The Commutative Property

Active learning helps students build a concrete understanding of the commutative property by allowing them to physically manipulate numbers and see relationships firsthand. Engaging with manipulatives and sorting activities moves beyond rote memorization, fostering deeper conceptual grasp.

• Year 2
• Mathematics
• 3 activities
• 10 min–20 min

Collaborative Problem-Solving

⏱ 15 min·Small Groups

Addition Commutativity with Manipulatives

Provide students with counters or blocks. Ask them to build towers representing 5 + 3, then 3 + 5. Have them compare the total height of each tower to see they are the same. Repeat with other small numbers.

Prepare & details

Explain why we can swap numbers in an addition sentence but not in a subtraction sentence.

Facilitation Tip: During Collaborative Problem-Solving for the Number Family Exploration, ensure students are taking turns in their defined roles to explore different combinations.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 10 min·Pairs

Subtraction Order Sort

Give pairs of students cards with subtraction problems like '7 - 2' and '2 - 7'. They must sort these into two piles: 'Answers are the same' and 'Answers are different'. Discuss why the answers differ.

Prepare & details

Analyze how knowing 7 + 3 helps us solve 10 minus 7 without counting.

Facilitation Tip: During Stations Rotation, monitor students at the Addition Commutativity with Manipulatives station to ensure they are accurately representing and comparing the two addition sentences.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 20 min·Small Groups

Number Family Exploration

Using a part-whole model (e.g., a circle with three sections), students choose three numbers that form an addition/subtraction family (e.g., 4, 5, 9). They write all four related number sentences and discuss which ones are commutative.

Prepare & details

Construct a part-whole model to show all the facts in a number family.

Facilitation Tip: During Stations Rotation, observe students at the Subtraction Order Sort station to see if they are discussing why certain pairs can or cannot be sorted into commutative pairs.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Focus on concrete representations before moving to abstract symbols. Using manipulatives in activities like Addition Commutativity with Manipulatives helps students visualize that combining sets results in the same total regardless of the order. Explicitly contrast this with subtraction, where order matters significantly.

What to Expect

Students will demonstrate understanding by correctly identifying and applying the commutative property in addition. They will be able to explain why the order of numbers doesn't change the sum and articulate how this differs from subtraction.

Watch Out for These Misconceptions

Common Misconception

During Addition Commutativity with Manipulatives, students may think subtraction is commutative because addition is.

What to Teach Instead

After modeling 8 - 3 and 3 - 8 with manipulatives, redirect students to compare the remaining counters. Discussing these concrete examples helps students distinguish between the properties of addition and subtraction.

Common Misconception

During Number Family Exploration, students might confuse the commutative property of addition with the associative property.

What to Teach Instead

Guide students to focus only on rearranging two numbers within their part-whole model, such as showing 2+3 and 3+2. Demonstrating how adding a third number in different orders, like (2+3)+1 versus 2+(3+1), can clarify the associative property without confusing it with commutativity.

Assessment Ideas

Quick Check

During Addition Commutativity with Manipulatives, observe students as they build towers and record their findings to check for accurate representation of both addition sentences.

Discussion Prompt

After Subtraction Order Sort, ask students to explain to a partner why '7 - 2' and '2 - 7' are not commutative, using examples from their sorted cards.

Exit Ticket

After Number Family Exploration, ask students to write two addition sentences that show the commutative property using numbers from their part-whole model.

Extensions & Scaffolding

• Challenge: Ask students to write an 'if...then...' statement about when the order of numbers in a calculation matters and when it does not.
• Scaffolding: For students struggling with Subtraction Order Sort, provide pre-made number bonds or equations that only involve small numbers (e.g., within 5).
• Deeper Exploration: Have students explore if the commutative property applies to other operations like multiplication or division.