Chapter 6 of 8

All 8 chapters
  1. 1Equal groups, readunderstood
  2. 2The 2, 5 and 10 times tables, readunderstood
  3. 3Sharing and grouping, readunderstood
  4. 4The 3, 4 and 8 times tables, readunderstood
  5. 5The 6, 7 and 9 times tables, readunderstood
  6. 6Times and divide facts together, readunderstood
  7. 7Remainders, readunderstood
  8. 8Times and divide with bigger numbers, readunderstood

M2.3.6 Chapter 6

Times and divide facts together

I know 3 × 7 = 21. Does that help me with 21 ÷ 3?

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

  • Write the four facts that link 3, 7 and 21
  • Use a times fact to answer a divide question
  • Show that 6 × 4 and 4 × 6 give the same answer

Look at 3 rows of 7 dots. There are 21 dots in all.

This one picture tells you four things. Watch.

Draw it first

Four facts from one array

  1. An array of 3 rows of 7 dots. Step 1 labels it 3 × 7 = 21. Step 2 adds 7 × 3 = 21. Step 3 adds 21 ÷ 3 = 7. Step 4 adds 21 ÷ 7 = 3.3 × 7 = 21

    1 Count 3 rows of 7. That is 3 × 7 = 21.

  2. An array of 3 rows of 7 dots. Step 1 labels it 3 × 7 = 21. Step 2 adds 7 × 3 = 21. Step 3 adds 21 ÷ 3 = 7. Step 4 adds 21 ÷ 7 = 3.3 × 7 = 217 × 3 = 21

    2 Turn the array on its side. You see 7 rows of 3. That is 7 × 3 = 21.

  3. An array of 3 rows of 7 dots. Step 1 labels it 3 × 7 = 21. Step 2 adds 7 × 3 = 21. Step 3 adds 21 ÷ 3 = 7. Step 4 adds 21 ÷ 7 = 3.3 × 7 = 217 × 3 = 2121 ÷ 3 = 7

    3 Share the 21 dots into 3 rows. Each row has 7. So 21 ÷ 3 = 7.

  4. An array of 3 rows of 7 dots. Step 1 labels it 3 × 7 = 21. Step 2 adds 7 × 3 = 21. Step 3 adds 21 ÷ 3 = 7. Step 4 adds 21 ÷ 7 = 3.3 × 7 = 217 × 3 = 2121 ÷ 3 = 721 ÷ 7 = 3

    4 Share them into 7 columns. Each column has 3. So 21 ÷ 7 = 3.

These four facts are a fact family. They all use the same three numbers.

The biggest number in the 3, 7 and 21 family is 21. It is the answer to the times facts. It comes first in the divide facts.

You can draw a family as a triangle. The biggest number goes at the top.

A triangle with 21 at the top corner, 3 at the bottom left corner and 7 at the bottom right corner.2137
Times uses the bottom two numbers to make the top. Divide starts at the top.

Worked example

Write the four facts for 4, 6 and 24.

  1. The biggest number is 24. It is the answer to the times facts.
  2. 4 × 6 = 24 and 6 × 4 = 24.
  3. Now divide 24. 24 ÷ 4 = 6 and 24 ÷ 6 = 4.

Answer: 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6, 24 ÷ 6 = 4

A times fact answers a divide question. For 42 ÷ 6, ask: 6 times what makes 42?

An array of 42 dots in 6 rows of 7, labelled 6 × 7 = 42. Under it, the label 42 ÷ 6 = 7.6 × 7 = 4242 ÷ 6 = 7
6 rows of 7 make 42. Share the 42 into 6 rows and each row has 7.

Worked example

Work out 42 ÷ 6.

  1. Ask: 6 times what makes 42?
  2. You know 6 × 7 = 42.
  3. So 42 ÷ 6 = 7.

Answer: 7

Times can swap places. Look at 6 × 4 and 4 × 6.

