## Narrative

As in grade 6, students start grade 7 by studying scale drawings, an engaging geometric topic that supports the subsequent work on proportional relationships in the second and fourth units. It also makes use of grade 6 arithmetic understanding and skill, without arithmetic becoming the major focus of attention at this point. Geometry and proportional relationships are also interwoven in the third unit on circles, where the important proportional relationship between a circle's circumference and its diameter is studied. By the time students reach the fifth unit on operations with rational numbers, both positive and negative, students have had time to brush up on and solidify their understanding and skill in grade 6 arithmetic. The work on operations on rational numbers, with its emphasis on the role of the properties of operations in determining the rules for operating with negative numbers, is a natural lead-in to the work on expressions and equations in the next unit. Students then put their arithmetical and algebraic skills to work in the last two units, on angles, triangles, and prisms, and on probability and sampling.

### Unit 1: Scale Drawings

Work with scale drawings in grade 7 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students’ work with geometric measurement began with length and continued with area. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as an array of unit squares, or rows or columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of a rectangle to include rectangles with fractional side lengths. In grade 6, students built on their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra.

In this unit, students study scaled copies of pictures and plane figures, then apply what they have learned to scale drawings, e.g., maps and floor plans. This provides geometric preparation for grade 7 work on proportional relationships as well as grade 8 work on dilations and similarity.

Students begin by looking at copies of a picture, some of which are to scale, and some of which are not. They use their own words to describe what differentiates scaled and non-scaled copies of a picture. As the unit progresses, students learn that all lengths in a scaled copy are multiplied by a scale factor and all angles stay the same. They draw scaled copies of figures. They learn that if the scale factor is greater than 1, the copy will be larger, and if the scale factor is less than 1, the copy will be smaller. They study how area changes in scaled copies of an image.

Next, students study scale drawings. They see that the principles and strategies that they used to reason about scaled copies of figures can be used with scale drawings. They interpret and draw maps and floor plans. They work with scales that involve units (e.g., “1 cm represents 10 km”), and scales that do not include units (e.g., “the scale is 1 to 100”). They learn to express scales with units as scales without units, and vice versa. They understand that actual lengths are products of a scale factor and corresponding lengths in the scale drawing, thus lengths in the drawing are the product of the actual lengths and the reciprocal of that scale factor. They study the relationship between regions and lengths in scale drawings. Throughout the unit, they discuss their mathematical ideas and respond to the ideas of others (MP3, MP6). In the culminating lesson of this unit, students make a floor plan of their classroom or some other room or space at their school. This is an opportunity for them to apply what they have learned in the unit to everyday life (MP4).

In the unit, several lesson plans suggest that each student have access to a geometry toolkit. Each toolkit contains tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor (clear protractors with no holes that show radial lines are recommended), and an index card to use as a straightedge or to mark right angles. Providing students with these toolkits gives opportunities for students to develop abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Note that the study of scaled copies is limited to pairs of figures that have the same rotation and mirror orientation (i.e. that are not rotations or reflections of each other), because the unit focuses on scaling, scale factors, and scale drawings. In grade 8, students will extend their knowledge of scaled copies when they study translations, rotations, reflections, and dilations.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as representing, generalizing, and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Represent

• a scaled copy for a given scale factor (Lessons 3 and 5)
• distances using different scales (Lesson 11)
• relevant features of a classroom with a scale drawing (Lesson 13)

Generalize

• about corresponding distances and angles in scaled copies (Lesson 4)
• about scale factors greater than, less than, and equal to 1 (Lesson 5)
• about scale factors and area (Lessons 6 and 10)
• about scale factors with and without units (Lesson 12)

Explain

• how to use scale drawings to find actual distances (Lessons 7 and 11)
• how to use scale drawings to find actual distances, speed, and elapsed time (Lesson 8)
• how to use scale drawings to find actual areas (Lesson 12)

In addition, students are expected to describe features of scaled copies, justify and critique reasoning about scaled copies, and compare how different scales affect drawings. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.1.1 scaled copy
original
polygon
7.1.2 corresponding
scale factor

figure
segment
measurement
distance
corresponding
scale factor

original
7.1.5 reciprocal
7.1.6 area
one-dimensional
two-dimensional
squared
7.1.7 scale drawing
scale

represent
actual
three-dimensional
scaled copy
7.1.8 estimate
travel
constant speed
scale
7.1.9 floorplan
7.1.10 appropriate
dimension
actual
represent
7.1.11 scale without units
___ to ___
scale drawing
7.1.12 equivalent scales ___ to ___

### Unit 2: Introducing Proportional Relationships

In this unit, students develop the idea of a proportional relationship out of the grade 6 idea of equivalent ratios. Proportional relationships prepare the way for the study of linear functions in grade 8.

