Formula to Solution

Overview

In this lesson, students learn how formulas relate to equations and how to rearrange them to solve practical problems. This builds toward solving linear equations and using them to model real situations.

Learning intentions

• Students will identify how a formula can represent a relationship between quantities.
• Students will express a formula as an equation and interpret what each variable means.
• Students will rearrange simple linear equations to solve for an unknown.
• Students will apply their rearranged equations to solve worded, practical problems.

Success criteria

• I can state what the symbols in a formula represent in context.
• I can rewrite a formula as an equation and choose the correct unknown to solve for.
• I can rearrange a linear equation using inverse operations correctly.
• I can check my solution by substituting back into the original formula.

Curriculum links

• Linear equations and their graphs — Solve practical problems involving linear equations.
• Linear equations and their graphs — Develop a linear equation from a description in words.
• Linear and non-linear relationships (Algebra) — Use technology to construct tables of values from a formula (used here as a quick check).
• Mathematics Methods — Solve mathematical problems by reasoning and applying algebraic methods to reach solutions.

Lesson structure (60 minutes)

1. 0–5 min · Starter (formula sense). Teacher displays two short contexts (e.g., distance–time and cost–number) and asks what a “rule” might look like as symbols. Students write 1–2 sentences describing what quantities are likely included and what the unknown could be.

2. 5–15 min · Direct teach: formula to equation. Teacher models: start with a formula (e.g., total cost (C = 2n + 5)) and rewrite it as an equation for different unknowns (solve for (n), then solve for (C)). Students follow a guided example on the board, circling the unknown and underlining the given information in the context.

3. 15–30 min · Mini-lesson: rearranging step by step. Teacher demonstrates rearranging a linear equation using inverse operations, highlighting one operation per step (e.g., from (C = 2n + 5) to (C-5 = 2n), then ((C-5)/2 = n)). Students complete a short guided set of 3 rearrangements where the equation remains linear and the target variable changes each time.

4. 30–45 min · Practical problem solving (worked then shared). Teacher provides two practical problems and solves the first together, focusing on: translating words to an equation, choosing the unknown, rearranging, then checking. Students work in pairs on the second problem, showing their algebra steps and performing a substitution check.

5. 45–55 min · Quick technology/representation check (table of values). Teacher shows how a quick table can confirm a rearranged solution (e.g., once (n) is found, calculate (C) using the original formula or try nearby values to see the pattern). Students complete a small table (3 values) and state whether their answer makes sense in context.

6. 55–60 min · Exit ticket (individual). Teacher gives one short worded problem requiring rearrangement of a simple linear formula. Students submit the equation they formed, the rearranged steps, and a one-line substitution check.

Resources

• Board and markers/whiteboard
• Student worksheets with 2 practical problems and a rearrangement practice set
• Graphing calculator or spreadsheet (optional) for the table-of-values check
• Printed formula cards (e.g., (C = 2n + 5), (d = vt), (A = lw))—choose ones matching today’s examples
• Sticky notes or small response slips for the exit ticket

Assessment

• Formative checks during guided rearrangements: teacher listens for correct inverse operations and correct algebra steps.
• During practical problem solving: teacher reviews word-to-equation translation and substitution checking.
• Exit ticket: equation formation, correct rearrangement, and at least one substitution check line.

Differentiation

• Support: provide a worked example template with headings (“Write equation”, “Rearrange”, “Solve”, “Check”). Include sentence starters for translating words into algebra.
• Support for algebra confidence: limit practice to equations of the form (ax+b=c) where (a) and (b) are integers or simple fractions; provide a “step bank” of inverse operations.
• Extension: ask students to compare two methods (rearrange algebraically vs check with a table) and explain which is faster and why.
• EAL/SEN considerations: allow students to highlight known quantities and unknowns with colour coding; use short, consistent contexts and read word problems aloud before independent work.