Mathematics · Year 7

Active learning ideas

Generalising Patterns with Variables

Active learning works for generalising patterns with variables because students need to physically construct and test expressions to see their power. Moving from numbers to symbols is abstract, so tactile and visual tasks like building shapes or matching cards make the abstraction concrete and memorable.

• ACARA Content Descriptions
• AC9M7A01

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• RAFT Writing
• 35 min · Pairs

Pattern Building: Growing Shapes

Provide interlocking cubes or dots for students to build visual patterns, such as squares adding layers. Pairs record the first five terms, then write an expression for the nth term. Test predictions by building the 10th term together.

Explain why a variable is more powerful than a specific number when describing a rule.

Facilitation Tip

During Pattern Building: Growing Shapes, ask each group to predict the 10th shape before building it, so students see the value of a general rule over counting each dot.

What to look for

Provide students with the sequence 7, 11, 15, 19. Ask them to: 1. Identify the common difference. 2. Write an algebraic expression for the nth term. 3. Calculate the 10th term using their expression.

• Apply
• Analyze
• Create
• Social Awareness
• Self-Awareness

Activity 02

• RAFT Writing
• 40 min · Small Groups

Expression Match-Up: Cards Game

Prepare cards with patterns (e.g., 2,5,8,...), tables, and expressions (e.g., 3n-1). Small groups sort and match in 5 minutes, then justify matches and create their own set. Discuss mismatches as a class.

Construct an algebraic expression to represent the nth term of a linear pattern.

Facilitation Tip

For Expression Match-Up: Cards Game, circulate and listen for students explaining their choices using terms like 'common difference' or 'starting value' to reinforce precise language.

What to look for

Display a visual pattern (e.g., growing squares made of dots). Ask students to sketch the next two stages of the pattern and then write an algebraic expression for the number of dots in the nth stage. Observe student work for understanding of pattern progression and variable representation.

• Apply
• Analyze
• Create
• Social Awareness
• Self-Awareness

Activity 03

• RAFT Writing
• 30 min · Whole Class

nth Term Challenge: Relay Race

Divide class into teams. Each student solves one step: identify pattern from sequence, write expression, check nth term. Pass baton with correct work to next teammate. First accurate team wins.

Compare different algebraic expressions that describe the same pattern.

Facilitation Tip

In the nth Term Challenge: Relay Race, insist teams show their work for each term before moving on, so errors in expression writing are caught early.

What to look for

Pose the question: 'Imagine you found a pattern where the rule is 5n + 3. Your friend found a rule 5(n+1) - 2 for the same pattern. Who is correct, and how can you prove it?' Facilitate a class discussion where students test values and justify their reasoning.

• Apply
• Analyze
• Create
• Social Awareness
• Self-Awareness

Activity 04

• RAFT Writing
• 25 min · Small Groups

Compare Expressions: Venn Diagram

Give two equivalent expressions for a pattern. Individuals list pros/cons, then small groups create Venn diagrams comparing them. Share with class, voting on clearest expression.

Explain why a variable is more powerful than a specific number when describing a rule.

Facilitation Tip

Use Compare Expressions: Venn Diagram to have students swap diagrams with another pair and defend one placement, deepening peer accountability.

What to look for

Provide students with the sequence 7, 11, 15, 19. Ask them to: 1. Identify the common difference. 2. Write an algebraic expression for the nth term. 3. Calculate the 10th term using their expression.

• Apply
• Analyze
• Create
• Social Awareness
• Self-Awareness

A few notes on teaching this unit

Teach this topic by letting students experience the frustration of counting dots in later stages of a pattern, then introducing variables as a solution. Avoid rushing to formal algebra; instead, anchor expressions in the physical or visual pattern first. Research shows that students need repeated exposure to equivalent expressions through substitution to internalise equivalence, so multiple activities should require them to test and prove their rules.

Students will confidently write algebraic expressions for sequences and visual patterns, and justify their rules using multiple terms. They will explain why different expressions can represent the same pattern, showing flexible understanding of variables as general rules.

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Watch Out for These Misconceptions

• During Pattern Building: Growing Shapes, watch for students treating n as a fixed position only, such as labeling the first shape as n=1, second as n=2, and so on without using n in an expression.

  Hand each group a strip of paper with the term numbers (1, 2, 3) and ask them to write an expression for the number of dots in the nth shape before building it. Then, have them test their expression on shapes 4 and 5 to confirm generality.

• During Expression Match-Up: Cards Game, watch for students assuming any expression with +2 matches the sequence 5, 8, 11, 14, ignoring the coefficient of n.

  Have students test each candidate expression on at least three terms from the sequence, recording the results on a mini whiteboard. If an expression fails for one term, it should be discarded, reinforcing that rules must fit all terms.

• During Compare Expressions: Venn Diagram, watch for students placing equivalent expressions like 3n + 2 and n + n + n + 2 in separate circles because they look different.

  Direct students to substitute the same term number (e.g., n=4) into both expressions and compare results. If they yield the same number, they belong in the overlap, prompting students to explain why different forms can describe the same pattern.

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Methods used in this brief

• RAFT Writing