Mathematics · Grade 8

Active learning ideas

Properties of Real Numbers

Students need to move beyond memorizing rules to truly grasp how properties of real numbers simplify calculations. Active learning lets them test these properties with their own hands, uncovering when and why each one works. This approach builds lasting intuition for mental math and algebraic reasoning, especially in mixed-ability classrooms where abstract explanations can feel distant.

• Ontario Curriculum Expectations
• Ontario Curriculum Mathematics 2020, Grade 8, Number B1.1: represent and compare rational numbers, including integers and positive and negative fractions and decimals
• Ontario Curriculum Mathematics 2020, Grade 8, Number B1.2: describe the relationship between the subsets of numbers that make up the set of rational numbers
• Ontario Curriculum Mathematics 2020, Grade 8, Number B1.3: use the concepts of square and square root to solve problems involving the side lengths and areas of squares

• 20–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Think-Pair-Share
• 25 min · Pairs

Pairs: Property Card Sort

Prepare cards with numerical expressions like 2+3 and 3+2. Pairs sort them into commutative, associative, or distributive piles, then create their own examples to test. Discuss why subtraction cards do not fit commutative.

Explain how the commutative property simplifies calculations in different contexts.

Facilitation Tip

During Property Card Sort, circulate and ask each pair to explain one choice before they record it, ensuring verbal reasoning drives the task.

What to look for

Present students with a series of expressions, such as 5 + (3 + 7) and (5 * 3) * 7. Ask them to identify which property (commutative or associative) allows them to rearrange the numbers and explain how it simplifies the calculation. For example, 'Which property allows you to calculate 5 + 10 first, and why is that easier?'

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 02

• Think-Pair-Share
• 35 min · Small Groups

Small Groups: Simplification Relay

Divide class into groups of four. Write a complex expression on the board. First student simplifies one step using a property, tags the next, until solved. Groups compare final answers and justifications.

Analyze the role of the distributive property in algebraic expressions.

Facilitation Tip

In Simplification Relay, set a visible timer for each round to keep energy high and prevent overthinking; students race against the clock, not each other.

What to look for

Give students the expression 3(x + 4). Ask them to use the distributive property to rewrite the expression and calculate its value if x = 2. Then, provide an expression like 12 + 5 + 8 and ask them to show two different ways to group the numbers using the associative property to find the sum.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 03

• Think-Pair-Share
• 40 min · Whole Class

Whole Class: Visual Property Builder

Use algebra tiles or drawings on chart paper. Class votes on steps to simplify expressions like 3(2x + 4) step-by-step, modeling distribution visually. Record class justifications on shared anchor chart.

Justify the application of specific properties to simplify numerical expressions.

Facilitation Tip

For Visual Property Builder, prepare two colors of tiles so students can physically model both sides of an equation to see equivalence.

What to look for

Pose the question: 'When might the commutative property be more useful than the associative property, and vice versa?' Facilitate a class discussion where students share examples of calculations where one property offers a clearer advantage for mental math or simplification.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 04

• Think-Pair-Share
• 20 min · Individual

Individual: Property Journal

Students list five real-world contexts, like shopping totals for commutativity. They write justifications and one counterexample per property, then share one with a partner for feedback.

Explain how the commutative property simplifies calculations in different contexts.

Facilitation Tip

With Property Journal, model the first entry with a think-aloud to demonstrate how to connect properties to personal calculation strategies.

What to look for

Present students with a series of expressions, such as 5 + (3 + 7) and (5 * 3) * 7. Ask them to identify which property (commutative or associative) allows them to rearrange the numbers and explain how it simplifies the calculation. For example, 'Which property allows you to calculate 5 + 10 first, and why is that easier?'

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

A few notes on teaching this unit

Teachers should start concrete, using tiles and cards to ground abstract names in tangible actions. Avoid rushing to symbols; let students name properties after they’ve discovered them through play. Research shows that misconceptions about subtraction and division persist when examples are limited to positive numbers, so include negative cases early and often. Encourage debate during group work to surface hidden misunderstandings before they calcify.

Students will correctly label commutative, associative, and distributive properties in varied expressions. They will justify choices with clear examples and apply properties to simplify calculations efficiently. Confidence in rearrangement and mental strategies signals successful learning.

---

Watch Out for These Misconceptions

• During Property Card Sort, watch for pairs labeling subtraction or division expressions as commutative without testing numerical examples.

  Prompt them to calculate both orders, like 7 - 2 versus 2 - 7, and ask if the results match. If not, guide them to reclassify the operation with a sticky note correction.

• During Simplification Relay, listen for groups claiming associativity applies to subtraction without testing regrouped expressions.

  Freeze the group and have them compute (10 - 4) - 2 versus 10 - (4 - 2). When they see the mismatch, ask them to adjust their relay cards and discuss why subtraction is different.

• During Visual Property Builder, observe students avoiding negative numbers in their tile models, limiting their understanding of the distributive property.

  Hand them two red tiles and two green tiles, then ask them to model 2(-1 + 3). When they see the tiles balance on both sides, they’ll recognize the property holds beyond positive values.

---

Methods used in this brief

• Think-Pair-Share