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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 7th Grade Units and Alignment to Common Core State Standards

Following is a recommendation from Calhoun Intermediate School District for teaching 7th 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

Stretching and Shrinking 25

Comparing and Scaling 20

Accentuate the Negative 24

Moving Straight Ahead 26

Filling and Wrapping 27

What do You Expect 23

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 Stretching and Shrinking

6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

7.RP.2 Recognize and represent proportional relationships between quantities.

7.G.1 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.

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

8.G.5 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.

Unit Alignment to Common Core State Standards by Lesson

Stretching and Shrinking

Problem CCSS Notes to the Teacher

1.1 7.G.1 Students use the known size of a small object in a photo of a mystery student to estimate the unknown actual size of a larger object.

Students are asked to express their informal ideas about the mathematical meaning of similarity.

1.2 7.G.1,

8.G.5

Students make a stretcher by tying 2 rubber bands together and use it to enlarge two different figures. They compare how general shape,

segment lengths, areas, perimeters and angles are affected. This focuses students' attention on the preservation of shape.

Note: Angle-angle similarity of triangles portion of CCSS 8.G.5 is introduced in this lesson.

1.3 7.G.1,

8.G.5

The context of copier size factors introduces students to the use of scale factors other than 2.

Note: Angle-angle similarity of triangles portion of CCSS 8.G.5 is taught in this lesson.


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2.1 7.G.1,

8.G.3

Students graph "Wumps" and "Imposters." They then compare corresponding parts to determine similarity.

Note: The dilation portion of CCSS 8.G.3 is touched on informally in this lesson. The vocabulary term “dilation” is not taught.

2.2 7.G.1,

8.G.3

Students graph a "Wump" hat and it's dilations, translations and distortions (non-similar) of the hat. They explore the effect of the rules

resulting in different "hats." Note: The dilation portion of CCSS 8.G.3 is touched on informally in this lesson. The vocabulary term

“dilation” is not taught.

2.3 7.G.1 Students continue to work with the Wump family as they investigate side lengths, angles, perimeters, and areas of similar rectangles and

triangles. Note: The vocabulary of scale factor is introduced.

3.1 7.G.1,

8.G.3

Students find scale factor from smaller to larger similar quadrilaterals. They also discover that the scale factor is multiplied by the side

lengths while the scale factor squared is multiplied by the area. Note: The dilation portion of CCSS 8.G.3 is touched on informally in this

lesson. The vocabulary term “dilation” is not taught.

3.2 7.G.1,

8.G.3

Students repeat the procedure in Problem 3.1 with triangles they also reverse the process by subdividing a triangle into four congruent

triangles, each similar to the original. Note: The dilation portion of CCSS 8.G.3 is touched on informally in this lesson. The vocabulary

term “dilation” is not taught.

3.3 7.G.1,

8.G.5

Students use a given scale factor to make a figure similar to a specific triangle or rectangle. They are required to find a scale factor and use

it to multiply and find missing side lengths. In the next investigation they will find missing lengths by using ratios.

Note: The angle-angle similarity of triangles portion of CCSS 8.G.5 is taught in Question C2.

4.1 7.G.1,

6.RP.1,

7.RP.2

Students first determine which parallelograms are similar and then they compare the ratios of corresponding lengths in one parallelogram

with the corresponding ratio of corresponding lengths in the other.

4.2 7.G.1,

7.RP.2,

8.G.5

Students repeat the process in problem 4.1 with triangles. Students also compare the information that is given by a scale factor or by the

ratios of corresponding sides.

Note: The angle-angle similarity of triangles portion of 8.G.5 is taught in this lesson.

4.3 7.G.1,

7.RP.2

Students use the information about the ratios of corresponding lengths within a figure to find missing measurements in two similar

polygons. Ratios formed this way for similar figures are equal.

5.1 7.G.1 Students use shadows to estimate the height of a real-world object.

5.2 7.G.1 Students use mirrors to estimate the height of an object.

5.3 7.G.1 Students use similar triangles to measure distances on the ground that cannot be measured directly.

(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 Comparing and Scaling 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in

the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This

recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5

per hamburger.”

