In this unit, students learn about negative numbers and ways to represent them on a number line and the coordinate plane. They write and graph simple inequalities in one variable and determine the greatest common factor and least common multiple of two whole numbers. In grade 7, students will perform arithmetic operations with signed numbers and write and solve more complex inequalities.

Students begin by considering situations involving temperature or elevation and interpreting what negative numbers mean in those contexts. They also plot points to represent positive and negative values and their opposites. 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).

Next, students use the symbols  and  to compare two values. They represent an unknown value with constraints as an inequality, and they graph the solutions on a number line. Students consider the inclusion or exclusion of boundary values and interpret solutions based on the context. In these grade 6 materials, inequality symbols in are limited to  and . However, in this unit students encounter situations that are best represented by both an inequality and an equation, such as “ or .”

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.

The last section of the unit returns to whole numbers. Students are introduced to common factors and common multiples. They determine the greatest common factor or the least common multiple of two numbers. They identify how these new concepts are involved in real-world situations and use their understanding to solve related problems.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as describing, interpreting, 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:

Describe and Interpret

• Situations involving negative numbers (Lesson 1).
• Features of a number line (Lessons 2, 4 and 6).
• Situations involving elevation (Lesson 7).
• Situations involving minimums and maximums (Lesson 8).
• Points on a coordinate plane (Lessons 11 and 14).
• Situations involving factors and multiples (Lesson 18).

Justify

• Reasoning about magnitude (Lesson 3).
• Reasoning about a situation involving negative numbers (Lesson 5).
• Reasoning about solutions to inequalities (Lesson 9).
• That all possible pairs of factors have been identified (Lesson 16).

Generalize

• The meaning of integers for a specific context (Lesson 5).
• Understanding of solutions to inequalities (Lesson 9).
• About the relationships between shapes (Lesson 10).
• About greatest common factors (Lesson 16).
• About least common multiples (Lesson 17).

In addition, students are expected to critique the reasoning of others, represent inequalities symbolically and in words, and explain how to order rational numbers and how to determine distances on the coordinate plane. Students also have opportunities 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.

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.

lesson	new terminology
	receptive	productive
6.7.1	positive number negative number temperature degrees Celsius elevation sea level	number line below zero
6.7.2	opposite (numbers) rational number location distance (away) from zero
6.7.3	sign inequality closer to 0 farther from 0	greater than less than
6.7.4	from least to greatest	temperature elevation sea level
6.7.5	positive change negative change context
6.7.6	absolute value	positive number negative number distance (away) from zero
6.7.7		closer to 0 farther from 0
6.7.8	maximum minimum
6.7.9	requirement solution to an inequality
6.7.10	unbalanced hanger	inequality
6.7.11	quadrant coordinate plane -coordinate -coordinate
6.7.12	(line) segment	axis
6.7.13	degrees Fahrenheit	degrees Celsius
6.7.14		absolute value -coordinate -coordinate
6.7.16	common factor greatest common factor (GCF)	factor
6.7.17	common multiple least common multiple (LCM)	multiple