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

Multiplying and Dividing Integers

Students will learn and apply the rules for multiplying and dividing integers, including understanding the sign rules.

• ACARA Content Descriptions
• AC9M7N02
• AC9M7N03

About This Topic

Multiplying and dividing integers centres on sign rules: same signs yield positive results, different signs yield negative. Students learn multiplication through patterns, such as building multiplication tables for -5 to 5, and justify why two negatives produce positive by linking to repeated addition on number lines. For division, they practise cases like -15 ÷ 3 = -5 and -15 ÷ -3 = 5, analysing how it reverses multiplication.

This content aligns with AC9M7N02 and AC9M7N03, extending positive number operations to integers. Students construct real-world problems, such as dividing negative temperatures or debts, and explain rules using diagrams. These tasks develop reasoning and connect to algebra, where integers underpin expressions and equations.

Active learning suits this topic because rules feel arbitrary without models. When students use two-colour counters to pair positives and negatives or collaborate on number line relays, they see why signs work. Physical manipulation and peer explanations turn memorisation into understanding, boosting confidence for complex problems.

Key Questions

1. Justify why the product of two negative integers is positive.
2. Analyze how integer multiplication relates to repeated addition or subtraction.
3. Construct a real-world problem that requires division of negative integers.

Learning Objectives

• Calculate the product of two integers using the established sign rules.
• Determine the quotient of two integers, applying correct sign conventions.
• Explain the rationale behind the sign rule for multiplying two negative integers.
• Construct a word problem that necessitates the division of negative integers to find a solution.

Before You Start

Multiplying and Dividing Positive Whole Numbers

Why: Students must be proficient with the basic operations of multiplication and division before extending these to include negative numbers.

Introduction to Integers and the Number Line

Why: Understanding the concept of negative numbers and their representation on a number line is fundamental to grasping integer operations.

Key Vocabulary

Integer	A whole number, which can be positive, negative, or zero. Examples include -3, 0, and 5.
Product	The result of multiplying two or more numbers. For example, the product of 4 and 3 is 12.
Quotient	The result of dividing one number by another. For example, the quotient of 12 divided by 3 is 4.
Sign Rule	A rule that determines the sign (positive or negative) of the result when multiplying or dividing integers. Same signs result in a positive, different signs result in a negative.

Watch Out for These Misconceptions

Common Misconception

The product of two negative integers is negative.

What to Teach Instead

Students often extend positive rules without patterns. Use two-colour counters in pairs to model pairing negatives, showing leftovers are positive. Group discussions compare models to reveal the rule.

Common Misconception

Division by a negative number keeps the same sign as the dividend.

What to Teach Instead

This stems from ignoring reciprocal multiplication. Number line relays in small groups let students test divisions visually, like -12 ÷ -3 by repeated subtraction. Peer explanations solidify sign flips.

Common Misconception

Integer rules do not relate to repeated addition or subtraction.

What to Teach Instead

Without connections, rules seem rote. Chip models and relays in pairs build repeated steps physically, helping students articulate links during whole-class shares.

Active Learning Ideas

Number Line Relay: Multiplication Rules

Mark a floor number line from -20 to 20 with tape. In small groups, one student starts at zero and jumps according to the multiplier (e.g., -2 × 3: jump left 2, three times). Group records the endpoint and sign pattern. Rotate roles for five problems.

35 min·Small Groups

Two-Colour Counters: Sign Exploration

Provide red chips for negatives, yellow for positives. Pairs model products like (-2) × 3 by making pairs of chips and removing opposites. Discuss why (-2) × (-3) leaves positives. Record rules in journals.

25 min·Pairs

Debt Division Cards: Real-World Problems

Distribute scenario cards (e.g., share -24 debt among 4 people). Small groups solve, draw models, and create their own problems. Share one per group with class justifications.

40 min·Small Groups

Integer Pattern Tables: Whole Class Build

Project a partially filled multiplication table for integers -5 to 5. Students suggest entries in turns, justifying with repeated addition. Class votes and corrects using number lines.

30 min·Whole Class

Real-World Connections

• Financial accounting uses integer multiplication and division to track profits and losses over time. For instance, calculating a company's total loss when it experiences a loss of $500 per day for 10 days (-$500 x 10 = -$5000).
• Temperature changes in weather forecasting involve integers. Dividing a total temperature drop of -20 degrees Celsius over 4 hours (-20 ÷ 4 = -5) helps determine the average hourly temperature decrease.
• Scuba diving depths are often represented by negative integers. Calculating the average depth change when a diver ascends 30 meters from a depth of -60 meters (-30 ÷ 2 = -15) can help monitor ascent rates.

Assessment Ideas

Exit Ticket

Provide students with three problems: 1. Calculate -7 x 8. 2. Calculate -45 ÷ -5. 3. Explain in one sentence why -3 x -4 = 12.

Quick Check

Display a number line. Ask students to show how repeated addition of -3, five times, results in -15. Then, ask them to write the corresponding multiplication sentence.

Discussion Prompt

Pose the question: 'Imagine you owe three friends $10 each. How can you represent this debt using integers? If you then found $30, how would you represent paying them all back using division?'

Frequently Asked Questions

Why is the product of two negative integers positive?

It follows from patterns in multiplication tables or repeated addition: -3 × -2 equals (-3) + (-3) but reversed as positives via subtraction logic on number lines. Model with counters: pair two negative groups, opposites cancel, leaving positive value. Students justify through diagrams, connecting to division checks like (-6) ÷ (-2) = 3.

How to teach sign rules for dividing integers?

Link division to its multiplication inverse: signs match multiplication rules. Use real-world contexts like sharing negative bank balances. Practice with tables and models ensures students verify, such as 12 ÷ -4 = -3 since (-4) × -3 = 12. Gradual scaffolding builds fluency.

What activities work for Year 7 integer multiplication?

Hands-on options include number line jumps for repeated addition and two-colour counters for signs. Card sorts matching products to models reinforce patterns. These keep engagement high while addressing AC9M7N02, with extensions to student-created problems for deeper reasoning.

How can active learning help with multiplying and dividing integers?

Active approaches make abstract signs concrete: physical number lines and counters let students manipulate to discover rules, reducing rote errors. Pair and group tasks encourage verbal justifications, aligning with key questions. This builds lasting intuition over worksheets, as collaborative models reveal patterns like negative pairs yielding positive.