Diving into Formulae and Equations

Overview

In this lesson (Lesson 3 of 9), students build on prior work with linear relationships by learning to identify equations from situations, rearrange them for a target variable, and solve practical problems using linear equations. Students will practise with realistic contexts and check reasonableness of solutions.

Learning intentions

• Students will recognise when a situation can be modelled using a linear equation.
• Students will rearrange a linear equation to make a chosen variable the subject.
• Students will solve linear equations using algebraic methods and verify solutions in context.
• Students will connect solutions to practical meaning (units, scales, and reasonableness).

Success criteria

• I can form a linear equation from a short description in words.
• I can rearrange an equation correctly (same operation on both sides) to isolate the required variable.
• I can solve for the unknown and substitute back to check the result.
• I can explain why my answer makes sense in the real-world context.

Curriculum links

• Students develop a linear equation from a description in words (QCAA General Mathematics — Unit 1 / Topic 5: Linear equations and their graphs).
• Students solve practical problems involving linear equations (QCAA General Mathematics — Unit 1 / Topic 5: Linear equations and their graphs).
• Students use linear relationships expressed algebraically and connect algebraic steps to graph/meaning (continuation within Topic 5: Linear equations and their graphs).

Lesson structure (60 minutes)

1. 0–5 min · Warm-up: equation from words. Teacher writes three quick scenarios on the board (e.g., “Total = base + hourly rate × hours”, “Distance = speed × time”) and asks students to choose which will form a linear equation; students justify briefly in pairs.

2. 5–15 min · Mini teach: rearranging safely. Teacher models isolating a variable using one worked example (e.g., start with 3x + 5 = 20, isolate x) and emphasises “do the same thing to both sides” plus order of operations; students copy a clean method chart: simplify → move terms → multiply/divide → check by substitution.

3. 15–25 min · Worked example: practical equation. Teacher guides a practical context (e.g., “A rideshare charges a base fee plus a per-km cost. The base fee is $8 and the per-km cost is $1.50. If the total bill is $38, how many km?”), forms the equation together, then rearranges to solve; students complete the final rearrangement step and check by substitution.

4. 25–45 min · Guided practice: rearrange and solve. Teacher circulates with a set of four “choose-the-subject” tasks where students must rearrange for a different variable each time (e.g., solve for time, solve for rate, solve for intercept term as given by a context). Students work in pairs/groups of 5 using a structured worksheet:

• Step 1: Write the equation
• Step 2: Rearrange to isolate the required variable
• Step 3: Solve
• Step 4: Substitute back and state the answer with units

5. 45–55 min · Error check challenge: spot and fix. Teacher displays two incorrect solution fragments (e.g., sign error when moving terms; dividing only one side) and asks students to identify the mistake and rewrite the corrected steps. Students submit corrected reasoning and final answer.

6. 55–60 min · Exit ticket: one context, one check. Teacher gives one new scenario requiring rearrangement (one unknown) and one requirement to “show substitution check”. Students complete independently for 5 minutes; teacher collects for quick review.

Resources

• Printed worksheet: “From words to linear equation” and “Rearrange and verify” tasks (4 guided + 1 exit ticket)
• Worked solution templates (equation → rearrange → solve → check)
• Whiteboard/marker or projector for teacher worked examples
• Calculation space (grid paper optional) and unit labels reminder sheet
• Marking checklist (for teacher use during circulation)
• Small cards or sticky notes for students to label “solve for x / solve for t / solve for rate”

Assessment

• Formative checks during guided practice: teacher listens for correct “same operation both sides” reasoning and correct substitution checks.
• Targeted questioning in the mini teach and error challenge: “What did you do to both sides? Where did the sign change?”
• Exit ticket assessed against: correct equation formation, correct rearrangement, correct solution, and a substitution/verification statement.

Differentiation

• Support: Provide sentence starters for modelling in words (e.g., “Let ___ be …; total equals …; therefore …”). Include a step-by-step rearrangement scaffold with blank lines for each algebra move.
• Support: Allow a “verification rule” card (substitute into original equation, confirm both sides match) and require students to circle their substitution.
• Extension: For the final guided task, include a “rearrange for a different variable” option so students practise isolating a chosen variable rather than only solving for x.
• Language/SEN/EAL: Use consistent context types (money, distance, time) and explicit unit prompts; highlight key words (total, difference, rate, per) and keep numbers manageable.

Extension (optional)

Skipped (not requested).