Multiplying and Dividing Integers

Active learning helps students move beyond memorizing rules by letting them physically model integer operations. Working with counters, number lines, and real-world scenarios makes abstract sign rules visible and discussable for all learners.

• Year 7
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

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

Stations Rotation

⏱ 35 min·Small Groups

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.

Prepare & details

Justify why the product of two negative integers is positive.

Facilitation Tip: During Number Line Relay, place masking tape number lines on the floor so students can physically step through multiplication as repeated jumps.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 25 min·Pairs

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.

Prepare & details

Analyze how integer multiplication relates to repeated addition or subtraction.

Facilitation Tip: For Two-Colour Counters, have students pair negative counters to model cancellation before recording results in a sign chart.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 40 min·Small Groups

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.

Prepare & details

Construct a real-world problem that requires division of negative integers.

Facilitation Tip: Use Debt Division Cards with play money so students can physically exchange bills to see how division reverses multiplication in real contexts.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 30 min·Whole Class

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.

Prepare & details

Justify why the product of two negative integers is positive.

Facilitation Tip: In Integer Pattern Tables, ask students to pause after each row to predict the next sign before completing calculations.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach this topic by starting with concrete models before moving to abstract rules. Avoid rushing to the sign rules—instead, let students discover patterns through repeated addition and subtraction. Research shows that students who build their own rules through modeling retain concepts longer and make fewer sign errors. Emphasize peer discussion to resolve contradictions during modeling phases.

What to Expect

Students will confidently apply sign rules to multiply and divide integers, explain why rules work using models, and connect operations to real-life contexts like debt and temperature changes. Look for clear explanations and correct calculations in both individual and group work.

Watch Out for These Misconceptions

Common Misconception

During Two-Colour Counters: Sign Exploration, watch for students who pair negative counters but then miscount the remaining counters as negative.

What to Teach Instead

Have students circle each pair of negative counters and label the leftover counters with a different-colored marker before recording the total sign. Ask them to explain why the leftover positives make sense in a debt scenario.

Common Misconception

During Number Line Relay: Multiplication Rules, watch for students who reverse the direction of jumps when multiplying by a negative number.

What to Teach Instead

Pause the relay and have students act out the jumps together, starting at zero and stepping backward for negative multipliers. Ask them to compare -3 x 4 with 3 x -4 to see the sign flip in action.

Common Misconception

During Integer Pattern Tables: Whole Class Build, watch for students who fill in the table without noticing the pattern in the signs.

What to Teach Instead

Have students pause after each row to predict the next sign based on the previous row. Ask them to explain their prediction using the counters or number lines they’ve used in other activities.

Assessment Ideas

Exit Ticket

After Integer Pattern Tables, give students three problems: 1. Calculate -7 x 8. 2. Calculate -45 ÷ -5. 3. Explain in one sentence why -3 x -4 = 12. Collect responses to check for correct calculations and clear reasoning.

Quick Check

During Number Line Relay, ask students to show how repeated addition of -3, five times, results in -15. Then, ask them to write the corresponding multiplication sentence on their whiteboards.

Discussion Prompt

After Debt Division Cards, 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?' Listen for correct integer representations and division equations like -30 ÷ -10 = 3.

Extensions & Scaffolding

• Challenge students to create their own real-world scenario cards involving multiplying or dividing integers, then trade with peers to solve.
• Scaffolding: Provide partially completed sign charts or counters pre-grouped for students who need a clearer starting point.
• Deeper: Ask students to write a short paragraph comparing how multiplication and division relate, using examples from at least two activities.

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.