In grade 6, students learned two ways of looking at equivalent ratios. First, if you multiply both values in a ratio $$a:b$$ by the same positive number $$s$$ (called the scale factor) you get an equivalent ratio $$sa:sb$$. Second, two ratios are equivalent if they have the same unit rate. A unit rate is the “amount per 1” in a ratio; the ratio $$a:b$$ is equivalent to $$\frac{a}{b}:1$$, and $$\frac{a}{b}$$ is a unit rate giving the amount of the first quantity per unit of the second quantity. You could also talk about the amount of the second quantity per unit of the first quantity, which is the unit rate $$\frac{b}{a}$$, coming from the equivalent ratio $$1:\frac{b}{a}$$.

In a table of equivalent ratios, a multiplicative relationship between the pair of rows is given by a scale factor. By contrast, the multiplicative relationship between the columns is given by a unit rate. Every number in the second column is obtained by multiplying the corresponding number in the first column by one of the unit rates, and every number in the first column is obtained by multiplying the number in the second column by the other unit rate. The relationship between pairs of values in the two columns is called a proportional relationship, the unit rate that describes this relationship is called a constant of proportionality, and the quantity represented by the right column is said to be proportional to the quantity represented by the left. (Although a proportional relationship between two quantities represented by $$a$$ and $$b$$ is associated with two constants of proportionality, $$\frac{a}{b}$$ and $$\frac{b}{a}$$, throughout the unit, the convention is if $$a$$ and $$b$$ are, respectively, in the left and right columns of a table, then $$\frac{b}{a}$$ is the constant of proportionality for the relationship represented by the table.)

For example, if a person runs at a constant speed and travels 12 miles in 2 hours, then the distance traveled is proportional to the time elapsed, with constant of proportionality 6, because $$\displaystyle \text{distance} = 6\boldcdot\text{time}.$$The time elapsed is proportional to distance traveled with constant of proportionality $$\frac16$$, because $$\displaystyle \text{time} = \frac16\boldcdot\text{distance}.$$Students learn that any proportional relationship can be represented by an equation of the form $$y = kx$$ where $$k$$ is the constant of proportionality, that its graph lies on a line through the origin that passes through Quadrant I, and that the constant of proportionality indicates the steepness of the line. By the end of the unit, students should be able to easily work with common contexts associated with proportional relationships (such as constant speed, unit pricing, and measurement conversions) and be able to determine whether a relationship is proportional or not.

Because this unit focuses on understanding what a proportional relationship is, how it is represented, and what types of contexts give rise to proportional relationships, the contexts have been carefully chosen. The first tasks in the unit employ contexts such as servings of food, recipes, constant speed, and measurement conversion, that should be familiar to students from the grade 6 course. These contexts are revisited throughout the unit as new aspects of proportional relationships are introduced.

Associated with the contexts from the grade 6 course are derived units: miles per hour; meters per second; dollars per pound; or cents per minute. In this unit, students build on their grade 6 experiences in working with a wider variety of derived units, such as cups of flour per tablespoon of honey, hot dogs eaten per minute, and centimeters per millimeter. The tasks in this unit avoid discussion of measurement error and statistical variability, which will be addressed in later units.

On using the terms quantity, ratio, proportional relationship, unit rate, and fraction. In these materials, a quantity is a measurement that is or can be specified by a number and a unit, e.g., 4 oranges, 4 centimeters, “my height in feet,” or “my height” (with the understanding that a unit of measurement will need to be chosen, MP6). The term ratio is used to mean a type of association between two or more quantities. A proportional relationship is a collection of equivalent ratios.

A unit rate is the numerical part of a rate per 1 unit, e.g., the 6 in 6 miles per hour. The fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are never called ratios. The fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are identified as “unit rates” for the ratio $$a : b$$. In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to $$a$$ to $$b$$, $$a:b$$, and $$\frac{a}{b}$$ as “ratios.”

In grades 6–8, students write rates without abbreviated units, for example as “3 miles per hour” or “3 miles in every 1 hour.” Use of notation for derived units such as $$\frac{\text{mi}}{\text{hr}}$$ waits for high school—except for the special cases of area and volume. Students have worked with area since grade 3 and volume since grade 5. Before grade 6, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation in grade 6, they also use $$\text{ cm}^2$$ and $$\text{ cm}^3$$.

