Chapter 6 of 8

All 8 chapters
  1. 1Multiplying and dividing in your head, readunderstood
  2. 2Short multiplication, readunderstood
  3. 3Long multiplication, readunderstood
  4. 4Short division, readunderstood
  5. 5Long division, readunderstood
  6. 6Order of operations, readunderstood
  7. 7Estimate, then check, readunderstood
  8. 8Choosing what to do, readunderstood

M3.2.6 Chapter 6

Order of operations

What is 3 + 4 × 2?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Work out 3 + 4 × 2 in the right order
  • Use brackets to change the order of working
  • Put in brackets to make 2 + 3 × 4 equal 20

Ana has 3 loose beans and 4 bags. Each bag holds 2 beans. She writes the calculation 3 + 4 × 2. How many beans does she have?

A calculation with more than one operation must give one answer, for everybody. So we agree on an order of operations. First, count the beans in the picture.

Draw it first

Count the beans, then write the calculation

  1. Three loose beans, labelled 3 beans, and four bags with 2 beans in each, labelled 4 bags of 2. Under them is the calculation 3 + 4 × 2 = 3 + 8 = 11.3 beans

    1 Ana has 3 loose beans.

  2. Three loose beans, labelled 3 beans, and four bags with 2 beans in each, labelled 4 bags of 2. Under them is the calculation 3 + 4 × 2 = 3 + 8 = 11.3 beans4 bags of 2

    2 She also has 4 bags with 2 beans in each. That is 4 × 2 = 8 beans in the bags.

  3. Three loose beans, labelled 3 beans, and four bags with 2 beans in each, labelled 4 bags of 2. Under them is the calculation 3 + 4 × 2 = 3 + 8 = 11.3 beans4 bags of 23 + 4 × 2 = 3 + 8 = 11

    3 Add the loose beans and the bags. 3 + 8 = 11. So 3 + 4 × 2 = 11.

The beans match the agreed order. The bags are counted as 4 × 2 first, and the loose beans are added after. So multiplying and dividing come before adding and taking away.

A table with the columns first, next and last. The one row reads brackets, × and ÷, and + and −.firstnextlastbrackets× and ÷+ and −
The order of working. Inside each box, work from left to right.

Brackets come first of all. Work out what is inside them before anything else. Multiplying and dividing share one step. Adding and taking away share the next. Inside a step, work from left to right.

Worked example

Work out 3 + 4 × 2.

  1. There are no brackets.
  2. Multiply first. 4 × 2 = 8.
  3. Now the calculation is 3 + 8. Add. 3 + 8 = 11.

Answer: 11

Worked example

Work out 20 − 12 ÷ 4 + 2.

  1. There are no brackets. Divide first. 12 ÷ 4 = 3.
  2. Now the calculation is 20 − 3 + 2. Subtracting and adding share a step, so work from left to right.
  3. 20 − 3 = 17.
  4. 17 + 2 = 19.

Answer: 19

See it three ways

With things

Put the 3 white beans and 4 black beans in one bag. Make 2 bags like that.

As a picture

Two bags, labelled bag 1 and bag 2. Each bag holds 3 white beans and 4 black beans, 14 beans in all.bag 1bag 2

In symbols

(3 + 4) × 2 = 7 × 2 = 14. The brackets say: add first.

Brackets change the order. 3 + 4 × 2 is 11, but (3 + 4) × 2 is 14. In the second calculation the adding is inside the brackets, so it happens first.

Worked example

Work out (8 − 2) × 3 + 12 ÷ 4.

  1. Brackets first. 8 − 2 = 6.
  2. The calculation is now 6 × 3 + 12 ÷ 4. Multiply and divide next.
  3. 6 × 3 = 18.
  4. 12 ÷ 4 = 3.
  5. Add. 18 + 3 = 21.

Answer: 21

Try it

Make 14 and make 20 from the same cards

You need: 3 small pieces of paper with 2, 3 and 4 written on them, a pencil.

  1. Write the calculation 2 + 3 × 4. Work it out in the right order. Use 3 bags of 4 beans and 2 loose beans if it helps.
  2. Now put brackets round 2 + 3, to make (2 + 3) × 4. Work it out.
  3. Compare the two answers.

