Your guess
Pick one before you read on. It is fine to be wrong.
Your guess By the end you can
Add two 2 digit numbers by adding the tens, then the ones Work out 46 + 29 as 46 + 30 − 1 You have 46 beans. A friend gives you 29 more. How many beans is that?
Do not try to add it all at once. Split each number into tens and ones. That means break it into parts.
46 splits into 40 and 6. 29 splits into 20 and 9.
Draw it first
Add the tens, then the ones Base ten blocks for 46 and 29 with a plus sign between them. 46 is 4 rods and 6 single squares. 29 is 2 rods and 9 single squares. Step 2 adds 40 + 20 = 60. Step 3 adds 6 + 9 = 15. Step 4 adds 60 + 15 = 75. 46 + 29 1 Show 46 and 29 as tens and ones.
Base ten blocks for 46 and 29 with a plus sign between them. 46 is 4 rods and 6 single squares. 29 is 2 rods and 9 single squares. Step 2 adds 40 + 20 = 60. Step 3 adds 6 + 9 = 15. Step 4 adds 60 + 15 = 75. 46 + 29 40 + 20 = 60 2 Add the tens. 40 + 20 = 60.
Base ten blocks for 46 and 29 with a plus sign between them. 46 is 4 rods and 6 single squares. 29 is 2 rods and 9 single squares. Step 2 adds 40 + 20 = 60. Step 3 adds 6 + 9 = 15. Step 4 adds 60 + 15 = 75. 46 + 29 40 + 20 = 60 6 + 9 = 15 3 Add the ones. 6 + 9 = 15.
Base ten blocks for 46 and 29 with a plus sign between them. 46 is 4 rods and 6 single squares. 29 is 2 rods and 9 single squares. Step 2 adds 40 + 20 = 60. Step 3 adds 6 + 9 = 15. Step 4 adds 60 + 15 = 75. 46 + 29 40 + 20 = 60 6 + 9 = 15 60 + 15 = 75 4 Join the answers. 60 + 15 = 75.
The ones made 15. That is 1 ten and 5 ones. So 60 + 15 is 75, not 6 015 . The answer to an add sum is the total .
Worked example
Work out 58 + 34 by adding the tens, then the ones.
Split. 58 is 50 and 8. 34 is 30 and 4. Add the tens. 50 + 30 = 80. Add the ones. 8 + 4 = 12. Join them. 80 + 12 = 92. Answer: 92
Base ten blocks: 1 rod and 2 single squares. The label says 12 ones make 1 ten and 2 ones. 12 ones make 1 ten and 2 ones 8 + 4 = 12. Ten of the ones make a new ten. 2 ones are left. A quicker way works when a number is close to a ten. 29 is close to 30.
Add 30 to 46. That gives 76. But 29 is 1 less than 30. You added 1 too many. Take 1 back.
Two number lines. The top line runs from 46 to 76. A long hop labelled +30 goes from 46 to 76. Under it the label says zoom in. The bottom line runs from 74 to 77, with the numbers 74, 75, 76 and 77 under it. A short hop labelled −1 goes back from 76 to 75. +30 46 76 zoom in 74 75 76 77 −1 46 + 29 is the same as 46 + 30 − 1. Both give 75. Worked example
Work out 53 + 38 by adding 40, then taking 2 back.
38 is close to 40. 38 is 2 less than 40. Add 40. 53 + 40 = 93. You added 2 too many. Take 2 back. 93 − 2 = 91. Answer: 91
Two number lines. The top line runs from 53 to 93. A long hop labelled +40 goes from 53 to 93. Under it the label says zoom in. The bottom line runs from 90 to 94, with the numbers 90, 91, 92, 93 and 94 under it. A short hop labelled −2 goes back from 93 to 91. +40 53 93 zoom in 90 91 92 93 94 −2 53 + 38 is the same as 53 + 40 − 2. Both give 91. See it three ways
With things
Make 38 with 3 bundles of ten sticks and 8 loose sticks. Make 25 with 2 bundles and 5 loose. Join the bundles: 5 bundles. Join the loose sticks: 13. Tie 10 into a bundle. You have 6 bundles and 3 loose.
As a picture
Base ten blocks for 38 and 25 with a plus sign between them. 38 is 3 rods and 8 single squares. 25 is 2 rods and 5 single squares. 38 + 25 In symbols
30 + 20 = 50 and 8 + 5 = 13. 50 + 13 = 63. So 38 + 25 = 63.
Try it
Add with bundles You need: about 80 small sticks, beans or stones, string or grass to tie bundles, a pencil and paper.
Make 46. Tie 4 bundles of 10. Keep 6 loose. Make 29. Tie 2 bundles of 10. Keep 9 loose. Push the bundles together. Count them. Write 6 tens. Push the loose ones together. Count 15. Tie 10 of them into a new bundle. Count again. Write the bundles and the loose ones. What you will see: You get 7 bundles and 5 loose. The ones made a new ten. That is 75.
If you cannot try it, this is what happens. Base ten blocks: 7 rods and 5 single squares. The label says 7 tens and 5 ones make 75. 7 tens and 5 ones make 75 A wrong idea many people have
For 46 + 29, add 30, then add 1 more. That gives 77.
Why it feels right: You changed 29 to 30. A change feels like it needs a fix. Adding 1 feels like the fix.
The test that breaks it: Try small numbers. Put down 5 beans. Add 9 more, one at a time. You count to 14. Now 5 + 10 is 15. So 5 + 9 is 1 less than 15. Check 46 + 29 the other way too. 60 + 15 = 75, not 77.
What is true: When you add a bigger number, take the extra back. 46 + 30 = 76. Then 76 − 1 = 75.
Key words
split break a number into parts, such as 46 into 40 and 6. total the number you get when you add numbers together. Back to your guess
The answer is B: Yes. Add the tens, then add the ones, then join them.
You can do it in your head. Split each number into tens and ones. Add the tens. Add the ones. Join the two answers. 46 + 29 = 75.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Add 34 + 27. Add the tens, then the ones.
See the answer 61
Tens: 30 + 20 = 50. Ones: 4 + 7 = 11. Then 50 + 11 = 61.
Ana adds 37 + 48. She gets 70 for the tens and 15 for the ones. What is the total?
Tap your answer. A 85 Yes, that is right. 70 + 15 = 85. Add the 15 ones to the 70. Do not write them beside it.
B 7 015 Not this one. You wrote 70 and 15 side by side. Join them with a sum: 70 + 15. Try another.
C 75 Not this one. You kept the 5 and lost the ten in 15. 15 is 1 ten and 5 ones. 70 + 15 = 85. Try another.
See the answer 85
70 + 15 = 85. Add the 15 ones to the 70. Do not write them beside it.
If you chose B: You wrote 70 and 15 side by side. Join them with a sum: 70 + 15.If you chose C: You kept the 5 and lost the ten in 15. 15 is 1 ten and 5 ones. 70 + 15 = 85.Add 64 + 39. Add 40 first, then take 1 back.
See the answer 103
64 + 40 = 104. 39 is 1 less than 40, so take 1 back. 104 − 1 = 103.
Kofi says 25 + 18 = 25 + 20 + 2 = 47. Is he right?
Tap your answer. A Yes. You add 2 to make up for it. Not this one. Test with 5 + 9. 5 + 10 = 15, but 5 + 9 = 14. The extra is taken back. Try another.
B No. 18 is 2 less than 20, so take 2 back. The total is 43. Yes, that is right. 25 + 20 = 45. 18 is 2 less than 20. You added 2 too many. 45 − 2 = 43.
