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# Percentage change

Number Foundations · Modelling

<!-- Speaker note: Students model percentage increases and decreases as multipliers, chain two changes, and explain why equal percentages up and down do not cancel (AC9M8N05, node 1). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** an increase of p% multiplies by (1 + p/100); a decrease multiplies by (1 − p/100). One change after another means multiplying the multipliers, so up 20% then down 20% is × 1.2 × 0.8 = × 0.96: a 4% fall overall, not back to the start.

- I can use a multiplier for a percentage increase or decrease.
- I can work out the result of two changes, one after the other.
- I can explain why up 20% then down 20% is not back to the start.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need percentages of amounts and increases and decreases (Number Foundations). The one idea: each change multiplies the current amount, so successive changes multiply. The common mistake is taking the second percentage of the original amount (Sam's mistake). The board draws the three amounts as bars with each multiplier between them; the Mark-up task uses the percentage bar. Wider: population or prices changing by the same percentage each year (repeated multipliers: informal exponential growth). -->

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## Why it matters

> A shop puts a price up 20% in March and has a 20% sale in April. The sign says 'back to the old price'. Is it?

<!-- Speaker note: A percentage change is a multiplication by a **multiplier**: - an **increase** of $p\%$: multiply by $1 + \frac{p}{100}$ (up 15%: × 1.15); - a **decrease** of $p\%$: multiply by $1 - \frac{p}{100}$ (down 15%: × 0.85). When one change follows another, the second is a percentage of the **new** amount, so multiply the multipliers: up 20% then down 20% is $1.2 \times 0.8 = 0.96$, a 4% decrease overall. Ask "a percentage of what?" every time. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A $100 price goes up 10%, then the new price goes down 10%. What is the final price?**

- $101
- $90
- $99
- $100

<!-- Speaker note: What happens: $99. Up 10% makes $110. Then 10% of $110 is $11, so the price falls to $99. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask '20% of what?' for the second change. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fpercentage-change%2Fstep-2&code=widget%3A+money%0Amode%3A+change%0Aq%3A+100%0Aslider+a%3A+-50%2C+50%2C+0%2C+5%0Aslider+b%3A+-50%2C+50%2C+0%2C+5%0Aunits%3A+%24%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/percentage-change/?full=step-2#step-2)

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask whether any pair of equal ups and downs gets back to $100. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* $99. Up 10% makes $110. Then 10% of $110 is $11, so the price falls to $99.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Multipliers

Here is the routine.

$$\text{up } 20\%$$

1) $\text{down } 20\%$ — \times 0.8
2) $1.2 \times 0.8 = 0.96$ — Multiply the multipliers.
3) $0.96 = 96\%$ — 4% less than the start: an overall decrease of 4%.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write up 20\% and say "× 1.2." Line 1: write down 20\% and say "× 0.8." Line 2: write 1.2 × 0.8 = 0.96 and say "Multiply the multipliers." Line 3: write 0.96 = 96\% and say "4% less than the start: an overall decrease of 4%." -->

<!-- Speaker note: Teacher: ask Sam what 10% of $220 is. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Multipliers

**What do you multiply by for a 15% increase?**

* Answer: 1.15

<!-- Speaker note: 100% + 15% = 115% = 1.15. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Sam works out the jacket's final price. Click the line where his working goes wrong.**

```
 1  A $200 jacket goes up 10%, then the new price goes down 10%.
 2  Up 10% adds 10% of $200, which is $20, so it costs $220.
 3  Down 10% takes off $20, so it costs $200 again.
```

* Answer: line 3

<!-- Speaker note: Line 3: the 10% decrease is 10% of the new price, $220, which is $22. The jacket costs $198. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Reading a multiplier

**A price is multiplied by 0.85. Has it gone up or down, and by how much?**

- Up 15%
- Down 15%
- Down 85%
- Up 85%

* Answer: Down 15%

<!-- Speaker note: 0.85 = 85%: 15% has been taken off. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Back to the start, and discounts



- **Undo it:** 
- **Two discounts:** 
- **Up 50, down 50:** 
- **Mark-up:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/percentage-change/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Finding a change that undoes another, comparing two discounts with one, and a mark-up show the multiplier model at work. Hints, in order: Up 25% is × 1.25. / What times 1.25 is 1? Try down 20%. For students who need more support: Write each multiplier under its arrow, and multiply with a calculator. For early finishers: A town's population grows 10% a year for 3 years. Is that a 30% increase? Find the true overall change. The hard part, on purpose: Asking 'a percentage of what?'. Ask students to name the amount each percentage is taken of before they calculate. When students get stuck: A student takes the second percentage of the original. Ask: What is the price just before the second change? Fix: Each percentage is of the current amount. / A student multiplies by 0.15 for a 15% increase. Ask: Have you found the new amount, or just the increase? Fix: New amount: × (1 + 0.15). / A student adds the percentages (−10 and −20 gives −30). Ask: What is 0.9 × 0.8? Fix: Multiply the multipliers: 0.72, a 28% fall. -->

<!-- Speaker note: Teacher: ask whether two 20% discounts are better or worse for the shopper than one of 40%. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Increase

**Increase $80 by 15%.**

* Answer: 92

<!-- Speaker note: 80 × 1.15 = $92. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Decrease multiplier

**What do you multiply by for a 35% decrease?**

* Answer: 0.65

<!-- Speaker note: 100% − 35% = 65% = 0.65. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Two steps

**A $250 bike goes up 20%, then has a 10% discount. What does it cost now, in dollars?**

* Answer: 270

<!-- Speaker note: 250 × 1.2 = 300, then 300 × 0.9 = $270. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: percentage of

**Review: what is 15% of 60?**

* Answer: 9

<!-- Speaker note: 0.15 × 60 = 9. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## GST

**A price is $80 before GST. GST adds 10%. What is the price with GST, in dollars?**

* Answer: 88

<!-- Speaker note: 80 × 1.1 = $88. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: an overall change

Then explain why the order of the two changes does not matter to the final price.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/percentage-change/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Searching for two changes with a given overall multiplier uses the product of multipliers as the model. Hints, in order: −28% overall means × 0.72. / 0.9 × 0.8 = 0.72. -->

<!-- Speaker note: Teacher: ask why the order does not matter (multiplying in either order). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A $60 item is discounted 25%, then the discounted price goes up 25%. What is the final price?**

* 60 × 0.75 × 1.25 = $56.25.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This starts AC9M8N05. Where it leads: money models (next lesson), compound interest and exponential growth (Years 9 and 10). -->