6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number

line diagrams, or equations.

a) Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on

the coordinate plane. Use tables to compare ratios.

b) Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how

many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

7.RP.2* Recognize and represent proportional relationships between quantities.

a) 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.

b) Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c) 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.

d) 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.3 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.

* CCSS 7.RP.2a and d are 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, examining and explaining the meaning of the points (0,0) and (1,r) to teach these standards.


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Comparing and Scaling


1.1 6.RP.1 Students informally explore strategies for making quantitative comparisons, including ratio comparisons.

1.2 6.RP.1 Students explore another set of survey data and analyze a set of statements from different points of view.

1.3 6.RP.1, 7.RP.2 Provided sets of data, students are asked to compose statements that make comparisons.

2.1 6.RP.1, 6.RP.2,

7.RP.2

Students look at part-to-whole relationship in a recipe for juice. They use fractions, decimals or percents to make comparisons. Students informally

explore rates and ratios.

2.2 6.RP.2, 7.RP.2 Students compare two table sizes and two amounts of pizza. They are challenged to determine whether the two arrangements allocate pizza to people

in a fair way.

2.3 6.RP.3, 7.RP.2,

7.RP.3

Students systematically explore scaling ratios up and down to help answer questions or solve problems. Students create a table to keep track of their

information and look for patterns in the table. Note: This lesson must be extended to include graphing of the ordered pairs in the rate table to meet

the requirements of CCSS 7.RP.2d.

3.1 6.RP.3b, 6.EE.2,

7.RP.2b,7.RP.2c

Students compare three different kinds of calculators with different prices. The use rate tables as a way of scaling rates up and down.

3.2 6.RP.2, 6.RP.3,

7.RP.2b

Students use the formula d=rt and compute the constant overall rate of speed for a bike trip.

3.3 6.RP.2, 6.RP.3b,

7.RP.2b, 7.RP.2c,

7.RP.3

Students use unit rates to compare 2 ways of buying CD's. This is actually a system of linear equations, but is solved informally.

3.4 6.RP.3a, 6.RP.3b,

7.RP.2b

Students learn the meaning of the inverse of a ratio. (For example: 2/3 of a pizza per person, or 1.5 people per pizza.)

4.1 7.RP.2 Students are introduced to proportions as setting two ratios equal to one another. They learn to write four versions of a proportion and to solve to find

a missing value. This lesson teaches students to set up and solve proportions of the form a/b = c/d with 1 unknown. Lesson does not focus on directly

proportional relationships in px=q, table or graph form as stated in CCSS 7.RP.2.

4.2 7.RP.2 Students practice setting up and solving proportions. This lesson teaches students to set up and solve proportions of the form a/b = c/d with 1

unknown. Lesson does not focus on directly proportional relationships in px=q, table or graph form as stated in CCSS 7.RP.2.

4.3 7.RP.2 Students analyze proportional thinking and solidify this method of finding a missing part of an equivalent ratio. This lesson teaches students to set up

and solve proportions of the form a/b = c/d with 1 unknown. Lesson does not focus on directly proportional relationships in px=q, table or graph form

as stated in CCSS 7.RP.2.



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Common Core State Standards in Accentuate the Negative

6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.6* Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a) Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b) Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c) Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.7 Understand ordering and absolute value of rational numbers.

a) Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b) 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.

c) Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d) 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.

Continued….


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7.NS.1 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 diagram.

a) 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.

b) 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.

c) 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.

d) Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

a) 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.

b) 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.

c) Apply properties of operations as strategies to multiply and divide rational numbers.

d) 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.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

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

* CCSS 6.NS.6b is not taught in Accentuate the Negative(7) , but Problem 2.5 could be extended to teach this concept.


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Accentuate the Negative


1.1 6.NS.5,

7.NS.3

Students solve integer addition and subtraction problems using informal arithmetic reasoning in the context of a game.

1.2 6.NS.5,

6.NS.6a,

6.NS.7a,

7.NS.1a,

7.NS.1b

Students develop informal strategies for ordering and locating a number and its opposite on a number line in the context of changes in

temperature.

Note: It is recommended that teachers supplement this lesson to meet the requirements of CCSS 6.NS.7a: Interpret statements of

inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a

statement that –3 is located to the right of –7 on a number line oriented from left to right. AND CCSS 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.

