Mathematical Explorers: Building Number and Space · 3rd Class

Active learning ideas

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.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Number - N.5
• NCCA: Junior Cycle - Number - N.6

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.

What to look for

Present students with three multiplication problems and three division problems involving positive and negative integers, such as (-5) x 4, 6 x (-3), (-7) x (-2), 15 / (-3), (-20) / 5, (-18) / (-6). Ask students to calculate the answer and write the sign rule they applied for each.

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

Activity 02

• 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.

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.

What to look for

Pose the question: 'Imagine you are managing a small business. Describe a situation where you might need to multiply or divide negative numbers. Explain your scenario and the calculation you would perform.' Facilitate a class discussion where students share their examples.

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

Activity 03

• 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.

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.

What to look for

Give each student a card with a statement like: 'Multiplying a negative number by a positive number results in a ____ number.' and 'Dividing a negative number by a negative number results in a ____ number.' Students fill in the blanks. Then, ask them to write one sentence comparing the outcome of 3 x (-4) to (-3) x (-4).

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

Activity 04

• 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.

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.

What to look for

Present students with three multiplication problems and three division problems involving positive and negative integers, such as (-5) x 4, 6 x (-3), (-7) x (-2), 15 / (-3), (-20) / 5, (-18) / (-6). Ask students to calculate the answer and write the sign rule they applied for each.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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

  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.

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Methods used in this brief

• Stations Rotation