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.

• Grade 3
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Demonstrate the commutative property of multiplication by rearranging arrays and writing corresponding number sentences.
2. Analyze how changing the order of factors in a multiplication problem affects the product using visual models.
3. Explain the role of the commutative property in simplifying multiplication calculations.
4. Construct examples to illustrate the associative property of multiplication by grouping factors differently.
5. Compare the products of multiplication problems solved with and without applying the commutative and associative properties.

Jigsaw

⏱ 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.

Prepare & details

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.

Setup: Flexible seating for regrouping

Materials: Expert group reading packets, Note-taking template, Summary graphic organizer

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Jigsaw

⏱ 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.

Prepare & details

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.

Setup: Flexible seating for regrouping

Materials: Expert group reading packets, Note-taking template, Summary graphic organizer

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Jigsaw

⏱ 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.

Prepare & details

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.

Setup: Flexible seating for regrouping

Materials: Expert group reading packets, Note-taking template, Summary graphic organizer

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Jigsaw

⏱ 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.

Prepare & details

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.

Setup: Flexible seating for regrouping

Materials: Expert group reading packets, Note-taking template, Summary graphic organizer

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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

Assessment Ideas

Exit Ticket

After Array Flip, provide students with two multiplication problems: 6 × 7 and 7 × 6. Ask them to draw an array for one problem and write the product for both. Then, have them write one sentence explaining why the products are the same.

Quick Check

After Group Shift, write the expression (2 × 5) × 3 on the board. Ask students to rewrite the expression showing a different grouping to solve it using the associative property. Then, have them calculate the product and explain how they regrouped the factors.

Discussion Prompt

During Real-World Arrays, pose the question: 'How does knowing that 3 × 8 is the same as 8 × 3 help you solve a problem like 8 × 6?' Facilitate a brief class discussion where students share strategies for using the commutative property to simplify calculations.

Extensions & Scaffolding

• Challenge pairs to create a three-factor multiplication problem using 2, 3, and 4, then find all possible groupings to prove the associative property holds.
• Scaffolding: Provide partially completed arrays with missing rows or columns for students to fill in before flipping or regrouping.
• Deeper exploration: Ask students to write a short paragraph explaining how the commutative and associative properties help them solve 3 × 12 × 2 more efficiently than calculating left to right.

Key Vocabulary

Commutative Property	This property states that the order of factors in a multiplication problem does not change the product. For example, 3 x 5 is the same as 5 x 3.
Associative Property	This property states that the way factors are grouped in a multiplication problem does not change the product. For example, (2 x 3) x 4 is the same as 2 x (3 x 4).
Array	An arrangement of objects in equal rows and columns, often used to visualize multiplication.
Factor	A number that is multiplied by another number to get a product.
Product	The answer when two or more numbers are multiplied together.