1.3 6.NS.5 Students use informal arithmetic reasoning to connect direction and distance to the number line and represent their reasoning with

number sentences.

1.4 Students connect the operations of addition and subtraction of integers to actions on chip board displays.

Note: This method of teaching students to add and subtract integers does not clearly link to any CCSS, but may be used as a strategy

to help students struggling with the concept.

2.1 7.NS.1b,

7.NS.1c,

7.NS.1d

Students use the set model and the number line model to generalize strategies for adding integers.

2.2 6.NS.7c,

7.NS.1b,

7.NS.1c,

7.NS.1d

Students use the set model (chip model) and the number line model to generalize strategies for subtracting integers.

2.3 7.NS.1c,

7.NS.1d

Students explore the inverse relationship between addition and subtraction with integers. They formalize the algorithm a-(-b)=a+b and

a-b=a+(-b).

2.4 7.NS.1c,

7.NS.1d

Students use fact families to explore addition and subtraction with integers. They solve equations for missing parts in addition and

subtraction problems.


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2.5 6.NS.6b,

6.NS.6c

This problem extends graphing to all four quadrants using integers.

Note: CCSS 6.NS.6b contains the following language: "recognize that when two ordered pairs differ only by signs, the locations of the

points are related by reflections across one or both axes." This lesson must be extended to address this part of the standard.

3.1 7.NS.2a,

7.NS.2c

Students study a relay race game with repeated constant running speeds, distance and time to help them develop algorithms for

multiplication. They use their intuition from the context to complete multiplication number sentences in cases involving negative

numbers. They then generalize that reasoning to context-free multiplication.

Note: This lesson only deals with multiplication, not division.

3.2 7.NS.2a,

7.NS.2c

Students look at patterns of integer multiplication as another way to confirm how the signs should behave in multiplication situations

with integers.

Note: This lesson only deals with multiplication, not division.

3.3 7.NS.2b,

7.NS.2c

This problem builds on the algorithm for multiplication and relates multiplication and division facts to develop algorithms for division of

negative numbers.

3.4 7.NS.2 Students play and analyze a game, getting more practice with multiplying and dividing integers.

4.1 6.EE.1,

7.NS.2c

Students discover a need for an agreed-upon order of operations to establish uniqueness of solution for computations.

4.2 7.NS.2a,

7.NS.2c,

7.EE.1

This problem introduces students to the Distributive Property with addition through finding expressions for areas of rectangles with

subdivided edges.

4.3 7.NS.2a,

7.NS.2c,

7.EE.1

This problem defines the Distributive Property for multiplication over addition and subtraction and examines how it can be used to

simplify calculations. Students both expand and factor expressions and solve contextualized problems.



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Common Core State Standards in Moving Straight Ahead

6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or

inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one

quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship

between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving

motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between

distance and time.

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

7.EE.3 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.

Continued….


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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.

a) 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?

b) 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.

8.EE.7** Solve linear equations in one variable.

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.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.

* CCSS 7.EE.4 is taught throughout Moving Straight Ahead Investigation 3, but MSA does not teach inequalities.

**CCSS 8.EE.7 is taught throughout MSA.


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Moving Straight Ahead


1.1 7.EE.4a Students determine their walking rate in meters per second. They answer questions about time and distance using their constant walking

rate and write an equation that represents the distance walked over time at their constant walking rate.

1.2 7.EE.4a Students explore the walking rate of three students and look at the walking rate and its effect on various representations - table, graph and

equation.

1.3 7.EE.4a A new situation furthers students' understanding of constant rate and helps students generalize the constant rate of change for linear

relationships across linear situations. The new rates are the cost(y) per kilometer(x) for a walkathon.

1.4 7.EE.4a This problem poses two situations of a decreasing linear situation for every unit change in c, y decreases by a constant amount or increases

by a constant negative integer. One situation is in graphical form and one is in tabular form.

2.1 7.EE.4a,

8.EE.8a,

8.EE.8c

Students look at the context of a boy challenging his older brother to a race. The older brother wants his brother to win, but in a close

race. Students explore the question, "How long should the race be so that the younger brother wins in a close race?"

