Chapter 2 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

M2.2.2 Chapter 2

Taking away in your head

What is 302 − 298? Do I have to write it down?

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

  • Take away by counting up from the smaller number
  • Find the difference between 302 and 298 without writing

Ravi has 302 beans. Mei has 298 beans. How many more beans has Ravi?

That is 302 − 298. Do not write it in columns. Look at the numbers first.

The difference between two numbers is how far apart they are.

To find it, count up from the smaller number. Hop up to the bigger one.

Draw it first

Count up from the smaller number

  1. A number line from 297 to 303. Dots mark 298 and 302. Step 2 adds a hop labelled +2 from 298 to 300. Step 3 adds a hop labelled +2 from 300 to 302, and the words 2 + 2 = 4.297298299300301302303

    1 Put 298 and 302 on a number line.

  2. A number line from 297 to 303. Dots mark 298 and 302. Step 2 adds a hop labelled +2 from 298 to 300. Step 3 adds a hop labelled +2 from 300 to 302, and the words 2 + 2 = 4.297298299300301302303+2

    2 Hop up to 300. That is 2.

  3. A number line from 297 to 303. Dots mark 298 and 302. Step 2 adds a hop labelled +2 from 298 to 300. Step 3 adds a hop labelled +2 from 300 to 302, and the words 2 + 2 = 4.297298299300301302303+2+22 + 2 = 4

    3 Hop on to 302. That is 2 more. 2 + 2 = 4.

The difference is 4. Ravi has 4 more beans than Mei.

You can also count back from 302 to 298. It is the same 4 steps.

A number line from 297 to 303. Dots mark 298 and 302. One hop labelled −4 goes back from 302 to 298.297298299300301302303−4
Count up or count back. The gap between 298 and 302 is 4.

When two numbers are close, count up. It is quicker than a column sum.

Worked example

Find the difference between 500 and 468.

  1. Start at the smaller number, 468.
  2. Hop up to 470. That is 2.
  3. Hop on to 500. That is 30.
  4. Add the hops. 2 + 30 = 32.

Answer: 32

Two number lines. The top line runs from 466 to 472, with the numbers 466, 467, 468, 469, 470, 471 and 472 under it. A short hop labelled +2 goes from 468 to 470. The bottom line runs from 470 to 500. A long hop labelled +30 goes from 470 to 500. The numbers 470 and 500 are written under it.466467468469470471472+2+30470500
468 to 470 is 2. 470 to 500 is 30. 2 + 30 = 32.

See it three ways

With things

Make a line of 10 beans. Make a line of 7 beans under it. Count the beans that stick out. That is the difference.

As a picture

Two rows of dots. The top row has 10 dots, labelled 10. The last 3 dots are filled in. The bottom row has 7 dots, labelled 7. Under the filled dots is the label difference: 3.107difference: 3

In symbols

10 − 7 = 3. Check: 7 + 3 = 10. The difference is 3.

Try it

Count up with stones

You need: about 15 small stones, beans or sticks, a pencil and paper.

  1. Make a line of 7 stones.
  2. Add stones one at a time. Say 8, 9, 10, 11 as you add each one.
  3. Stop at 11. Count the stones you added.
  4. Write 11 − 7 = and the number you counted.

What you will see: You added 4 stones. The difference between 7 and 11 is 4.

If you cannot try it, this is what happens.
A row of 11 dots. The first 7 are open and the last 4 are filled in. The label says 11 − 7 = 4.11 − 7 = 4

A wrong idea many people have

302 − 298 = 196. In each column, take the smaller digit from the bigger one.

Why it feels right: Each little step is true. 3 − 2 = 1. 9 − 0 = 9. 8 − 2 = 6. And you never have to cross anything out.

The test that breaks it: 302 and 298 are only a few apart. 196 is far too big. Count up from 298: 299, 300, 301, 302. That is 4 steps. Add back too: 298 + 196 = 494, not 302.

What is true: A take away works on the whole numbers, not on separate digits. 302 − 298 = 4.

Key words

difference
how far apart two numbers are.
count up
start at the smaller number and hop to the bigger one.

Back to your guess

The answer is B: It is a small answer. The two numbers are close together.

302 and 298 are only a few apart. Count up from 298. 2 steps reach 300. 2 more reach 302. The difference is 4.

Check yourself

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

  1. Find the difference between 51 and 47. Count up from 47.

    See the answer

    4

    47 to 50 is 3. 50 to 51 is 1. 3 + 1 = 4.

  2. Work out 403 − 398 in your head.

    See the answer

    5

    398 to 400 is 2. 400 to 403 is 3. 2 + 3 = 5.

  3. Tariq says 405 − 397 = 192. How can you tell he is wrong?

    Tap your answer.

