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# Recurring decimals

Irrational numbers and index laws · Recurring decimals

<!-- Speaker note: Students divide fractions out on a long-division board, watch the remainders, and see why decimals either stop or repeat, writing recurring decimals with bars and dots (AC9M8N03, 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** dividing a fraction out gives a decimal that either stops (terminates) or repeats a block of digits forever (recurs). In long division, a remainder of 0 means it stops; a remainder that comes back means it repeats. A bar or dots mark the repeating digits.

- I can tell whether a fraction's decimal terminates or recurs.
- I can write a recurring decimal with a bar or dots.
- I can explain why a remainder coming back means the digits repeat.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need fractions as decimals (Number Foundations). The one idea: the remainders are always less than the denominator, so either one is 0 (the decimal stops) or one comes back (the digits repeat). The common mistake is calling a recurring decimal irrational (Zoe's mistake): a recurring decimal is a fraction. The board shows the first 12 steps; longer blocks (1/17, 1/19, 1/23, 1/29) are written out in full on the decimal line. Wider: why 0.999… = 1; the longest blocks (1/7 has 6, 1/17 has 16: the block is at most one less than the denominator). -->

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## Why it matters

> Type 1 ÷ 7 into a calculator and the screen fills up. Does it ever stop? Long division can tell you, and it does not need a bigger screen.

<!-- Speaker note: A fraction is a division: $\frac{3}{8} = 3 \div 8$. Doing the division gives a decimal that does one of two things: - it **terminates** (stops): $\frac{3}{8} = 0.375$; - it **recurs** (repeats a block of digits forever): $\frac{1}{3} = 0.333\ldots$, $\frac{1}{7} = 0.142857142857\ldots$ Long division shows why. Each step has a **remainder**, and the remainder is always less than the denominator. If a remainder is 0, the decimal stops. If not, a remainder must come back sooner or later, and from then on the same digits repeat. The repeating block is marked with a **bar** over it, or **dots** over its first and last digits: $0.\overline{142857}$ or $0.\dot{1}4285\dot{7}$; $0.1\overline{6}$ or $0.1\dot{6}$ means $0.1666\ldots$ -->

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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

**Does the decimal for 1/3 ever stop?**

- No: the 3s go on forever
- Yes, after about 10 digits
- Yes: 0.33

<!-- Speaker note: What happens: No. Every step of 1 ÷ 3 leaves a remainder of 1, so every digit is 3, forever. 0.33 is an approximation. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask a 'yes' student to keep dividing 1 by 3. -->

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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%2Freal-numbers%2Frecurring-decimals%2Fstep-2&code=widget%3A+longdiv%0Aslider+a%3A+1%2C+20%2C+1%0Aslider+b%3A+2%2C+30%2C+3%0A&fresh=1}}

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

<!-- Speaker note: Change the numerator a and the denominator b. Watch the remainder column. -->

<!-- Speaker note: Teacher: ask why the remainder can never be 7 or more when dividing by 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>

## What happened?

* No. Every step of 1 ÷ 3 leaves a remainder of 1, so every digit is 3, forever. 0.33 is an approximation.

<!-- 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>

## Notation

Here is the routine.

$$1 \div 6$$

1) $40 \div 6 = 6 \text{ r } 4$ — Remainder 4 again.
2) $\tfrac{1}{6} = 0.1\overline{6}$ — The 6 repeats forever; the 1 does not.
3) $\tfrac{1}{8} = 0.125$ — A remainder of 0: it stops.

