Multiplication and Division of Integers

Active learning helps students grasp multiplication and division of integers by making abstract sign rules visible and concrete. When students move manipulatives, draw number lines, or craft problems, they build mental models that last beyond the lesson. These activities meet students where they are by addressing common confusions through hands-on practice.

• 3rd Class
• Mathematical Explorers: Building Number and Space
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate the product of two integers, including cases with negative integers, applying the correct sign rules.
2. Determine the quotient of two integers, including cases with negative integers, justifying the sign of the result.
3. Explain the mathematical rule for determining the sign of a product or quotient involving positive and negative integers.
4. Construct a word problem that requires multiplication or division of negative integers to find a solution.
5. Compare the results of multiplying an integer by a positive number versus multiplying it by a negative number.

Stations Rotation

⏱ 35 min·Small Groups

Manipulative Magic: Two-Color Counters

Provide red counters for negatives and yellow for positives. Students model problems like (-3) × 2 by making three pairs of red and yellow, then remove pairs to find the result. Extend to division by grouping counters evenly. Record sign patterns on charts.

Prepare & details

Explain the rules for determining the sign of a product or quotient of integers.

Facilitation Tip: During Manipulative Magic, ask students to model each problem twice: once with paired counters to show cancellation, and once without pairing to show unpaired negatives.

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·Whole Class

Number Line Jumps: Relay Race

Mark a floor number line from -10 to 10. Teams take turns jumping to represent steps, such as start at 0, jump -2 three times for (-2) × 3. Note landing spots to determine products. Switch to division by reversing jumps.

Prepare & details

Construct a real-world problem that involves multiplying or dividing negative numbers.

Facilitation Tip: In Number Line Jumps, require each team to verbalize the sign change aloud as they move backward for negative divisors.

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·Pairs

Story Swap: Problem Creation

In pairs, students write multiplication or division stories with negatives, like owing money or diving underwater. Swap with another pair, solve, and explain the sign rule used. Share one class example.

Prepare & details

Compare the effect of multiplying by a positive integer versus a negative integer.

Facilitation Tip: For Story Swap, provide a checklist of elements (who, what, why) so students craft problems that clearly connect to integer operations.

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

⏱ 20 min·Individual

Sign Sort: Card Challenges

Prepare cards with integer expressions. Students sort into positive or negative result piles individually, then justify in small groups using quick sketches. Time a class tournament for fastest accurate sorter.

Prepare & details

Explain the rules for determining the sign of a product or quotient of integers.

Facilitation Tip: Use Sign Sort to group problems by sign pattern first, then ask students to predict the result before calculating.

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

Start with clear sign rules, but quickly shift to hands-on activities to prevent over-reliance on memorization. Use peer discussions to address misconceptions in real time. Avoid teaching only procedures; instead, connect operations to real contexts so students see why sign rules matter. Research shows that when students explain their reasoning to peers, they refine their understanding and retain concepts longer.

What to Expect

Students will confidently apply sign rules and explain why products or quotients have particular signs. They will create accurate real-world problems, compare outcomes of multiplying by positive and negative integers, and correct peers' misconceptions during collaborative tasks. Success includes both computational accuracy and conceptual explanations.

Watch Out for These Misconceptions

Common Misconception

During Manipulative Magic, watch for students who assume two negatives always make a positive because they cancel each other out.

What to Teach Instead

Ask students to model (-3) × (-2) by pairing 3 sets of 2 negative counters and removing them three times, showing that paired negatives create positives only when removed in even groups.

Common Misconception

During Number Line Jumps, watch for students who treat dividing by a negative as merely moving forward.

What to Teach Instead

Have them model 12 ÷ (-3) by starting at 12 and making jumps of size 3 backward, counting how many jumps land them at zero to see the sign change.

Common Misconception

During Sign Sort, watch for students who believe the order of factors changes the sign outcome.

What to Teach Instead

Ask them to rearrange (-2) × 3 and 3 × (-2) in the same grouping and discuss why both yield negative results, using counters or number lines to confirm consistency.

Assessment Ideas

Quick Check

After Manipulative Magic, present students with three multiplication and three division problems. Ask them to solve and write the sign rule they applied for each, then share with a partner to compare explanations.

Discussion Prompt

During Story Swap, have students swap problems with peers and solve them, then discuss in pairs or small groups whether the problems clearly connect to the integer operations and if the calculations make sense in context.

Exit Ticket

After Sign Sort, give each student a card with statements to fill in about sign rules. Then ask them to write one sentence comparing the outcomes of 3 × (-4) and (-3) × (-4), using their sorted cards as a reference.

Extensions & Scaffolding

• Challenge students to write a step-by-step guide for multiplying three negative integers, including a visual model using counters or a number line.
• For students struggling with division, provide a scaffolded worksheet with number line diagrams where they fill in the jumps and signs.
• Deeper exploration: Have students research and present on how negative numbers appear in weather data, finance, or sports statistics, then create a set of problems based on their findings.

Key Vocabulary

Integer	A whole number that can be positive, negative, or zero. Examples include -3, 0, 5.
Product	The result of multiplying two or more numbers. For example, the product of 4 and 5 is 20.
Quotient	The result of dividing one number by another. For example, the quotient of 10 divided by 2 is 5.
Sign Rule	A mathematical rule that determines whether the result of multiplication or division will be positive or negative based on the signs of the numbers involved.