KS3 · Maths

Estimating numerical calculations

A calculator says 19.173 × 338 = 64,804.74. Round to 20 × 300 = 6,000 and you can see it is ten times too big — no calculator needed.

Maths · Estimating

One calculation, four sensible estimates

A calculator says 19.173 × 338 = 6,480.474 exactly. Four people estimate it by rounding in different ways: 19 × 300, 20 × 300, 20 × 350 and 19 × 350. Work each one out in your head, drag its tag to where it sits on the line, then drag Exact to where the real answer sits. Which estimate do you think lands closest?

Place ExactPlace 19×300Place 20×300Place 20×350Place 19×350

Maths · Rounding

What does 'one significant figure' really do?

Most quick estimates start by rounding every number to one significant figure — for example 892 × 176.9 ≈ 900 × 200. A significant figure is a digit that contributes to the accuracy of a number, and the first one is the first non-zero digit. So what happens when we round 36,789 to one significant figure?

Four learners give four answers. Which is closest to what you would write?

4 — the first digit is 3, and it rounds up to 4.30,000 — keep the 3 and swap everything else for zeros.37,000 — keep the first two digits, then round.40,000 — keep the 3, round it up, and keep the zeros.

How sure are you?

I'm sureI think soI'm guessing

4 — the first digit is 3, and it rounds up to 4.

This is a common misconception

You have found the right digit, which is the clever part: 3 is the first significant figure, and the 6 after it nudges it up to 4. But a lone 4 is worth four, not forty thousand — the number has shrunk. When you round, every digit keeps its place value: the 4 stays in the ten-thousands place and zeros fill the rest, giving 40,000. A place value table shows the magnitude of the number, so check any answer against it.

30,000 — keep the 3 and swap everything else for zeros.

This is a common misconception

Good news: you have kept the place value. But swapping the other digits for zeros chops the number instead of rounding it. Compare the distances: 36,789 is 3,211 away from 40,000 but 6,789 away from 30,000, so 40,000 is the closer one-significant-figure answer.

37,000 — keep the first two digits, then round.

This is a common misconception

37,000 is a sensible rounding — but of two significant figures, not one. Count them: the 3 and the 7 are two figures. One significant figure means keeping only the first non-zero digit, the 3, and rounding that.

40,000 — keep the 3, round it up, and keep the zeros.

This matches the evidence

Yes. The 3 is the first significant figure, 36,789 is nearer 40,000 than 30,000, and the zeros hold the size — it is still about forty thousand. A lone 4 would have lost the place value; 40,000 has not.

Rounding changes how precise a number is — never how big it is. Picture 6,523 as a ladder: 7,000 to one significant figure, 6,500 to two, 6,520 to three. Every rung is still about six and a half thousand; each extra figure just gets closer.

Worked example

Problem

Estimate (39.7 × 9.7) ÷ (0.779 − 0.601).

Maths · Dividing by less than 1

Does dividing always make things smaller?

What is 400 ÷ 0.5? Slide to your answer, then lock it in.

Your estimate

500

What is 400 ÷ 0.5? Slide to your answer, then lock it in. Estimate between 0 and 1000.

01000

Maths · Estimating in context

Your turn to fill the gaps

Two quick estimates. (A) Estimate the area of a floor measuring 844 cm by 596 cm. (B) Estimate how many books costing £6.89 each you can buy with £279.45. At each gap, choose the step that fits.

A floor measures 844 cm by 596 cm. To estimate its area we multiply the length by the width — but first we round each length.
missing step

Which line is step 2?

844 cm ≈ 800 cm and 596 cm ≈ 600 cm844 cm ≈ 900 cm and 596 cm ≈ 600 cm844 cm ≈ 8 cm and 596 cm ≈ 6 cm

Maths · Is it realistic?

Realistic or not? Let an estimate decide

Pick a claim, then choose where it belongs. Make a quick estimate in your head first — then read how the mistake probably happened.

Still to sort

Realistic (0)

About the size you would expect.

Not realistic (0)

Nowhere near the estimate.

Where the line is: If the claim is nowhere near a quick estimate, something has gone wrong — and the estimate often shows what.

7 of 7 still to sort.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A quick, deliberately rough answer that tells you how big the real one should be.