What you will see: The cards are the same, but the answers are different. 2 + 3 × 4 = 14. (2 + 3) × 4 = 20. The brackets changed the order.

If you cannot try it, this is what happens.
Two bars drawn to the same scale. The top bar has four parts of 5, labelled 2 + 3, and its total is (2 + 3) × 4 = 20. The bottom bar has one part of 2 and three parts of 4, and its total is 2 + 3 × 4 = 14.2 + 32 + 32 + 32 + 3(2 + 3) × 4 = 2024442 + 3 × 4 = 14

A wrong idea many people have

Always work from left to right.

Why it feels right: We read words from left to right, and the first signs you learned, + and −, are worked that way.

The test that breaks it: Try 3 + 4 × 2 from left to right. 3 + 4 = 7 and 7 × 2 = 14. But Ana's beans make 11, and the agreed order gives 11 too. Left to right gives 14, which does not match the beans.

What is true: Brackets first. Then multiply and divide. Then add and take away. Left to right works only inside a step.

Key words

order of operations
the agreed order for working out a calculation: brackets, then × and ÷, then + and −.
brackets
the marks ( ) round a part of a calculation that is worked out first.

Back to your guess

The answer is B: 11, because multiplying comes before adding: 4 × 2 = 8, then 3 + 8 = 11.

The answer is 11. Multiplying comes before adding, so 4 × 2 = 8 first, and then 3 + 8 = 11. Everyone uses the same order, so every calculation has just one answer.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is 5 + 3 × 4?

    A bar with four parts, labelled 5, 4, 4 and 4.5444
    Tap your answer.

    1. Not this one. You multiplied everything: 5 × 3 × 4. The first sign is +, so add 5 to the product 3 × 4. Go to M3.2.6. Try another.

    2. Not this one. You worked from left to right: 5 + 3 = 8, then 8 × 4 = 32. Multiply first: 3 × 4 = 12. Go to M3.2.6. Try another.

    3. Not this one. You added all the numbers: 5 + 3 + 4. The × sign means multiply, and that comes before adding. Go to M3.2.6. Try another.

    4. Yes, that is right. Multiply first. 3 × 4 = 12. Then add. 5 + 12 = 17.

    See the answer

    17

    Multiply first. 3 × 4 = 12. Then add. 5 + 12 = 17.

    • If you chose A: You multiplied everything: 5 × 3 × 4. The first sign is +, so add 5 to the product 3 × 4. Go to M3.2.6.
    • If you chose B: You worked from left to right: 5 + 3 = 8, then 8 × 4 = 32. Multiply first: 3 × 4 = 12. Go to M3.2.6.
    • If you chose C: You added all the numbers: 5 + 3 + 4. The × sign means multiply, and that comes before adding. Go to M3.2.6.

  2. What is 18 ÷ 3 × 2?

    Tap your answer.

    1. Yes, that is right. Divide and multiply share a step, so work from left to right. 18 ÷ 3 = 6. Then 6 × 2 = 12.

    2. Not this one. You multiplied first: 3 × 2 = 6, then 18 ÷ 6 = 3. Multiplying and dividing share a step, so work from left to right. Go to M3.2.6. Try another.

    3. Not this one. You multiplied 18 × 3 × 2. The ÷ sign means divide: 18 ÷ 3 = 6 first. Go to M3.2.6. Try another.

    See the answer

    12

    Divide and multiply share a step, so work from left to right. 18 ÷ 3 = 6. Then 6 × 2 = 12.

    • If you chose B: You multiplied first: 3 × 2 = 6, then 18 ÷ 6 = 3. Multiplying and dividing share a step, so work from left to right. Go to M3.2.6.
    • If you chose C: You multiplied 18 × 3 × 2. The ÷ sign means divide: 18 ÷ 3 = 6 first. Go to M3.2.6.

  3. What is (6 + 2) × 5?

    Tap your answer.

    1. Not this one. You added all the numbers. The sign outside the brackets is ×. Work out the brackets, 8, then multiply by 5. Go to M3.2.6. Try another.

