Swapping and grouping terms

Does the order in which we add terms change the answer?

Madhu flies a drone from a terrace. It goes 6 m up, then 4 m down. It ends 6 − 4 = 2 m above the terrace. As terms: 6 + (−4) = 2.

Swap the terms: (−4) + 6. Down 4 first, then up 6: still 2. The answer did not change.

You knew this for positive numbers. Now you see it works with negative terms too. (You can explain why with the Token Model of integers from Class 6.) So Term 1 + Term 2 = Term 2 + Term 1.

It works with more terms too. Add (−7) + 10 + (−11) in any order or grouping: (−7 + 10) + (−11), or −7 + (10 + (−11)), or (−7 + (−11)) + 10. Every way gives −8. So we can rearrange and regroup terms freely.

Real life isn't always like this! A cap and shoes can go on in either order. But socks and shoes can't: socks must go first. Adding terms is like the cap and shoes: the order never matters.