PROM/SE (Promoting Rigorous Outcomes in Math and...

Common Core State Standards Aligned to Connected Mathematics Project 2 by Calhoun ISD is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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PROM/SE (Promoting Rigorous Outcomes in Math and Science Education) and Calhoun ISD

Connected Mathematics Project 2 8th Grade Units and Alignment to Common Core State Standards

Following is a recommendation from Calhoun Intermediate School District for teaching 8th grade math using Connected Mathematics Project 2 (CMP2) curricular

materials. A team of experienced teachers and consultants have worked together to align the Common Core State Standards (CCSS) to the CMP2 materials. Any

recommendation of moving units to another grade level or omission of units has been discussed at length by our team with this focus:

What do students need to know and be able to do to meet the requirements of the Common Core State Standards?

Unit Title Number of Days

Samples and Populations 23

Thinking with Mathematical Models 18

Looking for Pythagoras 25

Growing, Growing, Growing (Investigation 4 and 5 only) 10

Kaleidoscopes, Hubcaps and Mirrors 30

Say It With Symbols 24

Shapes of Algebra (Investigations 2, 3 and 4 only) 15

Total # of Days 145*

*We recommend that the remaining days be used throughout the school year to spend additional time on concepts as data indicates student

needs. The team has also determined, with careful consideration, that there are a handful of Common Core State Standards that are not

sufficiently taught in CMP2. Those standards are listed at the end of this document. District teams should work together to determine their course

of action regarding these CCSS.


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Common Core State Standards in Samples and Populations

6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

7.SP.1 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.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.3 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.4 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.

8.SP.1 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.

8.SP.2 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.

8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 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.


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Unit Alignment to Common Core State Standards by Lesson

Samples and Populations

Problem CCSS Notes to the Teacher

1.1 6.SP.4, 7.SP.3, 7.SP.4

Students explore a data base of 37 peanut butters. In this problem, they focus on the quality ratings of natural vs. regular types of peanut butter. They learn how to make histograms to represent these data.

1.2 6.SP.4, 7.SP.3, 7.SP.4

Students compare the quality ratings of natural and regular types of peanut butter using histograms as tools to represent data. They develop strategies for comparing distributions that involve analyzing the variability in distributions represented using histograms and using statistics such as mean, median, and range.

1.3 6.SP.4, 7.SP.3, 7.SP.4

Students learn to construct and use box plots as a tool for comparing quality ratings or prices of natural or regular peanut butters and quality ratings of creamy or chunky types. They also learn how to identify outliers in a distribution.

1.4 6.SP.4, 7.SP.3, 7.SP.4

Students compare the quality ratings of peanut butters using each of the two remaining attributes (salted or unsalted and name or store brands) to identify which value of each attribute may indicate higher quality. They represent data using either histograms or box plots.

2.1 7.SP.1, 7.SP.2

Students consider the implications of making estimates about the entire U.S. population based on an internet survey involving a few thousand people. The survey raises issues about predictions made about an entire population using data collected from a sample.

2.2 7.SP.1, 7.SP.2

Students consider a variety of sampling methods, analyzing the advantages and disadvantages of each and determining which would produce a sample most useful in predicting characteristics of the population.

2.3 7.SP.1, 7.SP.2

Students explore variability as it relates to sample data and statistics. They select random samples of 30 from a database of 100, and they analyze their samples to help them draw conclusions about the population.

2.4 7.SP.1, 7.SP.2

Students explore and compare predictions made from samples of size 5, 10, and 30.They make distributions of sample medians (means) and discuss which size samples seem to have medians (means) that are most similar to the population median (mean).

3.1 7.SP.1, 7.SP.2

Students inspect tables of the measurements of Native American arrowheads found at six different archaeological sites. Scientists know the approximate time periods during which four of the sites were settled; the time periods for the two newly discovered sites are unknown. Students explore how data from the known sites may be used to make predictions about when each of the new sites was settled.

3.2 7.SP.1, 7.SP.2

Students employ a sampling procedure to investigate how many chocolate chips must be added to a batch of cookie dough to ensure that each cookie in a batch will contain at least five chips.

4.1 8.SP.1 Students explore the relationship between quality and price of the regular brands versus the natural brands of peanut butter. They investigate a scatter plot of the (quality rating, price) data.

