In this unit, students develop the idea of a proportional relationship. They work with proportional relationships that are represented in tables, as equations, and on graphs. This builds on grade 6 work with equivalent ratios and helps prepare students for the study of linear functions in grade 8.

Students begin by looking at tables. In a table of equivalent ratios, a multiplicative relationship between a pair of rows is given by a scale factor, while the multiplicative relationship between the columns is given by a unit rate. Students learn that the relationship between pairs of values in the two columns is called a "proportional relationship," and the unit rate that describes this relationship is called a "constant of proportionality."

Next, students use equations of the form \(y = kx\) to represent proportional relationships and solve problems. They determine whether given tables and equations could represent a proportional relationship.

Then students investigate graphs of proportional relationships. They recognize that the graph of a proportional relationship is a straight line through \((0, 0)\). They interpret points on the graph, including the point \((1, k)\). Here is an example of a graph, an equation, and a table that all represent the same proportional relationship.

Three representations of the linear function y = 7 fourths times x. Horizontal axis scale is 0 to 8 by 1’s. Vertical axis scale is 0 to 10 by 1’s. The graph of the function has the points (1 comma the fraction 7 over 4) and (4 comma 7). The table has rows (x comma y), (0 comma 0), (1 comma the fraction 7 over 4), (3 comma the fraction 21 over 4), (4 comma 7).  

By the end of the unit, students should be comfortable working with common contexts associated with proportional relationships (such as constant speed, unit pricing, and measurement conversions) and be able to determine whether or not a relationship is proportional. In a later unit, students will apply proportional reasoning to solve multi-step problems and to calculate more complex rates.

A note on using the terms "ratio," "proportional relationship," and "unit rate":

In these materials, the term "ratio" is used to mean a type of association between two or more quantities. A quantity is a measurement that can be specified by a number and a unit, for example 4 oranges, 4 centimeters, or “my height in feet.” A proportional relationship is a collection of equivalent ratios.

A unit rate is the numerical part of a rate per 1 unit, for example, 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 the study of ratios, rates, and proportional relationships—students discard the term “unit rate” and start 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.

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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students apply their knowledge of proportional relationships to the context of measuring circles. They learn the relationships between radius, diameter, circumference, and area of circles and use these relationships to solve problems. This builds on students’ work from previous grades with perimeter and area of polygons. Students will build on this work in grade 8 when they study the volume of spheres, cylinders, and cones.

The unit begins with activities designed to help build up students’ vocabulary for describing circles more precisely. The terms "center," "radius," "diameter," and "circumference" are introduced. Then students investigate the relationship between circumference and diameter and see that it is a proportional relationship. They apply this relationship to solve problems.

Next, students explore the area of circular regions. They see an informal derivation that shows where the formula \(A = \pi r^2\) comes from and then use this formula to solve problems. Finally, students solve problems that require deciding whether the situation relates to the circumference or area of a circle.

The first section of this unit, in which students recognize and apply proportional relationships involving circumference, serves as a bridge between the foundational work with proportional relationships in the previous unit and the more advanced applications in the following unit. The remaining sections of this unit, which deal with the area of circles, are preparation for the continued geometry work students will do later in this course.

A picture of three different circular objects. The leftmost object is a wagon wheel with a measuring tool starting from one point on the wheel, goes through the wheel center to a point on the other side of the wheel. The center object is a plane propellor with three identical propellor blades. A measuring tool starts from the center of the propellor and goes to the end of the blade. The third object is of a sliced orange. A measuring tool goes around the entire circular region of the orange.

A note on using the term "circle":

Strictly speaking, a circle is one-dimensional. It is the boundary of a two-dimensional region, rather than the region itself. The circular region is called a “disk.” Because students are not yet expected to make this distinction, these materials refer to both disks and the 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 the radius and the 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students deepen their understanding of proportional relationships and percentages. They solve multi-step problems and work with situations that involve fractional amounts. This builds on the work students did in grade 6 with ratios, rates, and percentages as well as previous units in grade 7 with proportional relationships. Students will build on this work in high school with exponential functions representing compounded percent increase and decrease.

Students begin the unit by revisiting constant rates, but this time the given values are fractional amounts. To determine the unit rate for the situation, students must compute the quotient of two fractions. Students also make sense of situations where an increase or decrease is expressed as a fraction of the initial amount. They create diagrams and apply the distributive property to generate expressions that represent these situations. They also use long division to write fractions as decimals, including their first introduction to repeating decimals.

Next, students make sense of situations where an increase or decrease is expressed as a percentage of the initial amount. They continue creating diagrams and writing equations to represent the situations. They solve for any one of the three quantities—the initial amount, the final amount, or the percentage of the change—given the other two quantities. They also reason about fractional percentages.

Then students apply percent increase and decrease to solve problems in a variety of real-world situations, such as tax, tip, interest, markup, discount, depreciation, and commission. Lastly, students make sense of situations where the difference between a correct measurement and an incorrect measurement is expressed as a percentage of the correct amount.

\(y = x + \frac14 x\)

\(y = (1 + \frac14)x\)

\(y = \frac54 x\)

\(y = (1 + 0.25)x\)

\(y = 1.25x\)

“a 25% increase”

​​​​​

\(y = x - \frac14 x\)

\(y = (1 - \frac14)x\)

\(y = \frac34 x\)

\(y = (1 - 0.25)x\)

\(y = 0.75x\)

“a 25% decrease”

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 (Lessons 8 and 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students perform operations on rational numbers, which are all numbers that can be written as a positive or negative fraction. This builds on grade 6 work with interpreting, comparing, and plotting rational numbers. It prepares students for a later unit when they will solve equations of the form \(px + q = r\) or \(p(x+q)=r\), where \(p\), \(q\), and \(r\) are rational numbers.

