Mathematics · 7th Grade · Rational Number Operations · Weeks 1-9

Integers and Absolute Value

Students will define integers, compare and order them, and understand the concept of absolute value.

• Common Core State Standards
• CCSS.Math.Content.7.NS.A.1

About This Topic

Adding and subtracting integers is a pivotal shift from whole-number arithmetic to the full number system. Students use number lines and the concept of absolute value to understand how positive and negative values interact. The Common Core standards focus on the idea that p + q is the number located a distance |q| from p, and that subtracting a number is the same as adding its additive inverse.

This topic is the 'gatekeeper' for algebra; students must be fluent with signed numbers to solve equations later. It connects to real world contexts like temperature changes, altitude, and financial debits. Students grasp this concept faster through structured discussion and physical modeling, such as walking on a giant number line, which helps them internalize the direction of movement for each operation.

Key Questions

1. Differentiate between natural numbers, whole numbers, and integers.
2. Analyze how absolute value represents distance from zero on a number line.
3. Construct a number line to represent and order a given set of integers.

Learning Objectives

• Classify numbers as natural, whole, or integers.
• Compare and order a given set of integers on a number line.
• Calculate the absolute value of any integer.
• Explain the relationship between an integer and its absolute value as distance from zero.
• Construct a number line to represent and order a given set of integers.

Before You Start

Introduction to Number Lines

Why: Students need to be familiar with the concept of a number line to represent and order integers and understand distance from zero.

Properties of Whole Numbers

Why: Understanding basic arithmetic with whole numbers provides a foundation for extending operations to include negative numbers.

Key Vocabulary

Integer	A whole number or its opposite, including zero. Examples are -3, 0, 5.
Natural Numbers	The counting numbers starting from 1. Examples are 1, 2, 3, 4.
Whole Numbers	The counting numbers including zero. Examples are 0, 1, 2, 3.
Absolute Value	The distance of a number from zero on the number line, always a non-negative value. It is written with two vertical bars, for example, |–5|.
Additive Inverse	A number that, when added to another number, results in zero. For example, the additive inverse of 7 is -7.

Watch Out for These Misconceptions

Common Misconception

Subtraction always results in a smaller number.

What to Teach Instead

In the world of integers, subtracting a negative actually increases the value. Using a number line simulation helps students see that 'taking away a debt' makes you 'richer,' moving you to the right on the line.

Common Misconception

The sign of the answer is always the sign of the first number.

What to Teach Instead

Students often ignore the magnitudes. Hands-on work with integer chips helps them see that the 'stronger' group (the one with the greater absolute value) determines the sign of the final sum.

Active Learning Ideas

Simulation Game: Human Number Line

Create a large number line on the floor. Students stand at a starting integer and 'walk' the addition or subtraction of another integer. For subtraction, they must physically turn around to face the opposite direction before moving, modeling the 'adding the opposite' rule.

30 min·Whole Class

Inquiry Circle: Integer Chips Battle

Pairs use two-colored counters (one color for positive, one for negative) to model expressions. They practice 'zeroing out' pairs to find the final sum. They then create their own 'puzzles' for other pairs to solve using only the chips.

25 min·Pairs

Think-Pair-Share: Real World Negatives

Students are given scenarios like 'a bird 20 feet up and a fish 5 feet down.' They independently write an expression to find the distance between them, pair up to compare if they used addition or subtraction, and share why 'distance' always results in a positive value.

15 min·Pairs

Real-World Connections

• Temperature readings in cities like Chicago or Moscow often involve negative integers to represent temperatures below freezing. Meteorologists use these values to track weather patterns and predict conditions.
• Financial transactions, such as bank account balances, use integers to represent deposits (positive) and withdrawals (negative). Accountants and personal finance managers track these values to maintain accurate records.

Assessment Ideas

Exit Ticket

Provide students with a list of numbers: -8, 5, 0, -3, 12. Ask them to: 1. Identify which numbers are integers. 2. Write the absolute value of -8 and 5. 3. Order the numbers from least to greatest.

Quick Check

Draw a number line from -10 to 10. Ask students to place five integer cards on the line. Then, pose questions like: 'Which number is farthest from zero?' or 'Which two numbers have the same absolute value?'

Discussion Prompt

Pose the scenario: 'A submarine is at -50 meters, and then it ascends 20 meters. What is its new depth? How does absolute value help us understand the submarine's position relative to sea level?' Facilitate a brief class discussion.

Frequently Asked Questions

Why does subtracting a negative feel like addition?

Think of it as removing a debt. If you owe someone $5 (-5) and that debt is taken away, your total value goes up by $5. On a number line, subtraction means 'change direction,' and a negative number means 'walk backward,' so you end up moving forward.

How can active learning help students understand integer operations?

Active learning strategies like the Human Number Line or using physical counters provide a tactile way to understand direction and magnitude. Instead of just memorizing 'keep-change-change,' students see the physical logic of why the signs change, leading to better long-term retention.

What is an additive inverse?

The additive inverse of a number is its opposite. When you add a number and its additive inverse (like 5 and -5), the sum is always zero. This is the foundation for the 'zero pair' strategy in modeling.

How do I explain absolute value in simple terms?

Absolute value is simply the distance a number is from zero, regardless of direction. Since distance can't be negative, the absolute value is always positive (or zero). It tells you 'how much' you have without worrying about the 'which way'.

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