In this unit, students learn about negative numbers and ways to represent them on a number line and the coordinate plane. They perform operations on rational numbers, which are all numbers that can be written as a positive or negative fraction or zero.

Students begin by considering situations involving temperature or elevation and interpreting what negative numbers mean in those contexts. Previously, when students worked only with nonnegative numbers, magnitude and order were indistinguishable. In this unit, when comparing two signed numbers, students learn to distinguish between the absolute value of a number (magnitude) and a number’s relative position on the number line (order).

• A represents 15 feet.
• B represents  feet.
• ​​​​​​C represents ​ feet.
• D represents -4 feet.

Then students 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 is equivalent to .

Then students use ordered pairs to describe pairs of numbers that include negative numbers. In grade 5, they plotted pairs of positive numbers on the coordinate grid. Here, they plot pairs of rational numbers in all four quadrants of the coordinate plane. They interpret the meanings of plotted points in given contexts and use coordinates to calculate horizontal or vertical distances between two points.

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 or , where and  are rational numbers. 

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

In these materials, an expression is built from numbers, variables, operation symbols (, , , ), parentheses, and exponents. (Exponents—in particular, negative exponents—are not a focus of this unit. Students work with integer exponents in a future course 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 .

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 negative numbers (Lessons 1 and 5).
• Graphs involving positive and negative numbers  (Lesson 12).
• Tables and situations involving signed numbers (throughout unit).

Represent

• Addition of signed numbers on a number line (Lesson 6).
• Situations involving signed numbers (Lessons 7, 10, and 16).
• Changes in elevation (Lesson 10).
• Position, speed, and direction (Lesson 14).

Generalize

• About subtracting and adding signed numbers (Lesson 9).
• About differences and magnitude (Lesson 10).
• About multiplying negative numbers (Lessons 14 and 15).
• About additive and multiplicative inverses (Lesson 20).

In addition, students are expected to use language to compare magnitudes of positive and negative numbers, compare features of ordered pairs, and compare appropriate axes for different sets of coordinates, Students are also expected to explain how to order rational numbers, how to determine distances on the coordinate plane, 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.