Chapter 4 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

M2.2.4 Chapter 4

Column taking away

How do I take 178 away from 524 on paper?

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 3 digit numbers in columns, exchanging when you need to
  • Take away across a zero, as in 403 − 167

Here is a take away: 524 − 178. Write it in columns.

The bigger number goes on top. Line up the places. Start at the right, with the ones.

You cannot take 8 from 4. There are not enough ones.

So exchange. That means swap 1 ten for 10 ones. The ten comes from the tens column.

Draw it first

Take away 178 from 524

  1. A column sum of 524 take away 178 with empty boxes for the answer. Step 2 crosses out digits, writes the new numbers 11 and 4 above them, writes a small 1 by the 4 to make 14, and fills in the answer 346. Step 3 adds the labels ones: 14 − 8 = 6, tens: 11 − 7 = 4 and hundreds: 4 − 1 = 3.425871−

    1 Write 524. Write 178 under it. Line up the places.

  2. A column sum of 524 take away 178 with empty boxes for the answer. Step 2 crosses out digits, writes the new numbers 11 and 4 above them, writes a small 1 by the 4 to make 14, and fills in the answer 346. Step 3 adds the labels ones: 14 − 8 = 6, tens: 11 − 7 = 4 and hundreds: 4 − 1 = 3.425871−1114643

    2 Start at the right. If a top digit is too small, cross it out and exchange.

  3. A column sum of 524 take away 178 with empty boxes for the answer. Step 2 crosses out digits, writes the new numbers 11 and 4 above them, writes a small 1 by the 4 to make 14, and fills in the answer 346. Step 3 adds the labels ones: 14 − 8 = 6, tens: 11 − 7 = 4 and hundreds: 4 − 1 = 3.425871−1114643ones: 14 − 8 = 6tens: 11 − 7 = 4hundreds: 4 − 1 = 3

    3 Read the three small sums. They are the working.

Ones: the 4 became 14. 14 − 8 = 6.

Tens: the 2 gave away 1 ten. Only 1 ten is left. You cannot take 7 from 1.

Exchange again. Take 1 hundred. It is 10 tens. Now you have 11 tens. 11 − 7 = 4.

Hundreds: 5 gave away 1, so 4 are left. 4 − 1 = 3. The answer is 346.

Now check by adding back. Add the answer to the number you took away.

346 + 178 = 524. It matches the top number, so the answer is right.

Worked example

Take away 631 − 254 in columns.

  1. Ones: 1 − 4 cannot be done. Exchange 1 ten. 11 − 4 = 7.
  2. Tens: 3 gave 1 away, so 2 are left. 2 − 5 cannot be done. Exchange 1 hundred. 12 − 5 = 7.
  3. Hundreds: 6 gave 1 away, so 5 are left. 5 − 2 = 3.
  4. The answer is 377. Check: 377 + 254 = 631.

Answer: 377

A column sum of 631 take away 254. The 6 is crossed out and 5 is written above it. The 3 is crossed out and 12 is written above it. A small 1 sits by the 1 to make 11. The answer is 377.136452−1125773
The 3 tens gave 1 ten away, so 2 were left. Then 1 hundred made them 12. The 6 hundreds became 5.

A zero in the middle needs care. Look at 403 − 167. There are 0 tens to give.

A column sum of 403 take away 167. The 4 is crossed out and 3 is written above it. The 0 is crossed out and 9 is written above it. A small 1 sits by the 3 to make 13. The answer is 236.304761−193632
The 4 hundreds became 3. The 0 tens became 9. The 3 ones became 13.

Worked example

Take away 403 − 167 in columns.

  1. Ones: 3 − 7 cannot be done. There are 0 tens to give.
  2. Exchange 1 hundred for 10 tens. The hundreds are now 3. The tens are 10.
  3. Exchange 1 ten for 10 ones. The tens are now 9. The ones are 13.
  4. Ones: 13 − 7 = 6. Tens: 9 − 6 = 3. Hundreds: 3 − 1 = 2.
  5. The answer is 236. Check: 236 + 167 = 403.

Answer: 236

See it three ways

With things

Make 52 with 5 bundles of ten and 2 loose sticks. Take away 18. You need 8 loose sticks but have 2. Untie 1 bundle. Now you have 12 loose. Take 8. Then take 1 bundle.

