Extension tasks — Algebraic equations

Topic: Algebraic equations | Objective: Solve multi-step equations and extend into reasoning and application | Audience: Middle school (advanced learners) | Time: 10–20 minutes | Context: During class while others complete core work

Opening line

“If you’re ready for a challenge — pick up the extension card. These aren’t just harder versions of what you’ve done. They ask you to think differently.”

Task 1 — The broken equation

Type: Challenge problem | Difficulty: Tier 1

Instructions: The equation below has been partially solved — but the working contains two errors. Find both errors, correct them, and write the right answer with clean working.

[Present a multi-step equation with a sign error and a distribution error embedded in the working.]

“You’re not just solving — you’re auditing. Write what went wrong at each step and why.”

Expected output: Corrected working with two errors identified and explained in words.

Task 2 — Design the problem

Type: Real-world application + creation | Difficulty: Tier 2

Instructions: A company charges a flat fee of $X plus $Y per hour for a service. A customer paid $Z in total. Write an algebraic equation that models this situation, solve it to find the number of hours, then change one of the values so the equation has no solution — and explain why no solution exists.

“You choose the numbers. The equation must require at least three steps to solve.”

Expected output: One written equation, solved working, one modified equation with written explanation of why it has no solution.

Task 3 — Open-ended exploration

Type: Open-ended reasoning | Difficulty: Tier 3

Instructions: Is it possible for a two-step linear equation to have more than one solution? Explore this by testing at least three equations of your own design. Write a conclusion — either prove it is possible with an example, or explain why it cannot happen.

“There’s a correct answer to this — but the goal is that you find it through testing, not by remembering a rule.”

Expected output: At least three tested equations with working, and a written conclusion with reasoning.

Difficulty progression

• Tier 1 (Task 1): Error identification — requires careful checking and algebraic fluency. No creativity required. Good starting point.
• Tier 2 (Task 2): Application and construction — requires modelling a real situation and understanding equation structure. Adds a creative constraint.
• Tier 3 (Task 3): Exploration and proof — requires conjecture, testing, and written mathematical argument. Highest independence required.

Hint ladder (for Task 3)

• Hint 1: “Start by writing three equations where you already know the answer. Pick easy numbers.”
• Hint 2: “What happens when you have a variable on both sides? Try one like that.”
• Hint 3: “Think about what happens if both sides of the equation are identical. What does that tell you about solutions?”

Facilitation tips

• Check in with one question, not an evaluation: “Explain your thinking so far.” Listen for the method, not just the answer.
• If a learner completes all three tasks: ask them to write a worked solution guide for Task 1 that a classmate could follow — this extends the task without adding new content
• If a learner stalls on Task 3: offer Hint 1 only. Do not offer Hint 2 until they have genuinely tried with Hint 1.
• If disengagement appears: switch to Task 2 — the real-world framing and freedom to choose numbers tends to re-engage learners who find pure algebra dry

Quick alternative task

If all three tasks feel inaccessible or the learner has used all hints: “Take any equation from today’s core work. Change it so it has no solution. Then change it again so it has infinitely many solutions. Write both versions and explain each.” This is concrete, self-directed, and requires genuine understanding without requiring the open-ended exploration of Task 3.