Mathematics · Grade 5

Active learning ideas

Generating and Analyzing Patterns

Active learning helps students move beyond memorizing steps to truly understanding how patterns grow and change. By manipulating materials and discussing ideas with peers, students build mental models that connect position and value in ways abstract rules alone cannot. The activities in this section give students multiple ways to see and describe patterns, reinforcing their understanding through movement, discussion, and concrete examples.

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

• 25–55 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Inquiry Circle
• 55 min · Small Groups

Inquiry Circle: Pattern Architects

Groups create a growing pattern using pattern blocks. They must record the first four terms, create a table of values, and write a secret 'rule' on a hidden card. Other groups visit the station and try to predict the 10th term based on the table.

Predict the next terms in a pattern given a rule.

Facilitation Tip

During Pattern Architects, circulate and ask students to explain their rules aloud while pointing to the visual representation of each step.

What to look for

Provide students with a table showing term numbers and term values for two patterns. Ask them to write the rule for each pattern and identify the relationship between the term values in the 5th position.

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Activity 02

• Gallery Walk
• 45 min · Small Groups

Gallery Walk: Visualizing the Rule

Students are given a rule like 'Start at 2 and add 3 each time.' They must represent this pattern in three ways: a drawing, a table, and a word problem. They display their work, and the class uses a gallery walk to find patterns that grow at the same rate but look different.

Analyze the relationship between two different patterns generated by distinct rules.

Facilitation Tip

During the Gallery Walk, provide sentence stems for discussion such as 'The pattern grows by... because...' to guide students' observations.

What to look for

Give students a rule, such as 'Add 3 to the term number'. Ask them to generate the first four terms of the pattern and then write one sentence explaining how they found the 10th term without listing all the previous ones.

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Activity 03

• Think-Pair-Share
• 25 min · Pairs

Think-Pair-Share: The 100th Term Challenge

The teacher presents a simple growing pattern (e.g., 2, 4, 6, 8...). Students discuss in pairs how they could find the 100th term without counting. This encourages them to look for a relationship between the term number and the term value (multiplicative thinking).

Construct a rule that describes the growth of a given numerical pattern.

Facilitation Tip

During The 100th Term Challenge, have students write their first three predictions separately before comparing with a partner to catch early missteps.

What to look for

Present two patterns: Pattern A (Rule: Multiply by 2, then add 1) and Pattern B (Rule: Add 3 to the term number). Ask students to compare the growth of these two patterns. Which pattern grows faster? How can you tell from the numbers?

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

A few notes on teaching this unit

Teachers should emphasize the difference between recursive and functional rules by modeling both approaches side by side. Avoid rushing to the 'answer'—instead, ask students to articulate their own rules before revealing the teacher’s version. Research shows that students benefit from multiple representations, so connect visual, numerical, and verbal explanations throughout the lessons. Plan for misconceptions by building in opportunities for students to explain their thinking, not just produce it.

Students will confidently describe pattern rules using precise language, distinguish between recursive and functional rules, and use tables to predict future terms. They will articulate how the term number relates to the term value, and they will justify their predictions with evidence from their work. Collaboration will help them refine their thinking and catch errors in their reasoning.

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Watch Out for These Misconceptions

• During Collaborative Investigation: Pattern Architects, watch for students who describe the pattern only by what changes from one step to the next without explaining the relationship between the term number and the term value.

  Prompt students to complete a two-column table labeled 'Position' and 'Value' during their investigation. Ask them to explain how to move from the left column to the right column, forcing them to articulate the functional rule rather than just observing recursive changes.

• During Gallery Walk: Visualizing the Rule, watch for students who assume all patterns grow by adding the same number each time, even when the visual or numerical data suggests otherwise.

  Provide a sorting sheet with columns for 'Additive Patterns' and 'Multiplicative Patterns.' Have students categorize examples during the walk, stopping to discuss why some patterns grow by doubling or other operations instead of constant addition.

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Methods used in this brief

• Inquiry Circle
• Gallery Walk
• Think-Pair-Share