4.2 8.SP.1, 8.SP.2, 8.SP.3

Students explore three different but related proportional relationships: height and arm span for people, body length and wingspan for airplanes, and body length and wingspan for birds. For each, they consider fitting a line to describe the pattern of the relationship and writing an equation to describe this relationship.

4.3 8.SP.2, 8.SP.3

Students explore the relationship between safe water and life expectancies for several different countries in the world. This relationship is different from the relationships in Problem 4.2.


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Common Core State Standards in Thinking with Mathematical Models

8.EE.5 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.8 Analyze and solve pairs of simultaneous linear equations.

8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 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.

8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 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.

8.F.4 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 (x, y) 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.

8.F.5 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.


Thinking with Mathematical Models


1.1 Students explore a linear relationship as they test how bridge thickness is related to strength. They display their collected data in a table and a graph, they look for patterns, and use the patterns to make predictions. Note: This lesson is a precursor for understanding CCSS 8.F.4 and should not be skipped.

1.2 Students explore a nonlinear relationship as they test how bridge length is related to strength. They look for patterns in their collected data and use the patterns to make predictions. Note: This lesson is a precursor for understanding CCSS 8.F.4 and should not be skipped.


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1.3 8.F.3 Students look for differences in the patterns of change for a linear relationship and a nonlinear relationship. Note: This lesson is a precursor for understanding CCSS 8.F.4 and should not be skipped.

2.1 8.F.4, 8.EE.5

Students are introduced to mathematical models for data sets. They find linear models—both graphs and equations—for data sets, and use the models to make predictions.

2.2 8.F.2, 8.F.4, 8.EE.5

Students review and extend their understanding and skill in writing linear equations to match conditions expressed in words, tables, and graphs.

2.3 8.F.4, 8.EE.5

Students use tables and graphs to estimate solutions to linear equations and inequalities and use symbolic reasoning to find exact solutions.

2.4 8.F.4, 8.EE.5, 8.EE.8

Students use their knowledge about linear models, equations, and inequalities to reason about related sets of linear data.

3.1 Students are introduced to inverse variation as they examine the relationship between length and width for rectangles with a fixed area. Note: The CCSS do not address the concept of inverse variation, but CCSS 8.F.3 requires that students give examples of functions that are not linear. Teachers may choose to skip Investigation 3 if they are teaching CCSS 8.F.3 elsewhere.

3.2 Students examine two more examples of inverse variation: the relationship between bridge length and breaking weight and the relationship between speed and time for a fixed distance. Students also contrast inverse variations with linear relationships. Note: The CCSS do not address the concept of inverse variation, but CCSS 8.F.3 requires that students give examples of functions that are not linear. Teachers may choose to skip Investigation 3 if they are teaching CCSS 8.F.3 elsewhere.

3.3 Students solve a problem involving a school trip with a fixed total cost. They examine the relationship between the number of students who go on the trip and the per-student cost. Here again, students compare inverse variations with linear relationships. Note: The CCSS do not address the concept of inverse variation, but CCSS 8.F.3 requires that students give examples of functions that are not linear. Teachers may choose to skip Investigation 3 if they are teaching CCSS 8.F.3 elsewhere.

(Regular type gives a brief description of the lesson. Bold-face type gives teachers additional information to consider before teaching a lesson.)


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Common Core State Standards in Looking for Pythagoras

8.NS.1 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.2 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.EE.2* 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.G.6 Explain a proof of the Pythagorean Theorem and its converse.

8.G.7** 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.8+ Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

* Cube roots are not addressed in this unit as required by CCSS 8.EE.2.

** CCSS 8.G.7 requires that students learn to apply the Pythagorean Theorem in 2 and 3 dimensions. Looking for Pythagoras (8) only requires 2 dimensions.

+ CCSS 8.G.8 is not taught in CMP2. However, Looking for Pythagoras (8) could easily be supplemented to teach this standard. One possible solution: After teaching Lesson 3.3, teach a lesson where students are given 2 coordinate points and required to find the distance using what they have learned about the Pythagorean Theorem.


Looking for Pythagoras


1.1 Students analyze a map of a fictitious city in which streets are laid out on a coordinate grid. They find driving distances from one location to another, making the connection between the coordinates of two points and the distance between them. They compare the driving and flying distances between two points. Note: This investigation sets up a need for the Pythagorean Theorem.