Two arrays of 24 dots. The left array has 6 rows of 4, labelled 6 × 4 = 24. The right array has 4 rows of 6, labelled 4 × 6 = 24.6 × 4 = 244 × 6 = 24
Turn the left array on its side. It becomes the right array. The number of dots does not change.

See it three ways

With things

Put 12 stones in 3 groups of 4. Share them out again into 4 groups of 3.

As a picture

Twelve stones in 3 groups. Each group has 4 stones.

In symbols

3 × 4 = 12 and 4 × 3 = 12. 12 ÷ 3 = 4 and 12 ÷ 4 = 3.

Try it

Make a fact family with stones

You need: 12 stones, beans or sticks, a pencil and paper.

  1. Make 3 rows of 4 stones. Write 3 × 4 = 12.
  2. Look at the stones from the side. You see 4 rows of 3. Write 4 × 3 = 12.
  3. Share the stones into 3 equal groups. Write 12 ÷ 3 = 4.
  4. Share them into 4 equal groups. Write 12 ÷ 4 = 3.

What you will see: Four facts come from three numbers: 3, 4 and 12. They make one fact family.

If you cannot try it, this is what happens.
An array of 3 rows of 4 dots, with four facts beside it: 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3.3 × 4 = 124 × 3 = 1212 ÷ 3 = 412 ÷ 4 = 3
Two pictures of stones. The top picture shows 12 stones in 4 groups of 3, labelled 12 ÷ 4 = 3. The bottom picture shows 4 stones, labelled 4 stones for 12 friends.12 ÷ 4 = 34 stones for 12 friends
12 stones between 4 friends: 3 each. 4 stones between 12 friends: not a whole one each.

A wrong idea many people have

Divide can swap places too. 12 ÷ 4 is the same as 4 ÷ 12.

Why it feels right: Times swaps places, so it seems that divide will too.

The test that breaks it: Share 12 stones between 4 friends. Each gets 3. Now share 4 stones between 12 friends. Nobody gets a whole stone.

What is true: Times can swap places. Divide cannot. 12 ÷ 4 = 3. The biggest number comes first.

Key words

fact family
times and divide facts that use the same three numbers.

Back to your guess

The answer is B: Yes. 21 ÷ 3 asks: 3 times what makes 21?

Times and divide are partners. 3 × 7 = 21, so 21 ÷ 3 = 7. One picture gives you four facts.

Check yourself

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

  1. Which fact is in the same family as 4 × 5 = 20?

    Tap your answer.

    1. Yes, that is right. The numbers are 4, 5 and 20. 20 ÷ 5 = 4 uses all three.

    2. Not this one. That uses 2 and 10. A family uses the same three numbers: 4, 5 and 20. Try another.

    3. Not this one. You added. A fact family uses times and divide, with the same three numbers. Try another.

    See the answer

    20 ÷ 5 = 4

    The numbers are 4, 5 and 20. 20 ÷ 5 = 4 uses all three.

    • If you chose B: That uses 2 and 10. A family uses the same three numbers: 4, 5 and 20.
    • If you chose C: You added. A fact family uses times and divide, with the same three numbers.

  2. Use 7 × 6 = 42 to answer 42 ÷ 6.

    See the answer

    7

    Ask: 6 times what makes 42? 7 × 6 = 42. So 42 ÷ 6 = 7.

  3. Write the four facts for 5, 8 and 40.

    See the answer

    5 × 8 = 40, 8 × 5 = 40, 40 ÷ 5 = 8, 40 ÷ 8 = 5

    40 is the biggest number. It is the answer to the times facts. It comes first in the divide facts.

  4. Ravi says 6 × 4 is more than 4 × 6. Is he right?

    Tap your answer.

    1. Not this one. Turn the array. 6 rows of 4 becomes 4 rows of 6. The dots do not change. Go to M2.3.1. Try another.

    2. Not this one. You added. 6 × 4 means 6 groups of 4, which is 24. Go to M2.3.1. Try another.

    3. Yes, that is right. Times can swap places. 6 × 4 = 24 and 4 × 6 = 24.

    See the answer

    No. They are both 24.

