Formula Makeover

Overview

In this seventh lesson of Algebra Detectives: Solve, Simplify, Share, students rearrange familiar formulae from area, perimeter and speed contexts. They build on substitution and inverse operations, then research, explain and verify a “formula makeover” collaboratively.

Learning intentions

• WALT rearrange known formulae using one or two inverse-operation steps.
• WALT identify the subject of a formula and preserve equality.
• WALT check that a rearranged formula is equivalent by substituting values.
• WALT explain mathematical reasoning clearly to others.

Success criteria

• I can identify the subject of a formula.
• I can use inverse operations on both sides without changing equality.
• I can rearrange a formula and show each step.
• I can test my new formula with values and explain whether it works.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that deepen mathematical understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: challenge connected to familiar textbook topics and prior learning.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: open-ended challenge and justification.
• Te Mātaiaho competencies: communicating reasoning, making connections and persevering with challenging problems.

Lesson structure (45 minutes)

1. 0–5 min · Hook and recall. Display a rectangle with (A=lw), ask “If the area and length are known, how could we find the width?” Open with the hook and learning-intention slides and invite students to think, pair and share. Students identify the known quantities, predict a rearrangement and explain what they already know about inverse operations.

2. 5–12 min · Model the makeover. Use a labelled rectangle and the perimeter and area formula mat as a visual reference. Model making (w) the subject of (A=lw): divide both sides by (l), giving (A/l=w), then rewrite as (w=A/l). Emphasise that equality is preserved by doing the same operation to both sides. Students annotate the worked example on the formula makeover worksheet and complete a one-step example such as (P=2l+2w) when finding (l) from known (P) and (w), with teacher guidance.

3. 12–17 min · Check and rehearse. Show a deliberately incorrect rearrangement on the worked-example and error-analysis slides. Students use substitution to decide whether it is equivalent, then rehearse the sentence frame: “I made ___ the subject by ___ both sides by ___.” Check responses from several students and address common errors, including changing only one side or reversing multiplication and division incorrectly.

4. 17–30 min · Formula card research. Place students in three groups of three and give each group one context: rectangle area, rectangle perimeter or speed (d=st). Provide the relevant information and questions on the formula research and practice pages. Students identify the subject, rearrange the formula in one or two steps, test it using sensible values, and prepare a before-and-after “formula makeover”. Each group must include working, a labelled diagram or context description, and one explanation of why the new formula is equivalent. Circulate, question and record evidence of reasoning.

5. 30–39 min · Share and critique. Open the presentation instructions and discussion-prompt slides. Each group gives a two-minute presentation showing its original formula, rearranged formula and numerical check. The audience records one “mathematical strength” and asks one respectful question, demonstrating Manaaki and Aroha. Prompt students to compare which inverse operations were used and whether the check proves equivalence for the chosen values.

6. 39–45 min · Exit check and reflection. Display the final prompts on the plenary and exit-ticket slides. Students complete the final section of the individual exit check: make (w) the subject of (A=lw), use (A=36) and (l=9) to check the result, and state one rule for preserving equality. Collect responses and invite students to rate their confidence from 1–4.

Resources

• the Formula Makeover slide deck
• the formula makeover worksheet
• the perimeter and area formula mat
• Whiteboard and markers
• Calculators, if routinely used
• Prepared context prompts for area, perimeter and speed
• A3 paper or mini-whiteboards for group presentations
• Coloured pens or highlighters

Assessment

• During modelling, check whether students can identify the subject and state the inverse operation required.
• During group research, conference with each group: ask students to justify each step and verify with substituted values.
• Use the exit check to identify whether students can rearrange (A=lw), calculate accurately and explain preservation of equality. Re-teach one-step rearrangement before the next lesson where needed.

Differentiation

• Support learners with the labelled diagrams, the formula mat, colour-coding for known and unknown quantities, and the sentence frame “To make ___ the subject, I ___ both sides by ___.”
• Begin with one-step examples before introducing two-step perimeter or speed tasks. Allow students to use a calculator after the algebraic rearrangement has been completed.
• Provide symbol and language supports: subject, inverse operation, equivalent, divide both sides, and substitute. Pair students strategically and accept oral explanation, annotated diagrams or written reasoning.
• Extend advanced learners by asking them to rearrange a formula involving fractions or two operations, such as (v=(d-t)/s) or (P=2l+2w), and test equivalence with two different sets of values. Ask them to explain why one numerical check is useful but does not replace showing the algebraic steps.