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1.2 Given two vertices, students find other vertices that define a square, a non-square rectangle, a right triangle, and a non-rectangular parallelogram.

1.3 Students find areas of irregular figures drawn on a dot grid.

2.1 8.EE.2 Students search for all the squares that can be drawn on a 5 dot-by-5 dot grid. In the process, they begin to see how the area of a square relates to its side length. Note: Cube roots are not addressed. Teachers must discuss cubes in order to fully meet the standard.

2.2 8.NS.2, 8.EE.2

Students are introduced to the concept of square root. They learn that the positive square root of a number is the side length of a square with that number as area. Note: Cube roots are not addressed. Teachers must discuss cubes in order to fully meet the standard.

2.3 8.EE.2 Students find the lengths of segments on a dot grid by drawing squares with the segment as the side length. The length of the segment is the square root of the square’s area. Note: Cube roots are not addressed. Teachers must discuss cubes in order to fully meet the standard.

3.1 8.G.6 Students collect information about the areas of the squares on the sides of right triangles and conjecture that the sum of the areas of the two smaller squares equals the area of the largest square.

3.2 8.G.6 Students investigate a puzzle that verifies that the sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse.

3.3 8.G.7, 8.G.8

Students use the Pythagorean Theorem to find distances between dots on a grid. Note: The Pythagorean Theorem is not applied in 3 dimensions as is required by CCSS 8.G.7. Also, in order to teach CCSS 8.G.8, teachers must give students points on a coordinate plane and have students use the Pythagorean Theorem to find the distance between the points.

3.4 8.G.7 Students explore the converse of the Pythagorean Theorem: If a, b, and c are the lengths of the sides of a triangle and a2 + b2 = c2, then the triangle is a right triangle. Note: The Pythagorean Theorem is not applied in 3 dimensions as is required by CCSS 8.G.7.

4.1 8.NS.1, 8.NS.2, 8.EE.2, 8.G.7

Students apply the Pythagorean Theorem to find the exact lengths of hypotenuses of right triangles. Then, they use a number-line ruler to estimate the lengths. Finally, they compare their ruler estimates to those made with a calculator. Note: Be sure to spend time on the "Did You Know?" (pg 48) in order to address CCSS 8.NS.1.

4.2 8.EE.2, 8.G.7

Students apply the Pythagorean Theorem to find distances on a baseball diamond. Note: The Pythagorean Theorem is not applied in 3 dimensions as is required by CCSS 8.G.7.

4.3 8.EE.2, 8.G.7

Students investigate properties of equilateral and 30-60-90 triangles by applying the Pythagorean Theorem. Note: The Pythagorean Theorem is not applied in 3 dimensions as is required by CCSS 8.G.7.

4.4 8.EE.2, 8.G.7

Students draw from their experiences in the previous three problems to find missing lengths and angles in a triangle made up of other triangles. Note: The Pythagorean Theorem is not applied in 3 dimensions as is required by CCSS 8.G.7.



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Common Core State Standards in Growing, Growing, Growing

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 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.

8.F.5 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.

The only standard taught in Growing, Growing, Growing that is not taught elsewhere in CMP2 is CCSS 8.EE.1. A teacher could choose to use other resources to teach CCSS 8.EE.1 instead of Growing, Growing, Growing.


Growing, Growing, Growing


1.1 This hands-on introduction to the unit offers a review of exponents. Students investigate the growth in the number of ballots created by repeatedly cutting a piece of paper in half.

1.2 Students investigate an exponential situation set in the fictitious ancient kingdom of Montarek. One coin is placed on the first square of a chessboard, two on the second square, four on the third square, and so on. Students explore patterns of change in this exponential relationship.

1.3 Students consider two variations on the previous problem. In the first, the number of coins is tripled on each square. In the second, the number is quadrupled. Students make tables and graphs for the variations, describe patterns, and write equations for them, looking for a general form for exponential equations. The term growth factor is introduced.

1.4 Students compare a linear relationship to the exponential relationships in the preceding problems.

2.1 Students read about a real situation in which a non-native plant spread rapidly and began to cover a lake. Students then solve a problem about a similar situation. In the problem, the area of the plant doubles each month, and the starting value is greater than 1.

2.2 Students are given an equation for a real-world exponential growth relationship. They find and interpret the y-intercept and growth factor, and use the equation to answer questions about the relationship.


