Level 2 · Mathematics M2.3

Times tables and sharing

Equal groups, the times tables, and sharing things fairly.

Learn the times tables by seeing equal groups. Then use them to share, to find what is left over, and to work with bigger numbers.

You are ready for this book if

You can count in 2s, 5s and 10s up to 50.

Written and drawn by attnom. Free to read, print and share.

M2.3.1 Chapter 1

Equal groups

There are 4 plates with 3 beans on each plate. How many beans is that?

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

Mei puts 3 beans on a plate. She does this for 4 plates.

Every plate has the same number of beans. The plates are equal groups. Equal groups all have the same number of things.

Draw it first

Write 4 groups of 3

  1. One group of 3 beans, labelled 1 group of 3. Then 4 equal groups of 3 beans, labelled 4 groups of 3. Last, the label 4 × 3 = 12.1 group of 3

    1 Draw one group of 3 dots or beans.

  2. One group of 3 beans, labelled 1 group of 3. Then 4 equal groups of 3 beans, labelled 4 groups of 3. Last, the label 4 × 3 = 12.4 groups of 3

    2 Draw 3 more groups the same.

  3. One group of 3 beans, labelled 1 group of 3. Then 4 equal groups of 3 beans, labelled 4 groups of 3. Last, the label 4 × 3 = 12.4 × 3 = 12

    3 Write it as 4 × 3 = 12. Say "4 times 3 is 12".

The word times means "groups of". We write it as ×. So 4 × 3 means 4 groups of 3.

To find the answer, add the groups. 3 + 3 + 3 + 3 = 12.

A number line from 0 to 12. Four hops of 3 land on 3, 6, 9 and 12. Each hop is labelled +3.+3+3+3+3036912
Each hop is one group of 3. Four hops land on 12, so 4 × 3 = 12.

Worked example

Write 3 groups of 5 beans as a times sum. Find the answer.

  1. Count the groups. There are 3.
  2. Count the beans in one group. There are 5.
  3. Write the groups first, then the beans in each: 3 × 5.
  4. Add the groups: 5 + 5 + 5 = 15.

Answer: 3 × 5 = 15

Now line the groups up under each other. Each group is a row. A row goes across.

Things set out in equal rows are an array. A line going down is a column. This array has 4 rows and 3 columns.

Two arrays of 12 dots. The left array has 4 rows of 3 dots, labelled 4 × 3 = 12. The right array has 3 rows of 4 dots, labelled 3 × 4 = 12.4 × 3 = 123 × 4 = 12
Turn the array on its side. The dots stay the same. So 4 × 3 and 3 × 4 are both 12.

See it three ways

With things

Put 12 stones in 4 rows. Put 3 stones in each row.

As a picture

Twelve stones in an array of 4 rows with 3 stones in each row.

In symbols

4 × 3 = 12. Also 3 × 4 = 12.

Try it

Make an array of beans

You need: 12 beans, or 12 small stones, a flat table.

  1. Put 3 beans in a line. That is one row.
  2. Make 3 more rows the same, one under the other.
  3. Count all the beans.
  4. Now walk to the side of the table. Look at the beans again.

What you will see: From the front you see 4 rows of 3. From the side you see 3 rows of 4. It is 12 beans both ways.

If you cannot try it, this is what happens.
Twelve beans in an array of 4 rows of 3, labelled 4 rows of 3.4 rows of 3
Two pictures of beans. The top picture shows 7 beans in one line, labelled 4 + 3 = 7. The bottom picture shows 4 groups of 3 beans, labelled 4 × 3 = 12.4 + 3 = 74 × 3 = 12
Seven beans is a small pile. Four groups of 3 is a bigger pile.

A wrong idea many people have

4 × 3 means 4 + 3.

Why it feels right: The numbers are the same. And the × sign looks like a + sign turned on its side.

The test that breaks it: Put 3 beans on each of 4 plates. Count all the beans. Is it 7?

What is true: 4 × 3 means 4 groups of 3, which is 12. Only 4 + 3 is 7.

Key words

equal groups
groups that all have the same number of things.
times
groups of. 4 × 3 is read "4 times 3".
row
a line of things going across.
array
things set out in equal rows.

Back to your guess

The answer is B: 12 beans, because it is 4 groups of 3.

There are 4 equal groups of 3. Add them: 3 + 3 + 3 + 3 = 12. We write this as 4 × 3 = 12.

