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.

• Year 6
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Formulate simple algebraic expressions to represent given numerical relationships.
2. Calculate unknown values using provided formulae for area, perimeter, or cost.
3. Justify the efficiency of using formulae for repetitive calculations compared to arithmetic methods.
4. Design a formula to model a real-world scenario, such as calculating the cost of multiple items.
5. Evaluate the suitability of a given formula for solving a specific problem, explaining any limitations.

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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).

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Quick Check

After the Lemonade Stand Formula Design, pose the cookie scenario and ask students to write C = 2 × n and calculate C for n = 7. Collect responses to check if they correctly use a variable and perform the calculation.

Discussion Prompt

During the Perimeter String Challenge, provide the two scenarios and ask: 'Which scenario benefits more from a general formula? Discuss how many different rectangles or apple totals you might need to calculate.' Listen for reasoning about repeated calculations and adaptability.

Exit Ticket

After the Formula Evaluation Relay, give students the perimeter formula P = 2l + 2w and ask them to define P, l, and w, then calculate P for l = 6 and w = 4. Review responses to assess symbol interpretation and order of operations.

Extensions & Scaffolding

• Challenge: Give students a formula like 'Score = 3 × goals + 2 × assists' and ask them to design a sport where this scoring system would encourage teamwork.
• Scaffolding: Provide partially written formulae with blanks for missing variables during the Savings Goal Creator so students focus on identifying what each letter stands for.
• Deeper exploration: Ask students to compare two different area formulae for parallelograms and triangles, then create a general polygon area formula using variables for sides and heights.

Key Vocabulary

Formula	A rule or a set of rules expressed in symbols, often using letters to represent unknown quantities, that shows how different quantities are related.
Variable	A symbol, usually a letter, that represents a quantity that can change or vary. For example, 'l' for length or 'c' for cost.
Expression	A combination of numbers, variables, and operation symbols that represents a mathematical relationship, but does not contain an equals sign.
Constant	A value that does not change, represented by a number or a symbol that always stands for the same quantity.