C No. 25 + 20 is 45, so the total is 45. Not this one. You only want to add 18. That is 2 less than 20, so take 2 back. 45 − 2 = 43. Try another.
See the answer No. 18 is 2 less than 20, so take 2 back. The total is 43.
25 + 20 = 45. 18 is 2 less than 20. You added 2 too many. 45 − 2 = 43.
If you chose A: Test with 5 + 9. 5 + 10 = 15, but 5 + 9 = 14. The extra is taken back.If you chose C: You only want to add 18. That is 2 less than 20, so take 2 back. 45 − 2 = 43.A farm has 48 hens and 35 ducks. How many birds are there?
See the answer 83
Tens: 40 + 30 = 70. Ones: 8 + 5 = 13. 70 + 13 = 83.
Hana adds 64 + 27 by splitting. Which two sums does she do?
Tap your answer. A 6 + 2 and 4 + 7 Not this one. The 6 and the 2 sit in the tens place. They stand for 60 and 20. So add 60 + 20, not 6 + 2. Try another.
B 60 + 4 and 20 + 7 Not this one. Those only split 64 and 27 again. Add the two tens, 60 + 20. Then add the two ones, 4 + 7. Try another.
C 60 + 20 and 4 + 7 Yes, that is right. 64 is 60 and 4. 27 is 20 and 7. Add the tens, 60 + 20. Add the ones, 4 + 7.
See the answer 60 + 20 and 4 + 7
64 is 60 and 4. 27 is 20 and 7. Add the tens, 60 + 20. Add the ones, 4 + 7.
If you chose A: The 6 and the 2 sit in the tens place. They stand for 60 and 20. So add 60 + 20, not 6 + 2.If you chose B: Those only split 64 and 27 again. Add the two tens, 60 + 20. Then add the two ones, 4 + 7.Look at 27 + 35. The 7 ones and the 5 ones are put together. What do they make?
Base ten blocks for 27 and 35 with a plus sign between them. 27 is 2 rods and 7 single squares. 35 is 3 rods and 5 single squares. 27 + 35 Tap your answer. A 1 ten and 2 ones Yes, that is right. 7 + 5 = 12 ones. 10 of them make 1 ten, and 2 ones are left.
B 2 tens and 1 one Not this one. 7 + 5 = 12. You swapped the digits. 12 is 1 ten and 2 ones. 2 tens and 1 one is 21. Try another.
C 12 tens Not this one. 12 tens would be 120. You have 12 single ones, and 10 ones make 1 ten. Try another.
See the answer 1 ten and 2 ones
7 + 5 = 12 ones. 10 of them make 1 ten, and 2 ones are left.
If you chose B: 7 + 5 = 12. You swapped the digits. 12 is 1 ten and 2 ones. 2 tens and 1 one is 21.If you chose C: 12 tens would be 120. You have 12 single ones, and 10 ones make 1 ten.Priya works out 37 + 49 by adding 50 first. The line shows her hop. What is 37 + 49?
A number line from 37 to 87. A long hop labelled +50 goes from 37 to 87. The numbers 37 and 87 are written under the line. +50 37 87 Tap your answer. A 88 Not this one. You added the extra 1. 49 is 1 less than 50. The hop of 50 is 1 too long. Take 1 back. Try another.
B 87 Not this one. 87 is 37 + 50. But 49 is 1 less than 50, so take 1 back. Try another.
C 86 Yes, that is right. 37 + 50 = 87. 49 is 1 less than 50, so you added 1 too many. 87 − 1 = 86.
See the answer 86
37 + 50 = 87. 49 is 1 less than 50, so you added 1 too many. 87 − 1 = 86.
If you chose A: You added the extra 1. 49 is 1 less than 50. The hop of 50 is 1 too long. Take 1 back.If you chose B: 87 is 37 + 50. But 49 is 1 less than 50, so take 1 back.Keep it
Copy this into your exercise book.
Split each number into tens and ones. Add the tens. Add the ones. Join them. Or add a near ten and take the extra back. 46 + 29 = 46 + 30 − 1 = 75.
Draw: A number line. Draw a long hop of +30. Draw a short hop back of −1. Write 46, 75 and 76 under it.
Your guess
Pick one before you read on. It is fine to be wrong.
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 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. 297 298 299 300 301 302 303
1 Put 298 and 302 on a number line.
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. 297 298 299 300 301 302 303 +2 2 Hop up to 300. That is 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. 297 298 299 300 301 302 303 +2 +2 2 + 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. 297 298 299 300 301 302 303 −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.
Start at the smaller number, 468. Hop up to 470. That is 2. Hop on to 500. That is 30. 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. 466 467 468 469 470 471 472 +2 +30 470 500 468 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. 10 7 difference: 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.
Make a line of 7 stones. Add stones one at a time. Say 8, 9, 10, 11 as you add each one. Stop at 11. Count the stones you added. 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.
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.
Work out 403 − 398 in your head.
See the answer 5
398 to 400 is 2. 400 to 403 is 3. 2 + 3 = 5.
Tariq says 405 − 397 = 192. How can you tell he is wrong?
Tap your answer. A The numbers are close, so the answer is small. It is 8. Yes, that is right. 397 to 400 is 3. 400 to 405 is 5. 3 + 5 = 8. And 397 + 8 = 405.
B 192 is right. Each column was taken away. Not this one. Add back. 397 + 192 = 589, not 405. Taking away is not done column by column. Try another.
C He should have added 405 and 397. 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.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.
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.
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. 56 57 58 59 60 61 62 63
Tap your answer. A At 62, the bigger number Not this one. From 62 you would count back down. To count up, start at 57 and hop up to 62. Try another.
B At 57, the smaller number 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.
C At 60, the next ten 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.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. 295 296 297 298 299 300 301 302
Tap your answer. A 4 Not this one. 296 to 300 is 4. But you must reach 301. One more hop makes 4 + 1 = 5. Try another.
B 6 Not this one. You counted the number you started on. Count the hops, not the numbers. 4 + 1 = 5. Try another.
C 5 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.What is 604 − 598?
Tap your answer. A 6 Yes, that is right. 598 to 600 is 2. 600 to 604 is 4. 2 + 4 = 6.
B 194 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.
C 2 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.Mei has 402 stones. Kofi has 396 stones. How many more stones has Mei?
Tap your answer. A 798 Not this one. 798 is 402 + 396. How many more asks for the gap. Count up from 396 to 402. Try another.
B 6 Yes, that is right. 396 to 400 is 4. 400 to 402 is 2. 4 + 2 = 6. Mei has 6 more stones.
C 7 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.
Your guess
Pick one before you read on. It is fine to be wrong.
Your guess By the end you can
Line up digits by place value before you add Add 3 digit numbers in columns, carrying when you need to Some sums are too big for your head. Try 358 + 276.
Write the numbers in a column . A column is a line of digits, one above the other.
Each column is one place. Ones go under ones. Tens go under tens. Hundreds go under hundreds.
A place value chart for 358 with three columns: hundreds, tens and ones. The digits are 3, 5 and 8. hundreds tens ones 3 5 8 In 358, the 3 is hundreds. The 5 is tens. The 8 is ones. Draw it first
Add 358 + 276 in columns A column sum of 358 plus 276 with empty boxes for the answer. Step 2 fills in the answer 634 and writes small carried 1s above the tens and hundreds. Step 3 adds the labels ones: 8 + 6 = 14, tens: 5 + 7 + 1 = 13 and hundreds: 3 + 2 + 1 = 6. 8 5 3 6 7 2 + 1 Write 358. Write 276 under it. Line up the places.