    2. Not this one. You ignored the brackets and multiplied first: 2 × 5 = 10, then 6 + 10. Brackets come first: 6 + 2 = 8. Go to M3.2.6. Try another.

    3. Yes, that is right. Brackets first. 6 + 2 = 8. Then 8 × 5 = 40.

    See the answer

    40

    Brackets first. 6 + 2 = 8. Then 8 × 5 = 40.

    • If you chose A: You added all the numbers. The sign outside the brackets is ×. Work out the brackets, 8, then multiply by 5. Go to M3.2.6.
    • If you chose B: You ignored the brackets and multiplied first: 2 × 5 = 10, then 6 + 10. Brackets come first: 6 + 2 = 8. Go to M3.2.6.

  4. Where do the brackets go to make 2 + 3 × 4 equal 20?

    Tap your answer.

    1. Not this one. Brackets round 3 × 4 change nothing, because multiplying already comes first. The answer stays 14. Go to M3.2.6. Try another.

    2. Yes, that is right. The adding must happen first, so the brackets go round 2 + 3. Then 5 × 4 = 20.

    3. Not this one. Brackets round the whole calculation change nothing. The order inside is the same, so the answer is 14. Go to M3.2.6. Try another.

    See the answer

    (2 + 3) × 4 = 20

    The adding must happen first, so the brackets go round 2 + 3. Then 5 × 4 = 20.

    • If you chose A: Brackets round 3 × 4 change nothing, because multiplying already comes first. The answer stays 14. Go to M3.2.6.
    • If you chose C: Brackets round the whole calculation change nothing. The order inside is the same, so the answer is 14. Go to M3.2.6.

  5. The picture shows 2 bags. Each bag holds 3 white beans and 4 black beans. Which calculation gives all the beans?

    Two bags, labelled bag 1 and bag 2. Each bag holds 3 white beans and 4 black beans.bag 1bag 2
    Tap your answer.

    1. Yes, that is right. Each bag holds 3 + 4 = 7 beans. Two bags hold 7 × 2 = 14 beans. The brackets show that 3 + 4 is done first.

    2. Not this one. That is 3 loose beans and 4 bags of 2. Here every bag holds both white and black beans. Go to M3.2.6. Try another.

    3. Not this one. Only the white beans were doubled. The 4 black beans are in both bags too, so add 3 and 4 first. Go to M3.2.6. Try another.

    See the answer

    (3 + 4) × 2 = 14

    Each bag holds 3 + 4 = 7 beans. Two bags hold 7 × 2 = 14 beans. The brackets show that 3 + 4 is done first.

    • If you chose B: That is 3 loose beans and 4 bags of 2. Here every bag holds both white and black beans. Go to M3.2.6.
    • If you chose C: Only the white beans were doubled. The 4 black beans are in both bags too, so add 3 and 4 first. Go to M3.2.6.

  6. Work out 8 + 6 × 3.

    See the answer

    26

    Multiply first. 6 × 3 = 18. Then 8 + 18 = 26.

  7. Work out (7 + 5) × 4.

    See the answer

    48

    Brackets first. 7 + 5 = 12. Then 12 × 4 = 48.

  8. Work out 30 − 12 ÷ 3 + 4.

    See the answer

    30

    Divide first. 12 ÷ 3 = 4. Then work from left to right. 30 − 4 = 26. Then 26 + 4 = 30.

  9. From an earlier chapter

    Work out 5 928 ÷ 24.

    See the answer

    247

    Start with 59. 2 × 24 = 48 and 59 − 48 = 11. Bring down the 2 to make 112. 4 × 24 = 96 and 112 − 96 = 16. Bring down the 8 to make 168. 7 × 24 = 168. The answer is 247, with 0 left over.

Keep it

Copy this into your exercise book.

Order: brackets first. Then × and ÷. Then + and −. Inside a step, work left to right. 3 + 4 × 2 = 11, but (3 + 4) × 2 = 14.

Draw: Draw 3 loose beans and 4 bags of 2 beans. Write 3 + 4 × 2 = 11 under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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