Add in any order

The idea

Since letter-numbers stand for numbers, the terms of an algebraic expression can be swapped and grouped, that is, added in any order, without changing its value.

Round the rectangle

A rectangle has length l and breadth b. Walking once round it gives the perimeter: l + b + l + b.

Would l + l + b + b (both lengths first) give the same total?

• Yes, the same total
• No, the order changes the total
• Only when l and b are equal

Swap the terms

Tap two terms to swap them. The numbers are in, so you can watch the value. Then put the same letters side by side.

Any order, any grouping

Letter-numbers stand for numbers, and numbers can be added in any order and grouped in any way. So the terms of an expression with letter-numbers can be too.

A term keeps its sign when it moves: 7p − 3q has the terms 7p and −3q, so −3q + 7p is the same.

Since letter-numbers stand for numbers, the terms of an algebraic expression can be swapped and grouped, that is, added in any order, without changing its value.

• Order | With l = 3 and b = 4
• l + b + l + b | 3 + 4 + 3 + 4 = 14
• l + l + b + b | 3 + 3 + 4 + 4 = 14
• b + l + b + l | 4 + 3 + 4 + 3 = 14

Notes

Since letter-numbers stand for numbers, the terms of an algebraic expression can be swapped and grouped: added in any order, without changing its value.

Check yourself

With *l* = 3 and *b* = 4, find the value of l + b + l + b.

Answer: 14

3 + 4 + 3 + 4 = 14.

Swap and group to find a + 9 + a + 1 when *a* = 5.

Answer: 20

5 + 9 + 5 + 1 = (5 + 5) + (9 + 1) = 10 + 10 = 20.

Which expression has the same value as p + q + p, whatever *p* and *q* stand for?

Swap the terms of 7p − 3q. Which expression is the same?

• p + q + q. That has two q’s, but the original has two p’s.
• p + p. That leaves out the term q.
• p + p + q — correct. Yes! The same terms, added in another order.
• 3q − 7p. That changes the signs of both terms.
• 7p + 3q. That changes −3q into +3q.
• −3q + 7p — correct. Yes! Each term keeps its sign when it moves.