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# Rounding decimals

Number Foundations · Place value and estimating

<!-- Speaker note: Students round decimals on a number line to a given number of decimal places, then choose how much to round for a context such as time, money, tins of paint and a ruler's accuracy (AC9M7N05, nodes 1 and 2). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 3.746 to 2 decimal places is 3.75: look at the next digit, 6, and round up. How much to round depends on what the number is for.

- I can round a decimal to a number of decimal places.
- I can say which digit decides, and why halfway rounds up.
- I can choose how much to round for the job, and explain my choice.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need decimal place value (the place value lesson and Year 6). The one idea: rounding picks the nearer of two numbers with the right number of places, and the digit straight after the place shows which. The common mistakes are cutting digits off (3.74), letting later digits push the deciding digit up (Kai's double rounding), and counting places wrongly (rounding to 1 place when 2 was asked). Node 2 is judgement: rounding to the cent for money, up for tins of paint, to the ruler's accuracy for measurements. Accept any justified answer in class discussion. -->

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## Why it matters

> A stopwatch says 12.846 seconds. Do you report 12.846, 12.85, 12.8 or 13? It depends who is asking, and what the timer can really measure.

<!-- Speaker note: To **round** a decimal to a number of **decimal places**, find the two numbers with that many places that it sits between, and pick the nearer one. The shortcut is to look at *one* digit, the one straight after the place you are rounding to: - 5 or more: round **up**. - 4 or less: round **down** (the digits stay). So 3.746 to 2 decimal places: the digit after the hundredths is 6, so round up to 3.75. A number exactly halfway rounds up. -->

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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

**Round 3.746 to 2 decimal places.**

- 3.7
- 3.74
- 3.75
- 3.8

<!-- Speaker note: What happens: 3.75. The digit after the hundredths is 6, so the 4 goes up to 5. 3.74 cuts the digits off. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 3.74 cut the digits off; 3.7 is one place too few. -->

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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%2Frounding%2Fstep-2&code=widget%3A+rounding%0Ax%3A+3.746%0Aslider+d%3A+0%2C+3%2C+3%0A&fresh=1}}

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

<!-- Speaker note: Change the decimal places and watch the line: the two ends are always the nearest numbers with that many places. -->

<!-- Speaker note: Teacher: slide the decimal places down and ask what the two ends of the line are each time. -->

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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?

* 3.75. The digit after the hundredths is 6, so the 4 goes up to 5. 3.74 cuts the digits off.

<!-- 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>

## Round it

**Round 8.372 to 1 decimal place.**

* Answer: 8.4

<!-- Speaker note: The digit after the tenths is 7, so round up: 8.4. -->

<!-- Speaker note: Teacher: ask what Kai's 2.35 and 2.4 would mean if 2.349 were a real length. -->

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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

**Kai rounds 2.349 to 1 decimal place. Click the line where the working goes wrong.**

```
 1  Round 2.349 to 1 decimal place.
 2  The 9 rounds the 4 up: 2.349 is about 2.35.
 3  The 5 rounds the 3 up: 2.35 is about 2.4.
 4  Answer: 2.4
```

* Answer: line 2

<!-- Speaker note: Line 2: only the digit straight after the tenths matters, and that is 4, so round down. Kai let the 9 push the 4 up, then rounded again. 2.349 to 1 decimal place is 2.3. -->

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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>

## Halfway

**Round 6.45 to 1 decimal place.**

- 6.0
- 6.4
- 6.45
- 6.5

* Answer: 6.5

<!-- Speaker note: The next digit is 5, exactly halfway, and halfway rounds up: 6.5. -->

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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>

## How much is enough?



- **The stopwatch:** 
- **Enough paint:** 
- **Dollars and cents:** 
- **The ruler:** 

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

<!-- Speaker note: What this step is for: The right amount of rounding depends on the job: a timer, a purchase and a ruler each suggest a different place, and sometimes 'enough' means always round up. For students who need more support: Keep the number line in view and say which two numbers it is between before choosing. For early finishers: Round at every step of 3.46 + 3.46 to the nearest whole number, then round only at the end. Do you get the same answer? Which should you trust? The hard part, on purpose: Deciding how much to round in context. Ask 'what is this number for, and how accurately could we have measured it?' before any digits are cut. When students get stuck: A student cuts digits off (3.74). Ask: Is 3.746 nearer 3.74 or 3.75? Where does it sit on the line? Fix: Use the number line: the next digit tells which end is nearer. / A student rounds 2.349 to 2.35 and then to 2.4 (Kai's slip). Ask: Which digit is straight after the tenths? Fix: Only that digit decides. The 9 is too far along to matter. / A student reports a ruler measurement to 4 decimal places. Ask: Could the ruler show that? Fix: Match the places to what the instrument can show. -->

<!-- Speaker note: Teacher: for the paint, ask what happens if she buys 3 tins. -->

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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 places

**Round 5.0763 to 3 decimal places.**

* Answer: 5.076

<!-- Speaker note: The digit after the thousandths is 3, so 5.076. -->

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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>

## The 9 does not count

**Round 7.0449 to 2 decimal places.**

* Answer: 7.04

<!-- Speaker note: The next digit is 4, so 7.04. -->

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## Share the bill

**Three friends share a $50 bill: $50 ÷ 3 = 16.666… Each pays this much, to the nearest cent. (Write it as 16.xx.)**

* Answer: 16.67

<!-- Speaker note: Cents are 2 decimal places. The next digit is 6, so 16.67. -->

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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: powers of 10

**Review: write $3 \times 10^2 + 5 \times 10^0$ as a number.**

* Answer: 305

<!-- Speaker note: 300 + 5 = 305. The tens column is 0. -->

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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: lands on 4.57

Then find the smallest number you can that still rounds to 4.57.

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

<!-- Speaker note: What this step is for: Finding a number that rounds to 4.57 shows a whole range of numbers, 4.565 up to just under 4.575, round to the same value. Hints, in order: It must be nearer 4.57 than 4.56 or 4.58. / Try 4.568. -->

<!-- Speaker note: Teacher: collect the smallest and largest numbers that round to 4.57. -->

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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 stick is measured as 83.46 cm with a ruler marked in millimetres. Round it sensibly, and say why.**

* 83.5 cm: the ruler shows millimetres, which is 1 decimal place of a centimetre.

<!-- 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: Estimating, using rounded numbers to check an answer. Where it leads: significant figures and accuracy in Year 8 and 9, and error in measurement. -->