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# Percentages of amounts

Number Foundations · Fractions and decimals

<!-- Speaker note: Students use a percentage bar to find a percentage of an amount, apply increases and decreases (in two steps and with one multiplier), and express one amount as a percentage of another (AC9M7N06, node 3). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 15% of $80 is 0.15 × 80 = $12. Take it off for a discount ($68, or 80 × 0.85 in one step), add it on for an increase ($92, or 80 × 1.15). And 18 out of 24 is 18 ÷ 24 = 75%.

- I can find a percentage of an amount, mentally and with a decimal.
- I can increase or decrease an amount by a percentage, including with one multiplier.
- I can write one amount as a percentage of another.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need percentages as decimals (Fractions, decimals and percentages) and multiplying by a fraction less than 1 (previous lesson). The one idea: a percentage is hundredths, and "of" means multiply. For a decrease, multiply by what is left (Grace multiplies by what is taken off). Varied structures matter: of, off, on, and out of. Calculators are fine for the decimals; the mental route (10%, 5%, 1%) builds number sense and checks calculator answers. -->

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## Why it matters

> A jacket is $80, and the sign says 15% off. Before you reach the counter, can you work out what you will pay?

<!-- Speaker note: A percentage is a number of hundredths: $15\% = \tfrac{15}{100} = 0.15$. So 15% **of** an amount is $0.15 \times$ the amount, the same "of means multiply" as for fractions. Three questions come up again and again: - **A percentage of an amount:** 15% of $80 is 0.15 × 80 = $12. - **An increase or decrease:** add it on or take it off, or multiply by what is left ($100\% - 15\% = 85\%$, so $\times 0.85$). - **What percentage one amount is of another:** divide, then $\times 100$. -->

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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 jacket costs $80. It is 15% off. What do you pay?**

- $68
- $65
- $12
- $92

<!-- Speaker note: What happens: $68. 15% of $80 is $12, and $80 − $12 = $68. $12 is the discount, not the price; $65 takes 15 dollars off, not 15%. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose $12 stopped at the discount; $65 took 15 dollars 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>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fpercentages%2Fstep-2&code=widget%3A+percent%0Aq%3A+80%0Aslider+p%3A+0%2C+50%2C+0%2C+5%0Amode%3A+decrease%0Aunits%3A+%24%0A&fresh=1}}

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

<!-- Speaker note: The orange part is taken off; the blue part is what you pay. -->

<!-- Speaker note: Teacher: ask what percentage the blue part is, and how that gives a one-step multiplier. -->

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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?

* $68. 15% of $80 is $12, and $80 − $12 = $68. $12 is the discount, not the price; $65 takes 15 dollars off, not 15%.

<!-- 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>

## Find it, then use it

Here is each kind with $80.

Now try one: 35% of 60.

$$10\% \text{ of } 80 = 8, \quad 5\% \text{ of } 80 = 4$$

1) $15\% \text{ of } 80 = 0.15 \times 80 = 12$ — Or as a decimal: 15% = 0.15.
2) $80 - 12 = 68 \quad \text{or} \quad 80 \times 0.85 = 68$ — A decrease: take it off, or keep 85% in one step.
3) $80 \times 1.15 = 92$ — An increase: 100% + 15% = 115%, so × 1.15.
4) $18 \div 24 = 0.75 = 75\%$ — What percentage: divide, then × 100.

