.SP.B.3
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 7.G.A.2
Make inferences about a population from a sample data set. - A7.DP.8.2
Draw comparative inferences about two populations using median and interquartile range (IQR). - A7.DP.8.3
Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional. - 7A.2a
Determine if a sample is representative of a population. - A7.DP.8.1
Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - 7.G.A.3
Informally derive the formula for area of a circle. - 7A.36a
Solve area and circumference problems in real-world and mathematical situations involving circles. - 7A.36b
Construct a function to model the linear relationship between two variables. - 7A.23
Understand that rewriting an expression in different forms in real-world and mathematical problems can reveal and explain how the quantities are related. - NY-7.EE.2
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - NY-7.EE.4a
Add, subtract, factor, and expand linear expressions with rational coefficients by applying the properties of operations. - NY-7.EE.1
Solve word problems leading to inequalities of the form px + q > r, px + q ≥ r, px + q ≤ r, or px + q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality on the number line and interpret it in the context of the problem. - NY-7.EE.4b
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - NY-7.EE.4
Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies. - NY-7.EE.3
Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms. - 7A.21
Represent constraints by equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear. - 7A.20
Mathematical Modeling: Use Sampling to Draw Inferences about Populations - A7.DP.8MM
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. - 7A.27
Examine a sample of a population to generalize information about the population. - 7A.26
Use a number from 0 to 1 to represent the probability of a chance event occurring, explaining that larger numbers indicate greater likelihood of the event occurring, while a number near zero indicates an unlikely event. - 7A.29
Make informal comparative inferences about two populations using measures of center and variability and/or mean absolute deviation in context. - 7A.28
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. - 8.NS.A.1
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi^2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. - 8.NS.A.2
Represent a proportional relationship using an equation. - NY-7.RP.2c
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. - NY-7.RP.2d
Decide whether two quantities are in a proportional relationship. - NY-7.RP.2a
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - NY-7.RP.2b
Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. - 7.G.6
Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - 7.G.3
Draw (freehand, with ruler and protractor, and with technology) geometric figures with given conditions. - 7.G.2
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - 7.G.5
Work with circles. - 7.G.4
Solve problems involving similar figures with right triangles, other triangles, and special quadrilaterals. - 7.G.1
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - NY-7.G.5
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. - 7.G.B.6
Apply the formulas for the area and circumference of a circle to solve problems. - NY-7.G.4
Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids.Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids.Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms. - NY-7.G.6
Given a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship. - 7A.42a
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. - NY-7.G.1
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. - 7.G.B.4
Describe the two-dimensional shapes that result from slicing three-dimensional solids parallel or perpendicular to the base. - NY-7.G.3
Draw triangles when given measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - NY-7.G.2
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - 7.G.B.5
Mathematical Modeling: Solve Problems Using Equations and Inequalities - A7.AF.6MM
Collect and use data to predict probabilities of events. - 7A.30a
Compare probabilities from a model to observed frequencies, explaining possible sources of discrepancy. - 7A.30b
Solve real-world and mathematical problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions. - 7.NS.3
Find the y-intercept of a graph and explain what it means. - A7.AF.7.8
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). - 8.EE.7a
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 8.G.C.9
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 8.EE.7b
Derive the equation y = mx + b. - A7.AF.7.9
Solve equations with variables on both sides of the equal sign. - A7.AF.7.2
Solve multistep equations and pairs of equations using more than one approach. - A7.AF.7.3
Solve equations that have like terms on one side. - A7.AF.7.1
Understand the slope of a line. - A7.AF.7.6
Write equations to describe linear relationships. - A7.AF.7.7
Determine the number of solutions an equation has. - A7.AF.7.4
Compare proportional relationships represented in different ways. - A7.AF.7.5
In a problem context, understand that rewriting an expression in an equivalent form can reveal and explain properties of the quantities represented by the expression and can reveal how those quantities are relateFor example, a discount of 15% (represented by p − 0.15p) is equivalent to (1 − 0.15)p, which is equivalent to 0.85p or finding 85% of the original price. - 7.EE.2