Check yourself

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

  1. Look at the picture. Which sum tells the story?

    Twelve beans in 3 equal groups of 4.
    Tap your answer.

    1. Yes, that is right. There are 3 groups with 4 beans in each. 4 + 4 + 4 = 12.

    2. Not this one. That adds the 3 groups to the 4 beans. A times sum means 3 groups of 4. Try another.

    3. Not this one. Each group has 4 beans, not 3. Count the beans in one group. Try another.

    See the answer

    3 × 4 = 12

    There are 3 groups with 4 beans in each. 4 + 4 + 4 = 12.

    • If you chose B: That adds the 3 groups to the 4 beans. A times sum means 3 groups of 4.
    • If you chose C: Each group has 4 beans, not 3. Count the beans in one group.

  2. Kofi has 5 bowls with 3 stones in each bowl. How many stones is that?

    Tap your answer.

    1. Not this one. That is 5 + 3. A times sum means 3 stones, five times. Try another.

    2. Not this one. That is the stones in one bowl. There are 5 bowls, so count 3 five times. Try another.

    3. Yes, that is right. 5 bowls of 3 is 5 × 3. Add: 3 + 3 + 3 + 3 + 3 = 15.

    See the answer

    15 stones

    5 bowls of 3 is 5 × 3. Add: 3 + 3 + 3 + 3 + 3 = 15.

    • If you chose A: That is 5 + 3. A times sum means 3 stones, five times.
    • If you chose B: That is the stones in one bowl. There are 5 bowls, so count 3 five times.

  3. Mei makes 2 rows of 6 dots. Tariq makes 6 rows of 2 dots. Who has more dots?

    Tap your answer.

    1. Not this one. Her rows are longer, but she has fewer of them. Count all the dots. Try another.

    2. Not this one. He has more rows, but his rows are shorter. Count all the dots. Try another.

    3. Yes, that is right. 2 × 6 = 12 and 6 × 2 = 12. Turn one array on its side and you get the other.

    See the answer

    They have the same number.

    2 × 6 = 12 and 6 × 2 = 12. Turn one array on its side and you get the other.

    • If you chose A: Her rows are longer, but she has fewer of them. Count all the dots.
    • If you chose B: He has more rows, but his rows are shorter. Count all the dots.

  4. Look at the picture. Write the array as a times sum.

    An array of 15 dots with 3 rows of 5 dots.
    See the answer

    3 × 5 = 15. You can also write 5 × 3 = 15.

    There are 3 rows with 5 dots in each row. 5 + 5 + 5 = 15. Turned on its side it is 5 rows of 3.

  5. Draw 5 groups of 2 dots. Write the times sum. How many dots are there?

    See the answer

    5 × 2 = 10. There are 10 dots.

    2 + 2 + 2 + 2 + 2 = 10. That is 5 groups of 2.

  6. Look at Ana's seeds. How many columns does the array have?

    An array of 12 seeds in 2 rows, labelled Ana's seeds.Ana's seeds
    Tap your answer.

    1. Not this one. That is the number of rows. A row goes across. A column goes down. Try another.

    2. Yes, that is right. There are 2 rows and 6 seeds in each row. A column goes down. Each row has 6 seeds, so there are 6 columns.

    3. Not this one. That is the number of seeds. A column is a line of seeds going down. Try another.

    See the answer

    6 columns

    There are 2 rows and 6 seeds in each row. A column goes down. Each row has 6 seeds, so there are 6 columns.

    • If you chose A: That is the number of rows. A row goes across. A column goes down.
    • If you chose C: That is the number of seeds. A column is a line of seeds going down.

  7. Which number sentence gives the same total as 5 × 2?

    Tap your answer.

    1. Yes, that is right. 5 × 2 = 10. Turn the array on its side. It is 2 rows of 5. 2 × 5 = 10 too.

    2. Not this one. You added. 5 × 2 means 5 groups of 2. Add 2 five times. Try another.

    3. Not this one. You used 5 twice. In 5 × 2 each group has 2, not 5. 2 + 2 + 2 + 2 + 2 = 10. Try another.

    See the answer

    2 × 5 = 10

    5 × 2 = 10. Turn the array on its side. It is 2 rows of 5. 2 × 5 = 10 too.