    Times can swap places. 6 × 4 = 24 and 4 × 6 = 24.

    • If you chose A: Turn the array. 6 rows of 4 becomes 4 rows of 6. The dots do not change. Go to M2.3.1.
    • If you chose B: You added. 6 × 4 means 6 groups of 4, which is 24. Go to M2.3.1.

  5. From an earlier chapter

    Work out 9 × 8. Use 10 × 8.

    See the answer

    72

    10 × 8 = 80. Take away one 8. 80 − 8 = 72.

  6. Which list has all four facts for 2, 9 and 18?

    Tap your answer.

    1. Yes, that is right. 18 is the biggest number. It is the answer to the times facts. It comes first in the divide facts.

    2. Not this one. The biggest number comes first in a divide fact. 2 ÷ 18 does not make 9. Try another.

    3. Not this one. That is a family for adding, with 2, 9 and 11. This family uses times and divide. Try another.

    See the answer

    2 × 9 = 18, 9 × 2 = 18, 18 ÷ 2 = 9, 18 ÷ 9 = 2

    18 is the biggest number. It is the answer to the times facts. It comes first in the divide facts.

    • If you chose B: The biggest number comes first in a divide fact. 2 ÷ 18 does not make 9.
    • If you chose C: That is a family for adding, with 2, 9 and 11. This family uses times and divide.

  7. 6 × 8 = 48. What is 48 ÷ 6?

    Tap your answer.

    1. Not this one. You took 6 away. Divide asks: 6 times what makes 48? Try another.

    2. Yes, that is right. Ask: 6 times what makes 48? 6 × 8 = 48. So 48 ÷ 6 = 8.

    3. Not this one. The 6 is what you divide by. Ask: 6 times what makes 48? Try another.

    See the answer

    8

    Ask: 6 times what makes 48? 6 × 8 = 48. So 48 ÷ 6 = 8.

    • If you chose A: You took 6 away. Divide asks: 6 times what makes 48?
    • If you chose C: The 6 is what you divide by. Ask: 6 times what makes 48?

  8. Share the dots between 3 friends. Which fact says how many each friend gets?

    An array of 24 dots in 4 rows of 6, labelled dots to share.dots to share
    Tap your answer.

    1. Not this one. Divide cannot swap places. The biggest number comes first. Try another.

    2. Not this one. You took 3 away. Sharing between 3 friends asks you to divide. Try another.

    3. Yes, that is right. The picture has 24 dots. Share them between 3 friends. Each gets 8, because 3 × 8 = 24.

    See the answer

    24 ÷ 3 = 8

    The picture has 24 dots. Share them between 3 friends. Each gets 8, because 3 × 8 = 24.

    • If you chose A: Divide cannot swap places. The biggest number comes first.
    • If you chose B: You took 3 away. Sharing between 3 friends asks you to divide.

  9. Is 20 ÷ 5 the same as 5 ÷ 20?

    Tap your answer.

    1. Not this one. Only times can swap places. 20 stones for 5 friends is not 5 stones for 20 friends. Try another.

    2. Not this one. Times can swap places. 6 × 4 and 4 × 6 are both 24. Divide cannot. Try another.

    3. Yes, that is right. 20 ÷ 5 = 4. Share 5 stones between 20 friends and nobody gets a whole stone.

    See the answer

    No. Times can swap places but divide cannot.

    20 ÷ 5 = 4. Share 5 stones between 20 friends and nobody gets a whole stone.

    • If you chose A: Only times can swap places. 20 stones for 5 friends is not 5 stones for 20 friends.
    • If you chose B: Times can swap places. 6 × 4 and 4 × 6 are both 24. Divide cannot.

Keep it

Copy this into your exercise book.

Three numbers make four facts. 3 × 7 = 21, 7 × 3 = 21, 21 ÷ 3 = 7, 21 ÷ 7 = 3. Times can swap. Divide cannot.

Draw: An array of 3 rows of 7 dots. Write the four facts beside it.

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

No internet tomorrow? Save the whole book as one file (905 KB). It opens on any phone.