Chapter 6 of 7

All 7 chapters
  1. 1Adding in your head, readunderstood
  2. 2Taking away in your head, readunderstood
  3. 3Column adding, readunderstood
  4. 4Column taking away, readunderstood
  5. 5Four digits and more, readunderstood
  6. 6Estimate and check, readunderstood
  7. 7Problems in two steps, readunderstood

M2.2.6 Chapter 6

Estimate and check

Ravi worked out 487 + 312 and got 7 990. How can you tell he is wrong?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Round numbers to estimate an answer before you work it out
  • Check a taking away answer by adding back

Ravi works out 487 + 312. He gets 7 990. Can that be right?

Before you add, make an estimate. An estimate is a quick answer that is near the right one.

To make one, you round each number. To round is to change it to the nearest ten or hundred.

Round to the nearest hundred. Look at the tens digit. If it is 5 or more, go up. If it is 4 or less, go down.

Draw it first

Estimate 487 + 312

  1. Two number lines. The first runs from 400 to 500 with ticks every 10, and a marker at 487. An arrow points to 500, with the note "nearer 500". The second runs from 300 to 400, with a marker at 312. An arrow points to 300, with the note "nearer 300". At the bottom: "500 + 300 = 800" and "Exact answer: 799".400500487

    1 Find 487 on the line between 400 and 500.

  2. Two number lines. The first runs from 400 to 500 with ticks every 10, and a marker at 487. An arrow points to 500, with the note "nearer 500". The second runs from 300 to 400, with a marker at 312. An arrow points to 300, with the note "nearer 300". At the bottom: "500 + 300 = 800" and "Exact answer: 799".400500487nearer 500

    2 It is nearer 500. Round it up to 500.

  3. Two number lines. The first runs from 400 to 500 with ticks every 10, and a marker at 487. An arrow points to 500, with the note "nearer 500". The second runs from 300 to 400, with a marker at 312. An arrow points to 300, with the note "nearer 300". At the bottom: "500 + 300 = 800" and "Exact answer: 799".400500487nearer 500300400312nearer 300

    3 Find 312 between 300 and 400. It is nearer 300. Round it down to 300.

  4. Two number lines. The first runs from 400 to 500 with ticks every 10, and a marker at 487. An arrow points to 500, with the note "nearer 500". The second runs from 300 to 400, with a marker at 312. An arrow points to 300, with the note "nearer 300". At the bottom: "500 + 300 = 800" and "Exact answer: 799".400500487nearer 500300400312nearer 300500 + 300 = 800Exact answer: 799

    4 Add the rounded numbers. 500 + 300 = 800. The exact answer is 799.

Worked example

Estimate 487 + 312. Then work it out.

  1. The tens digit of 487 is 8. Go up to 500.
  2. The tens digit of 312 is 1. Go down to 300.
  3. Add the rounded numbers. 500 + 300 = 800.
  4. Now work out the exact sum. 487 + 312 = 799.

Answer: About 800. The exact answer is 799.

The estimate and the exact answer are close. Ravi's 7 990 is nowhere near 800.

An estimate cannot tell you the last digit. It tells you if the answer is far off.

See it three ways

With things

Think of 48 sticks as nearly 5 bundles of ten. Think of 31 sticks as about 3 bundles.

As a picture

Five rods of ten, labelled "50", plus three rods of ten, labelled "30", make eight rods of ten, labelled "80".50+30=80

In symbols

48 + 31 is about 50 + 30 = 80. The exact answer is 79.

Now check a take away. To check is to test an answer a second way.

Add your answer to the number you took away. You must get the first number back.

A column take away, 403 minus 167, with the answer 236. The 4 is crossed out with a small 3 above. The 0 is crossed out with a small 9 above. A small 1 sits by the ones digit 3.304761−193632
403 − 167 = 236.
A column sum, 236 plus 167, with the answer 403. A small 1 is carried over the tens and a small 1 over the hundreds.632761+11304
236 + 167 = 403. You got the first number back.

Worked example

Check that 403 − 167 = 236.

  1. Add the answer to the number you took away.
  2. 236 + 167 = 403.
  3. That is the first number. So 236 is right.

Answer: 236 + 167 = 403, so the answer is right

One bar cut into two parts, labelled 236 and 167. A bracket over the whole bar is labelled 403.236167403
The answer and the number taken away make the whole. 236 + 167 = 403.

If the first number does not come back, work it out again.

Try it

Take away and put back

You need: 23 beans or small stones, a pencil and paper.

Be safe: Do not eat the beans.

  1. Count out 23 beans. Write 23.
  2. Take away 9 beans. Put them to one side.
  3. Count the beans left. Write 23 − 9 = 14.
  4. Put the 9 beans back with the 14. Count them all.

What you will see: You have 23 beans again. 14 + 9 = 23, so 23 − 9 = 14 is right.

If you cannot try it, this is what happens.
Fourteen beans, labelled "14 left", and nine beans, labelled "9 taken". Together they are 23 beans.14 left9 taken

A wrong idea many people have

An estimate is just a guess, or a wrong answer.

Why it feels right: An estimate is not exact, and people say "about".

The test that breaks it: Guess 487 + 312 with no rounding. Say 2 000. Now round and add. You get 800. The exact answer is 799.

What is true: An estimate is careful. You round, then add. It lands near the right answer and shows big mistakes.

Key words

estimate
a quick answer that is near the right one.
round
change a number to the nearest ten or hundred.
check
test an answer a second way.

Back to your guess

The answer is B: Round the numbers and add. The answer is about 800.

Round each number and work out a quick sum. 500 + 300 = 800. An answer near 800 is likely. 7 990 is far too big.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Estimate 396 + 213. Round each number to the nearest hundred.