2.2 7.EE.4a,

8.EE.8a,

8.EE.8c

Students compare data using tables, graphs and equations. The younger brother from problem 2.1 now gets a head start and this sets up a

situation to ask questions about the point where the line crosses the y-axis. The y-intercept is named and students interpret it as a point

on a line, entry in a table, or as the constant b, in the equation y=mx+b.

2.3 7.EE.4a,

8.EE.8a,

8.EE.8c

Students explore constant rate and y-Intercept in a new context and solve problems and make decisions about linear relationships using

information given in tables, graphs and symbolic expressions.

2.4 7.EE.4a Students investigate various pledge plans that are represented by equations. Students predict constant rate, decide whether the

relationship is decreasing or increasing, and begin to make the connections among points on a line, a pair of data points in a table, and a

solution to an equation.

3.1 6.EE.5,

7.EE.1,

7.EE.3

Students make the connection among points on a line, pair of data points in a table, and solutions to equations.


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3.2 7.EE.1,

7.EE.3

Students explore the properties of equality pictorially. This provides a transition into solving equations symbolically.

3.3 7.EE.1,

7.EE.3

Students translate the pictorial form of linear equations to linear equations with variables. They use the properties of equality to solve the

equations symbolically. Students are asked to find one of the variables given a value of the other variable.

3.4 7.EE.1 Students find the solution to linear equations using the properties of equality and then compare this method with graphical and tabular

methods for solving equations.

3.5 6.EE.5,

7.EE.1,

8.EE.8

Students find the point of intersection of two lines by setting the y-values equal and then solving for x. They use the x-value to find the y-

value for the coordinates of the point of intersection of the two lines.

4.1 8.F.3,

8.F.4

Students investigate the “steepness” of a set of stairs using carpenters’ guidelines. This ratio of “rise to run” informally introduces the

concept of slope.

4.2 8.F.3,

8.F.4

Students find the ratio of vertical change to horizontal change between two points on a line. The connection between this ratio and

constant rate of change is made explicit. Students find the slope of a line given two points on the line. They then find the y-intercept using

either a table or graph and write an equation of the form y = mx + b, where m is the slope and b is the y-intercept.

Note: This lesson begins to address CCSS 8.EE.6 but does not use similar triangles to explain slope.

4.3 8.F.3,

8.F.4

Students apply their knowledge of slope and lines to explore sets of lines—those with the same slope (parallel lines) and those whose

slopes are negative reciprocals of each other (perpendicular lines). Graphing calculators help students explore many lines before they make

their conjectures.

4.4 8.F.3,

8.F.4

Problem 4.4 provides two applications. The first situation gives information about the coordinates of two points on the line of a linear

relationship. The context helps students find the needed information in a variety of ways. In the second situation, students find an

equation that represents a relationship. Again two points of the relationship are given, one being the y-intercept. Students will most likely

find the slope of the line using the coordinates of the two points. Students continue to deepen their understanding of what it means for a

relationship to be linear and connect a constant rate of change to the slope of a line.



15

Common Core State Standards in Filling and Wrapping

6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge

lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V

= b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.

Apply these techniques in the context of solving real-world and mathematical problems.

7.G.6* 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.

8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

* CCSS 7.G.6 is taught throughout Filling and Wrapping (7).


Filling and Wrapping


1.1 6.G.4 Students design nets for cubic boxes, fold them into boxes, and consider their areas.

1.2 6.G.4 Students design nets for a given non-cubic box and consider their area as well as the number of unit cubes that would fill the box.

1.3 6.G.4 Students are given nets and asked to analyze the boxes the nets make.

1.4 6.G.4 Students are given a real box, for which they design a net. They then cut their boxes to unfold to match the net they drew.

2.1 6.G.2 Students study rectangular prisms with a common volume of 24 in3.

Note: The problems in FW(7) Investigation 2 do not have fractional edge lengths. In order to fully meet CCSS 6.G.2, teachers must add

some problems with fractional edge lengths.


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2.2 6.G.2,

6.G.4

Students make conjectures about minimizing surface areas for rectangular prisms and are encouraged to formulate a general strategy for

finding the surface area of a rectangular prism.

2.3 6.G.2,

6.G.4

By thinking about filling boxes in layers, students develop the formula for volume of a rectangular prism.

3.1 Students compare the volumes and surface areas of a variety of prisms with regular bases and a common height.

Note: This lesson is a precursor for meeting CCSS 8.G.9.