A fraction is a point on the number line that can be located by partitioning the segment between 0 and 1 into equal parts, then finding a point that is a whole number of those parts away from 0. A fraction can be written in the form $$\frac a b$$ or as a decimal.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as comparing, interpreting, and generalizing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Compare

• drink mixtures and figures (Lesson 1)
• approaches to solving problems involving proportional relationships (Lesson 6)
• proportional relationships with nonproportional relationships (Lesson 8)
• tables, descriptions, and graphs representing the same situations (Lesson 10)
• graphs of proportional relationships (Lesson 12)

Interpret

• representations showing equivalent ratios (Lesson 1)
• tables showing equivalent ratios (Lesson 2)
• situations involving proportional relationships (Lesson 6 and 9)
• how a graph represents features of a situation (Lesson 11)

Generalize

• about proportional relationships (Lesson 4)
• about equations that represent proportional relationships (Lesson 5)
• about how a constant of proportionality is represented by graphs and tables (Lesson 13)

In addition, students are expected to describe proportional relationships and constants of proportionality, explain how to determine whether or not a relationship is proportional and how to compare and represent situations with different constants of proportionality, justify whether or not a relationship is proportional, and represent proportional and nonproportional relationships in multiple ways.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.2.1 equivalent ratios
7.2.2 constant of proportionality
proportional relationship

value
equivalent ratios
row
column
7.2.3 ___ is proportional to ___
relate
constant
reciprocal
per
7.2.4 equation
quotient
___ is proportional to ___
situation
7.2.6 equation
quotient
7.2.7 constant of proportionality
proportional relationship
7.2.8 constant
7.2.10 origin
coordinate plane

plot
7.2.11 quantity
axes
coordinates
7.2.13 $$x$$-coordinate
$$y$$-coordinate
origin
7.2.14 axes
7.2.15 reasonable

### Unit 3: Measuring Circles

In this unit, students extend their knowledge of circles and geometric measurement, applying their knowledge of proportional relationships to the study of circles. They extend their grade 6 work with perimeters of polygons to circumferences of circles, and recognize that the circumference of a circle is proportional to its diameter, with constant of proportionality $$\pi$$. They encounter informal derivations of the relationship between area, circumference, and radius.

The unit begins with activities designed to help students come to a more precise understanding of the characteristics of a circle (MP6): a “circle” is the set of points that are equally distant from a point called the “center”; the diameter of a circle is a line segment that passes through its center with endpoints on the circle; the radius is a line segment with one endpoint on the circle and one endpoint at the center. Students identify these characteristics in a variety of contexts (MP2). They use compasses to draw circles with given diameters or radii, and to copy designs that involve circles. Using their newly gained familiarity with circumference and diameter, students measure circular objects, investigating the relationship between measurements of circumference and diameter by making tables and graphs.

The second section involves area. Students encounter two informal derivations of the fact that the area of a circle is equal to $$\pi$$ times the square of its radius. The first involves dissecting a disk into sectors and rearranging them to form a shape that approximates a parallelogram of height $$r$$ and width $$2\pi r$$. A second argument involves considering a disk as formed of concentric rings, “cutting” the rings with a radius, and “opening” the rings to form a shape that approximates an isosceles triangle of height $$r$$ and base $$2\pi \boldcdot 2r$$.

In the third and last section, students select and use formulas for the area and circumference of a circle to solve abstract and real-world problems that involve calculating lengths and areas. They express measurements in terms of $$\pi$$ or using appropriate approximations of $$\pi$$ to express them numerically. In grade 8, they will use and extend their knowledge of circles and radii at the beginning of a unit on dilations and similarity.

On using the term circle. Strictly speaking, a circle is one-dimensional—the boundary of a two-dimensional region rather than the region itself. Because students are not yet expected to make this distinction, these materials refer to both circular regions (i.e., disks) and boundaries of disks as “circles,” using illustrations to eliminate ambiguity.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as generalizing, justifying, and interpreting. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Generalize

• about categories for sorting circles (Lesson 2)
• about the relationship between circumference and diameter (Lesson 3)
• about circumference and rotation (Lesson 5)
• about the relationship between radius and area of a circle (Lesson 8)

Justify

• reasoning about circumference and perimeter (Lesson 4)
• estimates for the areas of circles (Lesson 7)
• reasoning about areas of curved figures (Lesson 9)
• reasoning about the cost of stained glass windows (Lesson 11)

Interpret

• situations involving circles (Lessons 5 and 8)
• floor plans and maps (Lesson 6)
• situations involving circumference and area (Lesson 10)

In addition, students are expected to critique reasoning about circles and circle measurements, explain reasoning, including about different approximations of pi, and describe features of graphs and of deconstructed circles.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.3.1 relationship
perimeter
diameter
circumference

center (of a circle)
circle
7.3.3 pi
7.3.4 half-circle
rotation
approximation
7.3.5 diameter
circumference
pi

travel
7.3.6 approximate
estimate
7.3.7 area of a circle
7.3.8 squared
formula
area of a circle
7.3.10 squared
center (of a circle)
formula
7.3.11 design

### Unit 4: Proportional Relationships and Percentages

Students began their work with ratios, rates, and unit rates in grade 6, representing them with expressions, tape diagrams, double number line diagrams, and tables. They used these to reason about situations involving color mixtures, recipes, unit price, discounts, constant speed, and measurement conversions. They extended their understanding of rates to include percentages as rates per 100, reasoning about situations involving whole-number percentages. They did not use the terms “proportion” and “proportional relationship” in grade 6.

