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 previous work with equivalent ratios and helps prepare students for the study of linear functions in later courses.

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." 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).  

Next, students apply their knowledge of proportional relationships to the context of measuring circles. This builds on students’ work from previous grades with perimeter and area of polygons. Students will build on this work in later courses when they study the volume of spheres, cylinders, and cones.

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 comes from and then use this formula to solve problems.

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 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.

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 comparing, justifying, 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

• Approaches to solving problems involving proportional relationships (Lesson 3).
• Proportional relationships with nonproportional relationships (Lesson 5).
• Tables, descriptions, and graphs representing the same situations (Lesson 7).
• Graphs of proportional relationships (Lesson 8).
• The relationships of square diagonals and perimeters to square diagonals and areas (Lesson 10).
• The relationships of diameters and circumferences to diameters and areas (Lesson 15).

Justify

• Reasoning about circumference and perimeter (Lesson 13).
• Estimates for the areas of circles (Lesson 15).
• Reasoning about areas of curved figures (Lesson 16).
• Whether or not a relationship is proportional (Lesson 17).
• Reasoning about the cost of stained-glass windows (Lesson 20).

Generalize

• About proportional relationships (Lesson 1).
• About equations that represent proportional relationships (Lesson 2).
• About categories for sorting circles (Lesson 11).
• About the relationships between circumference and diameter (Lesson 12).

In addition, students are expected to explain how to determine whether or not a relationship is proportional, how to use different approximations of , how to find the area of composite shapes, and how to compare and represent situations with different constants of proportionality. Students are also asked to interpret situations involving proportional relationships, floor plans and maps, situations involving circles, and situations involving circumference and area.

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.