A column sum of 358 plus 276 with empty boxes for the answer. Step 2 fills in the answer 634 and writes small carried 1s above the tens and hundreds. Step 3 adds the labels ones: 8 + 6 = 14, tens: 5 + 7 + 1 = 13 and hundreds: 3 + 2 + 1 = 6. 8 5 3 6 7 2 + 1 1 4 3 6
2 Start at the right. Write the answer, one column at a time. Carry 1 when a column makes 10 or more.
A column sum of 358 plus 276 with empty boxes for the answer. Step 2 fills in the answer 634 and writes small carried 1s above the tens and hundreds. Step 3 adds the labels ones: 8 + 6 = 14, tens: 5 + 7 + 1 = 13 and hundreds: 3 + 2 + 1 = 6. 8 5 3 6 7 2 + 1 1 4 3 6 ones: 8 + 6 = 14 tens: 5 + 7 + 1 = 13 hundreds: 3 + 2 + 1 = 6
3 Read the three small sums. They are the working.
Start at the right, with the ones. 8 + 6 = 14. That is 1 ten and 4 ones.
Write the 4. Carry the 1 ten. That means write it small above the tens column.
10 ones make 1 ten. So the carried 1 is a ten, not a one.
Now the tens. 5 + 7 + 1 = 13. Write 3 and carry 1 hundred.
Last, the hundreds. 3 + 2 + 1 = 6. The answer is 634.
A column sum of 352 plus 41. The 4 of 41 is under the 5 of 352 and the 1 is under the 2. The answer is 393. 2 5 3 1 4 + 3 9 3
41 has no hundreds. Its 4 sits under the 5 of 352. Worked example
Add 347 + 285 in columns.
Ones. 7 + 5 = 12. Write 2. Carry 1 ten. Tens. 4 + 8 + 1 = 13. Write 3. Carry 1 hundred. Hundreds. 3 + 2 + 1 = 6. Write 6. The answer is 632. Answer: 632
A column sum of 347 plus 285 with small carried 1s above the tens and hundreds. The answer is 632. 7 4 3 5 8 2 + 1 1 2 3 6
Both the ones and the tens carry. The answer is 632. See it three ways
With things
Make 47 with 4 bundles of ten and 7 loose sticks. Make 35 with 3 bundles and 5 loose. Join the loose sticks: 12. Tie 10 into a bundle. Join the bundles: 8 bundles and 2 loose.
As a picture
Base ten blocks for 47 and 35 with a plus sign between them. 47 is 4 rods and 7 single squares. 35 is 3 rods and 5 single squares. 47 + 35 In symbols
47 + 35. Ones: 7 + 5 = 12. Write 2, carry 1. Tens: 4 + 3 + 1 = 8. The answer is 82.
Try it
Carry with bundles You need: about 80 small sticks, beans or stones, string or grass to tie bundles, paper and a pencil.
Draw two boxes on paper. Write tens over one and ones over the other. Make 38. Put 3 bundles of ten in the tens box. Put 8 loose in the ones box. Add 27. Put 2 more bundles in the tens box. Put 7 loose in the ones box. Count the ones box. It holds 15. Tie 10 of them into a bundle. Move the new bundle to the tens box. That is the carry. Count both boxes. What you will see: The tens box has 6 bundles. The ones box has 5 loose. That is 65.
If you cannot try it, this is what happens. A column sum of 38 plus 27 with a small carried 1 above the tens. The answer is 65. 8 3 7 2 + 1 5 6
A wrong idea many people have
Line numbers up on the left. 352 + 41 means 352 + 410 = 762.
Why it feels right: We read and write words from the left. So lining up from the left feels natural.
The test that breaks it: 41 is 4 tens and 1 one. It has no hundreds. Add it to 352 and you get 393. 762 is more than twice 352. That is far too big.
What is true: Line up by place, from the right. Ones go under ones. Tens go under tens. 352 + 41 = 393.
Key words
column a line of digits, one above the other, in one place. carry move a new ten or hundred into the next column. Back to your guess
The answer is B: So ones add to ones, tens to tens, hundreds to hundreds.
Add each digit to a digit in the same place. Ones go with ones. Tens go with tens. Hundreds go with hundreds. The columns keep each place in its own line.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Add 245 + 132 in columns.
See the answer 377
Ones: 5 + 2 = 7. Tens: 4 + 3 = 7. Hundreds: 2 + 1 = 3. No column makes 10, so nothing is carried.
Add 467 + 285 in columns.
See the answer 752
Ones: 7 + 5 = 12. Write 2, carry 1. Tens: 6 + 8 + 1 = 15. Write 5, carry 1. Hundreds: 4 + 2 + 1 = 7.
Ravi adds 36 + 48 in columns. What is the total?
Tap your answer. A 84 Yes, that is right. Ones: 6 + 8 = 14. Write 4, carry 1. Tens: 3 + 4 + 1 = 8. The answer is 84.
B 714 Not this one. You wrote the whole 14 under the ones. Write 4 and carry the 1 ten. Try another.
C 74 Not this one. You forgot the carried ten. Tens: 3 + 4 + 1 = 8. Try another.
See the answer 84
Ones: 6 + 8 = 14. Write 4, carry 1. Tens: 3 + 4 + 1 = 8. The answer is 84.
If you chose B: You wrote the whole 14 under the ones. Write 4 and carry the 1 ten.If you chose C: You forgot the carried ten. Tens: 3 + 4 + 1 = 8.Amina adds 126 + 8. She puts the 8 under the 1 of 126. What is wrong?
Tap your answer. A 8 is only ones. It goes under the 6. Yes, that is right. The 6 is in the ones column. Ones go under ones. Then 126 + 8 = 134.
B Nothing is wrong. The 8 can go anywhere. Not this one. Under the 1, the 8 counts as 800. 126 + 800 = 926. But 126 + 8 = 134. Try another.
C The 8 should go under the 2. Not this one. The 2 is in the tens column. 8 is 8 ones, so it goes with the 6. Try another.
See the answer 8 is only ones. It goes under the 6.
The 6 is in the ones column. Ones go under ones. Then 126 + 8 = 134.
If you chose B: Under the 1, the 8 counts as 800. 126 + 800 = 926. But 126 + 8 = 134.If you chose C: The 2 is in the tens column. 8 is 8 ones, so it goes with the 6.A school has 258 girls and 267 boys. How many children are there?
See the answer 525
Ones: 8 + 7 = 15. Write 5, carry 1. Tens: 5 + 6 + 1 = 12. Write 2, carry 1. Hundreds: 2 + 2 + 1 = 5.
From an earlier chapter
Find the difference between 502 and 498. Count up.
See the answer 4
498 to 500 is 2. 500 to 502 is 2. 2 + 2 = 4.
Priya adds 352 + 41 in columns. Which two digits are in the tens column?
Tap your answer. A 3 and 4 Not this one. That lines the numbers up on the left. The 3 of 352 is in the hundreds. The tens column holds 5 and 4. Try another.
B 5 and 4 Yes, that is right. 352 is 3 hundreds, 5 tens and 2 ones. 41 is 4 tens and 1 one. So the tens column holds 5 and 4.
C 2 and 1 Not this one. The 2 and the 1 are the ones, the last digits. The tens come next, to the left of them. Try another.
See the answer 5 and 4
352 is 3 hundreds, 5 tens and 2 ones. 41 is 4 tens and 1 one. So the tens column holds 5 and 4.