As a picture

Base ten blocks. 52 is 5 rods and 2 single squares. A minus sign. 18 is 1 rod and 8 single squares. An equals sign. 34 is 3 rods and 4 single squares.52−18=34

In symbols

52 − 18 = 34. Ones: 2 − 8 cannot be done. Exchange 1 ten. 12 − 8 = 4. Tens: 4 − 1 = 3.

Try it

Exchange with bundles

You need: about 50 small sticks, beans or stones, string or grass to tie bundles, paper and a pencil.

  1. Draw two boxes on paper. Write tens over one and ones over the other.
  2. Make 42. Put 4 bundles of ten in the tens box. Put 2 loose in the ones box.
  3. Take away 17. You need to take 7 loose, but there are only 2.
  4. Untie 1 bundle. Put its 10 sticks in the ones box. It holds 12 now.
  5. Take 7 loose from the ones box. Take 1 bundle from the tens box.
  6. Count what is left in each box.

What you will see: You have 2 bundles and 5 loose. That is 25. The exchange moved 1 ten into the ones box.

If you cannot try it, this is what happens.
A column sum of 42 take away 17. The 4 is crossed out and 3 is written above it. A small 1 sits by the 2 to make 12. The answer is 25.2471−1352

A wrong idea many people have

403 − 167 = 364. In each column, take the smaller digit from the bigger one.

Why it feels right: Each little step is true. 7 − 3 = 4. 6 − 0 = 6. 4 − 1 = 3. And you never have to exchange.

The test that breaks it: Add back. 167 + 364 = 531. That is not 403. So 364 cannot be right.

What is true: Take the bottom digit from the top digit, never the other way. Exchange when the top is too small. 403 − 167 = 236.

Key words

exchange
swap 1 ten for 10 ones, or 1 hundred for 10 tens.
adding back
check a take away. Add the answer to the number you took away.

Back to your guess

The answer is B: When the top is too small, swap 1 ten for 10 ones.

You cannot take 8 ones from 4 ones. So swap 1 ten for 10 ones. That is an exchange. Then 14 − 8 = 6. The rest follows.

Check yourself

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

  1. Take away 685 − 243 in columns.

    See the answer

    442

    Ones: 5 − 3 = 2. Tens: 8 − 4 = 4. Hundreds: 6 − 2 = 4. Every top digit is big enough, so there is no exchange.

  2. Take away 732 − 456 in columns.

    See the answer

    276

    Ones: 2 − 6 cannot be done. Exchange. 12 − 6 = 6. Tens: 2 left. Exchange. 12 − 5 = 7. Hundreds: 6 left. 6 − 4 = 2. Check: 276 + 456 = 732.

  3. Take away 603 − 248 in columns.

    See the answer

    355

    There are 0 tens. Exchange 1 hundred for 10 tens. Then exchange 1 ten for 10 ones. Ones: 13 − 8 = 5. Tens: 9 − 4 = 5. Hundreds: 5 − 2 = 3. Check: 355 + 248 = 603.

  4. Lucía takes away 305 − 148. What is the answer?

    Tap your answer.

    1. Yes, that is right. Exchange 1 hundred, then 1 ten. Ones: 15 − 8 = 7. Tens: 9 − 4 = 5. Hundreds: 2 − 1 = 1. Check: 157 + 148 = 305.

    2. Not this one. You took the smaller digit from the bigger in each column. Add back: 148 + 243 = 391, not 305. Try another.

    3. Not this one. After the exchange the tens column is 9, not 10. 9 − 4 = 5, not 6. Try another.

    See the answer

    157

    Exchange 1 hundred, then 1 ten. Ones: 15 − 8 = 7. Tens: 9 − 4 = 5. Hundreds: 2 − 1 = 1. Check: 157 + 148 = 305.

    • If you chose B: You took the smaller digit from the bigger in each column. Add back: 148 + 243 = 391, not 305.
    • If you chose C: After the exchange the tens column is 9, not 10. 9 − 4 = 5, not 6.

  5. A bag holds 500 beans. 237 beans are used. How many are left?

    See the answer

    263

    Exchange 1 hundred for 10 tens, then 1 ten for 10 ones. Ones: 10 − 7 = 3. Tens: 9 − 3 = 6. Hundreds: 4 − 2 = 2. Check: 263 + 237 = 500.

  6. From an earlier chapter

    Add 276 + 348 in columns.