Use proportional relationships to solve multistep ratio and percent problems. - NY-7.RP.3
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 7.EE.1
Observe the relative frequency of an event over the long run, using simulation or technology, and use those results to predict approximate relative frequency. - 7A.31a
Compute unit rates associated with ratios of fractions. - NY-7.RP.1
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - 7.EE.4
Recognize and represent proportional relationships between quantities. - NY-7.RP.2
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 93/4 inches long in the center of a door that is 271/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. - 7.EE.3
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” - 7.EE.A.2
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 7.EE.A.1
Solve problems with rational numbers. - A7.NC.1.10
Use measures of center and measures of variability for quantitative data from random samples or populations to draw informal comparative inferences about the populations. - NY-7.SP.4
Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams, and determine the probability of an event by finding the fraction of outcomes in the sample space for which the compound event occurred. - 7A.32a
Design and use a simulation to generate frequencies for compound events. - 7A.32b
Construct and interpret box-plots, find the interquartile range, and determine if a data point is an outlier. - NY-7.SP.1
Informally assess the degree of visual overlap of two quantitative data distributions. - NY-7.SP.3
Represent events described in everyday language in terms of outcomes in the sample space which composed the event. - 7A.32c
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - 7.NS.A.1b
Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7.NS.A.1c
Apply properties of operations as strategies to add and subtract rational numbers. - 7.NS.A.1d
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. - 7.NS.A.1a
Determine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers). - 7A.21a
Represent and solve real-world and mathematical problems with equations and interpret each solution in the context of the problem. - 7A.21b
Given the formulas for the volume of cones, cylinders, and spheres, solve mathematical and real-world problems. - NY-8.G.9
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - NY-8.G.8
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - NY-8.G.7
Apply properties of operations as strategies to add and subtract rational numbers. - NY-7.NS.1d
Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - NY-7.NS.1c
Understand addition of rational numbers; p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - NY-7.NS.1b
Describe situations in which opposite quantities combine to make 0. - NY-7.NS.1a
Explore and understand the relationships among the circumference, diameter, area, and radius of a circle. - 7.G.4a
Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane. - NY-8.G.2
Verify experimentally the properties of rotations, reflections, and translations. - NY-8.G.1
Understand a proof of the Pythagorean Theorem and its converse. - NY-8.G.6
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. - NY-8.G.5
Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane. - NY-8.G.4
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - NY-8.G.3
Know and use the formulas for the area and circumference of a circle and use them to solve real-world and mathematical problems. - 7.G.4b
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - NY-7.NS.2a
Solve problems involving the area of a circle. - A7.G.10.6
Determine what the cross section looks like when a 3D figure is sliced. - A7.G.10.7
Solve problems involving angle relationships. - A7.G.10.4
Solve problems involving radius, diameter, and circumference of circles. - A7.G.10.5
Draw figures with given conditions. - A7.G.10.2
Find probabilities of compound events using organized lists, sample space tables, tree diagrams, and simulation. - NY-7.SP.8
Draw triangles when given information about their side lengths and angle measures. - A7.G.10.3
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? - 7.EE.4a
Use the key in a scale drawing to find missing measures. - A7.G.10.1
Solve word problems leading to inequalities of the form px +q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example, as a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. - 7.EE.4b
Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - 8.EE.A.2
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 × 3^-5 = 3^-3 = 1/33 = 1/27. - 8.EE.A.1
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology - 8.EE.A.4
Find the area and surface area of 2-dimensional composite shapes and 3-dimensional prisms. - A7.G.10.8
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10^8 and the population of the world as 7 times 10^9, and determine that the world population is more than 20 times larger. - 8.EE.A.3
Use the area of the base of a three-dimensional figure to find its volume. - A7.G.10.9
Use a sequence of translations, reflections, and rotations to show that figures are congruent. - A7.G.11.5
Dilate two-dimensional figures. - A7.G.11.6
Rotate a two-dimensional figure. - A7.G.11.3
Describe and perform a sequence of transformations. - A7.G.11.4
Translate two-dimensional figures. - A7.G.11.1
Convert a fraction to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. - NY-7.NS.2d
Reflect two-dimensional figures. - A7.G.11.2
Apply properties of operations as strategies to multiply and divide rational numbers. - NY-7.NS.2c