A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in grade 7. In previous grade 7 units, students worked with scale factors and scale drawings, and with proportional relationships and constants of proportionality. Although students have learned how to compute quotients of fractions in grade 6, these first units on scaling and proportional relationships do not require such calculations, allowing the new concept (scaling or proportional relationship) to be the main focus.

In this unit, students deepen their understanding of ratios, scale factors, unit rates (also called constants of proportionality), and proportional relationships, using them to solve multi-step problems that are set in a wide variety of contexts that involve fractions and percentages.

In the first section of the unit, students extend their use of ratios and rates to problems that involve computing quotients of fractions, and interpreting these quotients in contexts such as scaling a picture or running at constant speed (MP2). They use long division to write fractions presented in the form $$\frac a b$$ as decimals, e.g., $$\frac{11}{30} = 0.3\overline6$$.

The section begins by revisiting scale factors and proportional relationships, each of which has been the focus of a previous unit. Both of these concepts can be used to solve problems that involve equivalent ratios. However, it is often more efficient to view equivalent ratios as pairs that are in the same proportional relationship rather than seeing one pair as obtained by multiplying both entries of the other by a scale factor. From the scaling perspective, to see that one ratio is equivalent to another or to generate a ratio equivalent to a given ratio, a scale factor is needed—which may be different for each pair of ratios in the proportional relationship. From the proportional relationship perspective, all that is needed is the constant of proportionality—which is the same for every ratio in the proportional relationship.

The second section of the unit is about percent increase and decrease. Students consider situations for which percentages can be used to describe a change relative to an initial amount, e.g., prices before and after a 25% increase. They begin by considering situations with unspecified amounts, e.g., matching tape diagrams with statements such as “Compared with last year’s strawberry harvest, this year’s strawberry harvest increased by 25%”. They next consider situations with a specified amount and percent change, or with initial and final amounts, using double number line diagrams to find the unknown amount or percent change. Next, they use equations to represent such situations, using the distributive property to show that different expressions for the same amount are equivalent, e.g., $$x - 0.25x=0.75x$$. So far, percent change in this section has focused on whole-number rates per 100, e.g., 75%. The last lesson asks students to compute fractional percentages of given amounts.

In the third section of the unit, students begin by using their abilities to find percentages and percent rates to solve problems that involve sales tax, tip, discount, markup, markdown, and commission (MP2). The remaining lessons of the section continue the focus on situations that can be described in terms of percentages, but the situations involve error rather than change—describing an incorrect value as a percentage of the correct value rather than describing an initial amount as a percentage of a final amount (or vice versa).

The last section of the unit consists of a lesson in which students analyze news items that involve percent increase or decrease. In small groups, students identify important quantities in a situation described in a news item, use diagrams to map the relationship of of the quantities, and reason mathematically to draw conclusions (MP4). This is an opportunity to choose an appropriate type of diagram (MP5), to state the meanings of symbols used in the diagram, to specify units of measurement, and to label the diagram accurately (MP6). Each group creates a display to communicate its reasoning and critiques the reasoning shown in displays from other groups (MP3).

These materials follow specific conventions for the use of language around ratios, rates, and proportional relationships. Please see the unit narrative for the second unit to read about those conventions.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as interpreting, explaining, and representing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret

• situations involving constant speed (Lesson 2)
• concrete problems involving percent increase and decrease (Lesson 7)
• problems involving sales tax and tip (Lesson 10)
• concrete situations involving percent error (Lesson 14)

Explain

• how to solve concrete and abstract problems involving an amount plus (or minus) a fraction of that amount (Lesson 4)
• how to solve percent change problems (Lesson 6)
• strategies for solving percent problems with fractional percentages (Lesson 9)
• how to measure lengths and interpret measurement error (Lesson 13)
• strategies for solving percent error problems (Lesson 14)

Represent

• situations involving percent increase and decrease (Lesson 8)
• situations with percent error (Lesson 15)
• situations from the news involving percent change (Lesson 16)