If you chose A: That lines the numbers up on the left. The 3 of 352 is in the hundreds. The tens column holds 5 and 4.If you chose C: The 2 and the 1 are the ones, the last digits. The tens come next, to the left of them.Joseph adds 457 + 168 in columns. In the ones column, 7 + 8 = 15. What does he do next?
A column sum of 457 plus 168 with empty boxes for the answer. 7 5 4 8 6 1 + Tap your answer. A Write 15. Carry nothing. Not this one. Only one digit goes in a column. 15 is 1 ten and 5 ones. Write the 5 and carry the ten. Try another.
B Write 1. Carry 5. Not this one. 15 ones is 1 ten and 5 ones. The 5 ones stay in the ones column. The 1 ten moves to the tens. Try another.
C Write 5. Carry 1 ten. Yes, that is right. 15 ones is 1 ten and 5 ones. Write the 5 under the ones. Carry the 1 ten to the tens column.
See the answer Write 5. Carry 1 ten.
15 ones is 1 ten and 5 ones. Write the 5 under the ones. Carry the 1 ten to the tens column.
If you chose A: Only one digit goes in a column. 15 is 1 ten and 5 ones. Write the 5 and carry the ten.If you chose B: 15 ones is 1 ten and 5 ones. The 5 ones stay in the ones column. The 1 ten moves to the tens.Kofi adds 359 + 247 in columns. What is the total?
Tap your answer. A 596 Not this one. You forgot the carried ten. Ones: 9 + 7 = 16, so carry 1 ten. Tens: 5 + 4 + 1 = 10. Write 0 and carry 1 hundred. Try another.
B 606 Yes, that is right. Ones: 9 + 7 = 16. Write 6, carry 1. Tens: 5 + 4 + 1 = 10. Write 0, carry 1. Hundreds: 3 + 2 + 1 = 6.
C 506 Not this one. The tens made 10. Write 0 and carry 1 hundred. Hundreds: 3 + 2 + 1 = 6. Try another.
See the answer 606
Ones: 9 + 7 = 16. Write 6, carry 1. Tens: 5 + 4 + 1 = 10. Write 0, carry 1. Hundreds: 3 + 2 + 1 = 6.
If you chose A: You forgot the carried ten. Ones: 9 + 7 = 16, so carry 1 ten. Tens: 5 + 4 + 1 = 10. Write 0 and carry 1 hundred.If you chose C: The tens made 10. Write 0 and carry 1 hundred. Hundreds: 3 + 2 + 1 = 6.Keep it
Copy this into your exercise book.
Line up ones under ones, tens under tens. Add from the right. A column may make 10 or more. Write the ones. Carry the ten. 358 + 276 = 634.
Draw: Write 358 + 276 in columns. Add the small carried 1s above the tens and hundreds.
Your guess
Pick one before you read on. It is fine to be wrong.
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 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. 4 2 5 8 7 1 − 1 Write 524. Write 178 under it. Line up the places.
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. 4 2 5 8 7 1 − 1 11 4 6 4 3
2 Start at the right. If a top digit is too small, cross it out and exchange.
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. 4 2 5 8 7 1 − 1 11 4 6 4 3 ones: 14 − 8 = 6 tens: 11 − 7 = 4 hundreds: 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.
Ones: 1 − 4 cannot be done. Exchange 1 ten. 11 − 4 = 7. Tens: 3 gave 1 away, so 2 are left. 2 − 5 cannot be done. Exchange 1 hundred. 12 − 5 = 7. Hundreds: 6 gave 1 away, so 5 are left. 5 − 2 = 3. 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. 1 3 6 4 5 2 − 1 12 5 7 7 3
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. 3 0 4 7 6 1 − 1 9 3 6 3 2
The 4 hundreds became 3. The 0 tens became 9. The 3 ones became 13. Worked example
Take away 403 − 167 in columns.
Ones: 3 − 7 cannot be done. There are 0 tens to give. Exchange 1 hundred for 10 tens. The hundreds are now 3. The tens are 10. Exchange 1 ten for 10 ones. The tens are now 9. The ones are 13. Ones: 13 − 7 = 6. Tens: 9 − 6 = 3. Hundreds: 3 − 1 = 2. 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.
Draw two boxes on paper. Write tens over one and ones over the other. Make 42. Put 4 bundles of ten in the tens box. Put 2 loose in the ones box. Take away 17. You need to take 7 loose, but there are only 2. Untie 1 bundle. Put its 10 sticks in the ones box. It holds 12 now. Take 7 loose from the ones box. Take 1 bundle from the tens box. 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. 2 4 7 1 − 1 3 5 2
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.
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.
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.
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.
Lucía takes away 305 − 148. What is the answer?
Tap your answer. A 157 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.
B 243 Not this one. You took the smaller digit from the bigger in each column. Add back: 148 + 243 = 391, not 305. Try another.
C 167 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.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.
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.
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. A 7 − 3 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.
B 10 − 7 Not this one. The 3 ones are still there. 10 ones and 3 ones make 13. So work out 13 − 7. Try another.
C 13 − 7 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.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. 2 5 6 7 8 1 − Tap your answer. A 535 Not this one. You took the smaller digit from the bigger one in each column. Add back: 187 + 535 = 722, not 652. Try another.
B 465 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.
C 475 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.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. 0 0 8 5 4 3 − Tap your answer. A 455 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.
B 545 Not this one. You took the smaller digit from the bigger one in each column. Add back: 345 + 545 = 890, not 800. Try another.
C 465 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.Amina takes away 503 − 276 and gets 227. How can she check?
Tap your answer. A Take 227 away from 276. It should make 503. Not this one. 227 and 276 are both smaller than 503. Taking one from the other cannot make 503. Add them instead. Try another.
B Add 227 + 503. It should make 276. Not this one. 227 + 503 makes 730. Add the answer to the number you took away. That is 276. Try another.
C Add 227 + 276. It should make 503. 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.
Your guess
Pick one before you read on. It is fine to be wrong.
Your guess By the end you can
Add and take away 4 digit numbers in columns Add a list of three or four numbers You can add and take away in columns. Four digit numbers use the same steps.
There is one more column on the left, the thousands. Start at the ones, as always.
Draw it first
Add 3 465 + 2 879 in columns A column sum, 3 465 plus 2 879, with each digit under its place and a line under the numbers. Step 1 says "Line up the digits". Step 2 writes 4 under the ones, a small 1 over the tens, and says "Ones: 5 + 9 = 14". Step 3 writes 4 under the tens, a small 1 over the hundreds, and says "Tens: 1 + 6 + 7 = 14". Step 4 writes 3 under the hundreds, a small 1 over the thousands, and 6 under the thousands. It says "Hundreds: 1 + 4 + 8 = 13" and "Thousands: 1 + 3 + 2 = 6". The answer is 6 344. 5 6 4 3 9 7 8 2 + Line up the digits
1 Write the numbers so ones are under ones. Draw a line.
A column sum, 3 465 plus 2 879, with each digit under its place and a line under the numbers. Step 1 says "Line up the digits". Step 2 writes 4 under the ones, a small 1 over the tens, and says "Ones: 5 + 9 = 14". Step 3 writes 4 under the tens, a small 1 over the hundreds, and says "Tens: 1 + 6 + 7 = 14". Step 4 writes 3 under the hundreds, a small 1 over the thousands, and 6 under the thousands. It says "Hundreds: 1 + 4 + 8 = 13" and "Thousands: 1 + 3 + 2 = 6". The answer is 6 344. 5 6 4 3 9 7 8 2 + 4 1 Ones: 5 + 9 = 14
