Year 7 · Unit 1 Number systems and place value · Lesson 6 of 7

Which calculation goes first?

5 + 3 × 4 can only have one value if everyone reads it the same way. Step through an expression and see which operation each rule picks next.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. What is 3²?
    Answer9. Two factors of 3: 3 × 3.
  2. Write 36 as a product of primes in index form.
    Answer2² × 3². Split 36 = 4 × 9, then split 4 and 9.
  3. What is the lowest common multiple of 4 and 10?
    Answer20. The multiples of 10 are 10, 20, ... and 4 goes into 20.
Words used in this lesson
expression
numbers and operations written together, with no equals sign. For example: 5 + 3 × 4.
evaluate
work out the value. For example: evaluate 5 + 3 × 4 to get 17.
operation
something done to numbers: add, subtract, multiply, divide. For example: × is an operation.
brackets
symbols that show which part to do first. For example: (5 + 3) × 4.
power
repeated multiplication of a number by itself. For example: 3² = 9.
left to right
the order for operations on the same level. For example: 10 − 4 − 3 = 3.

One expression, one value

Entry to a fair costs 5 dirhams, and snacks cost 4 dirhams each. A group buys 3 snacks, so the total is 5 + 3 × 4. One person says 17 and another says 32. Only one can be right. Mathematicians agreed an order so that everyone reads an expression the same way:

  1. Brackets first (the inside of the innermost brackets first).
  2. Powers (indices, exponents).
  3. Multiplication and division, together, from left to right.
  4. Addition and subtraction, together, from left to right.

You may know this as BEDMAS, BIDMAS, BODMAS or PEMDAS. They are the same rule with different words. The pairing matters: multiplication and division share a level, and so do addition and subtraction.

Predict first: before pressing Next step on 6 + 8 ÷ 2 + 3 × 5, say which operation goes first and what the whole expression equals. Then check.

6 + 8 ÷ 2 + 3 × 5
0 of 4operations done
50a common wrong answer

6 + 8 ÷ 2 + 3 × 5: nothing has been done yet. Which operation goes first, and why?

Show me the steps
  1. 6 + 8 ÷ 2 + 3 × 5: divide first, 8 ÷ 2 = 4; multiply, 3 × 5 = 15; then add, 6 + 4 + 15 = 25. Working strictly left to right gives 50.
  2. (6 + 2) × 3: brackets first, 6 + 2 = 8, then 8 × 3 = 24. Ignoring the brackets gives 6 + 2 × 3 = 12, which is wrong.
  3. 20 − 4 × 2²: power first, 2² = 4; multiply, 4 × 4 = 16; subtract, 20 − 16 = 4. Working strictly left to right gives 64.
  4. 48 ÷ 6 × 2: the same level, so left to right: 48 ÷ 6 = 8, then 8 × 2 = 16. Not 48 ÷ 12 = 4.
  5. 30 − (2 + 3)² ÷ 5: bracket, 5; power, 25; divide, 5; subtract, 25. Subtracting before dividing, 30 − 25 = 5 and then 5 ÷ 5 = 1.
  6. 2 × (3 + 4) + 5²: bracket, 7; power, 25; multiply, 2 × 7 = 14; add, 14 + 25 = 39. Here the order of the levels makes no difference.

Work in this order: brackets, powers, multiply and divide, add and subtract. Within a level, work from left to right.

Brackets are the way to change the order. 5 + 3 × 4 = 17, but (5 + 3) × 4 = 32. A bracket says “do this part first”.

A common mistake: “multiplication always comes before division.” They share a level. 48 ÷ 6 × 2 is 8 × 2 = 16, going left to right. Doing the multiplication first gives 48 ÷ 12 = 4, which is wrong.

Why brackets change the value

Think of counters. 5 + 3 × 4 is 5 single counters and 3 groups of 4. (5 + 3) × 4 is 8 groups of 4. Same numbers, different groups, different total. Move the sliders and compare the two panels.

5
3
4
17a + b × c
32(a + b) × c

5 + 3 × 4: 5 single counters and 3 groups of 4 make 17. (5 + 3) × 4: 8 groups of 4 make 32.

Show me the steps
  1. a = 5, b = 3, c = 4: 5 + 3 × 4 = 5 + 12 = 17, but (5 + 3) × 4 = 8 × 4 = 32.
  2. a = 2, b = 2, c = 3: 2 + 2 × 3 = 8, but (2 + 2) × 3 = 12.
  3. a = 1, b = 4, c = 1: both are 5. When each group is a single counter, grouping makes no difference.

Worked example

Work out 5 × (8 − 3)² − 10 ÷ 2.

  1. Brackets: 8 − 3 = 5, so the expression is 5 × 5² − 10 ÷ 2.
  2. Powers: 5² = 25, so 5 × 25 − 10 ÷ 2.
  3. Multiply and divide, left to right: 5 × 25 = 125 and 10 ÷ 2 = 5, so 125 − 5.
  4. Subtract: 125 − 5 = 120.

Your go. The first two steps are given. Work out 4 + 12 ÷ (5 − 2) × 3².

  1. Brackets: 5 − 2 = 3, so 4 + 12 ÷ 3 × 3².
  2. Powers: 3² = 9, so 4 + 12 ÷ 3 × 9.
  3. Now do the multiplication and division from left to right, then the addition, then check.
Check the answer

12 ÷ 3 = 4, then 4 × 9 = 36, then 4 + 36 = 40.

Check before you go on

1. What is 5 + 3 × 4?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. What is 10 − 2 × 3?

3. What is (10 − 2) × 3?

Level 2 · Building

4. What is 36 ÷ 6 × 3?

5. Bilal has 4 loose biscuits and 2 trays. Each tray is a 3 by 3 square of biscuits, so the total is 4 + 3² × 2. How many biscuits is that?

6. Each of 4 friends buys a ticket costing 2 dirhams and a snack costing 3 dirhams. The total is written 2 + 3 × 4, which gives 14, not the right total. Where do the brackets go to make it 20?

Level 3 · Stretch

7. What is 7 + 2 × (9 − 4)² ÷ 10?

8. What is 100 − 5 × 2² + 6 ÷ 3?

Something to think about

Put brackets into 12 − 4 − 2 in two different ways. Do they give the same answer? What happens to 12 + 4 + 2 with the same brackets? What does that suggest about when brackets matter?

One way to think about it

(12 − 4) − 2 = 6 but 12 − (4 − 2) = 10, so the brackets change the answer. For 12 + 4 + 2 both ways give 18, and the same is true of a string of multiplications. Brackets round only additions, or only multiplications, change nothing; with subtraction or division they can, because those depend on which number comes first.

Exit ticket

9. Priya works out 24 ÷ 4 × 2 and gets 3. Which reply is right?