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = –p/q = p/–q. Interpret quotients of rational numbers by describing real-world contexts. - NY-7.NS.2b
Find the interior and exterior angle measures of a triangle. - A7.G.11.9
Use a sequence of transformations, including dilations, to show that figures are similar. - A7.G.11.7
Identify and find the measures of angles formed by parallel lines and a transversal. - A7.G.11.8
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.B.5
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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EE.B.6
Mathematical Modeling: Solve Problems Involving Geometry - A7.G.10MM
Solve equations resulting from proportional relationships in various contexts. - 7.2.4.2
Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context. - 7.2.4.1
Apply percent reasoning to solve simple interest problems. - A7.P.4.6
Solve problems involving percent markup and markdown. - A7.P.4.5
Solve problems involving percent change and percent error. - A7.P.4.4
Graph and describe translations and reflections of figures on a coordinate grid and determine the coordinates of the vertices of the figure after the transformation. - 7.3.2.4
Represent and solve percent problems using equations. - A7.P.4.3
Use proportions to solve percent problems. - A7.P.4.2
Understand, find, and analyze percents of numbers. - A7.P.4.1
Use proportions and ratios to solve problems involving scale drawings and conversions of measurement units. - 7.3.2.3
Use the Pythagorean Theorem to find the distance between two points in the coordinate plane. - A7.G.12.4
Use the Converse of the Pythagorean Theorem to identify right triangles. - A7.G.12.2
Use the Pythagorean Theorem to solve problems. - A7.G.12.3
Divide rational numbers. - A7.NC.1.9
Use the Pythagorean Theorem to find unknown sides of triangles. - A7.G.12.1
Divide integers. - A7.NC.1.8
Multiply rational numbers. - A7.NC.1.7
Multiply integers. - A7.NC.1.6
Graph proportional relationships. - 7A.5
Add and subtract rational numbers. - A7.NC.1.5
Determine whether a relationship between two variables is proportional or non-proportional. - 7A.4
Subtract integers. - A7.NC.1.4
Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems. - 7A.7
Add integers. - A7.NC.1.3
Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept. - 7A.6
Recognize rational numbers and write them in decimal form. - A7.NC.1.2
Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable. - 7A.9
Relate integers, their opposites, and their absolute values. - A7.NC.1.1
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. - 7.EE.B.4b
Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals. - 7A.8
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? - 7.EE.B.4a
Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions. - 7A.1
Represent a relationship between two quantities and determine whether the two quantities are related proportionally. - 7A.2
Determine whether a relationship is proportional and use representations to solve problems. - A7.P.3.6
Use a graph to determine whether two quantities are proportional. - A7.P.3.5
Use the constant of proportionality in an equation to represent a proportional relationship. - A7.P.3.4
Test for equivalent ratios to decide whether quantities are in a proportional relationship. - A7.P.3.3
Find unit rates with ratios of fractions and use them to solve problems. - A7.P.3.2
Use ratio concepts and reasoning to solve multi-step problems. - A7.P.3.1
Find the volumes of cones. - A7.G.13.3
Find the volume of a sphere and use it to solve problems. - A7.G.13.4
Find the surface areas of cylinders, cones, and spheres. - A7.G.13.1
Use scientific notation to write very large or very small quantities. - A7.NC.2.9
Use what I know about finding volumes of rectangular prisms to find the volume of a cylinder. - A7.G.13.2
Estimate large and small quantities using a power of 10. - A7.NC.2.8
Write a number with a negative or zero exponent a different way. - A7.NC.2.7
Use the properties of exponents to write equivalent expressions. - A7.NC.2.6
Solve equations involving squares or cubes. - A7.NC.2.5
Find square roots and cube roots of rational numbers. - A7.NC.2.4
Recognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give examples and show which of these possibilities is the case by successively transforming the given equation into simpler forms. - NY-8.EE.7a
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms. - NY-8.EE.7b
Compare and order rational and irrational numbers. - A7.NC.2.3
Identify a number that is irrational. - A7.NC.2.2
Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions. - 7.4.1.1
Write repeating decimals as fractions. - A7.NC.2.1
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 8.EE.C.7b
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). - 8.EE.C.7a
Use angle measures to determine whether two triangles are similar. - A7.G.11.10
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. - 7.NS.2d
Apply properties of operations as strategies to multiply and divide rational numbers. - 7.NS.2c
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then −(p/q) = (−p)/q = p/(−q). Interpret quotients of rational numbers by describing real-world contexts. - 7.NS.2b
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (−1)(−1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - 7.NS.2a
Expand expressions using the Distributive Property. - A7.AF.5.4
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 8.G.8
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - 8.G.A.4
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - 8.G.7