    See the answer

    624

    Ones: 6 + 8 = 14. Write 4, carry 1. Tens: 7 + 4 + 1 = 12. Write 2, carry 1. Hundreds: 2 + 3 + 1 = 6.

  7. Mei takes away 463 − 287. In the ones column, 3 − 7 cannot be done. She exchanges 1 ten. What does she work out now?

    Tap your answer.

    1. Not this one. That turns the sum round. The 7 must be taken away from the 3. Exchange 1 ten to get more ones. Try another.

    2. Not this one. The 3 ones are still there. 10 ones and 3 ones make 13. So work out 13 − 7. Try another.

    3. Yes, that is right. The ten gives 10 more ones. 10 + 3 = 13 ones. Then 13 − 7 = 6.

    See the answer

    13 − 7

    The ten gives 10 more ones. 10 + 3 = 13 ones. Then 13 − 7 = 6.

    • If you chose A: That turns the sum round. The 7 must be taken away from the 3. Exchange 1 ten to get more ones.
    • If you chose B: The 3 ones are still there. 10 ones and 3 ones make 13. So work out 13 − 7.

  8. Tariq takes away 652 − 187 in columns. What is the answer?

    A column sum of 652 take away 187 with empty boxes for the answer.256781−
    Tap your answer.

    1. Not this one. You took the smaller digit from the bigger one in each column. Add back: 187 + 535 = 722, not 652. Try another.

    2. Yes, that is right. Ones: 12 − 7 = 5. Tens: 4 are left. Exchange 1 hundred. 14 − 8 = 6. Hundreds: 5 − 1 = 4. Check: 465 + 187 = 652.

    3. Not this one. The first exchange took 1 ten from the tens column. Only 4 tens are left. 14 − 8 = 6, not 7. Try another.

    See the answer

    465

    Ones: 12 − 7 = 5. Tens: 4 are left. Exchange 1 hundred. 14 − 8 = 6. Hundreds: 5 − 1 = 4. Check: 465 + 187 = 652.

    • If you chose A: You took the smaller digit from the bigger one in each column. Add back: 187 + 535 = 722, not 652.
    • If you chose C: The first exchange took 1 ten from the tens column. Only 4 tens are left. 14 − 8 = 6, not 7.

  9. Sipho takes away 800 − 345 in columns. What is the answer?

    A column sum of 800 take away 345 with empty boxes for the answer.008543−
    Tap your answer.

    1. Yes, that is right. Exchange 1 hundred for 10 tens. Then 1 ten for 10 ones. Ones: 10 − 5 = 5. Tens: 9 − 4 = 5. Hundreds: 7 − 3 = 4. Check: 455 + 345 = 800.

    2. Not this one. You took the smaller digit from the bigger one in each column. Add back: 345 + 545 = 890, not 800. Try another.

    3. Not this one. After both exchanges the tens column is 9, not 10. One ten went to the ones. 9 − 4 = 5. Try another.

    See the answer

    455

    Exchange 1 hundred for 10 tens. Then 1 ten for 10 ones. Ones: 10 − 5 = 5. Tens: 9 − 4 = 5. Hundreds: 7 − 3 = 4. Check: 455 + 345 = 800.

    • If you chose B: You took the smaller digit from the bigger one in each column. Add back: 345 + 545 = 890, not 800.
    • If you chose C: After both exchanges the tens column is 9, not 10. One ten went to the ones. 9 − 4 = 5.

  10. Amina takes away 503 − 276 and gets 227. How can she check?

    Tap your answer.

    1. Not this one. 227 and 276 are both smaller than 503. Taking one from the other cannot make 503. Add them instead. Try another.

    2. Not this one. 227 + 503 makes 730. Add the answer to the number you took away. That is 276. Try another.

    3. Yes, that is right. Add the answer to the number you took away. 227 + 276 = 503. It matches the top number.

    See the answer

    Add 227 + 276. It should make 503.

    Add the answer to the number you took away. 227 + 276 = 503. It matches the top number.

    • If you chose A: 227 and 276 are both smaller than 503. Taking one from the other cannot make 503. Add them instead.
    • If you chose B: 227 + 503 makes 730. Add the answer to the number you took away. That is 276.

Keep it

Copy this into your exercise book.

Bigger number on top. Take away from the right. A top digit may be too small. Exchange 1 ten for 10 ones. Check by adding back. 524 − 178 = 346.

Draw: The sum 524 − 178 in columns. Cross out digits and write the new numbers above them.

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

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