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2.3 Exponential data is presented in the form of a graph. Students find and interpret the y-intercept and growth factor and use this information to write an equation. Students use the graph or equation to answer questions about the situation.

3.1 This problem is set in the context of a historical account of a rapidly multiplying rabbit population in Australia. Students use population data given in table form to write an equation to model population growth. The growth factor in this situation is not a whole number. Students also find the doubling time for a population.

3.2 Students examine patterns of change due to compound growth. Students learn the connection between growth rate (or percent change) and growth factor.

3.3 Students study the effects of the initial value (y-intercept) on the growth patterns of three different savings plans.

4.1 8.F.3, 8.F.5

Students revisit the paper-cutting activity of Investigation 1 with a new question in mind: How does the area of a ballot change with each successive cut?

4.2 8.F.3, 8.F.5

This problem focuses on the decreasing amount of active medicine in an animal’s blood in the hours following the initial dose.

4.3 8.F.3, 8.F.5

Students conduct an experiment to determine the rate at which a cup of water cools, a phenomenon that can be closely modeled by exponential decay.

5.1 Students examine patterns in the ones digits of powers. They use these patterns to predict ones digits for powers that would be tedious to find directly.

5.2 8.EE.1 Students use their work in the previous problem to develop rules for operating with numerical expressions with exponents. Note: Problem 5.2 GGG(8) and 5.3 GGG(8) are not enough practice with exponent rules. Be sure to use additional practice pages provided in the teacher resources to give students additional practice in meeting the requirements of CCSS 8.EE.1.

5.3 8.EE.1 Students use graphing calculators to study how the values of a and b affect the graph of y = a(bˆx). Note: Problem 5.2 GGG(8) and 5.3 GGG(8) are not enough practice with exponent rules. Be sure to use additional practice pages provided in the teacher resources to give students additional practice in meeting the requirements of CCSS 8.EE.1.



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Common Core State Standards in Kaleidoscopes, Hubcaps and Mirrors

8.G.1 Verify experimentally the properties of rotations, reflections, and translations: a) Lines are taken to lines, and line segments to line segments of the same length. b) Angles are taken to angles of the same measure. c) Parallel lines are taken to parallel lines.

8.G.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.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.G.4 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.


Kaleidoscopes, Hubcaps and Mirrors


1.1 8.G.1 Students explore ways for checking for reflection symmetry and search for examples of such symmetry in several designs.

1.2 8.G.1 Students analyze illustrations of hubcaps, pinwheels, and other designs to explore the properties of rotation symmetry and the concept of angle of rotation.

1.3 8.G.1 Students look for reflection and rotation symmetry in various kaleidoscope designs and are introduced to the idea of a basic design element.

1.4 8.G.1 Students begin to look at strip patterns and wallpaper patterns to understand translation symmetry.

2.1 8.G.1 Students observe properties that define a line reflection and would allow someone else to perform it. Line reflections match points of a figure to points of an image figure. The original figure and its image form a design that has reflection symmetry.

2.2 8.G.1 Students observe properties that define a rotation and would allow someone else to perform it. They find that the rotation of a figure produces an image that together with the original will not produce a design with rotation symmetry unless the rotation is 180°. However, the rotation of a figure will produce a symmetric design if the angle of rotation is a factor, k, of 360 and the design has 360/k figures.

2.3 8.G.1 Students observe properties that define a translation and would allow someone else to perform it. They find that if a translation of a figure is repeated infinitely, a strip design with translation symmetry is produced.


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2.4 Students examine tessellations to find basic elements of the design that can be used along with transformation of the element to reproduce the design. They also examine tessellation designs to describe the types of symmetry the design has.

3.1 8.G.2, 8.G.4

This problem relies on students’ informal sense of same shape and size to imagine the ways that two congruent polygons would be matched if corresponding congruent parts are face-to-face. Standard notation for expressing congruence of segments and angles is introduced, and students are asked to think about the symmetry transformations required to compare congruent figures.

3.2 8.G.2, 8.G.4

This problem develops students’ ability to recognize congruent triangles within diagrams that contain several adjacent and overlapping figures. They are asked to match corresponding parts of congruent triangles and to use what they know about symmetry of rectangles and properties of parallel lines and transversals to reason informally in support of their conjectures about congruent figures.