<!-- 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 1 ÷ 6 and say "Bring down a 0: 10 ÷ 6 = 1 remainder 4." Line 1: write 40 ÷ 6 = 6 r 4 and say "Remainder 4 again." Line 2: write 1 ÷ 6 = 0.16 and say "The 6 repeats forever; the 1 does not." Line 3: write 1 ÷ 8 = 0.125 and say "A remainder of 0: it stops." -->

<!-- Speaker note: Teacher: ask Zoe whether 1/7 is a fraction. -->

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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>

## Notation

**How should 0.272727… be written with a bar?**

- A bar over 2 only
- A bar over 27: 0.27 with the bar over both digits
- A bar over the last 7 only
- 0.27 with no bar

* Answer: A bar over 27: 0.27 with the bar over both digits

<!-- Speaker note: The repeating block is 27, so the bar goes over both digits. -->

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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

**Zoe decides whether 1/7 is a rational number (a fraction). Click the line where her reasoning goes wrong.**

```
 1  Zoe divides 1 by 7 on a calculator.
 2  The screen shows 0.1428571429.
 3  The digits go on past the end of the screen.
 4  A decimal that never stops is irrational, so 1/7 is irrational.
```

* Answer: line 4

<!-- Speaker note: Line 4: 1/7 is a fraction, so it is rational. Its decimal never stops, but it repeats (142857). Only a decimal that never stops and never repeats is irrational. -->

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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>

## Eleven

**Divide 1 by 11 (on paper, or on the Run board above). How many digits are in its repeating block?**

* Answer: 2

<!-- Speaker note: 1/11 = 0.090909…: the block 09 repeats. -->

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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>

## Stop, repeat, delay



- **Stops after 3 places:** 
- **A delay:** 
- **Thirteenths:** 
- **0.999…:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/recurring-decimals/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Finding fractions with a set kind of decimal: one that stops after 3 places, one with a delay before the repeat, a long block, and why 0.999… = 1. Hints, in order: Try denominators 2, 4, 8. / 1/8 = 0.125. / Try sixths and twelfths. For students who need more support: Do 1 ÷ 3 and 1 ÷ 4 on paper side by side before the board. For early finishers: Find the length of the repeating block of 1/17, 1/19 and 1/23. What is the longest a block can be for denominator b, and why? The hard part, on purpose: Trusting the remainders over the calculator. Ask students to predict the next remainder before the board shows it. When students get stuck: A student says a long calculator decimal never stops, so it is irrational. Ask: Do the digits repeat? Fix: A repeating decimal is a fraction: rational. / A student puts the bar over the wrong digits. Ask: Which digits come round again? Fix: The bar goes over the whole repeating block, and only that. / A student thinks 0.33 is exactly 1/3. Ask: What is 3 × 0.33? Fix: 0.99 is not 1: 0.33 is an approximation. -->

<!-- Speaker note: Teacher: ask which denominators gave decimals that stop. -->

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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>

## Stops or repeats

**Which fraction has a terminating decimal?**

- 2/7
- 1/3
- 3/8
- 5/6

* Answer: 3/8

<!-- Speaker note: 3/8 = 0.375: it stops. -->

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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>

## Sevenths

**How many digits are in the repeating block of 1/7?**

* Answer: 6

<!-- Speaker note: 1/7 = 0.142857142857…: 6 digits. -->

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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>

## Dot notation

**How is 0.1666… written with a dot?**

- 0.17
- 0.16 with a dot over the 1
- 0.16 with dots over the 1 and the 6
- 0.16 with a dot over the 6

* Answer: 0.16 with a dot over the 6

<!-- Speaker note: Only the 6 repeats: a dot over the 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: fractions to decimals

**Review: write 3/4 as a decimal.**

* Answer: 0.75

<!-- Speaker note: 3 ÷ 4 = 0.75. -->

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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>

## Make: a two-digit block

Then write your fraction's decimal with a bar over the repeating block.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/recurring-decimals/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Searching for a fraction with a given block length uses the board to test predictions about denominators. Hints, in order: Elevenths have a block of 2. / Try a denominator that is a multiple of 11. -->

<!-- Speaker note: Teacher: ask for the longest block they found. -->

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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

**Does 5/6 terminate or recur? Write it with a dot.**

* It recurs: 5/6 = 0.8333… = 0.83 with a dot over the 3.

<!-- 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 AC9M8N03. Where it leads: predicting from the denominator (next lesson), irrational numbers, and converting recurring decimals to fractions (Year 9 and 10). -->