2 Ones: 5 + 9 = 14. Write 4. Carry the 1 to the tens.
A column sum, 3 465 plus 2 879, with each digit under its place and a line under the numbers. Step 1 says "Line up the digits". Step 2 writes 4 under the ones, a small 1 over the tens, and says "Ones: 5 + 9 = 14". Step 3 writes 4 under the tens, a small 1 over the hundreds, and says "Tens: 1 + 6 + 7 = 14". Step 4 writes 3 under the hundreds, a small 1 over the thousands, and 6 under the thousands. It says "Hundreds: 1 + 4 + 8 = 13" and "Thousands: 1 + 3 + 2 = 6". The answer is 6 344. 5 6 4 3 9 7 8 2 + 4 1 4 1 Tens: 1 + 6 + 7 = 14
3 Tens: 1 + 6 + 7 = 14. Write 4. Carry the 1 to the hundreds.
A column sum, 3 465 plus 2 879, with each digit under its place and a line under the numbers. Step 1 says "Line up the digits". Step 2 writes 4 under the ones, a small 1 over the tens, and says "Ones: 5 + 9 = 14". Step 3 writes 4 under the tens, a small 1 over the hundreds, and says "Tens: 1 + 6 + 7 = 14". Step 4 writes 3 under the hundreds, a small 1 over the thousands, and 6 under the thousands. It says "Hundreds: 1 + 4 + 8 = 13" and "Thousands: 1 + 3 + 2 = 6". The answer is 6 344. 5 6 4 3 9 7 8 2 + 4 1 4 1 3 1 6 Hundreds: 1 + 4 + 8 = 13 Thousands: 1 + 3 + 2 = 6
4 Hundreds: 1 + 4 + 8 = 13. Write 3, carry 1. Thousands: 1 + 3 + 2 = 6.
When a column makes 10 or more, you carry . Write the last digit under the column.
To carry is to move the other digit one column left. Write it small above that column.
See it three ways
With things
Put 5 beans and 9 beans in the ones box. You have 14. Swap 10 beans for 1 stick in the tens box.
As a picture
Fourteen dots, labelled "14 ones", are equal to one rod of ten and four single squares, labelled "1 ten 4 ones". 14 ones = 1 ten 4 ones In symbols
5 + 9 = 14. Write 4. Carry 1.
Taking away uses the same columns. Sometimes the top digit is too small.
Then you exchange . Take 1 from the column on the left. It becomes 10 in this column.
If that column is 0, go on to the next one. Look at 5 002 − 2 457 .
A column take away, 5002 minus 2457, with the answer 2545. The 5 is crossed out and a small 4 is written above it. The two zeros are crossed out and each has a small 9 above it. A small 1 sits by the ones digit 2. 2 0 0 5 7 5 4 2 − 1 9 9 4 5 4 5 2
Two zeros in a row: go back to the 5. Each zero becomes 9. Worked example
Work out 5 002 − 2 457 .
In the ones, 2 is smaller than 7. The next two digits are 0. Go to the 5 thousands. Cross it out and write 4. Write 9 over the hundreds and 9 over the tens. Give the ones a ten. 2 becomes 12. Now take away. 12 − 7 = 5, 9 − 5 = 4, 9 − 4 = 5, 4 − 2 = 2. Answer: 2 545
You can add a list of numbers in columns too. Line up all the numbers.
Add down each column, starting at the ones.
A column sum of four numbers: 245, 130, 78 and 302, lined up by place with a line under them. A small 1 sits over the tens and a small 1 over the hundreds. The answer is 755. The notes say "Ones: 5 + 0 + 8 + 2 = 15", "Tens: 1 + 4 + 3 + 7 = 15" and "Hundreds: 1 + 2 + 1 + 3 = 7". 5 4 2 0 3 1 8 7 2 0 3 + 1 1 5 5 7 Ones: 5 + 0 + 8 + 2 = 15 Tens: 1 + 4 + 3 + 7 = 15 Hundreds: 1 + 2 + 1 + 3 = 7
The ones make 15. Write 5 and carry 1. Worked example
Work out 245 + 130 + 78 + 302.
Ones: 5 + 0 + 8 + 2 = 15. Write 5. Carry 1. Tens: 1 + 4 + 3 + 7 + 0 = 15. Write 5. Carry 1. Hundreds: 1 + 2 + 1 + 0 + 3 = 7. Write 7. Answer: 755
The ones can make 20 or more. Then you carry 2 or more.
Worked example
Work out 48 + 37 + 29 + 56.
Ones: 8 + 7 + 9 + 6 = 30. Write 0. Carry 3 to the tens. Tens: 3 + 4 + 3 + 2 + 5 = 17. Write 17. Answer: 170
Try it
Carry three tens with beans You need: 30 beans or small stones, 3 sticks, a pencil and paper.
Be safe: Do not eat the beans.
Draw two boxes on paper. Write "tens" over one and "ones" over the other. Put 8, 7, 9 and 6 beans in the ones box. Count them. You have 30. Take 10 beans out. Put 1 stick in the tens box. Do this 3 times. Look at the ones box. What you will see: No beans are left in the ones box. There are 3 sticks in the tens box. So you write 0 and carry 3.
If you cannot try it, this is what happens. Thirty beans, labelled "30 ones", are equal to three rods of ten, labelled "3 tens". 30 ones = 3 tens A wrong idea many people have
I can write 14 under the ones column.
Why it feels right: 5 + 9 really is 14, so writing 14 seems right.
The test that breaks it: Try 15 + 9 that way. You write 14 and then 1, which reads 114. Count on from 15 by 9. You reach 24.
What is true: Each column holds one digit. 14 ones is 1 ten and 4 ones. Write 4 and carry 1. So 15 + 9 = 24.
Key words
carry move the extra digit to the next column on the left. exchange take 1 from the column on the left. It becomes 10 in this column. Back to your guess
The answer is B: Write 4 under the ones and carry the 1 to the tens.
Each column holds one digit. 14 ones is 1 ten and 4 ones. Write the 4. Carry the 1 to the tens.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Work out 2 358 + 1 467 in columns.
See the answer 3 825
Ones: 8 + 7 = 15, write 5, carry 1. Tens: 1 + 5 + 6 = 12, write 2, carry 1. Hundreds: 1 + 3 + 4 = 8. Thousands: 2 + 1 = 3.
Work out 6 300 − 2 845 in columns.
See the answer 3 455
In the ones, 0 is smaller than 5. Exchange along the zeros. The hundreds, now 2, need an exchange too. 10 − 5 = 5, 9 − 4 = 5, 12 − 8 = 4, 5 − 2 = 3. Check: 3 455 + 2 845 = 6 300 .
Add 54 + 38 + 27 + 66. The ones column makes 25. What do you do?
Tap your answer. A Write 5 and carry 2 Yes, that is right. The ones make 4 + 8 + 7 + 6 = 25. That is 2 tens and 5 ones. The tens then make 5 + 3 + 2 + 6 + 2 = 18. The total is 185.
B Write 25 under the ones Not this one. A column holds one digit. Write the last digit, 5, and carry the rest. Try another.
C Write 2 and carry 5 Not this one. The digits are the wrong way round. 25 is 2 tens and 5 ones. The 5 stays. Try another.
See the answer Write 5 and carry 2
The ones make 4 + 8 + 7 + 6 = 25. That is 2 tens and 5 ones. The tens then make 5 + 3 + 2 + 6 + 2 = 18. The total is 185.
If you chose B: A column holds one digit. Write the last digit, 5, and carry the rest.If you chose C: The digits are the wrong way round. 25 is 2 tens and 5 ones. The 5 stays.Work out 126 + 85 + 308 + 49.