    • If you chose B: You added. 5 × 2 means 5 groups of 2. Add 2 five times.
    • If you chose C: You used 5 twice. In 5 × 2 each group has 2, not 5. 2 + 2 + 2 + 2 + 2 = 10.

Keep it

Copy this into your exercise book.

4 × 3 means 4 groups of 3. Add them: 3 + 3 + 3 + 3 = 12. An array sets the groups out in rows. 4 × 3 and 3 × 4 are both 12.

Draw: 12 dots in 4 rows of 3. Write 4 × 3 = 12 under them. Turn the page and write 3 × 4 = 12.

M2.3.2 Chapter 2

The 2, 5 and 10 times tables

The answers in the 5 times table end in 5 or 0. Why?

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

A times table is a list of answers. Each answer is for one more group of the same size.

The 5 times table starts 1 × 5, 2 × 5, 3 × 5. You get each answer by counting in 5s.

Draw it first

Count in 5s along a line

  1. A line from 0 to 30. Hops of 5 land on 5, 10, 15, 20, 25 and 30. The labels read 1 × 5 = 5, then 3 × 5 = 15, then 6 × 5 = 30.501 × 5 = 5

    1 Start at 0. Hop 5. You land on 5. That is 1 × 5.

  2. A line from 0 to 30. Hops of 5 land on 5, 10, 15, 20, 25 and 30. The labels read 1 × 5 = 5, then 3 × 5 = 15, then 6 × 5 = 30.5101503 × 5 = 15

    2 Hop 5 again, twice more. You land on 15. That is 3 × 5.

  3. A line from 0 to 30. Hops of 5 land on 5, 10, 15, 20, 25 and 30. The labels read 1 × 5 = 5, then 3 × 5 = 15, then 6 × 5 = 30.5101520253006 × 5 = 30

    3 Keep going. Six hops of 5 land on 30. That is 6 × 5.

The 10 times table counts in 10s. Each answer is a number of tens. 4 × 10 is 4 tens, which is 40.

Four rods of ten, labelled 4 × 10 = 40.4 × 10 = 40
Each rod is one ten. 4 rods make 40.

The 2 times table counts in 2s. Each answer is a double. 4 × 2 is 4 + 4, which is 8.

Four pairs of stones, 2 in each pair, labelled 4 × 2 = 8.4 × 2 = 8
Each pair is a group of 2. Four pairs make 8.
Three columns of answers for 1 to 10 groups. The first column, headed × 2, reads 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. The second, headed × 5, reads 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. The third, headed × 10, reads 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.× 22468101214161820× 55101520253035404550× 10102030405060708090100
The three tables side by side. Look down each column.

Now look at only the last digit of each answer. A digit is one of the figures 0 to 9.

The last digit of each answer. In the row 2s: 2, 4, 6, 8, 0, then again 2, 4, 6, 8, 0. In the row 5s: 5, 0, 5, 0, and so on. In the row 10s: all 0.2s24680246805s505050505010s0000000000
The last digits repeat. They never stop repeating.

The 2s end in 2, 4, 6, 8 or 0. These are even numbers. An even number shares into 2 equal groups with none left.

The 5s end in 5, then 0, then 5, then 0. The 10s all end in 0.

Two sets of rods. The first set is 8 rods, labelled 8 × 10 = 80. The second set is 4 rods, half as many, labelled 8 × 5 = 40.8 × 10 = 808 × 5 = 40
Take half of the rods. 4 rods are 40, so 8 × 5 = 40.

Worked example

Find 8 × 5 without counting in 5s.

  1. The 5 times table is half of the 10 times table.
  2. Find 8 × 10. That is 8 tens, which is 80.
  3. Take half of 80. Half of 80 is 40.

Answer: 8 × 5 = 40

See it three ways

With things

Put 20 stones in 4 rows of 5. Count along the rows in 5s.

As a picture

Twenty stones in an array of 4 rows with 5 stones in each row.

In symbols

4 × 5 = 20. Count: 5, 10, 15, 20.

Try it

Count in 5s with stones

You need: 15 small stones, or beans, a flat table.

  1. Put 5 stones in a pile. Say "5".
  2. Make a second pile of 5. Say the total so far, "10".
  3. Make a third pile of 5. Say the total, "15".
  4. Look at the last digit of each number you said.

What you will see: You said 5, 10, 15. The last digits go 5, 0, 5. The next is 20. The last digit goes back to 0.

