Mathematics · Year 8 · Proportional Reasoning and Multiplicative Relationships · Autumn Term

Percentage Increase and Decrease

Students will calculate percentage changes, including finding the new amount after an increase or decrease.

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

About This Topic

Percentage increase and decrease require students to calculate new amounts after applying a percentage change to an original value. They find the change as a percentage of the original, then add or subtract it, or use multipliers such as 1.15 for a 15% increase and 0.85 for a 15% decrease. These methods connect to real-life contexts like salary rises, price discounts, or population growth, helping students see mathematics in action.

This topic aligns with KS3 Mathematics standards in number and ratio, proportion, and rates of change, within the unit on proportional reasoning and multiplicative relationships. Students analyze why percentage increases and decreases have asymmetric effects, justify multipliers for efficiency over step-by-step addition, and predict results of successive changes, such as two 10% increases not equaling a 20% increase overall. These skills build fluency in multiplicative thinking.

Active learning benefits this topic because students often struggle with the counterintuitive effects of percentages. When they adjust prices on mock shopping lists in pairs or simulate successive changes on class timelines in small groups, they test predictions, debate discrepancies, and refine their understanding through shared calculations, turning potential confusion into confident mastery.

Key Questions

1. Analyze the impact of a percentage increase versus a percentage decrease on an original value.
2. Justify the use of a multiplier for efficient percentage change calculations.
3. Predict the outcome of successive percentage changes on an initial quantity.

Learning Objectives

• Calculate the new value after a given percentage increase or decrease, expressing the answer to an appropriate degree of accuracy.
• Compare the effect of a percentage increase versus a percentage decrease on an original quantity, explaining the difference in outcomes.
• Justify the use of a single multiplier to represent successive percentage changes, demonstrating efficiency over sequential calculations.
• Predict the final value of a quantity after multiple successive percentage changes, such as two consecutive discounts.

Before You Start

Calculating Percentages of Amounts

Why: Students must be able to find a percentage of a number before they can calculate an increase or decrease.

Fractions, Decimals, and Percentages Equivalence

Why: Understanding the relationship between these representations is crucial for converting percentages into multipliers.

Key Vocabulary

Percentage Increase	A calculation that determines how much a quantity has grown relative to its original value, expressed as a percentage.
Percentage Decrease	A calculation that determines how much a quantity has shrunk relative to its original value, expressed as a percentage.
Multiplier	A single number used to efficiently calculate a percentage change. For example, multiplying by 1.10 represents a 10% increase.
Successive Percentage Change	Applying one percentage change after another to a quantity, where the second change is calculated on the new, already changed value.

Watch Out for These Misconceptions

Common Misconception

A 10% increase followed by a 10% decrease returns exactly to the original amount.

What to Teach Instead

Show with £100: +10% to £110, then -10% of £110 is £11 off, leaving £99. Small group trials with varied originals reveal the pattern; discussions clarify percentages apply to the current amount, building proportional insight.

Common Misconception

To increase by 25%, multiply the original by 0.25 or 25.

What to Teach Instead

Correct multiplier is 1.25, as 0.25 is just the increase portion. Pairs practice on price cards, comparing methods, helps students visualize the full amount and avoid undercalculating through hands-on repetition.

Common Misconception

Successive percentage changes add up, like +10% then +20% equals +30%.

What to Teach Instead

Changes multiply: 1.10 × 1.20 = 1.32 or 32%. Relay activities let groups chain calculations, spot the error in addition, and justify multiplicative reasoning through prediction and verification.

Active Learning Ideas

Card Sort: Increase or Decrease Scenarios

Prepare cards with real-world problems like '£50 phone with 20% VAT increase' or '£80 jacket with 25% sale decrease'. In pairs, students sort into increase/decrease piles, calculate new amounts using multipliers, then swap and check calculations. Discuss efficiencies of the multiplier method.

30 min·Pairs

Multiplier Chain Relay

Divide small groups into teams. Each member applies one percentage change from a chain (e.g., +10%, -5%, +20%) to a starting amount, passes to the next. Teams predict and verify final amounts, then compare multiplier products. Extend to justify predictions.

40 min·Small Groups

Price Tracker Simulation

Provide item prices; small groups apply successive changes over 'months' (e.g., inflation +3%, then discount -2%). Record on tables, predict trends, and graph results. Whole class shares one surprising outcome and explains with multipliers.

45 min·Small Groups

Reverse Percentage Challenge

Individuals start with final prices and work backwards to originals using inverse multipliers (e.g., divide by 1.20 for 20% increase reversal). Pairs peer-review, then share strategies for successive reverses.

25 min·Individual

Real-World Connections

• Retailers use percentage discounts to attract customers, for example, offering 20% off all shoes or a 'buy one, get one 50% off' deal, impacting sales figures and profit margins.
• Financial advisors calculate compound interest on investments or loan repayments, where successive percentage increases or decreases significantly alter the final amount over time.
• Economists analyze inflation rates, which are percentage increases in the general price level of goods and services, affecting purchasing power and the cost of living for households.

Assessment Ideas

Quick Check

Present students with a scenario: 'A jacket costs $80. It is first discounted by 10%, then by an additional 20% off the sale price. Calculate the final price.' Ask students to show their working using multipliers and to write one sentence explaining why the final discount is not 30%.

Discussion Prompt

Pose the question: 'If you invest $1000 and it increases by 5% in year one and decreases by 5% in year two, is your final amount more than, less than, or equal to your original $1000? Explain your reasoning using calculations.' Facilitate a class discussion comparing different approaches and justifications.

Exit Ticket

Give each student a card with a different starting value and a percentage change (e.g., 'Increase 50 by 15%', 'Decrease 200 by 25%'). Ask them to calculate the new value using a multiplier and write down the multiplier they used.

Frequently Asked Questions

How do you calculate a percentage increase using multipliers?

Convert the percentage to a decimal and add 1: for 15% increase, use 1.15. Multiply the original amount by this factor. For £200 at 15%, 200 × 1.15 = £230. This skips finding the increase separately, saving time and reducing errors in multi-step problems.

Why does a percentage decrease after an increase not cancel out?

Percentages apply to the current value, not the original. A 20% increase on £100 makes £120; 20% of £120 is £24 off, leaving £96. Students grasp this asymmetry by tracking changes on familiar items like shop prices, reinforcing that decreases act on larger bases.

How can active learning help teach percentage increase and decrease?

Active approaches like group price adjustment simulations or multiplier relay races engage students in applying changes repeatedly. They predict outcomes, test with calculators, and discuss why results differ from intuition, such as successive changes not commuting. This hands-on practice solidifies multipliers and counters misconceptions through collaboration and real-time feedback.

What real-world examples illustrate successive percentage changes?

Salary reviews often compound: 5% raise then 3% bonus multiplies as 1.05 × 1.03. Shop sales chain discounts. Students model these on spreadsheets, predict finals, and verify, linking abstract math to career or consumer decisions while practicing justification.