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. 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. - 8.G.A.5
Use common factors and the Distributive Property to factor expressions. - A7.AF.5.5
Analyze and justify an informal proof of the Pythagorean Theorem and its converse. - 8.G.6
Write equivalent expressions for given expressions. - A7.AF.5.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - 8.G.A.2
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - 8.G.A.3
Use properties of operations to simplify expressions. - A7.AF.5.3
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. 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. - 8.G.5
Use an equivalent expression to find new information. - A7.AF.5.8
Add expressions that represent real-world problems. - A7.AF.5.6
Solve real-world and mathematical problems involving volumes of cones, cylinders, and spheres. - 8.G.9
Subtract expressions using properties of operations. - A7.AF.5.7
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (Include examples both with and without coordinates.) - 8.G.4
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - 8.G.3
Write and evaluate algebraic expressions. - A7.AF.5.1
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (Include examples both with and without coordinates.) - 8.G.2
Mathematical Modeling: Analyze and Solve Percent Problems - A7.P.4MM
Apply properties of operations as strategies to add and subtract rational numbers. - 7.NS.1d
Understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7.NS.1c
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - 7.NS.1b
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. - 7.NS.1a
Mathematical Modeling: Generate Equivalent Expressions - A7.AF.5MM
Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. - 8.1.1.4
Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved. - 8.1.1.5
Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. - 8.1.1.1
Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers. - 8.1.1.2
Determine rational approximations for solutions to problems involving real numbers. - 8.1.1.3
Use random numbers generated by a calculator or a spreadsheet or taken from a table to simulate situations involving randomness, make a histogram to display the results, and compare the results to known probabilities. - 7.4.3.1
Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. - 7.4.3.2
Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities. - 7.4.3.3
Use the Distributive Property to solve equations. - A7.AF.6.3
Solve inequalities using addition or subtraction. - A7.AF.6.4
Represent a problem with a two-step equation. - A7.AF.6.1
Solve a problem with a two-step equation. - A7.AF.6.2
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - 8.G.B.7
Solve inequalities that require multiple steps. - A7.AF.6.7
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 8.G.B.8
Perform operations with numbers in scientific notation. - A7.NC.2.10
Solve inequalities using multiplication or division. - A7.AF.6.5
Explain a proof of the Pythagorean Theorem and its converse. - 8.G.B.6
Write and solve two-step inequalities. - A7.AF.6.6
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. - 7.SP.C.7
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. - 7.SP.C.5
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. - 7.SP.C.6
Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them. - 7A.44
Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems. - 7A.41
Informally derive the formulas for the volume of cones and spheres by experimentally comparing the volumes of cones and spheres with the same radius and height to a cylinder with the same dimensions. - 7A.40
Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures. - 7A.43
Verify experimentally the properties of rigid motions (rotations, reflections, and translations): lines are taken to lines, and line segments are taken to line segments of the same length; angles are taken to angles of the same measure; and parallel lines are taken to parallel lines. - 7A.42
Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. - 8.2.2.1
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. - 7.RP.2c
Solve real-world and mathematical problems involving the four operations with rational numbers. - 7.NS.A.3
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. - 7.RP.2d
Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the 𝑦-intercept is zero when the function represents a proportional relationship. - 8.2.2.2
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. - 7.RP.2a
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - 7.RP.2b
Identify how coefficient changes in the equation f(x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects. - 8.2.2.3
Verify experimentally parallel lines are mapped to parallel lines. - NY-8.G.1c
Verify experimentally angles are mapped to angles of the same measure. - NY-8.G.1b
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language, e.g., “rolling double sixes," identify the outcomes in the sample space which compose the event. - 7.SP.8b
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? - 7.SP.8c
Understand that the graph of a proportional relationship is a line through the origin whose slope is the unit rate (constant of proportionality). Know how to use graphing technology to examine what happens to a line when the unit rate is changed. - 7.2.1.2
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane. - 7A.6a
