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Primary Maths · Year 4Chapter 2 · Adding and subtracting
Year 4 · Column Addition and Subtraction · Lesson 2.3

Year 4: Adding and subtracting in your head

Part of the Year 4 column addition and subtraction chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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2.3

Adding and subtracting in your head

Year 4
Mental addition and subtraction

A written method always works, but it is not always the quickest. If the numbers are friendly, you can add and subtract in your head. Look at the numbers first, then pick a good strategy. Here are five.

1. Partitioning. Split (partition) the numbers into thousands, hundreds, tens and ones. Add the matching parts, then add the totals. For 346 + 278: 300 + 200 = 500, 40 + 70 = 110, 6 + 8 = 14, and 500 + 110 + 14 = 624. To subtract, keep the first number whole and take away the parts of the second number one at a time.

2. Bridging. Jump to the next round number (10, 100 or 1000), then add or subtract what is left. For 396 + 47, add 4 to reach 400, then add the other 43.

396400443+4+43
Bridging through 100: 396 + 4 = 400, and 400 + 43 = 443.

3. Compensating. If you are adding or subtracting a number just below or above a round number (like 9, 99, 199 or 301), use the round number and then fix the answer. 546 + 99: add 100 (a bit too much) to get 646, then take away 1 to get 645. To take away 298, take away 300 and put 2 back.

4. Near doubles. If two numbers are very close, double the smaller number and add the small difference. 350 + 351: double 350 is 700, plus 1 is 701.

5. Counting up. To find the difference between two numbers that are close, start at the smaller and count up to the bigger. For 5,002 − 4,987, count from 4,987 up to 5,000 (13) and then on to 5,002 (2). The difference is 13 + 2 = 15.

4,9875,0005,002+13+2
Counting up from 4,987 to 5,002: 13 + 2 = 15.

Key words. To partition is to split a number into parts. To bridge is to jump to a round number. To compensate is to make up for a change you made. A double is a number added to itself.

Key ideas

  • Partition: split the numbers into thousands, hundreds, tens and ones, then add the matching parts.
  • Bridge: jump to the next 10, 100 or 1000, then do the rest.
  • Compensate: add or take away a round number, then fix the answer (add 99 = add 100, take away 1).
  • Near doubles: double the smaller number and add the difference.
  • Count up to find the difference when the numbers are close.
Worked example 2.3.1
Work out 346 + 278 in your head by partitioning.

Key words. To partition a number is to split it into parts, one part for each digit (for example 345 = 300 + 40 + 5).

Method. Split (partition) both numbers into thousands, hundreds, tens and ones. Add the parts that match, then add the totals.

  1. Partition both numbers. Write each number as the value of its digits.
    346 = 300 + 40 + 6
    278 = 200 + 70 + 8
  2. Add the matching parts. Thousands with thousands, hundreds with hundreds, and so on.
    hundreds: 300 + 200 = 500
    tens: 40 + 70 = 110
    ones: 6 + 8 = 14
  3. Add the totals together. Start with the biggest and keep a running total.
    500 + 110 = 610
    610 + 14 = 624
  4. Write the answer.
    346 + 278 = 624

Check. Use the opposite operation: 624 − 278 = 346. ✓

Watch out. Do not forget a part. Count the parts: every digit that is not 0 must be added.

Worked example 2.3.2
Work out 745 − 278 in your head by partitioning the second number.

Key words. To partition a number is to split it into parts, one part for each digit.

Method. Keep the first number whole. Split (partition) the number you are taking away, then take the parts away one at a time, biggest first.

  1. Partition the number you take away. Only the second number is split up.
    278 = 200 + 70 + 8
  2. Take the parts away one at a time. Start with the biggest part. Use the new number as the start of the next step.
    745 − 200 = 545
    545 − 70 = 475
    475 − 8 = 467
  3. Write the answer. The last number is the answer.
    745 − 278 = 467

Check. Use the opposite operation: 467 + 278 = 745. ✓

Watch out. Take the parts away from the first number, one after another. Do not take them away from each other.

Worked example 2.3.3
Work out 396 + 47 by bridging through 100.

Key words. To bridge through 100 is to use a jump that lands exactly on a multiple of 100, then jump again.

Method. Bridge through 100. Add enough to reach the next 100, then add what is left.

  1. How far is 396 from the next 100? 400 is the next multiple of 100.
    400 − 396 = 4
  2. Split 47 into 4 and 43. The first part takes us exactly to 400.
    47 = 4 + 43
  3. Add the first part to reach 400. This is the bridge.
    396 + 4 = 400
  4. Add the second part. This is easy because we are on a round number.
    400 + 43 = 443
  5. Show it on a number line.
    396400443+4+43

Check. Use the opposite operation: 443 − 47 = 396. ✓

Watch out. Make sure the two parts add back to 47: 4 + 43 = 47.

Worked example 2.3.4
Work out 1,203 − 8 by bridging through 100.

Method. Bridge through 100. Take away enough to land on the 100 below, then take away what is left.

