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Math · Number Sense & Mental Strategies
Mental Addition & Subtraction: When Is a Shortcut Worth It?
Choose when to make ten, split numbers, double, compensate or use straightforward arithmetic; explain why an adjustment is valid.
The idea
A shortcut is useful only if it makes the thinking clearer or easier. The goal is a reliable answer, not a fancy trick.
Making ten works when a number is close to ten, twenty, thirty or another friendly multiple of ten. Compensation changes a number temporarily and then corrects exactly what changed.
For subtraction, subtracting more than intended makes the result too small, so add back the extra amount. For addition, adding more than intended makes the total too large, so subtract the extra amount.
When neither number offers an easy adjustment, split into tens and ones or use written arithmetic. A good strategy is one you can explain and check.
Worked examples
Make a ten
Find 38 + 7.
- Move 2 of the 7 to the 38: 38 + 2 = 40.
- 5 remain, so 40 + 5 = 45.
- Check by splitting: 38 + 5 + 2 = 45.
45
Compensate for addition
Find 47 + 29.
- 29 is one less than 30.
- 47 + 30 = 77.
- Take away the 1 extra: 77 − 1 = 76.
76
Compensate for subtraction
Find 64 − 19.
- Subtract 20 instead of 19: 64 − 20 = 44.
- You subtracted 1 too much, so add 1 back.
- 44 + 1 = 45.
45
Use a direct split instead
Find 53 − 24.
- 24 is not especially close to 10 or 20 for everyone.
- Subtract 20: 53 − 20 = 33.
- Subtract the remaining 4: 33 − 4 = 29.
- Check: 29 + 24 = 53.
29
Spot a wrong adjustment
A learner writes 28 + 19 = 28 + 20 + 1.
- Adding 20 adds one more than adding 19.
- Correct by subtracting 1, not adding it.
- 28 + 20 − 1 = 47.
- The incorrect expression equals 49 and is not equivalent.
47
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: For 26 + 19, changing 19 to 20 makes a sum one too large; subtract 1.
Why this fails: For 60 − 29, subtracting 30 already took away one extra; add 1 back.
Why this fails: Breaking into tens and ones can be safer and faster when there is no nearby friendly number.
Why this fails: Estimate and use the inverse: for 54 − 26 = 28, check 28 + 26 = 54.
Foundation · Make friendly numbers
Foundation · Make friendly numbers practice
Use a ten or a round number without changing the total.
1. Calculate 8 + 7. Show one reliable strategy.
2. Calculate 29 + 6. Show one reliable strategy.
3. Calculate 37 + 5. Show one reliable strategy.
4. Calculate 46 + 14. Show one reliable strategy.
5. Calculate 58 + 12. Show one reliable strategy.
6. Calculate 23 + 19. Show one reliable strategy.
7. Calculate 31 + 28. Show one reliable strategy.
Subtraction · Compensate carefully
Subtraction · Compensate carefully practice
Subtract a close round number, then correct the difference.
8. Calculate 51 − 9. Show one reliable strategy.
9. Calculate 70 − 19. Show one reliable strategy.
10. Calculate 64 − 29. Show one reliable strategy.
11. Calculate 83 − 38. Show one reliable strategy.
12. Calculate 60 − 27. Show one reliable strategy.
13. Calculate 72 − 35. Show one reliable strategy.
14. Calculate 45 − 16. Show one reliable strategy.
Choose an efficient strategy
Choose an efficient strategy practice
Decide whether to compensate, split, double or calculate directly.
15. Calculate 49 + 8. Show one reliable strategy.
16. Calculate 34 + 23. Show one reliable strategy.
17. Calculate 46 + 27. Show one reliable strategy.
18. Calculate 29 + 29. Show one reliable strategy.
19. Calculate 25 + 25. Show one reliable strategy.
20. Calculate 52 − 18. Show one reliable strategy.
21. Calculate 64 − 21. Show one reliable strategy.
Explain and catch misleading shortcuts
Explain and catch misleading shortcuts practice
Check the direction of each adjustment; a shortcut is optional, not always better.
22. Calculate 47 + 36. Show one reliable strategy.
23. Calculate 38 + 27. Show one reliable strategy.
24. Calculate 54 − 26. Show one reliable strategy.
25. Calculate 42 + 19. Show one reliable strategy.
26. Calculate 61 − 38. Show one reliable strategy.
27. Calculate 9 + 8 + 1. Show one reliable strategy.
28. Calculate 72 − 36. Show one reliable strategy.
Answer key
- 1. 15
Move 2 from 7 to 8: 10 + 5 = 15. - 2. 35
29 needs 1 to reach 30; 30 + 5 = 35. - 3. 42
37 + 3 = 40, then add 2. - 4. 60
46 needs 4 to reach 50; 50 + 10 = 60. - 5. 70
58 + 2 = 60; 60 + 10 = 70. - 6. 42
23 + 20 − 1 = 42. - 7. 59
31 + 30 − 2 = 59. - 8. 42
51 − 10 + 1 = 42 because 10 is one too many. - 9. 51
70 − 20 + 1 = 51. - 10. 35
64 − 30 + 1 = 35. - 11. 45
83 − 40 + 2 = 45. - 12. 33
Subtract 20, then 7: 60 − 20 − 7 = 33. - 13. 37
72 − 30 − 5 = 37. - 14. 29
45 − 10 − 6 = 29. - 15. 57
Use 50 + 7 = 57; give 1 from 8 to 49. - 16. 57
Split 23 into 20 and 3: 34 + 20 + 3 = 57. - 17. 73
46 + 30 − 3 = 73. - 18. 58
Double 30 and take away 2: 60 − 2 = 58. - 19. 50
Doubling 25 is straightforward: 50. - 20. 34
52 − 20 + 2 = 34. - 21. 43
64 − 20 − 1 = 43. - 22. 83
47 + 30 + 6 = 83; a simple split may beat a complicated shortcut. - 23. 65
38 + 30 − 3 = 65, since adding 30 is 3 too much. - 24. 28
54 − 30 + 4 = 28. Compensate by adding back 4. - 25. 61
42 + 20 − 1 = 61; the adjustment must be subtracted. - 26. 23
61 − 40 + 2 = 23; subtracting 40 took away 2 too much. - 27. 18
Pair 9 and 1 to make 10; then add 8. - 28. 36
72 − 30 − 6 = 36. Direct subtraction is also valid.