What you need to know

An estimate is a calculation done with rounded numbers, usually rounded to one significant figure, so the answer is close to — but not the same as — the exact answer.
Rounding keeps place value: 36,789 to one significant figure is 40,000, never 4.
Use ≈ for an estimate. It means 'approximately equal', not '='.
The order of operations still applies, and a long dividing line works like brackets.
Dividing by a number less than 1 is the same as multiplying by its reciprocal, so dividing by 0.5 is not halving.

The big picture

Estimating means rounding each number so a calculation becomes easy, keeping every digit's place value and the order of operations, then checking the result is sensible. There is more than one acceptable estimate: some are quicker, some land closer.

Key points

1Round each number so the sum is easy — usually to one significant figure — and keep every digit's place value.

2The first significant figure is the first non-zero digit. 6,523 is 7,000 to one significant figure, 6,500 to two and 6,520 to three.

3There is more than one acceptable estimate. Rounding to one significant figure is quicker; rounding more finely tends to be closer but takes longer.

4Keep the order of operations when you estimate: brackets, then powers and roots, then multiply and divide, then add and subtract.

5Dividing by 0.5 is the same as multiplying by 2, dividing by 0.2 is multiplying by 5, and dividing by 0.1 is multiplying by 10.

6Use an estimate to check an answer is realistic, and to work out how a wrong answer came about.

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Worked example

Problem

Tickets for a school show cost £7.95 each. A learner works out the cost of 41 tickets and gets £3,260. Use an estimate to decide whether that is realistic.

⚠ Watch out

Writing 36,789 to one significant figure as 4. That loses the size: it should be 40,000. And dividing by 0.5 is not halving — it is the same as multiplying by 2.

🧠

Memory hook

Round it, keep the zeros, write ≈, then ask: does that look sensible?

✓

Check yourself

Say your estimate out loud before you press equals. Then ask: is the answer about that size? If not, what slip explains it?

Flashcards

(14)

What is an estimate?

A deliberately rounded calculation. Its answer is close to the exact answer, but not the same as it.

Which digit is the first significant figure?

The first non-zero digit, reading from the left.

Round 6,523 to 1, 2 and 3 significant figures.

7,000 (1 s.f.), 6,500 (2 s.f.) and 6,520 (3 s.f.).

Why is 36,789 to one significant figure 40,000 and not 4?

Rounding keeps place value. The 4 stays in the ten-thousands place, with zeros filling the rest.

What does the ≈ sign mean?

'Approximately equal' — the two sides are about the same but not equal. Use it when you estimate.

How are quick estimates usually made?

By rounding each number to one significant figure, for example 892 × 176.9 ≈ 900 × 200.

Is there only one correct estimate of a calculation?

No. There is more than one acceptable estimate. For 19.173 × 338, the estimate 19 × 350 is closest and 19 × 300 is furthest away.

Rounding to one significant figure versus rounding more finely: what is the trade-off?

One significant figure is quicker. Rounding more finely tends to be closer to the exact answer but takes longer.

Does the order of operations still apply when you estimate?

Yes: brackets, then powers and roots, then multiply and divide, then add and subtract.

What does a long dividing line do?

It acts like brackets. Work out the top and the bottom separately first, then divide.

Dividing by 0.5 is the same as multiplying by what?

2. Dividing by 0.2 is multiplying by 5, and dividing by 0.1 is multiplying by 10. Dividing by 0.5 is not halving.

In a word problem, how do you decide whether to multiply or divide?

Ask what the context needs. Finding a total, such as an area, uses multiplication. Asking how many fit into an amount uses division.

A calculation gives an answer far from your estimate. What now?

Look for how the error came about. For example, £371.02 at £4.82 each is about 400 ÷ 5 = 80 books, so 2,000 came from multiplying instead of dividing.

Give three everyday sizes you can use to estimate.

A hand is about 10 cm wide, a person is about 1.5 m tall, and a car's petrol tank holds about 55 litres.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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One calculation, four sensible estimates

What does 'one significant figure' really do?

Does dividing always make things smaller?

Your turn to fill the gaps

Realistic or not? Let an estimate decide

More KS3 Maths topics