Expressions that look alike but are not

Are 5u and 5 + u equal? They look alike, but 5u means 5 times u and 5 + u means 5 more than u.

Test them:

Equal expressions must match for every value. These don't, so they are not equal.

Brackets matter too. 10y − 3 is 3 less than 10 times y; 10(y − 3) is 10 times (3 less than y). For y = 2: 17 against −10.

But simplifying gives expressions that really are equal. 4(x + y) − y = 4x + 4y − y = 4x + 3y.

A picture shows eight 3s and two each of r and s. Add it three ways: row by row, (4 × 3) + (r + s) + (r + s) + (4 × 3); like with like, (8 × 3) + (r + r) + (s + s); or the top half doubled, 2 × (4 × 3 + r + s). They look different, but all simplify to 2r + 2s + 24.

Formula Detective: a number machine takes two numbers at the top of a Y and gives one at the bottom, the same way every time. With 5 and 2 it gives 8. Guess: twice the first minus the second, 2a − b: 2 × 5 − 2 = 8 ✓. Test your guess on every other pair the machine was given.