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 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 . They interpret points on the graph, including the point . 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 and are never called ratios. The fractions and are identified as “unit rates” for the ratio . In high school—after the study of ratios, rates, and proportional relationships—students discard the term “unit rate” and start referring to to , , and 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 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.