In addition, students are expected to compare measurements, scale factors, and decimal and fraction representations, compare representations of an increase (or decrease) of an amount by a fraction or decimal, generalize about using constants of proportionality to solve problems efficiently and about relationships with percent increase and decrease, and justify why specific information is needed to solve percent change problems.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.4.1 percentage
7.4.2 unit rate
7.4.4 (a fraction) more than
(a fraction) less than
initial / original amount
final / new amount
tape diagram
distributive property
7.4.5 repeating decimal
long division

decimal representation
7.4.6 percent increase
percent decrease
(a fraction) more than
(a fraction) less than
7.4.7 discount initial / original amount
final / new amount
7.4.10 sales tax
tax rate
tip
percent increase
7.4.11 interest
commission
markup
markdown
percent decrease
7.4.12 percentage
discount
7.4.13 measurement error
7.4.14 percent error
temperature
degrees Fahrenheit

### Unit 5: Rational Number Arithmetic

In grade 6, students learned that the rational numbers comprise positive and negative fractions. They plotted rational numbers on the number line and plotted pairs of rational numbers in the coordinate plane. In this unit, students extend the operations of addition, subtraction, multiplication, and division from fractions to all rational numbers, written as decimals or in the form $$\frac a b$$.

The unit begins by revisiting ideas familiar from grade 6: how signed numbers are used to represent quantities such as measurements of temperature and elevation, opposites (pairs of numbers on the number line that are the same distance from zero), and absolute value.

In the second section of the unit, students extend addition and subtraction from fractions to all rational numbers. They begin by considering how changes in temperature and elevation can be represented—first with tables and number line diagrams, then with addition and subtraction expressions and equations. Initially, physical contexts provide meanings for sums and differences that include negative numbers. Students work with numerical addition and subtraction expressions and equations, becoming more fluent in computing sums and differences of signed numbers. Using the meanings that they have developed for addition and subtraction of signed numbers, they write equivalent numerical addition and subtraction expressions, e.g., $$\text-8 + \text-3$$ and $$\text-8 – 3$$; and they write different equations that express the same relationship.

The third section of the unit focuses on multiplication and division. It begins with problems about position, direction, constant speed, and constant velocity in which students represent quantities with number line diagrams and tables of numerical expressions with signed numbers. This allows products of signed numbers to be interpreted in terms of position and direction, using the understanding that numbers that are additive inverses are located at the same distance but opposite sides of the starting point. These examples illustrate how multiplication of how multiplication of fractions extends to rational numbers. The third lesson of this section focuses on computing products of signed numbers and is optional. In the fourth lesson, students use the relationship between multiplication and division to understand how division extends to rational numbers. In the process of solving problems set in contexts (MP4), they write and solve multiplication and division equations.

In the fourth section of the unit, students work with expressions that use the four operations on rational numbers, making use of structure (MP7), e.g., to see without calculating that the product of two factors is positive because the values of the factors are both negative. They extend their use of the “next to” notation (which they used in expressions such as $$5x$$ and $$6 (3 + 2)$$ in grade 6) to include negative numbers and products of numbers, e.g., writing $$\text-5x$$ and $$(\text-5) (\text-10)$$ rather than $$(\text-5)\boldcdot (x)$$ and $$(\text-5)\boldcdot (\text-10)$$. They extend their use of the fraction bar to include variables as well as numbers, writing $${\text-8.5}\div{x}$$ as well as $$\frac{\text-8.5}{x}$$. They solve problems that involve interpreting negative numbers in context, for instance, when a negative number represents a rate at which water is flowing (MP2).

In the fifth section of the unit, students begin working with linear equations in one variable that have rational number coefficients. The focus of this section is representing situations with equations (MP4) and what it means for a number to be a solution for an equation, rather than methods for solving equations. Such methods are the focus of a later unit.

The last section of the unit is a lesson in which students use rational numbers in the context of stock-market situations, finding values of quantities such as the value of a portfolio or changes due to interest and depreciation (MP4).

Note. In these materials, an expression is built from numbers, variables, operation symbols ($$+$$, $$-$$, $$\cdot$$, $$\div$$), parentheses, and exponents. (Exponents—in particular, negative exponents—are not a focus of this unit. Students work with integer exponents in grade 8 and non-integer exponents in high school.) An equation is a statement that two expressions are equal, thus always has an equal sign. Signed numbers include all rational numbers, written as decimals or in the form $$\frac a b$$.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as interpreting, representing, and generalizing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret

• situations involving signed numbers (throughout Unit)
• tables with signed numbers (Lesson 3)
• bank statements with signed numbers (Lesson 4)

