Work left to right
Add and subtract two- and three-digit numbers in your head by starting with the biggest part, the way you read the number, instead of the paper method school taught you.
THE IDEA
Biggest part first
47 is really 40 + 7, and 38 is 30 + 8. That's place value. And because addition gives the same result in any order, you're free to add the tens first and the ones after. So start where the money is: the tens.
47 + 38: the tens give 40 + 30 = 70. The ones give 7 + 8 = 15. Together, 70 + 15 = 85.
Why school taught it the other way
Right to left (paper)
Start at the ones, write a digit, carry the rest. Great on paper, because the paper remembers the carries for you. In your head, you hold carries and half-built digits, and the answer comes out backwards: 5… then 8.
Left to right (head)
Start at the tens, the way you say the number. Your first step, 70, is already a good estimate. Every later step only adjusts it. If you get interrupted, you still know it's 'seventy-something'.
47 + 38
tens: 40 + 30 = 70
ones: 7 + 8 = 15
join: 70 + 15 = 8585The ones gave 15, which is more than 9. That's fine: 15 is just 10 + 5, so 70 becomes 80 and 5 is left over.
Check yourself
Do it in your head, tens first: 56 + 27 = ?
- 73
- 83
- 93
- 81
Show the answer
83
Yes. 50 + 20 = 70, then 6 + 7 = 13, and 70 + 13 = 83.
Even lighter: keep the first number whole
- Start with 368, untouched
Only split the number you're adding: 457 = 400 + 50 + 7. Now there is just one running total to hold in your head.
- Add the hundreds: 368 + 400 = 768
Only the hundreds digit moves. 68 stays as it is.
- Add the tens: 768 + 50 = 818
60 + 50 = 110, so you cross into the 800s. Say the new total to yourself: 'eight eighteen'.
- Add the ones: 818 + 7 = 825
Done. Three small steps, and at every moment you held only one number.
Check yourself
Work each one out left to right, then match it to its answer
Show the answer
- 34 + 25 → 59
- 68 + 24 → 92
- 76 + 47 → 123
- 135 + 48 → 183
SAME IDEA
Subtraction, big part first
Keep the first number whole and take the second one away in pieces, biggest piece first. Taking away 37 is the same as taking away 30 and then 7.
82 − 37: first 82 − 30 = 52, then 52 − 7 = 45. For the last step, think 52 − 2 = 50, then 5 more off: 45.
Check yourself
Negar has 93 in her wallet and pays 58 for lunch. How much is left?
- 35
- 25
- 45
- 33
Show the answer
35
Right. 93 − 50 = 43, then 43 − 8 = 35.
Check yourself
In 586 + 347 the hundreds add up to 800, so the final answer must be somewhere in the 800s.
Show the answer
False
False. The leftovers are 86 + 47 = 133, which is more than 100, so they push the total to 933. The first step gives you a floor, not a ceiling: here the answer had to land between 800 and 1,000.
Make it a habit
- Say the running total to yourself after every step: 'seven forty-eight… twelve forty-eight…'. A number you've said is a number you won't drop.
- Keep the first number whole; split only the second.
- Practise on real numbers around you today: two prices, two page numbers, the minutes left on two timers.
Check yourself
The hard one. Omid's two invoices are 748 and 586. In your head: 748 + 586 = ?
- 1,234
- 1,324
- 1,344
- 1,334
Show the answer
1,334
Exactly. 748 + 500 = 1,248, then + 80 = 1,328, then + 6 = 1,334.
Lesson recap
- Right-to-left with carrying is a paper method. In your head, go left to right, biggest part first.
- It works because of place value (47 = 40 + 7) and because addition can be done in any order.
- Keep the first number whole and add or subtract the second one in pieces: 82 − 37 is 82 − 30, then − 7.
- Your first step is a floor for the answer, and saying each running total keeps you from losing it.