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.

• Year 7
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Construct algebraic expressions to represent the nth term of given linear numerical and visual patterns.
2. Compare and contrast different algebraic expressions that describe the same linear pattern, justifying their equivalence.
3. Explain, using examples, why a variable is a more powerful tool than a specific number for describing general rules.
4. Analyze a given linear pattern and identify the constant difference and the starting value to formulate its algebraic rule.

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.

Prepare & details

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.

Setup: Standard classroom seating, individual or paired desks

Materials: RAFT assignment card, Historical background brief, Writing paper or notebook, Sharing protocol instructions

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

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.

Prepare & details

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.

Setup: Standard classroom seating, individual or paired desks

Materials: RAFT assignment card, Historical background brief, Writing paper or notebook, Sharing protocol instructions

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

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.

Prepare & details

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.

Setup: Standard classroom seating, individual or paired desks

Materials: RAFT assignment card, Historical background brief, Writing paper or notebook, Sharing protocol instructions

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

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.

Prepare & details

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.

Setup: Standard classroom seating, individual or paired desks

Materials: RAFT assignment card, Historical background brief, Writing paper or notebook, Sharing protocol instructions

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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.

What to Teach Instead

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.

Common Misconception

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.

What to Teach Instead

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.

Common Misconception

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.

What to Teach Instead

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.

Assessment Ideas

Exit Ticket

After Pattern Building: Growing Shapes, give students the sequence 4, 7, 10, 13 and ask them to: 1. Find the common difference. 2. Write an algebraic expression for the nth term. 3. Use their expression to find the 15th term.

Quick Check

During Expression Match-Up: Cards Game, display a visual pattern (e.g., L-shapes made of squares) and ask students to sketch the next two stages and write an expression for the nth stage. Collect one expression per pair to check for correct coefficients and constants.

Discussion Prompt

After Compare Expressions: Venn Diagram, pose the question: 'Two students wrote 2n + 5 and 5 + 2n for the same pattern. Are both correct? How can you prove it?' Facilitate a class discussion where students substitute values and justify their reasoning.

Extensions & Scaffolding

• Challenge: Provide a non-linear pattern (e.g., squares: 1, 4, 9, 16) and ask students to write two different expressions for the nth term, justifying equivalence.
• Scaffolding: For students struggling with variables, have them write the rule in words first (e.g., 'three times the term number plus two') before translating to symbols.
• Deeper exploration: Ask students to create their own visual pattern with a rule involving two operations (e.g., 2n + 3) and trade with a peer to find and justify the rule.

Key Vocabulary

Variable	A symbol, usually a letter like 'n', that represents a number that can change or vary, often used to represent the position of a term in a sequence.
Term	A single number or element in a sequence or pattern. For example, in the sequence 3, 6, 9, 12, each number is a term.
nth term	A formula or expression that describes any term in a sequence based on its position (n) in the sequence. It allows calculation of any term without listing all preceding terms.
Linear pattern	A sequence where the difference between consecutive terms is constant. This constant difference is often called the common difference.
Algebraic expression	A mathematical phrase that contains variables, numbers, and operation symbols, used to represent a rule or relationship.