See the answer 568
Ones: 6 + 5 + 8 + 9 = 28, write 8, carry 2. Tens: 2 + 2 + 8 + 0 + 4 = 16, write 6, carry 1. Hundreds: 1 + 1 + 0 + 3 + 0 = 5.
From an earlier chapter
Ravi works out 403 − 167 and gets 364. He took the smaller digit from the bigger one each time. Is he right?
Tap your answer. A Yes, he is right. Not this one. In the ones, 3 is smaller than 7. You cannot take 7 from 3, so you must exchange. Go to M2.2.4 . Try another.
B No. The answer is 236. Yes, that is right. Exchange 1 hundred for 10 tens, then 1 ten for 10 ones. 13 − 7 = 6, 9 − 6 = 3, 3 − 1 = 2. Check: 236 + 167 = 403.
C No. The answer is 570. Not this one. 570 is 403 + 167. A take away gives less than 403. Try another.
See the answer No. The answer is 236.
Exchange 1 hundred for 10 tens, then 1 ten for 10 ones. 13 − 7 = 6, 9 − 6 = 3, 3 − 1 = 2. Check: 236 + 167 = 403.
If you chose A: In the ones, 3 is smaller than 7. You cannot take 7 from 3, so you must exchange. Go to M2.2.4 .If you chose C: 570 is 403 + 167. A take away gives less than 403.Leila adds 4 763 + 1 859 in columns. What is the total?
A column sum of 4763 plus 1859 with empty boxes for the answer. 3 6 7 4 9 5 8 1 + Tap your answer. A 5 622 Not this one. The hundreds made 16. Write 6 and carry 1 to the thousands. Thousands: 4 + 1 + 1 = 6. Try another.
B 6 622 Yes, that is right. Ones: 3 + 9 = 12, write 2, carry 1. Tens: 6 + 5 + 1 = 12, write 2, carry 1. Hundreds: 7 + 8 + 1 = 16, write 6, carry 1. Thousands: 4 + 1 + 1 = 6.
C 6 612 Not this one. The ones made 12. Carry 1 ten. Tens: 6 + 5 + 1 = 12, not 11. Try another.
See the answer 6 622
Ones: 3 + 9 = 12, write 2, carry 1. Tens: 6 + 5 + 1 = 12, write 2, carry 1. Hundreds: 7 + 8 + 1 = 16, write 6, carry 1. Thousands: 4 + 1 + 1 = 6.
If you chose A: The hundreds made 16. Write 6 and carry 1 to the thousands. Thousands: 4 + 1 + 1 = 6.If you chose C: The ones made 12. Carry 1 ten. Tens: 6 + 5 + 1 = 12, not 11.Hana takes away 4 000 − 1 235 in columns. What is the answer?
A column sum of 4000 take away 1235 with empty boxes for the answer. 0 0 0 4 5 3 2 1 − Tap your answer. A 3 235 Not this one. You took the smaller digit from the bigger one in each column. Add back: 1 235 + 3 235 = 4 470 , not 4 000 . Try another.
B 3 765 Not this one. The 4 thousands gave 1 away. Only 3 are left. 3 − 1 = 2, not 3. Try another.
C 2 765 Yes, that is right. Exchange along the zeros. The thousands are 3. The hundreds and tens are 9. The ones are 10. 10 − 5 = 5, 9 − 3 = 6, 9 − 2 = 7, 3 − 1 = 2. Check: 2 765 + 1 235 = 4 000 .
See the answer 2 765
Exchange along the zeros. The thousands are 3. The hundreds and tens are 9. The ones are 10. 10 − 5 = 5, 9 − 3 = 6, 9 − 2 = 7, 3 − 1 = 2. Check: 2 765 + 1 235 = 4 000 .
If you chose A: You took the smaller digit from the bigger one in each column. Add back: 1 235 + 3 235 = 4 470 , not 4 000 .If you chose B: The 4 thousands gave 1 away. Only 3 are left. 3 − 1 = 2, not 3.Mei adds 36 + 48 + 19 + 27 in columns. The ones make 30. What is the total?
Tap your answer. A 130 Yes, that is right. Ones: 6 + 8 + 9 + 7 = 30. Write 0, carry 3. Tens: 3 + 4 + 1 + 2 + 3 = 13. The total is 130.
B 110 Not this one. You carried only 1. The ones made 30, which is 3 tens. Carry 3. Tens: 3 + 4 + 1 + 2 + 3 = 13. Try another.
C 100 Not this one. You forgot the 3 tens from the ones. Tens: 3 + 4 + 1 + 2 + 3 = 13. Try another.
See the answer 130
Ones: 6 + 8 + 9 + 7 = 30. Write 0, carry 3. Tens: 3 + 4 + 1 + 2 + 3 = 13. The total is 130.
If you chose B: You carried only 1. The ones made 30, which is 3 tens. Carry 3. Tens: 3 + 4 + 1 + 2 + 3 = 13.If you chose C: You forgot the 3 tens from the ones. Tens: 3 + 4 + 1 + 2 + 3 = 13.Look at 5 002 − 2 457 . The hundreds digit is 0. What does it become after the exchange?
Tap your answer. A 10 Not this one. The 0 gets 10 from the thousands. Then it gives 1 to the tens. That leaves 9, not 10. Try another.
B 9 Yes, that is right. The 0 gets 10 from the thousands. Then it gives 1 to the tens. 10 − 1 = 9.
C 4 Not this one. The 4 is in the thousands. It was 5 and gave 1 away. The hundreds 0 became 9. Try another.
See the answer 9
The 0 gets 10 from the thousands. Then it gives 1 to the tens. 10 − 1 = 9.
If you chose A: The 0 gets 10 from the thousands. Then it gives 1 to the tens. That leaves 9, not 10.If you chose C: The 4 is in the thousands. It was 5 and gave 1 away. The hundreds 0 became 9.Keep it
Copy this into your exercise book.
Line up the digits. Start at the ones. When a column makes 10 or more, carry. Write the last digit. A list can carry 2 or more.
Draw: Write 3 465 + 2 879 in columns. Show each carry as a small 1.
Your guess
Pick one before you read on. It is fine to be wrong.
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 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". 400 500 487
1 Find 487 on the line between 400 and 500.
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". 400 500 487 nearer 500 2 It is nearer 500. Round it up to 500.
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". 400 500 487 nearer 500 300 400 312 nearer 300 3 Find 312 between 300 and 400. It is nearer 300. Round it down to 300.
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". 400 500 487 nearer 500 300 400 312 nearer 300 500 + 300 = 800 Exact answer: 799 4 Add the rounded numbers. 500 + 300 = 800. The exact answer is 799.
Worked example
Estimate 487 + 312. Then work it out.
The tens digit of 487 is 8. Go up to 500. The tens digit of 312 is 1. Go down to 300. Add the rounded numbers. 500 + 300 = 800. 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. 3 0 4 7 6 1 − 1 9 3 6 3 2
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. 6 3 2 7 6 1 + 1 1 3 0 4
236 + 167 = 403. You got the first number back. Worked example
Check that 403 − 167 = 236.
Add the answer to the number you took away. 236 + 167 = 403. 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. 236 167 403 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.
Count out 23 beans. Write 23. Take away 9 beans. Put them to one side. Count the beans left. Write 23 − 9 = 14. 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 left 9 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.
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.
Amina works out 624 + 275 and gets 1 899 . Which estimate shows she is wrong?
Tap your answer. A About 600 + 300 = 900, so 1 899 is too big. 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.
