Mathematics · Year 7 · The Language of Number · Term 1

Adding and Subtracting Integers

Students will practice addition and subtraction of integers using number lines and conceptual models.

• ACARA Content Descriptions
• AC9M7N02
• AC9M7N03

About This Topic

Adding and subtracting integers builds on students' whole number skills by introducing negatives to represent opposites, such as gains and losses or temperatures above and below zero. Year 7 students use number lines to plot movements: right for positive addition or subtracting negatives, left for negative addition or subtracting positives. Conceptual models like two-color counters demonstrate cancellation of opposites and doubling for same-sign sums. This addresses standards AC9M7N02 and AC9M7N03, focusing on key questions like explaining why subtracting a negative equals adding a positive, predicting signs for mixed operations, and designing models for two negative integers.

In the Australian Curriculum's Number strand, this topic strengthens computational fluency and conceptual understanding, forming the base for rational numbers and algebra. Students connect operations to real contexts, like bank balances or elevation changes, fostering flexible thinking over memorization.

Active learning benefits this topic greatly because integers challenge intuition; manipulatives and group tasks make rules visible and debatable. When students jump on human number lines or build counter models collaboratively, they test predictions, explain reasoning to peers, and correct errors through shared evidence, leading to lasting proficiency.

Key Questions

1. Explain why subtracting a negative integer is equivalent to adding a positive integer.
2. Predict the sign of the result when adding integers with different signs.
3. Design a visual model to demonstrate the sum of two negative integers.

Learning Objectives

• Calculate the sum and difference of two integers using a number line model.
• Explain the relationship between adding a negative integer and subtracting a positive integer.
• Design a visual representation to demonstrate the result of adding two integers with unlike signs.
• Predict the sign of the sum when adding integers with different signs, justifying the prediction.
• Compare the results of adding and subtracting integers to verify the equivalence of operations.

Before You Start

Whole Number Operations

Why: Students must be proficient in adding and subtracting whole numbers before introducing negative numbers.

Introduction to Negative Numbers

Why: Students need a basic understanding of what negative numbers represent (e.g., below zero, debt) to grasp integer operations.

Key Vocabulary

Integer	A whole number or its opposite, including zero. Integers can be positive, negative, or zero.
Number Line	A visual tool representing numbers in order. Movements to the right indicate addition or subtracting a negative, while movements to the left indicate subtraction or adding a negative.
Opposite Integers	Two integers that are the same distance from zero on the number line but in opposite directions, such as 5 and -5. Their sum is always zero.
Additive Inverse	A number that, when added to another number, results in zero. For any integer 'a', its additive inverse is '-a'.

Watch Out for These Misconceptions

Common Misconception

Subtracting a negative integer means subtracting a positive integer.

What to Teach Instead

A number line shows subtracting -4 from 3 as moving right four units to 7, equivalent to adding 4. Pair demonstrations with counters reinforce this by removing a negative pair, which adds to the positive side. Active modeling helps students verbalize the 'double negative' rule through trial and peer critique.

Common Misconception

The sum of two negative integers is positive.

What to Teach Instead

Two-color counters or number lines reveal -3 + (-5) as further left to -8, deepening the negative. Small group races on number lines expose this error quickly, as peers challenge incorrect jumps and rebuild consensus visually.

Common Misconception

When adding integers with different signs, always take the larger absolute value.

What to Teach Instead

Students overlook equal magnitudes that cancel to zero, like 5 + (-5). Whole-class story voting highlights this, prompting debates that clarify direction and magnitude through collective number line trials.

Active Learning Ideas

Pairs: Two-Color Counter Drills

Provide red counters for negatives and yellow for positives. Pairs take turns posing integer problems; the other models the operation by adding or removing counters, then states the result. Switch roles after five problems and discuss any sign prediction errors.

25 min·Pairs

Small Groups: Human Number Line Challenges

Mark a floor number line from -20 to 20 with tape. Groups of four solve addition/subtraction problems by having one student start at zero and jump left or right while others predict and verify the landing spot. Rotate roles and record results on mini-whiteboards.

35 min·Small Groups

Whole Class: Real-World Integer Stories

Project scenarios like temperature changes or debt payments. Class votes on operation signs, then volunteers demonstrate on a shared number line. Discuss predictions as a group and adjust based on collective evidence.

30 min·Whole Class

Individual: Model Design Task

Students design and draw a visual model for a given problem, such as -3 + (-5), using number lines or counters. They label steps and explain the result in writing, then share one with a partner for feedback.

20 min·Individual

Real-World Connections

• Financial analysts use integer addition and subtraction to track bank account balances, calculating net profit or loss from various transactions and investments.
• Meteorologists use integers to represent temperatures above and below freezing point, calculating daily temperature fluctuations and average temperatures for weather forecasts.
• Pilots use integers to represent altitude changes, adding positive values for climbing and subtracting negative values for descending to maintain safe flight levels.

Assessment Ideas

Exit Ticket

Provide students with three problems: 1) -5 + 3, 2) 7 - (-2), 3) -4 + (-6). Ask them to solve each problem using a number line and write one sentence explaining their strategy for problem 2.

Quick Check

Display a scenario: 'A submarine is at -50 meters. It ascends 20 meters, then descends 30 meters. What is its final depth?' Have students write the integer expression and solve it on mini-whiteboards.

Discussion Prompt

Pose the question: 'If you have $10 and spend $15, what is your balance? Explain how this relates to adding integers with different signs.' Facilitate a class discussion where students share their reasoning and connect it to the concept of owing money.

Frequently Asked Questions

How do you explain why subtracting a negative is adding a positive?

Use a number line: subtracting -3 from 5 means not taking away a debt but adding value, so move right from 5 to 8. Two-color counters show removing a negative pair increases positives. Practice with pairs modeling 10 problems builds fluency; students explain back to solidify the debt analogy common in Australian banking contexts.

What are common errors when predicting signs in integer addition?

Students often ignore direction, assuming larger absolute value always wins, or treat negatives like positives. Address with quick whole-class polls on mixed-sign problems, followed by human number line demos. Track patterns on a board to reteach, ensuring 80% class accuracy before advancing.

How can active learning help students master adding and subtracting integers?

Active approaches like manipulatives and group challenges make abstract signs concrete. Pairs with counters debate cancellations, while floor number lines let students physically test predictions. These reveal misconceptions instantly, encourage peer teaching, and boost retention by 30-40% per studies, outperforming worksheets alone.

How to differentiate integer operations activities for Year 7?

Offer tiered problems: basic same-sign for support, mixed-sign challenges for proficient, and story designs for extension. Pair strong with emerging learners during counter drills. Use individual model tasks for assessment; provide scaffolds like pre-drawn lines. Rotate groupings to build confidence across abilities.