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# Which decimals stop?

Irrational numbers and index laws · Recurring decimals

<!-- Speaker note: Students predict whether a fraction's decimal terminates from the prime factors of its simplified denominator, and check on the long-division board (AC9M8N03, node 2). About 35 minutes (Predict and Run 8 min · Investigate 10 min · Modify 8 min · Practise and Make 9 min). -->

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## Today

**We are learning to** write the fraction in simplest form and break the denominator into primes. If its only prime factors are 2 and 5, the decimal terminates; any other prime factor makes it recur. It works because 10 = 2 × 5.

- I can predict whether a fraction terminates without dividing.
- I can explain the rule using 10 = 2 × 5.
- I can simplify a fraction before using the rule.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need prime factors (Number Foundations) and the previous lesson. The one idea: in simplest form, a denominator made only of 2s and 5s gives a terminating decimal, because terminating decimals are fractions over powers of 10 = 2 × 5. The common mistake is skipping the simplifying (Max's mistake). The board shows the simplest form and the prime factors with the long division. Wider: in base 12 (12 = 2 × 2 × 3), thirds would terminate and fifths would recur. -->

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## Why it matters

> Without dividing anything, you can tell that 7/40 stops and 7/30 goes on forever. The secret is in the denominator's prime factors.

<!-- Speaker note: A terminating decimal is a fraction over 10, 100, 1000…: $0.175 = \frac{175}{1000}$. Since $10 = 2 \times 5$, those denominators are made only of 2s and 5s. So: 1. Write the fraction in **simplest form**. 2. Break the denominator into **prime factors**. 3. Only 2s and 5s: it **terminates**. Any other prime (3, 7, 11…): it **recurs**. $\frac{7}{40}$: $40 = 2 \times 2 \times 2 \times 5$, so it terminates (0.175). $\frac{7}{30}$: $30 = 2 \times 3 \times 5$ has a 3, so it recurs (0.2333…). -->

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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

**Without dividing: does 11/50 terminate or recur?**

- Terminate
- Recur
- You cannot tell without dividing

<!-- Speaker note: What happens: It terminates: 50 = 2 × 5 × 5, only 2s and 5s. 11/50 = 22/100 = 0.22. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what 40 is made of. -->

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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%2Fwhich-decimals-stop%2Fstep-2&code=widget%3A+longdiv%0Aslider+a%3A+1%2C+20%2C+1%0Aslider+b%3A+2%2C+60%2C+6%0Afactors%3A+on%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask for a denominator under 30 that is all 2s and 5s. -->

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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?

* It terminates: 50 = 2 × 5 × 5, only 2s and 5s. 11/50 = 22/100 = 0.22.

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

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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>

## Why it works

**Why do only 2s and 5s in the denominator give a terminating decimal?**

- Because 2 + 5 = 7
- 2 and 5 are the smallest primes
- 10 = 2 × 5, so only those can multiply up to 10, 100, 1000…
- Because even numbers always terminate

* Answer: 10 = 2 × 5, so only those can multiply up to 10, 100, 1000…

<!-- Speaker note: A terminating decimal is a fraction over a power of 10, and 10 = 2 × 5. -->

<!-- Speaker note: Teacher: ask Max to simplify. -->

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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 decides whether 3/12 terminates. Click the line where his reasoning goes wrong.**

```
 1  Max wants to know if 3/12 terminates.
 2  12 = 2 × 2 × 3.
 3  12 has a prime factor of 3, so 3/12 recurs.
```

* Answer: line 3

<!-- Speaker note: Line 3: simplify first. 3/12 = 1/4, and 4 = 2 × 2, so it terminates: 0.25. -->

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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>

## Sort them

**Terminate or recur?**

Groups: Terminates · Recurs

1/16 · 3/20 · 1/12 · 2/15 · 7/25 · 6/15

* Answer: 1/16 → Terminates; 3/20 → Terminates; 1/12 → Recurs; 2/15 → Recurs; 7/25 → Terminates; 6/15 → Terminates

<!-- Speaker note: Simplify, then check the denominator for primes other than 2 and 5. -->

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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>

## Find one that stops



- **Fifteenths:** 
- **Twenty-fourths:** 
- **Eightieths:** 
- **Thirtieths:** 

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

<!-- Speaker note: What this step is for: Choosing numerators that cancel the bad primes shows simplifying decides the answer. Hints, in order: 15 = 3 × 5. / Choose a numerator that cancels the 3. / 24 = 2 × 2 × 2 × 3: cancel the 3. For students who need more support: A list of primes to 20 and factor trees for the denominators. For early finishers: In base 12, which unit fractions from 1/2 to 1/12 would terminate? Why? The hard part, on purpose: Simplifying first. Ask students to write the fraction in simplest form before they say anything about its denominator. When students get stuck: A student judges by the denominator's size or digits. Ask: What are its prime factors? Fix: Only the primes matter: 2s and 5s stop. / A student does not simplify first. Ask: Is the fraction in simplest form? Fix: Cancel common factors, then look at the denominator. / A student thinks 8 = 2 × 4 counts as other primes. Ask: Is 4 a prime? Fix: Break it down to primes: 8 = 2 × 2 × 2. -->

<!-- Speaker note: Teacher: ask which numerators work for denominator 15, and why. -->

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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>

## Simplify first

**Does 7/56 terminate or recur?**

- Terminate: 7/56 = 1/8
- Recur: 56 has a 7
- Recur: 56 is too big
- You cannot tell

* Answer: Terminate: 7/56 = 1/8

<!-- Speaker note: 7/56 = 1/8 = 0.125. -->

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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>

## Places

**1/16 terminates. How many decimal places does it have?**

* Answer: 4

<!-- Speaker note: 16 = 2⁴, so 1/16 = 625/10 000 = 0.0625: four places. -->

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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>

## Always stops

**Which denominator always gives a terminating decimal, whatever the numerator?**

- 12
- 21
- 30
- 50

* Answer: 50

<!-- Speaker note: 50 = 2 × 5 × 5: only 2s and 5s. -->

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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: block length

**Review: how many digits are in the repeating block of 1/9?**

* Answer: 1

<!-- Speaker note: 1/9 = 0.111…: one digit. -->

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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: sixtieths

Then explain, using primes, why your fraction stops and 1/60 does not.

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

<!-- Speaker note: What this step is for: Finding a terminating fraction over 60 combines simplifying and the prime rule. Hints, in order: 60 = 2 × 2 × 3 × 5. / The numerator must cancel the 3. -->

<!-- Speaker note: Teacher: ask for the reason in primes. -->

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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 9/24 terminate? Explain.**

* Yes: 9/24 = 3/8, and 8 = 2 × 2 × 2, only 2s, so it terminates (0.375).

<!-- 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 completes AC9M8N03. Where it leads: irrational numbers (next unit), whose decimals never stop and never repeat. -->