Mathematics · Year 6

Active learning ideas

Writing Simple Formulae

Active learning works for writing simple formulae because students must physically manipulate quantities and symbols to see how they relate. Moving from concrete measurements to abstract notation strengthens their understanding of variables and operations. These tasks make the purpose of formulae clear: they turn repeated calculations into reusable tools.

• National Curriculum Attainment Targets
• KS2: Mathematics - Algebra

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Problem-Based Learning
• 30 min · Pairs

Pairs: Perimeter String Challenge

Pairs use string and tape measures to enclose classroom objects, recording lengths and widths. They derive and write the perimeter formula, then test it on three different shapes. Partners swap roles to verify calculations.

Justify why it is useful to have a universal formula for calculating things like area or perimeter.

Facilitation Tip

During the Perimeter String Challenge, circulate with a metre stick and string so pairs can physically measure and adjust lengths as they test their perimeter formula.

What to look for

Present students with a scenario: 'A baker sells cookies for $2 each. Write a formula to calculate the total cost (C) of buying 'n' cookies.' Ask students to write the formula and then calculate the cost of 7 cookies.

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

Activity 02

• Problem-Based Learning
• 45 min · Small Groups

Small Groups: Lemonade Stand Formula Design

Groups brainstorm costs for a class lemonade stand, identifying variables like cups sold and price per cup. They write a total profit formula and adjust it for extras like cups cost. Groups present and test each other's formulae with sample data.

Design a simple formula to represent a real-life scenario.

Facilitation Tip

In the Lemonade Stand Formula Design, ask each group to present their formula on a large sheet and invite others to spot missing variables or unclear steps.

What to look for

Provide students with two scenarios: 1) Calculating the area of a rectangle with length 5cm and width 3cm. 2) Calculating the total cost of 4 apples costing $0.50 each. Ask: 'Which scenario is more likely to benefit from a general formula? Explain why, considering how many different calculations you might need to do.'

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

Activity 03

• Problem-Based Learning
• 35 min · Whole Class

Whole Class: Formula Evaluation Relay

Divide class into teams. Project scenarios; one student per team writes a formula at the board, next justifies it, third evaluates with numbers. Teams discuss improvements before rotating.

Evaluate the effectiveness of a given formula for a specific problem.

Facilitation Tip

For the Formula Evaluation Relay, ensure every student contributes one step in the calculation so no one waits passively for their turn.

What to look for

Give students a simple formula, e.g., 'P = 2l + 2w' for perimeter. Ask them to define what 'P', 'l', and 'w' represent in this formula and then calculate the perimeter of a rectangle with l=6cm and w=4cm.

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

Activity 04

• Problem-Based Learning
• 25 min · Individual

Individual: Savings Goal Creator

Students design a formula for reaching a savings goal, using weekly amount and weeks as variables. They solve for different goals and reflect on formula strengths in journals.

Justify why it is useful to have a universal formula for calculating things like area or perimeter.

What to look for

Present students with a scenario: 'A baker sells cookies for $2 each. Write a formula to calculate the total cost (C) of buying 'n' cookies.' Ask students to write the formula and then calculate the cost of 7 cookies.

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

A few notes on teaching this unit

Teachers should model the process of turning a real situation into a formula by thinking aloud as they write. Avoid rushing to abstract notation; anchor each step in a concrete context first. Research shows students benefit from seeing multiple worked examples, so share both correct and flawed attempts and discuss why each works or not.

Students will confidently write formulae using variables to represent changing quantities. They will justify why a formula fits a scenario and adjust it when conditions change. Most importantly, they will explain how their formula saves time and reduces errors in repeated calculations.

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Watch Out for These Misconceptions

• During the Perimeter String Challenge, watch for students who treat letters as fixed numbers instead of variables that can change.

  Ask them to swap the length and width on their rectangle and recalculate using the same formula to see that the letters represent any value, not just the current measurement.

• During the Lemonade Stand Formula Design, watch for students who ignore the order of operations when writing their cost formula.

  Have them write the formula both ways (with and without brackets) and calculate the cost of 5 cups at $1.20 each to show the difference between 5 × 1.20 and 5 × (1 + 0.20).

• During the Formula Evaluation Relay, watch for students who believe a formula is only useful in one fixed situation.

  After the relay, ask groups to change one variable, such as the price per lemonade cup, and adjust their original formula to fit the new scenario.

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Methods used in this brief

• Problem-Based Learning