Represent

• addition of signed numbers on a number line (Lesson 2)
• situations involving signed numbers (Lessons 3 and 11)
• changes in elevation (Lesson 6)
• position, speed, and direction (Lesson 8)

Generalize

• about differences and magnitude (Lesson 6)
• about multiplying negative numbers (Lesson 9)

In addition, students are expected to justify reasoning about distances on a number line and about negative numbers, account balances, and debt. Students are also expected to explain how to determine changes in temperature, how to find information using inverses, and how to model situations involving signed numbers.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.5.1 absolute value
degrees Celsius
vertical
elevation
sea level
positive number
negative number
7.5.2 signed numbers temperature
number line
7.5.3 sum
opposite
expression
7.5.4 deposit
withdrawal

account balance
debt
7.5.6 difference distance
7.5.7 absolute value
$$x$$-coordinate
$$y$$-coordinate
7.5.8 velocity
7.5.11 solution (to an equation)
factor
7.5.13 rational number sum
difference
7.5.15 variable
multiplicative inverse
opposite
solution (to an equation)
7.5.16 operation
7.5.17 increase
decrease

### Unit 6: Expressions, Equations, and Inequalities

In this unit, students solve equations of the forms $$px+q=r$$ and $$p(x+q)=r$$, and solve related inequalities, e.g., those of the form $$px+q > r$$ and $$px+q \geq r$$, where $$p$$, $$q$$, and $$r$$ are rational numbers.

In the first section of the unit, students represent relationships of two quantities with tape diagrams and with equations, and explain correspondences between the two types of representations (MP1). They begin by examining correspondences between descriptions of situations and tape diagrams, then draw tape diagrams to represent situations in which the variable representing the unknown is specified. Next, they examine correspondences between equations and tape diagrams, then draw tape diagrams to represent equations, noticing that one tape diagram can be described by different (but related) equations. At the end of the section, they draw tape diagrams to represent situations in which the variable representing the unknown is not specified, then match the diagrams with equations. The section concludes with an example of the two main types of situations examined, characterized in terms of whether or not they involve equal parts of an amount or equal and unequal parts of an amount, and as represented by equations of different forms, e.g., $$6(x+8) =72$$ and $$6x+8 =72$$. This initiates a focus on seeing two types of structure in the situations, diagrams, and equations of the unit (MP7).

In the second section of the unit, students solve equations of the forms $$px+q=r$$ and $$p(x+q)=r$$, then solve problems that can be represented by such equations (MP2). They begin by considering balanced and unbalanced “hanger diagrams,” matching hanger diagrams with equations, and using the diagrams to understand two algebraic steps in solving equations of the form $$px+q=r$$: subtract the same number from both sides, then divide both sides by the same number. Like a tape diagram, the same balanced hanger diagram can be described by different (but related) equations, e.g., $$3x + 6 = 18$$ and $$3(x+2) = 18$$. The second form suggests using the same two algebraic steps to solve the equation, but in reverse order: divide both sides by the same number, then subtract the same number from both sides. Each of these algebraic steps and the associated structure of the equation is illustrated by hanger diagrams (MP1, MP7).

So far, the situations in the section have been described by equations in which $$p$$ is a whole number, and $$q$$ and $$r$$ are positive (and frequently whole numbers). In the remainder of the section, students use the algebraic methods that they have learned to solve equations of the forms $$px+q=r$$ and $$p(x+q)=r$$ in which $$p$$, $$q$$, and $$r$$ are rational numbers. They use the distributive property to transform an equation of one form into the other (MP7) and note how such transformations can be used strategically in solving an equation (MP5). They write equations in order to solve problems involving percent increase and decrease (MP2).

In the third section of the unit, students work with inequalities. They begin by examining values that make an inequality true or false, and using the number line to represent values that make an inequality true. They solve equations, examine values to the left and right of a solution, and use those values in considering the solution of a related inequality. In the last two lessons of the section, students solve inequalities that represent real-world situations (MP2).

In the last section of the unit, students work with equivalent linear expressions, using properties of operations to explain equivalence (MP3). They represent expressions with area diagrams, and use the distributive property to justify factoring or expanding an expression.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as comparing, explaining, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Compare

• stories with corresponding tape diagrams (Lesson 2)
• tape diagrams with corresponding equations (Lesson 3)
• hanger diagrams and equations (Lesson 7)
• solution pathways (especially Lesson 10)
• descriptions of situations with corresponding inequalities (Lesson 16)

Explain

• strategies for using hanger diagrams to solve equations (Lesson 8)
• different strategies for solving equations (Lesson 9) and inequalities (Lesson 14)
• reasoning about situations, tape diagrams, and equations (Lesson 12)
• strategies for identifying and writing equivalent expressions (Lesson 22)

