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# Estimating to check

Number Foundations · Place value and estimating

<!-- Speaker note: Students estimate by rounding each number to its first digit, compare the estimate with the exact answer, and use it to catch slips such as a misplaced decimal point (AC9M7N05, node 3). 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** round each number to its first digit, then calculate: 38 × 52 ≈ 40 × 50 = 2000. If the calculator says 197.6 or 19 760, something slipped.

- I can estimate a calculation by rounding each number to its first digit.
- I can compare an estimate with the exact answer and say whether it is reasonable.
- I can use an estimate to find a slip, such as a decimal point in the wrong place.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need rounding (previous lesson) and multiplying by multiples of 10 (Year 6). The one idea: round to the leading digit and keep the size, calculate the easy numbers, and compare with the exact answer. The commonest mistakes are losing the size (Lina rounds 19.6 to 2), losing a zero in the calculation, and trusting a calculator answer because it looks plausible. Estimation is only worth teaching if it is used. In class, make students estimate before pressing = for the whole lesson. -->

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## Why it matters

> You type 3.2 × 48 and the calculator shows 15.36. Looks plausible. But before you trust it, how big should the answer be?

<!-- Speaker note: An **estimate** is a quick answer that is close, found by rounding the numbers to something easy. A good habit: round each number to its **first digit** (keep its size), do the easy calculation, and compare with the real answer. $$38 \times 52 \approx 40 \times 50 = 2000$$ If the exact answer is near the estimate, it is **reasonable**. If it is ten times too big or too small, check the decimal point. -->

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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 best estimate for $38 \times 52$?**

- 90
- 200
- 2000
- 20 000

<!-- Speaker note: What happens: 2000. 38 is about 40 and 52 is about 50, and 40 × 50 = 2000. 90 is 38 + 52. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 200 lost a zero; 90 added the numbers. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Festimating%2Fstep-2&code=widget%3A+rounding%0Amode%3A+estimate%0Aslider+a%3A+10%2C+99%2C+27%2C+1%0Aslider+b%3A+10%2C+99%2C+63%2C+1%0Aop%3A+%C3%97%0A&fresh=1}}

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

<!-- Speaker note: The orange bar is the estimate, the blue bar the exact answer. They are nearly the same length. -->

<!-- Speaker note: Teacher: ask whether the exact answer is more or less than the estimate, 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>

## What happened?

* 2000. 38 is about 40 and 52 is about 50, and 40 × 50 = 2000. 90 is 38 + 52.

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

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## Round, then calculate

**Estimate $6.8 \times 31$ by rounding each number to its first digit.**

* Answer: 210

<!-- Speaker note: 6.8 ≈ 7 and 31 ≈ 30, so 7 × 30 = 210. -->

<!-- Speaker note: Teacher: have students write their estimate before the calculator answer is shown. -->

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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

**Lina estimates 19.6 × 4.1. Click the line where her working goes wrong.**

```
 1  Estimate 19.6 × 4.1.
 2  Round each number to its first digit: 19.6 → 2, 4.1 → 4.
 3  2 × 4 = 8
 4  So 19.6 × 4.1 is about 8.
```

* Answer: line 2

<!-- Speaker note: Line 2: 19.6 is about 20, not 2: rounding keeps the size. 20 × 4 = 80, so 19.6 × 4.1 is about 80. -->

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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>

## Is it reasonable?

**You type $3.2 \times 48$ and the calculator shows 15.36. Is that reasonable?**

- No: 3 × 50 = 150, so the decimal point is in the wrong place
- Yes: a calculator is always right
- Yes: 3 × 5 = 15 is close
- No: the answer should be a whole number

* Answer: No: 3 × 50 = 150, so the decimal point is in the wrong place

<!-- Speaker note: 3.2 × 48 ≈ 3 × 50 = 150. The real answer is 153.6: the calculator shows 15.36 only if the 3.2 was typed as 0.32. -->

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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>

## Other operations, other sizes



- **Add them:** 
- **Keep the size:** 
- **Divide:** 
- **Bigger or smaller?:** 

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

<!-- Speaker note: What this step is for: Estimating works for adding, dividing and with decimals; an estimate can be above or below the exact answer. For students who need more support: Round to the nearest ten or hundred, and write the rounded numbers before multiplying. For early finishers: Round one number up and the other down so that the estimate is closer to the exact answer. When is that a good idea? The hard part, on purpose: Keeping the size when rounding: 19.6 is about 20, not 2. Ask 'how many zeros should the estimate have?' before computing. When students get stuck: A student rounds 19.6 to 2 (Lina's slip). Ask: Is 19.6 nearer 2 or 20? Fix: Round to the first digit and keep its size: 20. / A student computes 5 × 80 = 40. Ask: How many zeros in 5 × 8, and how many in 80? Fix: Count the zeros: 40 for 5 × 8, then one more for 80, so 400. / A student says any calculator answer must be reasonable. Ask: What if a decimal point was typed in the wrong place? Fix: Estimate first, then compare. -->

<!-- Speaker note: Teacher: ask when rounding both numbers up gives an estimate that is too big. -->

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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>

## Two numbers

**Estimate $58 \times 19$.**

* Answer: 1200

<!-- Speaker note: 58 ≈ 60 and 19 ≈ 20, so 60 × 20 = 1200. -->

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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 decimal times

**Estimate $7.9 \times 4.2$.**

* Answer: 32

<!-- Speaker note: 7.9 ≈ 8 and 4.2 ≈ 4, so 8 × 4 = 32. -->

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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>

## Check the calculator

**The calculator shows $504 \div 6.3 = 8$. Is that reasonable?**

- Yes: 50 ÷ 6 is about 8
- Yes: 504 ÷ 6 is about 8
- No: 500 ÷ 6 is about 80, so 8 has lost a zero
- No: you cannot divide by a decimal

* Answer: No: 500 ÷ 6 is about 80, so 8 has lost a zero

<!-- Speaker note: 504 ÷ 6.3 = 80. Estimating 500 ÷ 6 ≈ 80 shows 8 has lost a zero. -->

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## Review: rounding

**Review: round 6.45 to 1 decimal place.**

* Answer: 6.5

<!-- Speaker note: The next digit is 5, so 6.5. -->

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## Make: close but under

Then find a pair where the exact answer is more than 2000, and write which rounding you used and why.

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

<!-- Speaker note: What this step is for: Finding numbers whose estimate is exactly 2000 but whose exact product is less shows an estimate is a close value, not a bound. Hints, in order: The estimate must be 40 × 50, so a is near 40 and b near 50. / Try a = 38 and b = 52. -->

<!-- Speaker note: Teacher: collect pairs of numbers; check which have an exact answer above and below 2000. -->

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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

**Estimate 4.8 × 31. The calculator shows 148.8. Is it reasonable?**

* 5 × 30 = 150, so yes.

<!-- 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: Ratios, where estimating also helps check sharing and scaling. Where it leads: estimating in modelling problems (money, measurement), and significant figures in Year 8 and 9. -->