Example 1 · Grades 1–2
Pairs that make ten
What is 6 + 7 + 8 + 2 + 3 + 4?
- Look for pairs that make 10. Here there are three of them, 6 + 4, 7 + 3 and 8 + 2.
- Three tens make 30.
- So 6 + 7 + 8 + 2 + 3 + 4 = 30, with no carrying at all.
Answer 30.
Numbers with a trickGrades 1–6
Arithmetic with a trick
What is 19 + 23 + 81 + 77? Added in order, it is three sums with carrying; paired as 19 + 81 and 23 + 77, it is 100 + 100. In these tasks the arithmetic is easy once the right order is found, and “I thought of a number” puzzles are solved by undoing the steps from the end. The skill is to look before counting.
- The main move
- Look for the easy order before counting
- What it teaches
- To regroup a sum or a product, and to undo a chain of steps from the end
(19 + 81) + (23 + 77) 100 + 100
The rules of clever counting
Any order
Numbers can be added or multiplied in any order and grouped any way: 25 × 9 × 4 is 25 × 4 × 9, which is 100 × 9.
Round numbers
Look for pairs that make 10, 100 or 1000: 36 + 64, 25 × 4, 125 × 8. A round number is easy to carry on with.
Undo from the end
In “I thought of a number”, go back from the result and undo each step: what was added is taken away, and what was doubled is halved.
Look before you count
The same sum can be long or short. Before counting, look for the pairs and groups that make it round.
| The task | The easy way | The answer |
|---|---|---|
| 19 + 23 + 81 + 77 | (19 + 81) + (23 + 77) | 200 |
| 4 × 19 × 25 | (4 × 25) × 19 | 1900 |
| 1 + 2 + 3 + … + 10 | five pairs of 11, such as 1 + 10 and 2 + 9 | 55 |
How to solve: look for round numbers
Before counting, look at the whole task. Pair numbers that make round ones, put factors that give 10 or 100 next to each other, and in a chain of steps start from the end and undo them one by one. The examples below go from the simplest to an olympiad's.
- Look at the whole task first
- Regroup into round numbers, or undo from the end
- Check the answer another way
Example 2 · Grades 3–4
A number to find
Masha thought of a number. She doubled it, added 7 and divided the result by 3. She got 9. What number did Masha think of?
- Go from the end and undo each step, the last one first.
- She divided by 3 and got 9, so before that she had 9 × 3 = 27. She added 7, so before that she had 27 − 7 = 20.
- She doubled the number, so it was 20 : 2 = 10. Check: 10 × 2 + 7 = 27, and 27 : 3 = 9.
Answer 10.
Example 3 · Grades 5–6
Two big products
What is 2026 × 2026 − 2027 × 2025? Find it without multiplying the big numbers.
- Both products hide 2026 × 2025: 2026 × 2026 is 2026 × 2025 and one more 2026.
- And 2027 × 2025 is 2026 × 2025 and one more 2025.
- So the difference is 2026 − 2025 = 1, and nothing big was multiplied.
Answer 1.
Where children go wrong
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
- Trap
Stopped halfway through the solution
The work stops before the end: two of the three pairs are added and the third forgotten, or one step of the chain is undone and its result given as the answer.
What to say
“Is every number used, and every step undone?”
- Trap
Used the wrong operation
A step is undone with the same action instead of the opposite one: 7 is added again instead of taken away, or 9 is divided by 3 instead of multiplied.
What to say
“What undoes adding? What undoes dividing?”
- Trap
Took a number from the question as the answer
A number from the task becomes the answer: the 9 Masha got, or the 7 she added, instead of the number she thought of.
What to say
“Is this the number she got, or the one she thought of?”
How to help at home
Pairs to ten
Write six or eight small numbers on cards and ask the child to find the pairs that make 10 before adding them all up.
Guess my number
Play “I thought of a number” both ways: you give the steps and the result and the child finds the number, then the child makes up the steps.
Clever shopping
In a shop, add up prices in a clever order: 99 is 100 less 1, and 49 and 51 make 100. Round first, put right after.