3.2 8.G.9 Students extend their thinking about layers in rectangular prisms to develop a strategy for finding the volume of a cylinder.

3.3 7.G.6 Students consider nets for prisms and cylinders to determine the surface areas of these solids.

3.4 7.G.6 Students design a rectangular box with the same volume as a given cylinder.

Note: This lesson really synthesizes learning about volume. Although it does not fully address a specific standard, it is important to lead

students through this type of thinking so it is recommend you do not skip this lesson.

4.1 8.G.9 Students compare the volumes of a sphere and a cylinder by manipulating modeling clay.

4.2 8.G.9 Students compare the volumes of a cone and a cylinder to develop a formula for the volume of a cone. They repeat the experiment to

compare the volumes of a pyramid and a prism to develop a formula for the volume of a pyramid.

Note: In order to meet the requirements of 8.G.9 it is important that the students know and use formulas for the volume of a sphere,

cylinder, cone and pyramid.

4.3 8.G.9 Students compare the volumes of cones, cylinders, and spheres in an application problem.

5.1 Students consider how to double the volume of a rectangular prism and examine how other measures change as a result.

Note: CCSS do not have any standards that address the impact of changing 1 dimension of a prism on the volume of that prism.

However, this is a valuable way for kids to continue to study volume. This lesson should not be omitted.

5.2 Students study the effects of applying scale factors to the dimensions of rectangular prisms to create similar prisms. Note: CCSS do not

have any standards that address the impact of changing the dimensions of a prism by a common scale factor on the volume of that

prism. However, this is a valuable way for kids to continue to study volume. This lesson should not be omitted.

5.3 Students apply their knowledge of similarity and scale factors to explore the relationships between a model of a ship and the actual ship.



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Common Core State Standards in What do you Expect

7.SP.5 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.6 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.7* 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.

a) 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.

b) 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.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

a) 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.

b) 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.

c) 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?

* CCSS 7.SP.7 is taught throughout What Do You Expect (7).


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What do you Expect


1.1 7.SP.5, 7.SP.8

Students spin a spinner twice and look at the outcomes of a match/no-match, assigning points for each outcome. Students determine whether the game is fair.

1.2 7.SP.5, 7.SP.8

Students choose one colored marble from each of two buckets. If a blue and a red are chosen, the contestant wins. Students consider the long-term average result of the game.

1.3 7.SP.5, 7.SP.6

Students roll two number cubes and determine the product of the two numbers. Player A wins if the product is odd, and Player B wins if the product is even. Students must decide if this is a fair game and how many points each player can expect after 36 plays, 100 plays, etc.

2.1 7.SP.7, 7.SP.8

Students are introduced to an area model for analyzing probabilities with two-stage outcomes.

2.2 7.SP.7, 7.SP.8

In a new context, students continue to study multi-stage outcomes. Students design simulations to determine the experimental probabilities of ending up in each of two rooms at the end of a series of paths with forks.

2.3 7.SP.7, 7.SP.8

Students again use an area model to find theoretical probabilities in a novel context. In the previous problems, the second stage has been similar to the first stage, spinning the pointer of a second spinner or making a second choice along a path. In this case, the two stages of the outcome are of a different nature from each other.

3.1 7.SP.6 Students investigate two-stage outcomes in the context of a one-and-one free-throw situation. After determining experimental probabilities that the player will get a score of 0, 1, or 2, students find the theoretical probability by using an area model.

3.2 7.SP.6 Students determine the long-term average (expected value) for the one-and-one free-throw situation in the previous problem.

3.3 7.SP.6 Students use expected value to make decisions in a variety of different probability settings.

4.1 7.SP.5, 7.SP.8

Students are introduced to binomial outcomes by taking a four-item true/false quiz on which they must generate their answers at random.

4.2 7.SP.5, 7.SP.8

Students analyze all the combinations of boys and girls in families in the fictitious town of Ortonville. Each family has exactly five children.

4.3 7.SP.5, 7.SP.8

Students analyze the last five games in a baseball series where each team has a 50% chance of winning for each game and consider the probabilities of the series ending in various numbers of games.



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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?

prom/se (promoting rigorous outcomes in math and...

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