If you cannot try it, this is what happens.
Three piles of 5 stones. The piles are labelled 5, 10 and 15.51015

A wrong idea many people have

The answers in the 5 times table all end in 5.

Why it feels right: The first answers are 5, 15 and 25. Many of them do end in 5.

The test that breaks it: Count in 5s with stones: 5, 10, 15, 20. Look at the last digit of each number.

What is true: They end in 5, then 0, then 5, then 0. 4 × 5 = 20 ends in 0.

Key words

times table
a list of answers for more and more equal groups.
digit
one of the figures 0 to 9. In 35, the last digit is 5.
even
a number that shares into 2 equal groups with none left.

Back to your guess

The answer is B: Each step adds 5. Two steps add 10, so the last digit comes back.

Each step adds 5. Two steps add 10, which does not change the last digit. So the last digits go 5, 0, 5, 0.

Check yourself

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

  1. Which number is not in the 5 times table?

    Tap your answer.

    1. Not this one. Count 7 fives and you reach 35. Try another.

    2. Not this one. 40 ends in 0, as many 5 table answers do. Count 8 fives and you reach 40. Try another.

    3. Yes, that is right. Answers in the 5 times table end in 5 or 0. 52 ends in 2.

    See the answer

    52

    Answers in the 5 times table end in 5 or 0. 52 ends in 2.

    • If you chose A: Count 7 fives and you reach 35.
    • If you chose B: 40 ends in 0, as many 5 table answers do. Count 8 fives and you reach 40.

  2. Write the answers to 6 × 2, 6 × 5 and 6 × 10.

    See the answer

    12, 30 and 60

    6 × 2 is the double of 6, which is 12. Six fives are 30. Six tens are 60.

  3. From an earlier chapter

    Ravi has 4 bags with 5 stones in each bag. Which sum tells how many stones?

    Tap your answer.

    1. Not this one. That puts 4 and 5 together. Times means 4 groups of 5. Go to M2.3.1. Try another.

    2. Yes, that is right. 4 groups of 5 is 4 × 5. Count in 5s: 5, 10, 15, 20.

    See the answer

    4 × 5 = 20

    4 groups of 5 is 4 × 5. Count in 5s: 5, 10, 15, 20.

    • If you chose A: That puts 4 and 5 together. Times means 4 groups of 5. Go to M2.3.1.

  4. Tariq says 45 is in the 2 times table. It ends in 5. Is he right?

    Tap your answer.

    1. Not this one. Count in 2s: 44, 46. You jump over 45. Every answer in the 2 times table is even. Try another.

    2. Not this one. Even depends on the last digit only. 45 ends in 5, so it is odd. Try another.

    3. Yes, that is right. 45 ends in 5, so it is not even. It is in the 5 times table instead: 9 × 5 = 45.

    See the answer

    No. Those answers end in 0, 2, 4, 6 or 8.

    45 ends in 5, so it is not even. It is in the 5 times table instead: 9 × 5 = 45.

    • If you chose A: Count in 2s: 44, 46. You jump over 45. Every answer in the 2 times table is even.
    • If you chose B: Even depends on the last digit only. 45 ends in 5, so it is odd.

  5. Joseph has 9 pairs of sandals. How many sandals is that? Use the 2 times table.

    See the answer

    18 sandals

    A pair is 2. 9 pairs is 9 × 2, which is 9 + 9 = 18.

  6. What is 7 × 5?

    Tap your answer.

    1. Not this one. You added 7 + 5. Times means 7 groups of 5. Go to M2.3.1. Try another.

    2. Not this one. That is 6 fives. Count one more 5 to reach 7 fives. Try another.

    3. Yes, that is right. Count 7 fives: 5, 10, 15, 20, 25, 30, 35. 7 × 5 = 35.

    See the answer

    35

    Count 7 fives: 5, 10, 15, 20, 25, 30, 35. 7 × 5 = 35.

    • If you chose A: You added 7 + 5. Times means 7 groups of 5. Go to M2.3.1.
    • If you chose B: That is 6 fives. Count one more 5 to reach 7 fives.

  7. Look at the rods. Each rod is 10. What is the total?

    Some rods of ten, labelled rods of ten.rods of ten
    Tap your answer.