3.3 8.G.2 This problem asks students to play a matching game that requires constructing triangles congruent to given figures and, by analyzing strategies that help one to win that matching game, to discover minimal sets of information that guarantee congruence of two triangles. These are the familiar Side-Angle-Side and Angle-Side-Angle conditions.

3.4 This problem connects student ideas about properties that give triangles their shapes (from the Shapes and Designs unit of year one) to the problem of using minimal information about triangle sides to decide whether triangles are congruent. This leads to the familiar Side-Side-Side congruence condition. One question about drawing congruent quadrilaterals highlights the special property of triangles that makes them rigid figures.

4.1 Here students use the ideas of congruence developed in Investigation 3 to solve a problem involving congruent triangles in a story context. In so doing, they see the power of inference based on known facts.

4.2 Students use what they have learned about symmetry and congruence to solve problems involving polygons. In some cases, students are asked to argue from what is given to find measures of other angles and sides of polygons. In other problems, students reason from what is given and what they can deduce to tell what kind of polygon the shape is. (See extended notes in CMP Teacher's Guide)

5.1 8.G.3 Students write coordinate rules for transforming a point (x, y) to its image point using selected reflections. The explorations are set in the context of writing computer instructions for drawing line segments, which focuses attention on what happens to the coordinates of a point under the desired transformation.

5.2 8.G.3 Students write coordinate rules for transforming a point (x, y) to its image point using selected translations.

5.3 8.G.3 Students write coordinate rules for transforming a point (x, y) to its image point under selected rotations.

5.4 8.G.3 Here students get a taste of what is involved in specifying a single transformation that produces the same result as one transformation followed by another. Using coordinate rules can help in keeping track of changes to the location of points.



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Common Core State Standards in Say It With Symbols

6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the

equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of

operations to y + y + y to produce the equivalent expression 3y.

7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.2 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.”

8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and

collecting like terms.

8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 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.

8.F.3* Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the

function A = s2

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.

8.F.5 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.

* CCSS 8.F.3 is taught throughout Say It With Symbols.


Say It With Symbols


1.1 6.EE.3, 7.EE.2

Students write equations to represent the number of unit tiles N that surround a square pool of side length s. They justify the equivalence of two or more symbolic expressions for the same situation.

1.2 6.EE.3, 7.EE.2

This problem serves as a summary for Problem 1.1. It provides several equations that students have used to represent the relationships between the number of unit border tiles and the side length of a square pool. Students use the logic underlying each equation or tables and graphs to show equivalence.


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1.3 Students interpret a symbolic expression, A= (πx2/8)+x2 + 8x + (πx2/4) that represents the surface area of a community center pool, part of which is inside and part of which is outside the community center. Note: This lesson is all about quadratics. It can be skipped.

1.4 6.EE.3, 7.EE.1, 7.EE.2

This problem revisits the area model for the Distributive Property and then applies the Distributive Property to write equivalent expressions with or without parentheses and to show that two or more expressions are equivalent. Note: Use this lesson to focus on Distributive Property with the use of area models. This lesson is not intended to teach quadratic functions.

2.1 6.EE.3, 7.EE.1, 7.EE.2

Problem 2.1 revisits a walkathon from Moving Straight Ahead (7). Students add several expressions and apply the Distributive and Commutative properties to write equivalent expressions. Some of the expressions contain parentheses with which students have not had much experience. This is an opportunity for teachers to assess what understanding of the Distributive Property students bring to the problem. Note: For question C change to “linear or non-linear.”

2.2 7.EE.1, 8.EE.7b

Students use two given equations to write one equation by substituting an equivalent expression for one variable in the other equation. They then write a simpler equivalent expression and compare the information that each equation represents in the situation. The equations are used to predict the profit for a given probability of rain and conversely to predict the probability given a certain profit.

2.3 8.F.2 Students substitute equivalent expressions for quantities in a given equation in order to write equations for area and for profit. To find the profit equation, students substitute one equivalent linear expression for a variable in a quadratic expression. In the previous problem, an equivalent linear expression was substituted for a variable in a linear expression. In these two contexts, the resulting equations for profit and area are the same quadratic relationship, showing that one equation can model two different situations.

3.1 8.EE.7b Students are introduced to solving equations that involve parentheses and negative signs.

3.2 8.EE.7b Problem 3.2 presents two linear equations with parentheses that represent the costs for tiles needed to surround a pool. The students are asked to find the number of tiles when the costs are equal without using tables or graphs.

