Mathematics · Year 11

Active learning ideas

Reverse Percentages

Active learning works especially well for reverse percentages because students must repeatedly apply the same logic in varied contexts. By handling calculations as physical tasks or peer debates, students convert abstract multipliers into concrete actions, reducing reliance on memorized rules and building proportional reasoning.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Number
• GCSE: Mathematics - Ratio, Proportion and Rates of Change

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Problem-Based Learning
• 25 min · Pairs

Pair Relay: Multiplier Chains

Pairs start with a final amount and percentage change, calculate the original, then pass to the next pair who applies another change and reverses it. Include 8-10 chained problems on cards. Debrief as a class to verify chains.

Justify the method for finding an original amount after a percentage change.

Facilitation Tip

During Pair Relay: Multiplier Chains, provide each pair with a strip of paper showing one step in a chain so they physically pass it to the next student after each calculation.

What to look for

Present students with a scenario: 'A laptop costs £720 after a 10% discount. What was the original price?' Ask students to show their calculation using multipliers and justify why they divided by 0.9.

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

Activity 02

• Problem-Based Learning
• 35 min · Small Groups

Error Hunt Stations: Common Mistakes

Set up four stations with worked examples containing one error each, like wrong multipliers or addition instead of division. Small groups identify errors, explain corrections, and rewrite correctly. Rotate every 7 minutes.

Predict common errors when solving reverse percentage problems.

Facilitation Tip

In Error Hunt Stations, place incorrect calculations on walls and give students colored stickers to mark the exact step where the error occurs before they write the correct multiplier on the back.

What to look for

Pose the question: 'If a price increased by 20% and then decreased by 20%, is the final price the same as the original? Why or why not?' Facilitate a discussion where students use examples and explain their reasoning.

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

Activity 03

• Problem-Based Learning
• 30 min · Small Groups

Problem Construction Carousel: Real-World Scenarios

In small groups, students create reverse percentage problems from contexts like discounts or price hikes, swap with another group to solve, then discuss solutions. Provide templates for salary, VAT, or sale items.

Construct a real-world problem that requires the use of reverse percentages.

Facilitation Tip

For Problem Construction Carousel, give students blank cards and ask them to craft a scenario with a realistic percentage change before swapping with another pair to solve.

What to look for

Give students a card with a problem like: 'After a 5% pay rise, Sarah's new salary is £31,500. What was her salary before the rise?' Students write their answer and one sentence explaining their method.

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

Activity 04

• Problem-Based Learning
• 20 min · Individual

Individual Whiteboard Blitz: Quick Reversals

Project 10 final amounts with percentage changes; students work individually on whiteboards, hold up answers after 2 minutes each. Follow with whole-class pairing to justify one method.

Justify the method for finding an original amount after a percentage change.

Facilitation Tip

Run Individual Whiteboard Blitz by flashing a scenario for 10 seconds, then reset clocks so only the multiplier and answer show for peer checking.

What to look for

Present students with a scenario: 'A laptop costs £720 after a 10% discount. What was the original price?' Ask students to show their calculation using multipliers and justify why they divided by 0.9.

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

A few notes on teaching this unit

Teach reverse percentages by first building the multiplier concept through fraction to decimal conversion. Avoid rushing to shortcuts; ensure students derive the multiplier from the percentage change each time. Research shows that students who construct their own multipliers retain the method longer than those who receive them pre-made.

Successful learning looks like students confidently selecting the correct multiplier, explaining its meaning in context, and spotting errors in others’ work. They should justify each step without prompting and connect changes to real-world situations.

---

Watch Out for These Misconceptions

• During Error Hunt Stations, watch for students who subtract the percentage amount from the final value to find the original.

  Have students test their method with a known original price like £100 after a 20% decrease. When they see £80 and subtract £16 to get £64, ask them to divide £80 by 0.8 instead and compare answers, leading to the correct multiplier of 1.25.

• During Pair Relay: Multiplier Chains, watch for students adding the percentage back for a reverse decrease.

  Ask pairs to model the discount on mini whiteboards by first writing the original amount, then applying the percentage decrease. This shows why adding back doesn’t return to the original and reinforces division by the correct multiplier.

• During Problem Construction Carousel, watch for students who confuse multipliers for increases and decreases.

  Give students cards with percentage changes written as words (increase by 15%, decrease by 30%) and ask them to match these to pre-cut multiplier cards (1.15, 0.70) before constructing their scenario.

---

Methods used in this brief

• Problem-Based Learning