Justify

• reasoning about inequalities (Lesson 13)
• reasoning about solutions to inequalities (Lesson 15)
• the need for specific information in order to write and solve inequalities (Lesson 17)
• reasoning about the distributive property (Lesson 19)
• whether different sequences of calculations give the same result (Lesson 23)

In addition, students are expected to interpret solutions to equations, interpret and represent non-proportional situations with constant rates of change, represent non-proportional situations using tape diagrams, describe the structure of equations and tape diagrams, critique reasoning of peers about expressions and corresponding diagrams, critique reasoning about solving equations, critique reasoning about equivalent expressions, and generalize about solving equations and about when expressions are equivalent.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow were it was first introduced.

lesson new terminology
receptive productive
7.6.2 unknown amount
7.6.3 equivalent expressions
commutative (property)
7.6.4 unknown amount
relationship
7.6.6 variable
7.6.7 balanced hanger
each side (of an equation)
7.6.8 equivalent expression
each side (of an equation)
7.6.9 operation
solve
7.6.10 distribute
substitute
7.6.13 inequality
less than or equal to
greater than or equal to
open / closed circle
less than
greater than
7.6.14 solution to an inequality
boundary
direction (of an inequality)
less than or equal to
greater than or equal to
substitute
7.6.15 open / closed circle
7.6.16 solution to an inequality
7.6.17 inequality
7.6.18 term
7.6.19 factor (an expression)
expand (an expression)
7.6.20 combine like terms term
commutative (property)
7.6.21 distribute
7.6.22 associative property factor (an expression)
expand (an expression)

### Unit 7: Angles, Triangles, and Prisms

In this unit, students investigate whether sets of angle and side length measurements determine unique triangles or multiple triangles, or fail to determine triangles. Students also study and apply angle relationships, learning to understand and use the terms “complementary,” “supplementary,” “vertical angles,” and “unique” (MP6). The work gives them practice working with rational numbers and equations for angle relationships. Students analyze and describe cross-sections of prisms, pyramids, and polyhedra. They understand and use the formula for the volume of a right rectangular prism, and solve problems involving area, surface area, and volume (MP1, MP4). Students should have access to their geometry toolkits so that they have an opportunity to select and use appropriate tools strategically (MP5).

Note: It is not expected that students memorize which conditions result in a unique triangle, are impossible to create a triangle, or multiple possible triangles. Understanding that, for example, SSS information results in zero or exactly one triangle will be explored in high school geometry. At this level, students should attempt to draw triangles with the given information and notice that there is only one way to do it (or that it is impossible to do).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as critiquing, explaining, interpreting, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• reasoning about measuring angles (Lesson 1)
• reasoning about decomposition of prisms (Lesson 13)
• reasoning about surface area of prisms (Lesson 14)

Explain

• how to measure angles (Lesson 2)
• how to find unknown angle measurements (Lessons 4 and 5)
• how to find the volume of prisms (Lessons 12 and 13)
• how to find the surface area of prisms (Lesson 14)

Interpret

• situations involving intersecting lines in order to form a conjecture (Lesson 3)
• which information is relevant to answer questions (Lesson 4)
• equations representing angle measurements (Lesson 5)
• situations involving volume and surface area (Lesson 15 and 16)

Justify

• whether or not shapes are identical copies (Lesson 6)
• whether or not measurements determine identical copies (Lesson 9)
• whether or not measurements determine unique triangles (Lesson 10)

In addition, students are expected to use language to compare angle measurements, compare triangles in a set, compare cross sections of figures, describe characteristics of pattern blocks, describe positioning and movement of side lengths and angles, and describe cross sections of prisms and pyramids. Students also have opportunities to generalize about patterns of angle measurements, about categories for unique triangles, and about categories for cross sections.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.7.1 straight angle

degree
right angle
7.7.2 supplementary
complementary

angle measure
protractor
measurement error
degrees
7.7.3 vertical angles
intersect
vertex (of an angle)
7.7.4 supplementary
vertical angles
7.7.5 perpendicular complementary
7.7.6 identical copy
condition
angle measure
side length
7.7.7 compass
different triangle
intersect
identical copy
segment
7.7.8 condition
different triangle
7.7.9 unique triangle
parallel
7.7.10 protractor
compass
7.7.11 cross section
base (of a prism or pyramid)

vertex (of a pyramid)
face
prism
pyramid

perpendicular
parallel
7.7.12 volume
cross section
base (of a prism or pyramid)
7.7.14 face
perimeter
7.7.15 surface area