Students begin by revisiting how signed numbers are used to represent quantities above and below a reference point, such as measurements of temperature and elevation. They use tables and number line diagrams to represent changes in temperature or elevation. They extend addition and subtraction from fractions to all rational numbers. And they see that \(a - b\) is equivalent to \(a + (\text-b)\).

Next, students examine multiplication and division. They work with constant velocity, which is a signed number that indicates direction and speed. This allows products of signed numbers to be interpreted in terms of position, direction of movement, and time before or after a specific point. Students use the relationship between multiplication and division to understand how division extends to rational numbers.

Then students work with expressions that use the four operations on rational numbers. They also solve problems that involve interpreting negative numbers in context. They solve linear equations of the form \(p+x=q\) or \(px=q\), where \(p\) and \(q\) rational numbers. The focus of these lessons is representing situations with equations 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.

Four vertical thermometers measured in degrees Celsius. There are 16 evenly spaced tick marks and starting from the bottom of the thermometer, negative 5 is on the first tick mark, zero on the sixth, 5 on the eleventh, and 10 on the sixteenth. The first thermometer is shaded starting from the bottom of the thermometer to the tenth tickmark. The second thermometer is shaded starting from the bottom of the thermometer to the third tickmark. The third thermometer is shaded starting from the bottom of the thermometer to between the eleventh and twelfth tickmark. The fourth thermometer is shaded starting from the bottom of the thermometer to between the fourth and fifth tickmark.

A note on using the terms "expression," "equation," and "signed number":

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 noninteger exponents in high school.) An equation is a statement that two expressions are equal, thus it 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 subtracting and adding signed numbers (Lesson 5).
• About differences and magnitude (Lesson 6).
• About multiplying negative numbers (Lesson 9).
• About additive and multiplicative inverses (Lesson 15).

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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students deepen their algebraic reasoning as they write and solve equations of the forms \(px+q=r\) and \(p(x+q)=r\) and inequalities of the forms \(px+q>r\) and \(p(x+q). Students also work with equivalent expressions that are more complex than what they have seen previously. This builds on grade 6 work with equations of the form \(p+x=q\) or \(px=q\) and with simpler equivalent expressions. Students will build on this work in grade 8 when they solve equations that have a variable on both sides of the equal sign and when they work with systems of equations.

Students begin the unit by making sense of situations that involve both multiplication and addition. They represent such situations with tape diagrams and with equations. They see that different diagrams and equations can represent the same situation, and they use diagrams to find solutions to equations.

Next, students consider hanger diagrams as another way to represent equations. The diagrams help students understand solving equations in terms of “doing the same thing to each side of the equation.” Students examine different pathways for solving the same equation and consider whether one method is more efficient than another.

Balanced hanger. Left side, circle labeled x and square labeled 2, the square appears to be loose from the hanger. Right side, rectangle labeled 4 and square labeled 2, the square appears to be loose from the hanger. To the side, an equation says x = 4.

Then students apply what they have learned about equations to inequalities. They write inequalities to represent situations and solve inequalities by reasoning about the related equation. The inequality symbols \(\geq\) and \(\leq\) are introduced.

Lastly, students work with equivalent linear expressions that are more complex due to having more terms, more parentheses, and negative rational numbers. Students use properties of operations to justify why the expressions are equivalent.

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 21).

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 22).

In addition, students are expected to interpret solutions to equations, interpret and represent nonproportional situations with constant rates of change, represent nonproportional 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

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.” 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. Students should have access to their geometry toolkits so that they have an opportunity to select and use appropriate tools strategically.

Note: It is not expected that students memorize which conditions result in a unique triangle, an impossible-to-create triangle, or multiple possible triangles. Understanding that, for example, side-side-side (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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students work with probability and sampling. They use their understanding of basic chance experiments to quantify how likely events are to happen and develop a working understanding of probability. Then they design and use simulations to further understand probability as the frequency of the event occurring when repeating an experiment many times. Students represent sample spaces using tables, tree diagrams, and lists, and use the number of outcomes in a sample space to calculate an expected probability.

Two tree diagrams. The leftmost tree diagram has three branches for the first choice, labeled “A,” “B”, and “C.” Choices “A”, “B”, and “C” each have four branches labeled with a different number from 1 through 4. The rightmost tree diagram has four branches for the first choice, labeled 1, 2, 3, and 4. Choices 1, 2, 3, and 4 each have three branches, labeled with a different letter “A,” “B,” or “C.”

Next, students examine different ways to collect data from samples within a population to understand why random selection is useful. Then students generate samples and estimate information about the population from sample data. Finally, students compare two groups by examining the measures of center and measures of variability calculated from sample data representing each group.

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 probability 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 sample 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this optional unit, students use concepts and skills from previous units to solve problems. The first section focuses on calculating or estimating quantities associated with running a restaurant. The second section explores a variety of different contexts, such as population density, Fermi problems, measurement error, and deforestation. In the last section, students build a trundle wheel and design a five-kilometer race course.

All related standards in this unit have been addressed in prior units. These sections provide an optional opportunity for students to go more deeply and make connections between domains.

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 

• Choices and predictions in the context of running a restaurant (Lesson 1).
• Reasoning about length, area, and volume in the context of a restaurant (Lesson 3).
• Reasoning about the forested area on a map (Lesson 8).

Represent 

• Costs of ingredients in a spreadsheet (Lesson 1).
• Situations using expressions and equations (Lesson 6).
• A map of a designed race course (Lesson 12).

Critique 

• Peer reasoning about calculations of age, heart beats, and hairs (Lesson 5).
• Peer reasoning about percent error in length, area, and volume measurement (Lesson 7).
• Peer methods of measuring distance (Lesson 9).

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

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.