Mathematics · Grade 3

Active learning ideas

Commutative and Associative Properties

Students need to see multiplication properties in action to trust the math, not just memorize rules. With concrete tools like tiles and counters, they can build, flip, and regroup to witness why the order and grouping of factors do not change the product. This hands-on proof builds lasting confidence in the reliability of these properties across all numbers.

• Ontario Curriculum Expectations
• 3.OA.B.5

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

Activity 01

• Numbered Heads Together
• 20 min · Pairs

Array Flip: Commutative Check

Pairs build a 3 × 4 array with counters, count the total units, then flip to 4 × 3 and recount. They draw both arrays and note the product stays the same. Discuss why order does not matter.

Explain how the commutative property helps simplify multiplication problems.

Facilitation Tip

During Array Flip, remind students to count rows and columns aloud before and after flipping to highlight the identical total.

What to look for

Provide students with two multiplication problems: 6 x 7 and 7 x 6. Ask them to draw an array for one problem and write the product for both. Then, ask them to write one sentence explaining why the products are the same.

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

Activity 02

• Numbered Heads Together
• 30 min · Small Groups

Group Shift: Associative Relay

Small groups get factors like 2, 3, 4 on cards. Multiply one grouping like (2 × 3) × 4, then shift to 2 × (3 × 4), compare results. Record steps on chart paper.

Analyze how changing the order of factors affects the product.

Facilitation Tip

In Group Shift, circulate to listen for language like 'split' or 'move' as students regroup, reinforcing the associative concept.

What to look for

Write the expression (2 x 5) x 3 on the board. Ask students to rewrite the expression showing how they would group the factors differently to solve it using the associative property. Then, have them calculate the product.

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

Activity 03

• Numbered Heads Together
• 25 min · Individual

Property Puzzle: Card Sort

Individuals sort equation cards into commutative, associative, or neither piles. Pairs check each other's sorts and justify with quick array sketches. Share one example class-wide.

Construct an example to demonstrate the associative property of multiplication.

Facilitation Tip

For Property Puzzle, model how to justify choices by pointing to matching arrays on the cards to anchor the reasoning.

What to look for

Pose the question: 'How does knowing that 3 x 8 is the same as 8 x 3 help you solve a problem like 8 x 6?' Facilitate a brief class discussion where students share strategies for using the commutative property to simplify calculations.

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

Activity 04

• Numbered Heads Together
• 35 min · Pairs

Real-World Arrays: Partner Build

Pairs use grid paper to model real items like 2 rows of 3 boxes of 4 cookies. Regroup as (2 × 3) × 4 or 2 × (3 × 4), calculate both ways, discuss totals.

Explain how the commutative property helps simplify multiplication problems.

Facilitation Tip

In Real-World Arrays, circulate with a checklist to note which pairs use the commutative property correctly in their designs.

What to look for

Provide students with two multiplication problems: 6 x 7 and 7 x 6. Ask them to draw an array for one problem and write the product for both. Then, ask them to write one sentence explaining why the products are the same.

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

A few notes on teaching this unit

Use the commutative property to build fluency first, since students can see it clearly with arrays. Teach the associative property next by connecting it to grouping in real-world contexts like packaging or stacking. Avoid rushing to abstract symbols; let students verbalize their observations before introducing formal notation. Research shows that students who articulate their own proofs internalize the concepts more deeply than those who only hear explanations.

By the end of these activities, students will explain why 4 × 6 equals 6 × 4 and why (2 × 3) × 5 equals 2 × (3 × 5) using visual models and clear language. They will apply these properties to simplify multiplication problems and share their reasoning with peers in partner and whole-group discussions.

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Watch Out for These Misconceptions

• During Array Flip, watch for students who believe 3 × 4 and 4 × 3 produce different products because the arrangement looks different.

  Remind students to count the total units in each array aloud and compare the counts side-by-side to see the equality before writing the products.

• During Group Shift, listen for students who claim grouping only works for addition because they see counters grouped differently but not matched by product.

  Have students calculate the partial products for each grouping and compare totals, using the same counters to see how 2 × (3 × 4) equals (2 × 3) × 4.

• During Real-World Arrays, note students who generalize that these properties only work for small factors seen in class examples.

  Challenge pairs to extend their arrays to larger factors like 5 × 2 × 3, counting and regrouping to confirm the pattern holds for bigger numbers.

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Methods used in this brief

• Numbered Heads Together