    1. Yes, that is right. 397 to 400 is 3. 400 to 405 is 5. 3 + 5 = 8. And 397 + 8 = 405.

    2. Not this one. Add back. 397 + 192 = 589, not 405. Taking away is not done column by column. Try another.

    3. Not this one. Adding gives 802. Count up from 397 to find the gap. Try another.

    See the answer

    The numbers are close, so the answer is small. It is 8.

    397 to 400 is 3. 400 to 405 is 5. 3 + 5 = 8. And 397 + 8 = 405.

    • If you chose B: Add back. 397 + 192 = 589, not 405. Taking away is not done column by column.
    • If you chose C: Adding gives 802. Count up from 397 to find the gap.

  4. Maya has a rope of 126 cm. Joseph has a rope of 118 cm. How much longer is Maya's rope?

    See the answer

    8 cm

    118 to 120 is 2. 120 to 126 is 6. 2 + 6 = 8. So Maya's rope is 8 cm longer.

  5. From an earlier chapter

    Add 57 + 38. Add the tens, then the ones.

    See the answer

    95

    Tens: 50 + 30 = 80. Ones: 7 + 8 = 15. 80 + 15 = 95.

  6. Sipho finds the difference between 62 and 57 by counting up. Where does he start?

    A number line from 56 to 63. Dots mark 57 and 62. The numbers 56 to 63 are written under the line.5657585960616263
    Tap your answer.

    1. Not this one. From 62 you would count back down. To count up, start at 57 and hop up to 62. Try another.

    2. Yes, that is right. You hop up, so you start low. Start at 57 and hop up to 62. 57 to 60 is 3. 60 to 62 is 2. The difference is 5.

    3. Not this one. 60 is a stepping stone on the way, not the start. Start at 57. 57 to 60 is 3. 60 to 62 is 2. Try another.

    See the answer

    At 57, the smaller number

    You hop up, so you start low. Start at 57 and hop up to 62. 57 to 60 is 3. 60 to 62 is 2. The difference is 5.

    • If you chose A: From 62 you would count back down. To count up, start at 57 and hop up to 62.
    • If you chose C: 60 is a stepping stone on the way, not the start. Start at 57. 57 to 60 is 3. 60 to 62 is 2.

  7. Amina counts up from 296 to 301. What is the difference between them?

    A number line from 295 to 302. Dots mark 296 and 301. The numbers 295 to 302 are written under the line.295296297298299300301302
    Tap your answer.

    1. Not this one. 296 to 300 is 4. But you must reach 301. One more hop makes 4 + 1 = 5. Try another.

    2. Not this one. You counted the number you started on. Count the hops, not the numbers. 4 + 1 = 5. Try another.

    3. Yes, that is right. 296 to 300 is 4. 300 to 301 is 1. 4 + 1 = 5.

    See the answer

    5

    296 to 300 is 4. 300 to 301 is 1. 4 + 1 = 5.

    • If you chose A: 296 to 300 is 4. But you must reach 301. One more hop makes 4 + 1 = 5.
    • If you chose B: You counted the number you started on. Count the hops, not the numbers. 4 + 1 = 5.

  8. What is 604 − 598?

    Tap your answer.

    1. Yes, that is right. 598 to 600 is 2. 600 to 604 is 4. 2 + 4 = 6.

    2. Not this one. That takes the smaller digit from the bigger one in each column. 604 and 598 are close, so the answer is small. Count up from 598. Try another.

    3. Not this one. 598 to 600 is 2. But you must go on to 604. 2 + 4 = 6. Try another.

    See the answer

    6

    598 to 600 is 2. 600 to 604 is 4. 2 + 4 = 6.

    • If you chose B: That takes the smaller digit from the bigger one in each column. 604 and 598 are close, so the answer is small. Count up from 598.
    • If you chose C: 598 to 600 is 2. But you must go on to 604. 2 + 4 = 6.

  9. Mei has 402 stones. Kofi has 396 stones. How many more stones has Mei?

    Tap your answer.

    1. Not this one. 798 is 402 + 396. How many more asks for the gap. Count up from 396 to 402. Try another.

    2. Yes, that is right. 396 to 400 is 4. 400 to 402 is 2. 4 + 2 = 6. Mei has 6 more stones.

    3. Not this one. You counted the 396 as well. Count the hops. 396 to 400 is 4. 400 to 402 is 2. 4 + 2 = 6. Try another.

    See the answer

    6

    396 to 400 is 4. 400 to 402 is 2. 4 + 2 = 6. Mei has 6 more stones.

    • If you chose A: 798 is 402 + 396. How many more asks for the gap. Count up from 396 to 402.
    • If you chose C: You counted the 396 as well. Count the hops. 396 to 400 is 4. 400 to 402 is 2. 4 + 2 = 6.

Keep it

Copy this into your exercise book.

The difference is how far apart two numbers are. Count up from the smaller number. 298 to 300 is 2. 300 to 302 is 2. The difference is 4.

Draw: A number line. Mark 298 and 302. Draw a hop of +2 and another hop of +2.

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.