Understand that a relationship between two variables, 34𝑥 and 34𝑥𝑦, is proportional if it can be expressed in the form 34𝑥𝑦𝑦/34𝑥𝑦𝑦𝑥 = k or 34𝑥𝑦𝑦𝑥𝑦 = k34𝑥𝑦𝑦𝑥𝑦𝑥. Distinguish proportional relationships from other relationships, including inversely proportional relationships (xy = k or y = k/x). - 7.2.1.1
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - 7.SP.8a
Recognize and represent proportional relationships between quantities. - 7.RP.2
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2) /(1/4) miles per hour, equivalently 2 miles per hour. - 7.RP.1
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. - 7.RP.3
Graph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value. - 7A.6c
Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line. - 7A.6b
Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts. - 7A.6d
Verify experimentally lines are mapped to lines, and line segments to line segments of the same length. - NY-8.G.1a
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. - 7.SP.7a
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? - 7.SP.7b
Interpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope. - 7A.5a
Lines are taken to lines, and line segments are taken to line segments of the same length. - 8.G.1a
Understand that a function is linear if it can be expressed in the form f(x) = mx + b or if its graph is a straight line. - 8.2.1.3
Angles are taken to angles of the same measure - 8.G.1b
Parallel lines are taken to parallel lines. - 8.G.1c
Use proportional reasoning to solve problems involving ratios in various contexts. - 7.1.2.5
Look for and make use of structure. - A7.MP.7
Look for and express regularity in repeated reasoning. - A7.MP.8
Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. - 7.1.2.4
Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value. - 7.1.2.6
Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents. - 7.1.2.1
Construct viable arguments and critique the reasoning of others. - A7.MP.3
Model with mathematics. - A7.MP.4
Understand that calculators and other computing technologies often truncate or round numbers. - 7.1.2.3
Use appropriate tools strategically. - A7.MP.5
Attend to precision. - A7.MP.6
Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense. - 7.1.2.2
Interpret Results: Draw logical conclusions and make generalizations from the data based on the original question. (GAISE Model, step 4) - 7.SP.2d
Make sense of problems and persevere in solving them. - A7.MP.1
Reason abstractly and quantitatively. - A7.MP.2
Formulate Questions: Recognize and formulate a statistical question as one that anticipates variability and can be answered with quantitative datFor example, “How do the heights of seventh graders compare to the heights of eighth graders?” (GAISE Model, step 1) - 7.SP.2a
Collect Data: Design and use a plan to collect appropriate data to answer a statistical question. (GAISE Model, step 2) - 7.SP.2b
Analyze Data: Select appropriate graphical methods and numerical measures to analyze data by displaying variability within a group, comparing individual to individual, and comparing individual to group. (GAISE Model, step 3) - 7.SP.2c
Solve multi-step problems involving proportional relationships in numerous contexts. - 7.2.2.2
Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses. - 7A.8a
Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations. - 7.2.2.1
Explain subtraction of rational numbers as addition of additive inverses. - 7A.8c
Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts. - 7A.8b
Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers. - 7.2.2.4
Use knowledge of proportions to assess the reasonableness of solutions. - 7.2.2.3
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - 7A.18a
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality, and interpret it in the context of the problem. - 7A.18b
Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved. - 7A.8e
Use a number line to demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7A.8d
Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. - 7A.8g
Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number. - 7A.8f
Differentiate between a sample and a population. - 7.SP.1a
Understand that conclusions and generalizations about a population are valid only if the sample is representative of that population. Develop an informal understanding of bias. - 7.SP.1b
Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. - 8.2.4.2
Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line. - 8.2.4.3
Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships. - 8.2.4.1
Use the relationship between square roots and squares of a number to solve problems. - 8.2.4.9
Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws. - 7.2.3.1
Apply understanding of order of operations and grouping symbols when using calculators and other technologies. - 7.2.3.3
Evaluate algebraic expressions containing rational numbers and whole number exponents at specified values of their variables. - 7.2.3.2
Calculate the volume and surface area of cylinders and justify the formulas used. - 7.3.1.2
Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is π. Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts. - 7.3.1.1
Summarize quantitative data sets in relation to their context by using mean absolute deviation (MAD), interpreting mean as a balance point. - 7.SP.3a
Informally assess the degree of visual overlap of two numerical data distributions with roughly equal variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot (line plot), the separation between the two distributions of heights is noticeable. - 7.SP.3b




List of all Files Validated:


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