Commutative Multiplication

Overview

Students explore the commutative property of multiplication by comparing number sentences and tape diagrams, recognising that changing the order of factors does not change the total. They represent equal groups as repeated addition and then use dice to generate and check commutative multiplication examples.

Learning intentions

• Students will understand that swapping the order of numbers in multiplication does not change the total.
• Students will represent multiplication using repeated addition and tape diagrams (equal groups).
• Students will create and compare multiplication number sentences, including examples such as 14 × 4 and 4 × 14.
• Students will justify their answers using diagrams and reasoning.

Success criteria

• I can use a tape diagram to show equal groups for a multiplication problem.
• I can write the repeated addition that matches a multiplication sentence.
• I can explain why 4 × 5 equals 5 × 4 using my diagram or number sentence.
• I can check my commutative example with a partner and correct it if needed.

Curriculum links

• Number — VC2M3N05: multiply and divide one- and two-digit numbers; represent problems using number sentences, diagrams and arrays; use a variety of calculation strategies.
• Number — VC2M3N05: use part-part-whole and comparative models; represent multiplicative relationships visually and connect multiplication to repeated addition.
• Number — VC2M3N05: match or create problem scenarios represented by multiplication and division number sentences (focus here is multiplication/comparison).

Lesson structure (60 minutes)

1. 0–5 min · Hook (visual comparison). Teacher displays two side-by-side examples: “4 + 4 + 4 + 4 + 4” and “5 × 4”, then “14 × 4” and “4 × 14”. Students do a quick think: “What stays the same? What changes?”

2. 5–15 min · Explicit mini lesson (commutative idea + representations). Teacher models with repeated addition first:

• For 5 × 4, write 4 + 4 + 4 + 4 + 4.
• Build a tape diagram with 5 equal sections (each section labelled 4). Then show commutativity with swapped factors:
• 4 × 5 can be shown as 5 added 4 times (4 sections labelled 5, or alternatively 4 groups of 5).
• Emphasise the key message: the total number of units is the same even when factor order changes. Teacher states: “When we multiply, the order of factors changes, but the total number of things does not.”

3. 15–25 min · Guided practice (shared examples). On the board, teacher writes two pairs: “5 × 4 and 4 × 5” and “14 × 4 and 4 × 14”. Teacher draws one tape diagram for 5 × 4, then asks students to partner-suggest the matching repeated addition and what the swapped tape diagram would look like (no full drawing required from all students). Students share one explanation each.

4. 25–40 min · Independent task (dice + tape diagram + number sentence). Students work individually, then check with a partner for one minute at the end.

• Roll dice to choose two factors (units of 4,5,6,7… etc depending on dice used).
• Write a multiplication sentence: a × b.
• Draw a tape diagram showing a equal groups of size b (or b equal groups of size a).
• Write the repeated addition that matches the diagram.
• Then write the commutative sentence: b × a, and predict the total (without re-drawing if time is tight). Differentiation: use standard dice; for some students teacher provides 6-sided, others 9-sided or 20-sided dice to reach manageable factor ranges.

5. 40–48 min · Enable task (support scaffold). Students who need support complete a short “commutative matching” sheet:

• Given a tape diagram for one multiplication sentence, choose the correct repeated addition and the correct swapped sentence.
• Sentence starters provided: “The total stays the same because…” and “In the commutative example, the groups are swapped.”

6. 48–55 min · Extension task (deepen reasoning). Students choose one challenge card:

• Create two different commutative multiplication sentences that both show the same repeated addition total (e.g., using factors that multiply to 24 or 36). Draw a tape diagram for one and only a number sentence for the other, explaining why it must match.

7. 55–60 min · Exit ticket + debrief (quick). Exit ticket collected immediately. Students complete the debrief questions verbally with the teacher, then submit.

Resources

• Dice options: 6-sided, 9-sided, 20-sided
• Dice cups or trays (optional)
• “Tape diagram” printable template (equal sections with labels space)
• Independent task worksheet: roll → sentence → tape diagram → repeated addition → commutative sentence
• Enable task matching sheet with sentence starters
• Extension challenge cards
• Whiteboard/marker, visual examples prepared on slides/board
• Coloured pencils for students to shade equal groups (optional)
• Timer for partner check and task transitions

Assessment

• Teacher formative check during independent task: listen for correct linking between multiplication, repeated addition, and the tape diagram.
• Quick partner verification: students point to where “equal groups” appear in the swapped commutative diagram.
• Exit ticket (individual): correctness of totals plus explanation using a commutative statement.

Differentiation

• Support: enable task sheet with sentence starters and partially labelled tape diagrams (fewer factors or provided factors).
• Access: allow students to choose which diagram direction to draw (a groups of size b, or b groups of size a).
• Extension: 20-sided dice for larger two-digit products and challenge cards requiring two commutative pairs with the same total.
• For students needing extra structure: provide a multiplication fact “bank” for factors used today (e.g., focus on multiplying by 4,5,6,7,8 depending on dice outcomes).

Exit ticket

Answer these and show reasoning with words or a simple tape diagram:

1. Complete: 14 × 4 = ______ × ______
2. Write the repeated addition for 4 × 5.
3. True or false: 5 × 4 and 4 × 5 give the same total. Explain in one or two sentences.

Debrief questions

• What stays the same when you swap the order in a multiplication sentence?
• How does the tape diagram show “equal groups”?
• In your work, how did repeated addition help you prove the commutative property?
• What common mistake might someone make with commutative multiplication, and how can we check it?