3.3 Students revisit factoring quadratic expressions, which was first done in Frogs, Fleas, and Painted Cubes(8). Note: As quadratic functions is not a CCSS topic, it is recommended that you skip this lesson.

3.4 The connections between a factored form of ax2 + bx + c, the x-intercepts of the graph of the equation y = ax2 + bx + c, and the solutions to 0 = ax2 + bx + c are explored. The two equivalent forms for a quadratic expression are used to predict the x- and y-intercepts, the maximum or minimum point, and the line of symmetry of the graph of a quadratic equation. Note: As quadratic functions is not a CCSS topic, it is recommended that you skip this lesson.

4.1 8.F.3, 8.F.5

Problem 4.1 uses linear equations to represent the amount of water w in a pool that is emptied after t hours. Students use the equations to answer questions about the rate at which water is being pumped out each hour, as well as the amount of water at the beginning of the pumping and the number of hours that it takes to empty the pool.

4.2 8.F.3 Students write equations representing a linear, an exponential, and a quadratic function given two points on the graph of each function. This problem provides the opportunity to revisit situations from previous units, but more important to investigate how symbolic statements capture the patterns of change in these three functions. Note: The CCSS do not require that students write equations for exponential and quadratic functions, but that they can give examples of functions that are not linear.

4.3 8.F.3 Students learn to predict the type of relationship the equation of a function represents and the characteristics a graph of the function would have. Note: For question A, students could sort functions into linear and nonlinear. Questions B2 and C can be skipped.

5.1 Students explore why a familiar number trick works. Students use symbolic statements and properties of equality and numbers to show why the trick works.

5.2 Students explore algebraic expressions that represent even and odd integers. They use these expressions to examine the sums and products of two evens, two odds, or an even and an odd. This conjecture was first explored in Prime Time.

5.3 Students explore the patterns that emerge from squaring an odd number and then subtracting one. An interesting connection back to triangular numbers (or the number of handshakes) from Frogs, Fleas, and Painted Cubes is found in the pattern.



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Common Core State Standards in Shapes of Algebra

6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

7.EE.2 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.4 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.

8.EE.8 Analyze and solve pairs of simultaneous linear equations.

a) 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.

b) Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c) Solve real-world and mathematical problems leading to two linear equations in two variables. 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.

8.F.4 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 (x, y) 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.


Shapes of Algebra


1.1 This lesson begins by asking students to think about patterns of coordinates for points on and inside of circles. The problem deals only with the case of circles centered at the origin of a coordinate system and uses the Pythagorean Theorem to lead to a general formula x

2 + y

2 = r

2.To give interior points,

we write x2 + y

2< r

2. Note: This lesson does not relate to any 8

th grade CCSS. It can be skipped.

1.2 Explores patterns in coordinates of points that determine parallelograms and rectangles. To start the exploration, students look at crop circle designs made of polygons inscribed in circles in various ways. Students construct rectangles and parallelograms on a coordinate grid, find the equations for the sides of those figures, and connect geometric slope to equations. The result should be a reminder that parallel lines have the same slope and perpendicular lines have slopes that are inverse reciprocals of each other. This should connect with and extend work in the Looking for Pythagoras unit done earlier. Note: This lesson does not relate to any 8

th grade CCSS. It can be skipped.


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1.3 Focuses on the special properties of midpoints and lines that connect them in quadrilaterals. Students establish the beautiful result that the figure formed by connecting (in order) midpoints of sides of any quadrilateral is a parallelogram. Note: This lesson does not relate to any 8

th grade CCSS. It

can be skipped.

2.1 8.F.4, 8.EE.8

Prepares students for the geometric interpretation of algebraic systems that will follow in the next investigation.

2.2 7.EE.4b Builds on the system work in 2.1 to raise questions that involve linear inequalities in one variable. It develops ideas about reasoning with inequalities and their use in finding solutions of inequality problems such as ax + b<cx + d. This connects to work with linear equations.

2.3 6.EE.8, 7.EE.4b

This problem directs students’ attention to systematic methods for solving linear inequalities in one variable and number line graphs of those solutions. The problem also connects to graphs of y = mx + b as an alternative way of visualizing solutions for inequalities in one variable.

