Mathematics · Year 8

Active learning ideas

Reverse Percentages

Active learning turns abstract multiplier rules into concrete reasoning by letting students test, correct, and refine their own mental models. When students manipulate real prices, wages, and discounts, they see why dividing by 1.20 (not subtracting 20%) finds the original amount. These hands-on steps build the proportional thinking needed for later topics like compound interest and VAT calculations.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number
• KS3: Mathematics - Ratio, Proportion and Rates of Change

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Problem-Based Learning
• 35 min · Small Groups

Shopping Challenge: Reverse Discounts

Provide groups with final sale prices and discount percentages. Students calculate originals, then apply discounts forward to check. They create a class display of findings with real shop examples.

Explain why simply reversing a percentage change does not return to the original value.

Facilitation Tip

For the Shopping Challenge, ask each pair to record both the discounted price and the original price on a mini whiteboard so the class can scan all answers at once.

What to look for

Present students with a scenario: 'A jacket is on sale for £36 after a 25% discount. What was the original price?' Ask students to show their calculation using multipliers and write down the original price.

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

Activity 02

• Problem-Based Learning
• 25 min · Pairs

Multiplier Match: Pairs Puzzle

Pairs sort cards showing final amounts, percentages, and originals. They explain matches using multipliers, then solve new problems. Switch pairs to verify solutions.

Construct a method for finding the original amount after a percentage change.

Facilitation Tip

In Multiplier Match, give each pair only half the cards so they must explain their reasoning before combining the set.

What to look for

Pose the question: 'If a price increased by 10% and then decreased by 10%, does it return to the original price? Why or why not?' Facilitate a class discussion where students use examples to justify their answers.

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

Activity 03

• Problem-Based Learning
• 30 min · Whole Class

Error Detective: Whole Class Relay

Display common wrong workings on board. Teams race to spot errors, correct with multipliers, and justify. Debrief as class to reinforce methods.

Evaluate the common errors made when solving reverse percentage problems.

Facilitation Tip

During Error Detective, insist teams write the exact miscalculation on a sticky note before offering the correct multiplier, forcing verbal correction.

What to look for

Give each student a card with a different reverse percentage problem (e.g., 'A salary increased to £30,000 after a 5% rise. What was the original salary?'). Students must write the multiplier used, show the calculation, and state the original salary.

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

Activity 04

• Problem-Based Learning
• 40 min · Individual

Wage Rise Workshop: Individual to Groups

Students start individually reversing salary increases from news articles, then share methods in groups to refine and peer-assess.

Explain why simply reversing a percentage change does not return to the original value.

What to look for

Present students with a scenario: 'A jacket is on sale for £36 after a 25% discount. What was the original price?' Ask students to show their calculation using multipliers and write down the original price.

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

A few notes on teaching this unit

Teach the multiplier method as a tool for undoing operations, not as a new formula. Start with a concrete scenario (£100 then +20%), model dividing by 1.20, and ask students to explain why 1.20 undoes the 1.20 multiplication. Avoid shortcuts like ‘just add or subtract the percentage’—they trap students in additive thinking that fails on compound steps. Research shows that alternating between forward and reverse calculations strengthens proportional schemas faster than isolated drill.

By the end of the activities, students confidently identify whether to divide by 1 + p/100 or 1 - p/100, justify their choice, and apply it quickly in context. They catch their own errors through paired checks and explain why a 20% increase cannot be reversed by a 20% decrease.

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Watch Out for These Misconceptions

• During Shopping Challenge, watch for students who subtract the percentage of the sale price instead of dividing by the multiplier.

  Circulate with a mini whiteboard showing £120 after 20% increase and ask pairs to calculate 20% of £120 (£24), then compare £120 – £24 = £96 with the expected £100. Ask them to adjust their method to match the correct original price.

• During Multiplier Match, watch for students who treat a 20% increase and a 20% decrease as reversible steps using the same 20% figure.

  Hand each pair the two matching cards (e.g., 20% increase and 20% decrease) and demand they test both on a £100 starting price, recording final values. The £120 vs £96 discrepancy reveals why the multipliers differ.

• During Error Detective, watch for teams that confuse the multipliers for increases and decreases.

  Require teams to verbalize the rule before claiming a match: ‘Increase uses 1 + p/100, decrease uses 1 – p/100.’ If they swap them, the relay card returns to them for redoing with peer explanation.

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Methods used in this brief

• Problem-Based Learning