Foundations of Mathematical Thinking · Junior Infants

Active learning ideas

Operations with Integers: Multiplication & Division

Active learning helps students move beyond memorizing sign rules by letting them see patterns emerge. When students manipulate models and work through problems in real time, abstract rules become concrete. This approach builds confidence and reduces reliance on rote memory for operations with integers.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Number - N.1.2

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Case Study Analysis
• 35 min · Small Groups

Chip Model: Sign Patterns

Provide two-color counters (red for negative, yellow for positive). Students model multiplication like (-3)×2 by pairing 3 red with 2 yellow groups, flipping pairs to positives. Discuss results, then extend to division by separating into equal groups. Record patterns in journals.

Explain the pattern that emerges when multiplying two negative integers.

Facilitation Tip

During Chip Model: Sign Patterns, circulate and ask pairs to explain why removing negative pairs results in positive totals.

What to look for

Present students with three multiplication problems: (-4) x (-5), 6 x (-3), and (-7) x 2. Ask them to write the answer and briefly explain the sign rule used for each.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 02

• Case Study Analysis
• 25 min · Small Groups

Number Line Relay: Mixed Operations

Mark number lines on floor with tape. Teams solve multiplication/division problems by jumping to represent integers, e.g., start at -4, multiply by -2 to reach 8. Relay passes marker; first accurate team wins. Debrief sign rules as class.

Analyze how the sign rules for division relate to those for multiplication.

Facilitation Tip

For Number Line Relay: Mixed Operations, ensure each team member verbalizes the sign change before moving the counter.

What to look for

Give students a card with the problem: 'A company lost €1000 over 5 days. What was the average daily loss?' Ask them to write the calculation using integers and state the answer.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 03

• Case Study Analysis
• 45 min · Pairs

Contextual Problem Stations

Set up stations with scenarios: debts, elevations, temperatures. Students solve using rules, draw models, and create their own problems. Rotate stations, share solutions whole class. Emphasize pattern application.

Construct a real-world scenario where dividing negative integers is necessary.

Facilitation Tip

In Contextual Problem Stations, provide calculators only after students have set up the equation using integer rules.

What to look for

Pose the question: 'How are the rules for multiplying integers similar to the rules for dividing integers?' Facilitate a class discussion, encouraging students to use examples to support their reasoning.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 04

• Case Study Analysis
• 30 min · Pairs

Pattern Hunt Cards

Distribute cards with integer pairs and products/quotients. Pairs sort into pattern groups (++, +-, --, -+), justify rules. Create posters displaying findings for class gallery walk.

Explain the pattern that emerges when multiplying two negative integers.

What to look for

Present students with three multiplication problems: (-4) x (-5), 6 x (-3), and (-7) x 2. Ask them to write the answer and briefly explain the sign rule used for each.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start with visual models to establish the foundations of sign rules, then transition to number lines to reinforce consistency. Avoid rushing to abstract rules before students see the patterns for themselves. Research shows that students retain integer operations better when they construct the rules through guided discovery rather than direct instruction.

Successful learning looks like students using models to justify their answers, not just stating them. They should connect multiplication and division rules through shared patterns and apply these rules accurately in contextual problems. Discussions should include clear explanations of why signs behave as they do.

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Watch Out for These Misconceptions

• During Chip Model: Sign Patterns, watch for students who assume the rules are the same as for addition and subtraction without testing the model.

  Ask students to physically remove negative pairs and observe the positive outcome before recording the rule. Have them explain the visual transformation to the group.

• During Number Line Relay: Mixed Operations, watch for students who treat division as unrelated to multiplication.

  After each relay round, have teams present how reversing a multiplication step led to the division result, emphasizing the shared sign pattern.

• During Contextual Problem Stations, watch for students who treat zero as a negative or positive number in context.

  Use zero-pair counters during the activity to model how adding and subtracting zero does not change the value, reinforcing the identity property for integers.

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Methods used in this brief

• Case Study Analysis