B About 600 + 300 = 900, so 1 899 is right. Not this one. Look again. 1 899 is about twice 900. The answer should be near 900. Try another.
C An estimate cannot show a mistake. 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.Tariq says 702 − 458 = 354. Check by adding back. Is he right?
Tap your answer. A Yes. 354 + 458 = 702. Not this one. Add again. 354 + 458 = 812, not 702. Try another.
B No. 354 + 458 = 812. The answer is 244. Yes, that is right. Adding back should give 702. It gives 812. The right answer is 244, because 244 + 458 = 702.
C You cannot tell without working it out again. 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.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 .
From an earlier chapter
Which is 568 + 275?
Tap your answer. A 843 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.
B 733 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.
C 743 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 .Hana estimates 748 + 231. She rounds each number to the nearest hundred. What is her estimate?
Tap your answer. A 1 000 Not this one. You rounded 748 up to 800. The tens digit is 4, so go down to 700. Try another.
B 979 Not this one. 979 is the exact answer. An estimate rounds first. 700 + 200 = 900. Try another.
C 900 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.Mei works out 905 − 378 and gets 527. Which sum checks her answer?
Tap your answer. A 527 + 378 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.
B 905 + 378 Not this one. That makes 1 283 . Add the answer to the number you took away. Try another.
C 527 − 378 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.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. 5 6 4 5 4 3 + Tap your answer. A His answer is right, because 810 is close to 800. Not this one. Adding back must give the first number exactly. 810 is 10 more than 800, so there is a mistake. Try another.
B His answer is wrong. Adding back must give the first number. Yes, that is right. Adding back gives 810, not 800. So 465 is wrong. The right answer is 455, because 455 + 345 = 800.
C His answer is right, because 465 is near the estimate of 500. 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.
Your guess
Pick one before you read on. It is fine to be wrong.
Your guess By the end you can
Draw a bar model to decide whether to add or take away Solve a problem that needs two steps Find how many more you need to get from 37 to 50 Some problems do not say add or take away. A picture can help you choose.
A bar model is a picture of a problem. It uses bars for the numbers.
The whole is the long bar on top. It is all of the thing.
A part is one piece of the whole. The parts go side by side under it.
A farmer has 58 hens and 35 ducks. How many birds are there?
Draw it first
Draw the hens and ducks A bar model. Step 1 draws a long bar on top, shaded with dots and labelled "?", next to the word "whole". Step 2 draws two bars side by side under it, labelled "58" and "35", next to the word "parts". Step 3 writes "58 + 35 = 93". ? whole 1 Draw a long bar for the whole. You do not know it, so write ?.
A bar model. Step 1 draws a long bar on top, shaded with dots and labelled "?", next to the word "whole". Step 2 draws two bars side by side under it, labelled "58" and "35", next to the word "parts". Step 3 writes "58 + 35 = 93". ? whole 58 35 parts 2 Under it, draw the parts side by side. Write 58 and 35.
A bar model. Step 1 draws a long bar on top, shaded with dots and labelled "?", next to the word "whole". Step 2 draws two bars side by side under it, labelled "58" and "35", next to the word "parts". Step 3 writes "58 + 35 = 93". ? whole 58 35 parts 58 + 35 = 93 3 You know the parts and want the whole. So add. 58 + 35 = 93.
Now the farmer has 93 birds. 58 are hens. How many are ducks?
A bar model. The long bar on top is labelled "93". Under it are two bars side by side. The first is labelled "58". The second is shaded with dots and labelled "?". 93 58 ? You know the whole and one part. So take away. 93 − 58 = 35. Worked example
There are 93 birds. 58 are hens. How many are ducks?
The whole is 93. One part is 58. You want the other part. Take away. 93 − 58 = 35. Answer: 35 ducks
See it three ways
With things
Put 12 beans in one pile and 7 beans in another. Push the piles together. Count them.
As a picture
A bar model. The long bar on top is labelled "19". Under it are two bars side by side, labelled "12" and "7". 19 12 7 In symbols
12 + 7 = 19. Take away one part to find the other: 19 − 7 = 12.
Some problems need two steps. Do one step, then the next.
A storeroom has 245 books. A van brings 130 more. Then 78 books go to a village. How many are left?
Two bar models, one under the other. The first is for step 1, add. The long bar is shaded with dots and labelled "?". Its parts are labelled "245" and "130". The second is for step 2, take away. The long bar is labelled "375". Its parts are labelled "78" and "?". The words are "Step 1: add" and "Step 2: take away". Step 1: add ? 245 130 Step 2: take away 375 78 ? Step 1 finds the whole, 375. Step 2 takes 78 away from it. Worked example
Work out how many books are left.
Step 1: the parts are 245 and 130. Add. 245 + 130 = 375. Step 2: the whole is 375. One part is 78. Take away. 375 − 78 = 297. Answer: 297 books
Some problems go the other way. Take away first, then add.
A tank has 500 litres of water. Ana takes out 185. Rain adds 130. How much is in it now?
Worked example
Work out how many litres are in the tank now.
Step 1: the whole is 500. One part is 185. Take away. 500 − 185 = 315. Step 2: the parts are 315 and 130. Add. 315 + 130 = 445. Answer: 445 litres
You need 50 beads for a necklace. You have 37. How many more beads do you need?
The beads you still need are the missing part . Take the 37 you have from the 50 you need.
A bar model. The long bar on top is labelled "50". Under it are two bars side by side. The first is labelled "37". The second is shaded with dots and labelled "?". 50 37 ? You know the whole and one part. The beads you need are the missing part. You can also count up from 37 to 50.
A number line from 37 to 50 with a mark at every number. The numbers 37, 40 and 50 are written under it. An arrow from 37 to 40 is labelled "+3". An arrow from 40 to 50 is labelled "+10". +3 +10 37 40 50 3 makes 40. 10 more makes 50. You need 3 + 10 = 13 more. Worked example
You need 50 beads and have 37. How many more do you need?
The whole is 50. One part is 37. Take away: 50 − 37. Count up from 37. 3 more makes 40. From 40 to 50 is 10 more. Add the jumps. 3 + 10 = 13. Answer: 13 beads
Try it
Two steps with beans You need: 21 beans or small stones, a pencil and paper.
Be safe: Do not eat the beans.
Make a pile of 12 beans and a pile of 9 beans. Push them together. Count them. Write 12 + 9 = 21. Take 6 beans away. Count what is left. Write 21 − 6 = 15. Draw a bar model for each step. What you will see: Step 1 gave the whole, 21. Step 2 used 21 as the new whole.
If you cannot try it, this is what happens. Two bar models, one under the other. The first is for step 1. The long bar is labelled "21". Its parts are labelled "12" and "9". The second is for step 2. The long bar is labelled "21". Its parts are labelled "6" and "15". The words are "Step 1" and "Step 2". Step 1 21 12 9 Step 2 21 6 15 A wrong idea many people have
A word in the story tells you the sum. "More" always means add.
Why it feels right: Many stories say "more" when something is added.
The test that breaks it: Kofi has 50 beans. Ana has 37. How many more has Kofi? Lay 37 beans in a row. Lay 50 under them, one to one. 13 have no partner.
What is true: Draw the bar model, not the words. The whole is 50 and one part is 37. Take away: 50 − 37 = 13.
Key words
bar model a picture of a problem. The whole is on top. Its parts are under it. whole all of the thing. It is the long bar on top. part one piece of the whole. missing part the part you do not know. You find it by taking away. Back to your guess
The answer is C: Draw the whole and its parts. See which one is missing.