<!-- 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 10\% of 80 = 8, 5\% of 80 = 4 and say "Mental: 10% is a tenth, 5% is half of that." Line 1: write 15\% of 80 = 0.15 × 80 = 12 and say "Or as a decimal: 15% = 0.15." Line 2: write 80 - 12 = 68 or 80 × 0.85 = 68 and say "A decrease: take it off, or keep 85% in one step." Line 3: write 80 × 1.15 = 92 and say "An increase: 100% + 15% = 115%, so × 1.15." Line 4: write 18 ÷ 24 = 0.75 = 75\% and say "What percentage: divide, then × 100." -->

<!-- Speaker note: Teacher: compare the mental route and the decimal route for 35% of 60. -->

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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 it, then use it

**Find 35% of 60.**

* Answer: 21

<!-- Speaker note: 10% of 60 is 6, so 30% is 18, and 5% is 3: 21. Or 0.35 × 60 = 21. -->

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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

**Grace finds the sale price of a $50 game that is 20% off. Click the line where her working goes wrong.**

```
 1  A $50 game is 20% off. Find the sale price.
 2  20% = 0.2
 3  Sale price = 50 × 0.2
 4  = $10
```

* Answer: line 3

<!-- Speaker note: Line 3: 50 × 0.2 is the discount ($10), not the price. What is left is 80%: 50 × 0.8 = $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>

## Out of

**Ari scored 18 out of 24. What percentage is that?**

* Answer: 75

<!-- Speaker note: 18 ÷ 24 = 0.75 = 75%. -->

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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>

## One change at a time



- **GST:** Each task changes one thing.
- **Increase:** 
- **Decrease:** 
- **What percentage?:** 
- **Up, then down:** 

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

<!-- Speaker note: What this step is for: One change at a time: an increase on the board, an increase by a multiplier, a decrease, a percentage of a group, then increase then decrease. Hints, in order: GST in Australia is 10%. For students who need more support: Use the bar for every problem; start from 10% and build. For early finishers: A shop increases a price by 25%, then offers 20% off the new price. Is the final price more, less or the same as the start? Explain with multipliers. The hard part, on purpose: Choosing the multiplier for a decrease. Ask what percentage is left after the discount before students touch the calculator. When students get stuck: A student multiplies by the percentage taken off for a decrease (Grace's mistake). Ask: Is your answer the price or the saving? Fix: Multiply by what is left: 100% − 20% = 80%. / A student takes off the number, not the percentage (80 − 15 = 65). Ask: 15 what: dollars or hundredths? Fix: Find 15% of 80 first. / A student gives a decimal for 'what percentage' (0.75). Ask: Out of what? Fix: × 100 for a percentage. -->

<!-- Speaker note: Teacher: for up then down, take a vote before working it out. -->

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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>

## A friendly percentage

**Find 12.5% of 64. (12.5% is $\tfrac{1}{8}$.)**

* Answer: 8

<!-- Speaker note: 12.5% = 1/8, and 64 ÷ 8 = 8. -->

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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>

## A multiplier

**What do you multiply by to decrease an amount by 30%?**

* Answer: 0.7

<!-- Speaker note: 100% − 30% = 70% = 0.7. -->

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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>

## Growth

**A plant 120 mm tall grows 5% taller. How tall is it now, in mm?**

* Answer: 126

<!-- Speaker note: 5% of 120 = 6, so 126 mm. Or 120 × 1.05 = 126. -->

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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: multiplying fractions

**Review: work out $\tfrac{2}{3} \times \tfrac{3}{4}$ in simplest form.**

* Answer: 1/2

<!-- Speaker note: 6/12 = 1/2. -->

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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: save $15

Then find the pair with the lowest sale price.

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

<!-- Speaker note: What this step is for: Finding a price and a discount that save exactly $15 shows that many pairs work, and that the saving is percentage × price. Hints, in order: 10% of $150 is $15. Find another. / Try 25% of something, or 50%. -->

<!-- Speaker note: Teacher: collect pairs on the board; order them by sale price. -->

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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 $40 shirt is 35% off. Find the sale price.**

* 35% of $40 = $14, so $26. Or 40 × 0.65 = 26.

<!-- 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: Next: Choosing a strategy (node 4), choosing between fractions, decimals and percentages. Where it leads: financial maths (profit, GST, discounts in AC9M7N09), and percentage change with repeated multipliers (Year 9 and 10). -->