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# Fractions, decimals and percentages

Number Foundations · Fractions and decimals

<!-- Speaker note: Students use a fraction bar and a hundred grid to see one amount as a fraction, a decimal and a percentage, convert in every direction, and meet a recurring decimal (AC9M7N04, node 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** one amount, three ways to write it. A percentage is out of 100, so 3/4 = 75/100 = 75% = 0.75. Divide the top by the bottom for a decimal; put a percentage over 100 and simplify for a fraction.

- I can convert between fractions, decimals and percentages.
- I can explain why 0.5 = 50%, not 5%.
- I can compare amounts written in different forms.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need Equivalent fractions (the lesson before) and place value with decimals (Year 6). The one idea: percent means out of 100, so hundredths connect all three. The common mistake is reading the digits after the point as the percentage (0.5 = 5%); it is planted in Max's working and named in the success criteria. Decimal answers are marked by form (3/8 is told to write a decimal); a percentage can be typed with or without the % sign. -->

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## Why it matters

> A 25% discount, a quarter off, 0.25 of the price: the same saving, written three ways. Which is easiest depends on the job.

<!-- Speaker note: In Equivalent fractions you wrote one amount as different fractions. **Percent** means "out of 100", so a percentage is a fraction with 100 parts: $75\% = \tfrac{75}{100}$. A **decimal** shows tenths, hundredths and thousandths: $0.75$ is 75 hundredths. In this lesson you move between all three, using a hundred grid to see why they match. -->

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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

**Which is the same amount as $\tfrac{3}{4}$?**

- $3.4\%$
- $34\%$
- $75\%$
- $0.34$

<!-- Speaker note: What happens: 75%. 3/4 = 75/100 (× 25 on the top and the bottom), and out of 100 is a percentage: 75%. As a decimal, 0.75. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 34% are reading digits. -->

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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%2Ffractions-decimals-percentages%2Fstep-2&code=widget%3A+bars%0Aa%3A+3%0Ab%3A+4%0Aslider+c%3A+0%2C+100%2C+0%0Ahundred%3A+on%0Ashow%3A+percent%0A&fresh=1}}

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

<!-- Speaker note: $c$ is the number of squares shaded, out of 100. -->

<!-- Speaker note: Teacher: ask how many squares one quarter is, then three. -->

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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?

* 75%. 3/4 = 75/100 (× 25 on the top and the bottom), and out of 100 is a percentage: 75%. As a decimal, 0.75.

<!-- 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>

## Three ways

Here is how to move between the three.

Try one: write $\tfrac{3}{8}$ as a decimal.

$$\tfrac{3}{4} = \tfrac{75}{100} = 75\%$$

1) $\tfrac{3}{8} = 3 \div 8 = 0.375$ — Fraction to decimal: divide the top by the bottom.
2) $0.375 \times 100 = 37.5\%$ — Decimal to percentage: multiply by 100.
3) $16\% = \tfrac{16}{100} = \tfrac{4}{25}$ — Percentage to fraction: put it over 100, then simplify.
4) $45\% = 0.45$ — Percentage to decimal: divide by 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 3 ÷ 4 = 75 ÷ 100 = 75\% and say "Fraction to percentage: make it out of 100 (× 25 on the top and the bottom)." Line 1: write 3 ÷ 8 = 3 ÷ 8 = 0.375 and say "Fraction to decimal: divide the top by the bottom." Line 2: write 0.375 × 100 = 37.5\% and say "Decimal to percentage: multiply by 100." Line 3: write 16\% = 16 ÷ 100 = 4 ÷ 25 and say "Percentage to fraction: put it over 100, then simplify." Line 4: write 45\% = 0.45 and say "Percentage to decimal: divide by 100." -->

<!-- Speaker note: Teacher: model each line of the worked example with the grid beside it. -->

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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>

## Three ways

**Write $\tfrac{3}{8}$ as a decimal.**

* Answer: 0.375

<!-- Speaker note: 3 ÷ 8 = 0.375. As a percentage: 37.5%. -->

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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

**Max writes $0.5$ as a percentage. Click the line where his working goes wrong.**

```
 1  Write 0.5 as a percentage.
 2  Percent means out of 100.
 3  0.5 has a 5 after the point.
 4  So 0.5 = 5%.
```

* Answer: line 4

<!-- Speaker note: Line 4: 0.5 is 5 tenths, which is 50 hundredths. Out of 100: 50%. Multiply a decimal by 100 for the percentage. -->

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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>

## Percentage to fraction

**Write $16\%$ as a fraction in its simplest form.**

* Answer: 4/25

<!-- Speaker note: 16% = 16/100 = 4/25 (÷ 4). -->

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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



- **Fifths:** Each task changes one thing.
- **Decimal to percentage:** 
- **Percentage to decimal:** 
- **Fraction to percentage:** 
- **A decimal that never ends:** 

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

<!-- Speaker note: What this step is for: One change at a time: fifths on the grid, each direction of conversion, then the edge case of a decimal that never ends. Hints, in order: One fifth of 100 is 20. For students who need more support: Use the hundred grid for every conversion; for eighths, allow half squares. For early finishers: Which fractions with a bottom number up to 12 give decimals that stop, and which recur? What do the stopping ones have in common? (In simplest form, bottoms made only of 2s and 5s.) When students get stuck: A student writes 0.5 as 5% (Max's mistake). Ask: How many hundredths is 0.5? Shade it on the grid. Fix: × 100: 0.5 = 50%. / A student writes 3/4 as 3.4 or 0.34. Ask: What does the fraction bar mean: 3 of 4 parts. What is 3 ÷ 4? Fix: Divide the top by the bottom: 0.75. / A student cannot simplify a percentage as a fraction. Ask: What do 16 and 100 have in common? Fix: Over 100, then divide by the highest common factor. -->

<!-- Speaker note: Teacher: before the 1/3 task, ask students to shade 1/3 of the grid exactly. -->

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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>

## Decimal to fraction

**Write $0.45$ as a fraction in its simplest form.**

* Answer: 9/20

<!-- Speaker note: 0.45 = 45/100 = 9/20. -->

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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>

## Fraction to percentage

**Write $\tfrac{3}{5}$ as a percentage.**

* Answer: 60

<!-- Speaker note: 3/5 = 60/100 = 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>

## Order them

**Which is smallest: $0.6$, $\tfrac{2}{3}$ or $65\%$?**

- 0.6
- 2/3
- 65%

* Answer: 0.6

<!-- Speaker note: Write each as a decimal: 0.6, 0.666… and 0.65. The smallest is 0.6. -->

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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: simplest form

**Review: write $\tfrac{9}{12}$ in its simplest form.**

* Answer: 3/4

<!-- Speaker note: ÷ 3: 3/4. -->

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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: eighths that are whole percentages

Which eighths cannot be shaded exactly on the grid? Why?

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

<!-- Speaker note: What this step is for: Searching eighths for whole-number percentages shows which fractions fit hundredths exactly, and that the rest (3/8 = 37.5%) need a decimal. Hints, in order: 1/8 = 12.5%: not a whole number. / Try 2/8: simplify it first. -->

<!-- Speaker note: Teacher: ask why odd numbers of eighths cannot be shaded exactly. -->

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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

**Write 3/5 as a decimal and as a percentage, and 0.08 as a percentage.**

* 0.6 and 60%; 0.08 = 8%.

<!-- 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: Fractions and decimals on the line (node 3). Where it leads: percentage of an amount, percentage change and discounts (Year 8), and probability. -->