3.1 7.EE.2 Problem 3.1 introduces linear equations in “standard form,” ax + by = c. Students discover that graphs of solutions for such equations also turn out to be straight lines.

3.2 7.EE.2 Connects the y = mx + b and ax + by = c forms of linear equations. Useful strategies for connecting these two forms are revealed by asking students to analyze sample student work. The main algebraic properties needed in these cases are the distributive property and the general principle that equals can be added to, subtracted from, multiplied by, or divided into equals without changing the equality relationship.

3.3 8.EE.8 Introduces the concept of a system of linear equations and the graphic image of solving that system. Symbol-based algorithms for solving linear systems come in the next investigation.

4.1 8.EE.8 Addresses the familiar and simplest case in which both equations express y as a function of x and the solution results from setting those two expressions equal to each other and solving the single variable equation.

4.2 8.EE.8 Addresses the cases of linear systems in which it is relatively easy to transform each of the given equations (in the form ax + by = c) into “y = . . .” or “x = . . .” form and then proceed as in Problem 4.1.

4.3 8.EE.8 Addresses the cases in which one of the equations can be transformed into “y = . . .” or “x = . . .” and the result substituted into the other equation to yield an equation in one variable.

4.4 8.EE.8 Addresses the standard linear combinations method in which multiples of one or both given equations are combined by addition or subtraction in a way that again eliminates one of the unknowns.

5.1 Uses two questions about the relationship between automobile mileage and air pollution to introduce the notion that the solution graph for a linear inequality in two variables will be a half-plane. It addresses the simplest case in which there is a single linear inequality of the form x + y ≤ 1,000. Students first plot possibilities for solutions of a linear inequality in two variables and then think about how to represent all possible answers.

5.2 Extends understanding of solutions and graphs for linear inequalities while introducing the notion of compound inequality conditions formed using “and” and “or.”This sort of compound inequality condition is developed further in Problem 5.4.

5.3 Generalizes the notion of linear inequality to a variety of examples given in symbolic and graphic domains to positive values of the variables. It also gives students a chance to match inequalities and their graphs with a view toward developing strategies for graphing inequalities.

5.4 Formalizes the notion of a system of linear inequalities and its graph.



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6th through 8th Grade Common Core State Standards Not Addressed in CMP2

6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3° C > –7° C to express the fact that –3° C is warmer than –7° C.

6.NS.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Note: This concept is taught in Looking for Pythagoras(8). However, most of the distances are oblique rather than vertical or horizontal. Absolute value is taught in ATN(7).

6.EE.2a Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.

6.EE.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. Note: The concept of distance on a coordinate grid is taught in LFP(8).1.1. However, LFP goes more in depth than this standard. A teacher could create a new lesson using same context (street map.)

7.RP.1 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. Note: Unit rates are taught in Comparing and Scaling (7) Problems 3.1, 3.2, 3.3 and 3.4. However, they do not meet the requirement of the standard that they are ratios of fractions. The ratios in these problems are between whole numbers and decimals.

7.RP.2a 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. Note: This standard is not specifically taught in Comparing and Scaling (7). However, Problems 2.3, 3.1, 3.4 could easily include graphing the values on the rate tables and drawing the line which passes through the origin to teach this standard.


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7.RP.2d 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. Note: This standard is not specifically taught in Comparing and Scaling (7). However, Problems 2.3, 3.1 and 3.4 could easily include graphing the values on the rate tables, examining and explaining the meaning of the points (0,0) and (1,r) to teach this standard.

7.EE.2 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.” Note: Properties of operations are used in solving equations, but not in generating equivalent expressions for a single expression. This standard is taught in Say It With Symbols (8).

7.G.3 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.5 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. Note: There are resources in Shapes and Designs (6) for this standard.

8.EE.3 Use numbers expressed in the form of a single digit times a whole-number 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 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. Note: There are some ACE questions in Growing, Growing, Growing that address this CCSS.

8.EE.4 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. Note: There are some ACE questions in Growing, Growing, Growing that address this CCSS.

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.

8.EE.7a 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). Note: This CCSS should be taught with the unit Shapes of Algebra.

8.F.1 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.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Note: Looking for Pythagoras could easily be supplemented to teach this standard. One possible solution is to teach LFP(8).3.3 and then teach a lesson where students are given 2 coordinate points and required to find the distance using what they have learned about the Pythagorean Theorem.

8.SP.4 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. 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?

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