    See the answer

    600

    396 rounds up to 400, because the tens digit is 9. 213 rounds down to 200. 400 + 200 = 600. The exact answer is 609.

  2. Amina works out 624 + 275 and gets 1 899. Which estimate shows she is wrong?

    Tap your answer.

    1. Yes, that is right. 624 rounds to 600. 275 rounds to 300. 600 + 300 = 900. The exact answer is 899. Her answer is about twice as big.

    2. Not this one. Look again. 1 899 is about twice 900. The answer should be near 900. Try another.

    3. Not this one. It can. A big gap between the estimate and the answer shows a mistake. Try another.

    See the answer

    About 600 + 300 = 900, so 1 899 is too big.

    624 rounds to 600. 275 rounds to 300. 600 + 300 = 900. The exact answer is 899. Her answer is about twice as big.

    • If you chose B: Look again. 1 899 is about twice 900. The answer should be near 900.
    • If you chose C: It can. A big gap between the estimate and the answer shows a mistake.

  3. Tariq says 702 − 458 = 354. Check by adding back. Is he right?

    Tap your answer.

    1. Not this one. Add again. 354 + 458 = 812, not 702. Try another.

    2. Yes, that is right. Adding back should give 702. It gives 812. The right answer is 244, because 244 + 458 = 702.

    3. Not this one. You can tell by adding back. 354 + 458 = 812, which is not 702. Try another.

    See the answer

    No. 354 + 458 = 812. The answer is 244.

    Adding back should give 702. It gives 812. The right answer is 244, because 244 + 458 = 702.

    • If you chose A: Add again. 354 + 458 = 812, not 702.
    • If you chose C: You can tell by adding back. 354 + 458 = 812, which is not 702.

  4. Work out 5 000 − 2 618. Then check by adding back.

    See the answer

    2 382

    Exchange along the zeros. 5 000 − 2 618 = 2 382. Check: 2 382 + 2 618 = 5 000.

  5. From an earlier chapter

    Which is 568 + 275?

    Tap your answer.

    1. Yes, that is right. Ones: 8 + 5 = 13, write 3, carry 1. Tens: 1 + 6 + 7 = 14, write 4, carry 1. Hundreds: 1 + 5 + 2 = 8.

    2. Not this one. You did not carry. 8 + 5 = 13 and 6 + 7 = 13 each carry a 1. Go to M2.2.3. Try another.

    3. Not this one. You carried once. The tens make 6 + 7 + 1 = 14, so carry again. Go to M2.2.3. Try another.

    See the answer

    843

    Ones: 8 + 5 = 13, write 3, carry 1. Tens: 1 + 6 + 7 = 14, write 4, carry 1. Hundreds: 1 + 5 + 2 = 8.

    • If you chose B: You did not carry. 8 + 5 = 13 and 6 + 7 = 13 each carry a 1. Go to M2.2.3.
    • If you chose C: You carried once. The tens make 6 + 7 + 1 = 14, so carry again. Go to M2.2.3.

  6. Hana estimates 748 + 231. She rounds each number to the nearest hundred. What is her estimate?

    Tap your answer.

    1. Not this one. You rounded 748 up to 800. The tens digit is 4, so go down to 700. Try another.

    2. Not this one. 979 is the exact answer. An estimate rounds first. 700 + 200 = 900. Try another.

    3. Yes, that is right. 748 rounds down to 700, because the tens digit is 4. 231 rounds down to 200. 700 + 200 = 900.

    See the answer

    900

    748 rounds down to 700, because the tens digit is 4. 231 rounds down to 200. 700 + 200 = 900.

    • If you chose A: You rounded 748 up to 800. The tens digit is 4, so go down to 700.
    • If you chose B: 979 is the exact answer. An estimate rounds first. 700 + 200 = 900.

  7. Mei works out 905 − 378 and gets 527. Which sum checks her answer?

    Tap your answer.

    1. Yes, that is right. Add the answer to the number you took away. 527 + 378 = 905. That is the first number, so she is right.

    2. Not this one. That makes 1 283. Add the answer to the number you took away. Try another.

    3. Not this one. Taking 378 from 527 does not give you back 905. Add instead. Try another.

    See the answer

    527 + 378

    Add the answer to the number you took away. 527 + 378 = 905. That is the first number, so she is right.

    • If you chose B: That makes 1 283. Add the answer to the number you took away.
    • If you chose C: Taking 378 from 527 does not give you back 905. Add instead.

  8. Joseph works out 800 − 345 and gets 465. He adds back. 465 + 345 = 810. What does that tell him?

    A column sum of 465 plus 345 with empty boxes for the answer.564543+
    Tap your answer.

    1. Not this one. Adding back must give the first number exactly. 810 is 10 more than 800, so there is a mistake. Try another.

    2. Yes, that is right. Adding back gives 810, not 800. So 465 is wrong. The right answer is 455, because 455 + 345 = 800.

    3. Not this one. An estimate shows only big mistakes. 465 is near 500, but adding back shows it is off. Try another.

    See the answer

    His answer is wrong. Adding back must give the first number.

    Adding back gives 810, not 800. So 465 is wrong. The right answer is 455, because 455 + 345 = 800.

    • If you chose A: Adding back must give the first number exactly. 810 is 10 more than 800, so there is a mistake.
    • If you chose C: An estimate shows only big mistakes. 465 is near 500, but adding back shows it is off.

Keep it

Copy this into your exercise book.

Round, then work out a quick sum. 487 + 312 is about 500 + 300 = 800. To check a take away, add your answer back. You must get the first number.

Draw: Draw a line from 400 to 500. Mark 487 and draw an arrow to 500. Write 500 + 300 = 800.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (699 KB). It opens on any phone.