Which operation first?

The idea

Without a context, an expression with more than one operation can be evaluated in more than one way, giving different values, so brackets and terms are used to specify how it is to be evaluated.

Shalini and the toys

Read these two sentences. They have the same words. Only a comma is different. (a) Shalini sat next to a friend with toys. (b) Shalini sat next to a friend, with toys.

Do they mean the same thing?

• Yes, the same
• No, the comma changes who has the toys
• We cannot tell

Choose the order

Each expression has two operations. Tap the one you want to do first. Then find a different answer by choosing the other order.

The order matters

30 + 5 × 4 gave 140 (add first) or 50 (multiply first). 100 − 15 + 56 gave 141 or 29. Without the story we cannot tell which order is meant. Mallesh’s story says 50, and Irfan’s change must be 29.

An expression with more than one operation can be evaluated in more than one way, giving different values. So mathematics uses brackets and terms to say exactly how to evaluate it, like punctuation in language.

Notes

An expression with more than one operation can be evaluated in more than one way, with different values. Brackets and terms tell us exactly how.

Check yourself

Purna got 140 and Mallesh got 50 for 30 + 5 × 4. Why could both get an answer?

Work out 10 − 4 + 2 by doing the + first (4 + 2, then take it from 10). What do you get?

Answer: 4

4 + 2 = 6 and 10 − 6 = 4. Doing the − first gives 8. The same expression, two answers.

Which expression can be worked out in only one way?

• One of them added wrongly.. Both calculated correctly: 35 × 4 = 140 and 30 + 20 = 50. They did the operations in different orders.
• The expression does not say which operation to do first. — correct. Yes! (30 + 5) × 4 = 140 and 30 + (5 × 4) = 50 are both ways of reading it.
• Adding and multiplying always give the same answer.. They do not: 140 and 50 are different.
• 7 + 8 — correct. Yes! It has only one operation, so there is nothing to order.
• 6 + 2 × 3. Add first gives 24. Multiply first gives 12. There are two ways.
• 20 − 5 + 3. Subtract first gives 18. Add first gives 12. There are two ways.