Properties of operations
The idea
Numbers follow a few friendly rules, and knowing them is like having permission slips to rearrange your work. Order does not change a sum or a product: 3 + 8 equals 8 + 3, and 3 × 8 equals 8 × 3 — these are often called turn-around facts. Grouping does not matter either: adding three numbers, you may combine whichever pair you like first. And you can break a number apart, work on the pieces, then put the results together, like finding 4 × 27 as 4 × 20 plus 4 × 7.
The point of these rules is not to memorize names — it is to turn hard arithmetic into easy arithmetic. Faced with 38 + 25 + 12, a sharp eye spots that 38 and 12 make a tidy 50, and the problem collapses. But be careful where the permission slips apply: they work for addition and multiplication only. Subtraction and division refuse to be turned around — 9 − 5 is not the same as 5 − 9 — so never reorder those without thinking.
Worked example
Add 38 + 25 + 12 in your head by choosing a smarter order. What is the total?
- Scan for numbers that fit together nicely. The ones digits of 38 and 12 are 8 and 2, and 8 + 2 = 10, so 38 + 12 makes a tidy 50.
- Order does not change a sum, so you are allowed to add 38 + 12 first even though 25 sits between them: 38 + 12 = 50.
- Finish with the easy part: 50 + 25 = 75.
- Check the slow way, left to right: 38 + 25 = 63, then 63 + 12 = 75. Same total — rearranging changed the work, not the answer.
Answer. The total is 75, found faster by pairing 38 with 12 first.
Check your understanding
- Why is it safe to add numbers in any order but not to subtract them in any order?
- How does breaking 6 × 14 into 6 × 10 and 6 × 4 make the problem easier, and why does it still give the right answer?
- What pairs of numbers do you hunt for when you want to make a long addition easy in your head?
- How would you explain a turn-around fact like 3 × 8 = 8 × 3 using a picture of rows and columns?