  1. How far is 1,203 above the 100 below it? 1,200 is the multiple of 100 just below 1,203.
    1,203 − 1,200 = 3
  2. Split 8 into 3 and 5. The first part takes us exactly to 1,200.
    8 = 3 + 5
  3. Take away the first part to reach 1,200. This is the bridge.
    1,203 − 3 = 1,200
  4. Take away the second part. This is easy because we start from a round number.
    1,200 − 5 = 1,195
  5. Show it on a number line. We count back from the right.
    1,2031,2001,195−3−5

Check. Use the opposite operation: 1,195 + 8 = 1,203. ✓

Watch out. Make sure the two parts add back to 8: 3 + 5 = 8.

Worked example 2.3.5
Work out 546 + 99 by compensating.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: add the nearby round number 100 instead, then fix the answer.

  1. Spot a nearby round number. 100 is much easier to work with than 99.
    99 is 1 less than 100:  99 = 100 − 1
  2. Add 100 instead of 99. Round numbers are easy.
    546 + 100 = 646
  3. Fix the answer. We added 1 too many, so we must take 1 away.
    646 − 1 = 645

Check. Use the opposite operation: 645 − 99 = 546. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Worked example 2.3.6
Work out 1,204 − 298 by compensating.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: take away the nearby round number 300 instead, then fix the answer.

  1. Spot a nearby round number. 300 is much easier to work with than 298.
    298 is 2 less than 300:  298 = 300 − 2
  2. Take away 300 instead of 298. Round numbers are easy.
    1,204 − 300 = 904
  3. Fix the answer. We took away 2 too many, so we must put 2 back.
    904 + 2 = 906

Check. Use the opposite operation: 906 + 298 = 1,204. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Worked example 2.3.7
Work out 350 + 351 using a near double.

Key words. A double is a number added to itself. A near double is a sum of two numbers that are almost the same.

Method. Near doubles: if two numbers are close, double the smaller one, then add the small difference.

  1. Spot the near double. The numbers are very close together.
    351 is 1 more than 350, so 351 = 350 + 1
  2. Double 350. Double each part of the number, then add.
    double 300 = 600
    double 50 = 100
    600 + 100 = 700
  3. Add the extra 1. 351 is 1 more than 350, so the total is 1 more than the double.
    700 + 1 = 701

Check. Use the opposite operation: 701 − 351 = 350. ✓

Watch out. Double the SMALLER number and add the difference. Do not add it to the bigger number.

Worked example 2.3.8
Work out 5,002 − 4,987 by counting up.

Key words. The difference between two numbers is how far apart they are. Counting up is a good way to find a difference when the numbers are close.

Method. To find the difference between two numbers that are close, start at the smaller number and count up to the bigger one. Add up the jumps.

  1. Start at the smaller number. The difference is how far we have to count up.
    Start: 4,987.   Finish: 5,002.
  2. Count up in easy jumps. Jump to a round number first, then add the rest.
    4,987 + 13 = 5,000
    5,000 + 2 = 5,002
    4,9875,0005,002+13+2
  3. Add up the jumps. The total of all the jumps is the difference.
    13 + 2 = 15

Check. Use addition: 4,987 + 15 = 5,002. ✓

Watch out. Add up the jumps. The answer is the size of the jumps, not the number you finish on.

Watch out

Compensating has two steps. People often do the first step (add 100) and then forget to fix the answer. 546 + 99 is not 646. Also be careful with the direction: when you take away too much, you must put some back.

Exercise 2C  Mental strategies

Non-calculator
1
FAdd by partitioning. Split both numbers, add the matching parts, then add the totals.
(a)254 + 131
(b)1,250 + 2,340
(c)527 + 186
(d)3,468 + 1,257
2
FSubtract by partitioning the second number. Take the parts away one at a time.
(a)879 − 435
(b)4,675 − 1,230
(c)924 − 368
(d)5,408 − 2,153
3
FAdd by bridging through the next 10 or 100.
(a)78 + 25
(b)496 + 47
(c)1,985 + 28
(d)2,970 + 85
4
FSubtract by bridging. Take away enough to land on a round number first.
(a)302 − 6
(b)4,102 − 15
(c)5,010 − 25
(d)3,004 − 19
5
FAdd by compensating. Use a nearby round number, then fix the answer.
(a)347 + 9
(b)586 + 99
(c)2,415 + 199
(d)4,356 + 999
6
FSubtract by compensating. Use a nearby round number, then fix the answer.
(a)83 − 9
(b)645 − 99
(c)1,480 − 198
(d)5,206 − 301
7
FUse a near double to add.
(a)35 + 36
(b)160 + 162
(c)450 + 451
(d)1,200 + 1,201
8
FFind each difference by counting up from the smaller number.
(a)5,000 − 4,988
(b)3,002 − 2,994
(c)2,000 − 1,875
(d)6,100 − 5,950
9
RMatch each calculation to the best strategy:   counting up,   compensating,   a near double,   partitioning. Then work out each answer.
10
PSA comic costs £99 and a game costs £149. How much do they cost together? Work it out in your head.
11
PSJack has a rope that is 304 cm long. He cuts off 8 cm. How long is the rope now?
12
RLeo works out 546 + 99. He thinks, “Add 100 to get 646, then add 1 to get 647.” Spot Leo’s mistake and write the correct answer.
13
RShow that 4,999 + 3,999 = 8,998 without using the column method.
14
RThe sum of two numbers that are next to each other (like 7 and 8) is 1,001. What are the two numbers?

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