### Unit 8: Probability and Sampling

In this unit, students understand and use the terms “event,” “sample space,” “outcome,” “chance experiment,” “probability,” “simulation,” “random,” “sample,” “random sample,” “representative sample,” “overrepresented,” “underrepresented,” “population,” and “proportion.” They design and use simulations to estimate probabilities of outcomes of chance experiments and understand the probability of an outcome as its long-run relative frequency. They represent sample spaces (that is, all possible outcomes of a chance experiment) in tables and tree diagrams and as lists. They calculate the number of outcomes in a given sample space to find the probability of a given event. They consider the strengths and weaknesses of different methods for obtaining a representative sample from a given population. They generate samples from a given population, e.g., by drawing numbered papers from a bag and recording the numbers, and examine the distributions of the samples, comparing these to the distribution of the population. They compare two populations by comparing samples from each population.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, explaining, justifying, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• observations and predictions during a game (Lesson 1)
• patterns observed in repeated experiments (Lesson 4)
• chance experiments to model situations (Lessons 6 and 7)
• a simulation used to model a situation (Lesson 10)
• observations about data sets (Lessons 11 and 17)

Explain

• predictions (Lesson 2)
• how to determine which events are more likely (Lesson 3)
• possible differences in experimental and theoretical probability (Lesson 5)
• how to use simulations to estimate probability (Lesson 7)
• how to use a simulation to answer questions about the situation (Lesson 10)

Justify

• whether situations are surprising and possible (Lesson 4)
• which samples are or are not representative of a larger population (Lesson 13)
• which samples correspond with each show, which show is most appropriate for a commercial, and whether a movie is eligible for an award (Lesson 15)
• reasoning about samples and populations (Lesson 16)
• whether or not differences between samples are meaningful (Lesson 18, 19, and 20)

Compare

• sample spaces and probably of outcomes for different spinners (Lesson 5)
• methods for writing sample spaces (Lesson 8)
• heights of two groups (Lesson 11)
• measures of center with samples (Lesson 13)
• sampling methods (Lesson 14)
• populations based on samples (Lessons 18 and 20)

In addition, students are expected to critique predictions about the mean of random samples, and generalize about samples spaces, predictions, sampling, and fairness. Students also have opportunities to use language to represent data from repeated experiments, represent probabilities and sample spaces, and interpret situations involving sample spaces, probability, and populations.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.8.1 more likely
less likely
7.8.2 event
chance experiment
outcome

equally likely as not
likely
unlikely
impossible
certain
7.8.3 probability
random
sample space
outcome
7.8.5 simulation probability
random
7.8.7 event
simulation
7.8.8 tree (diagram) sample space
7.8.9 tree (diagram)
distribution
very different
overlap
mean
median

dot plot
7.8.12 population
sample

survey
7.8.13 representative sample
measure of center
distribution
center (of a distribution)
7.8.14 random sample
7.8.15 interquartile range (IQR)
measure of variability
box plot
population
sample

random sample
symmetric
7.8.16 proportion representative sample
7.8.17 interquartile range (IQR)
measure of variability
7.8.18 meaningful difference overlap
measure of center
7.8.20 meaningful difference

### Unit 9: Putting it All Together

In this optional unit, students use concepts and skills from previous units to solve three groups of problems. In calculating or estimating quantities associated with running a restaurant, e.g., number of calories in one serving of a recipe, expected number of customers served per day, or floor space, they use their knowledge of proportional relationships, interpreting survey findings, and scale drawings. In estimating quantities such as age in hours and minutes or number of times their hearts have beaten, they use measurement conversions and consider accuracy of their estimates. Estimation of area and volume measurements from length measurements introduces considerations of measurement error. In designing a five-kilometer race course for their school, students use their knowledge of measurement and scale drawing. They select appropriate tools and methods for measuring their school campus, build a trundle wheel and use it to make measurements, make a scale drawing of the course on a map or a satellite image of the school grounds, and describe the number of laps, start, and finish of the race.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as justifying, representing, and critiquing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Justify

• reasoning about the nutritional value of recipes (Lesson 1)
• choices and predictions in the context of running a restaurant (Lesson 2)
• reasoning about length, area, and volume in the context of a restaurant (Lesson 4)

Represent

• costs of ingredients in a spreadsheet (Lesson 2)
• situations using expressions and equations (Lesson 7)
• a map of a designed race course (Lesson 13)

Critique

• peer reasoning about calculations of age, heart beats, and hairs (Lesson 6)
• peer reasoning about percent error in length measurement (Lesson 8)
• peer reasoning about percent error in area and volume measurement (Lesson 9)
• peer methods of measuring distance (Lesson 10)

In addition, students are also expected to describe methods for measuring distance, including how to build and use a trundle wheel, and to compare advantages and disadvantages of different methods.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
7.9.1 serving