Group the terms, add in any order

The idea

(Term 1 + Term 2) + Term 3 = Term 1 + (Term 2 + Term 3), the associative property of addition; with swapping, this means the terms of an expression having only additions can be added in any order.

Three terms

Look at (−7) + 10 + (−11). You can add the first two and then the third. Or add the last two first, and then add that to the first.

Do you think both ways give the same sum?

• Yes, always the same
• No, they will differ
• It depends on the numbers

Every order, every grouping

Swap neighbours with ⇄ to change the order. Choose which two terms to group. Find all 6 orders and both groupings.

Always the same

(−7 + 10) + (−11) = 3 + (−11) = −8 (−7) + (10 + (−11)) = (−7) + (−1) = −8 Grouping did not change the sum. Together with swapping, the terms can be added in any order.

(Term 1 + Term 2) + Term 3 = Term 1 + (Term 2 + Term 3). This is the associative property of addition. With swapping, the terms of an expression with only additions can be added in any order.

Notes

(Term 1 + Term 2) + Term 3 = Term 1 + (Term 2 + Term 3), the associative property of addition. With swapping: the terms of an expression with only additions can be added in any order.

Check yourself

Add in any order you like: (−3) + 8 + (−5) = ?

Answer: 0

Any order gives 0. For example 8 + (−3) = 5, then 5 + (−5) = 0.

Which order makes 25 + 48 + 75 easiest to add in your head?

Add in any order: 8 + (−3) + 2 + (−7) = ?

Answer: 0

8 + 2 = 10 and (−3) + (−7) = −10, so the total is 0.

• (25 + 48) + 75. It works, but 25 + 48 = 73 is not a nice number to carry on with.
• (25 + 75) + 48 — correct. Yes! 25 + 75 = 100, and 100 + 48 = 148. Terms can be added in any order.
• The value changes with the order, so we must go left to right. The value never changes, so you can pick the easiest order.