A picture shows what you know and what is missing. If the whole is missing, add. If a part is missing, take away.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
There are 84 children in a hall. 39 are girls. Which sum finds the boys?
Tap your answer. A 84 − 39 = 45 Yes, that is right. The whole is 84. One part is 39. Take away. 84 − 39 = 45 boys, and 45 + 39 = 84.
B 84 + 39 = 123 Not this one. 84 is the whole. 39 is one part. You want the other part, so take away. Try another.
C You cannot tell. The story needs a third number. Not this one. You know the whole, 84, and one part, 39. That is enough to find the other part. Try another.
See the answer 84 − 39 = 45
The whole is 84. One part is 39. Take away. 84 − 39 = 45 boys, and 45 + 39 = 84.
If you chose B: 84 is the whole. 39 is one part. You want the other part, so take away.If you chose C: You know the whole, 84, and one part, 39. That is enough to find the other part.One sack holds 376 beans. Another holds 248. Draw a bar model. How many beans in all?
See the answer 624
You know the two parts and want the whole. Add. 376 + 248 = 624.
Kofi picks 468 mangoes in the morning and 259 in the afternoon. He gives away 600. How many are left?
See the answer 127
Step 1: 468 + 259 = 727. Step 2: 727 − 600 = 127.
A water drum holds 50 litres. It has 28 litres in it. How many more litres fill it?
See the answer 22 litres
50 − 28 = 22. Or count up: 2 makes 30, then 20 makes 50. 2 + 20 = 22.
From an earlier chapter
Mei says 727 − 389 = 348. Add back to check. Is she right?
Tap your answer. A Yes, she is right. Not this one. Add 348 + 389. You get 737, not 727. Adding back shows a mistake. Go to M2.2.6 . Try another.
B No. The answer is 338. Yes, that is right. 348 + 389 = 737, not 727. The right answer is 338, because 338 + 389 = 727.
C No. The answer is 1 116 . Not this one. 1 116 is 727 + 389. This is a take away, so the answer is less than 727. Try another.
See the answer No. The answer is 338.
348 + 389 = 737, not 727. The right answer is 338, because 338 + 389 = 727.
If you chose A: Add 348 + 389. You get 737, not 727. Adding back shows a mistake. Go to M2.2.6 .If you chose C: 1 116 is 727 + 389. This is a take away, so the answer is less than 727.A rope is 120 m long. Sipho cuts off 75 m. How long is the rest?
A bar model. The long bar on top is labelled "120 m". Under it are two bars side by side. The first is labelled "75 m". The second is shaded with dots and labelled "?". 120 m 75 m ? Tap your answer. A 195 m Not this one. 120 m is the whole rope before the cut. The rest is a part, so take away. Try another.
B 45 m Yes, that is right. The whole is 120 m. One part is 75 m. Take away. 120 − 75 = 45. Check: 45 + 75 = 120.
C 155 m Not this one. You took the smaller digit from the bigger one in each column. Add back: 155 + 75 = 230, not 120. Try another.
See the answer 45 m
The whole is 120 m. One part is 75 m. Take away. 120 − 75 = 45. Check: 45 + 75 = 120.
If you chose A: 120 m is the whole rope before the cut. The rest is a part, so take away.If you chose C: You took the smaller digit from the bigger one in each column. Add back: 155 + 75 = 230, not 120.A tank has 250 litres of water. Ana adds 135 litres. Then she uses 180 litres. How many litres are left?
Tap your answer. A 565 litres Not this one. You added all three numbers. Ana uses 180 litres, so take them away at step 2. Try another.
B 70 litres Not this one. You left out the 135 litres Ana added. Add first: 250 + 135 = 385. Then take away 180. Try another.
C 205 litres Yes, that is right. Step 1: 250 + 135 = 385. Step 2: 385 − 180 = 205.
See the answer 205 litres
Step 1: 250 + 135 = 385. Step 2: 385 − 180 = 205.
If you chose A: You added all three numbers. Ana uses 180 litres, so take them away at step 2.If you chose B: You left out the 135 litres Ana added. Add first: 250 + 135 = 385. Then take away 180.Priya has a book of 400 pages. She has read 156 pages. Then she reads 85 more. How many pages are left?
Tap your answer. A 159 pages Yes, that is right. Step 1: 156 + 85 = 241 pages read. Step 2: 400 − 241 = 159 pages left.
B 244 pages Not this one. You forgot the 85 pages. Step 1: she has read 156 + 85 = 241. Step 2: 400 − 241. Try another.
C 241 pages Not this one. 241 is how many pages she has read. The question asks how many are left. Take 241 from 400. Try another.
See the answer 159 pages
Step 1: 156 + 85 = 241 pages read. Step 2: 400 − 241 = 159 pages left.
If you chose B: You forgot the 85 pages. Step 1: she has read 156 + 85 = 241. Step 2: 400 − 241.If you chose C: 241 is how many pages she has read. The question asks how many are left. Take 241 from 400.A minibus has 62 seats. 48 are taken. Hana counts up from 48 to 62. Her first hop is 2, to 50. How long is the next hop?
A number line from 48 to 62. A short hop labelled +2 goes from 48 to 50. The numbers 48, 50 and 62 are written under the line. +2 48 50 62 Tap your answer. A 14 Not this one. 14 is the whole gap, both hops together. The next hop goes from 50 to 62. Try another.
B 62 Not this one. 62 is where the hop ends. The hop is how far it goes, from 50 to 62. Try another.
C 12 Yes, that is right. The next hop goes from 50 to 62. 62 − 50 = 12. The two hops make 2 + 12 = 14 free seats.
See the answer 12
The next hop goes from 50 to 62. 62 − 50 = 12. The two hops make 2 + 12 = 14 free seats.
If you chose A: 14 is the whole gap, both hops together. The next hop goes from 50 to 62.If you chose B: 62 is where the hop ends. The hop is how far it goes, from 50 to 62.Keep it
Copy this into your exercise book.
Draw the whole on top and its parts under it. Know the parts? Add. Know the whole and one part? Take away. Missing part: 50 − 37 = 13.
Draw: A long bar labelled 50. Under it, two bars: 37 and ?.
Key words, A to Z Every key word in this book, what it means, and the chapter that teaches it.
adding back check a take away. Add the answer to the number you took away. M2.2.4 bar model a picture of a problem. The whole is on top. Its parts are under it. M2.2.7 carry move a new ten or hundred into the next column. M2.2.3 , M2.2.5 check test an answer a second way. M2.2.6 column a line of digits, one above the other, in one place. M2.2.3 count up start at the smaller number and hop to the bigger one. M2.2.2 difference how far apart two numbers are. M2.2.2 estimate a quick answer that is near the right one. M2.2.6 exchange swap 1 ten for 10 ones, or 1 hundred for 10 tens. M2.2.4 , M2.2.5 missing part the part you do not know. You find it by taking away. M2.2.7 part one piece of the whole. M2.2.7 round change a number to the nearest ten or hundred. M2.2.6 split break a number into parts, such as 46 into 40 and 6. M2.2.1 total the number you get when you add numbers together. M2.2.1 whole all of the thing. It is the long bar on top. M2.2.7 About this book For learners who can add and take away numbers up to 100, such as 46 + 7 and 50 − 8. In many schools this fits children of about 8 to 10, and older learners who get lost in column sums can start here too. Activities use sticks, beans and paper. A learner can work alone: every exercise has its answer and the reason for it.
Written and drawn by attnom. Edition 1.0, 2026-11-09.
This book is free. You may read it, print it, copy it and share it with anyone.
The newest copy is at https://attnom.com/academy/adding-and-taking-away-big-numbers .