    1. Yes, that is right. 6 rods of 10 is 6 × 10. Count in 10s: 10, 20, 30, 40, 50, 60.

    2. Not this one. You added 6 and 10. Each rod is worth 10, so count in 10s. Go to M2.3.1. Try another.

    3. Not this one. That is how many rods there are. Each rod is worth 10, so 6 rods are 6 tens. Try another.

    See the answer

    60

    6 rods of 10 is 6 × 10. Count in 10s: 10, 20, 30, 40, 50, 60.

    • If you chose B: You added 6 and 10. Each rod is worth 10, so count in 10s. Go to M2.3.1.
    • If you chose C: That is how many rods there are. Each rod is worth 10, so 6 rods are 6 tens.

  8. Think of 5, 10, 15 and 20. Which list shows the last digit of each?

    Tap your answer.

    1. Not this one. That is the wrong idea that every answer ends in 5. 2 × 5 = 10 ends in 0. Try another.

    2. Yes, that is right. The last digit of 5 is 5. Of 10 it is 0. Of 15 it is 5. Of 20 it is 0.

    3. Not this one. Those are the first digits. The last digit is on the right: 15 ends in 5. Try another.

    See the answer

    5, 0, 5, 0

    The last digit of 5 is 5. Of 10 it is 0. Of 15 it is 5. Of 20 it is 0.

    • If you chose A: That is the wrong idea that every answer ends in 5. 2 × 5 = 10 ends in 0.
    • If you chose C: Those are the first digits. The last digit is on the right: 15 ends in 5.

  9. Which sentence is true for every answer in the 2 times table?

    Tap your answer.

    1. Not this one. 5 × 2 = 10 ends in 0. The last digits go 2, 4, 6, 8, 0 and repeat. Try another.

    2. Not this one. That is the 5 times table. The 2 times table ends in 2, 4, 6, 8 or 0. Try another.

    3. Yes, that is right. Each answer is a double. A double shares into 2 equal groups with none left. That is what even means.

    See the answer

    It is always even.

    Each answer is a double. A double shares into 2 equal groups with none left. That is what even means.

    • If you chose A: 5 × 2 = 10 ends in 0. The last digits go 2, 4, 6, 8, 0 and repeat.
    • If you chose B: That is the 5 times table. The 2 times table ends in 2, 4, 6, 8 or 0.

Keep it

Copy this into your exercise book.

The 2 times table gives even numbers. The 5 times table ends in 5, 0, 5, 0. The 10 times table always ends in 0. Count in 2s, 5s or 10s to find an answer.

Draw: Write 5, 10, 15, 20, 25, 30 along a line. Circle the last digit of each number.

M2.3.3 Chapter 3

Sharing and grouping

I have 12 beans for 3 friends. How many beans does each friend get?

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

Amina has 12 beans. She wants to share them fairly with 2 friends. That makes 3 people in all.

She gives one bean to each person in turn. She keeps going until none are left. This is sharing.

Draw it first

Share 12 beans between 3 people

  1. Twelve loose beans, labelled 12 beans. Then the same beans in 3 equal groups of 4, labelled 4 each. Last, the label 12 ÷ 3 = 4.12 beans

    1 Start with 12 beans in a pile.

  2. Twelve loose beans, labelled 12 beans. Then the same beans in 3 equal groups of 4, labelled 4 each. Last, the label 12 ÷ 3 = 4.4 each

    2 Deal them out, one at a time, to 3 people. Each person gets 4.

  3. Twelve loose beans, labelled 12 beans. Then the same beans in 3 equal groups of 4, labelled 4 each. Last, the label 12 ÷ 3 = 4.12 ÷ 3 = 4

    3 Write it as 12 ÷ 3 = 4. Say "12 divided by 3 is 4".

The sign ÷ means divide. To divide is to split into equal groups.

Division and times go together. 4 × 3 = 12, so 12 ÷ 3 = 4.

An array of 12 beans in 3 rows of 4, labelled 3 rows of 4. Under it, the labels 3 × 4 = 12 and 12 ÷ 3 = 4.3 rows of 43 × 4 = 1212 ÷ 3 = 4
Times puts the rows together. Divide splits the 12 back into 3 rows.

There is a second way to divide. Kofi has 20 stones. He puts them in groups of 4. How many groups can he make?

Here you do not ask how many go in each group. The size of the group is 4 already. You ask how many groups there are. This is grouping.

Twenty stones in 5 equal groups of 4, labelled 20 stones in groups of 4. Three groups are on the top line and two groups are on the bottom line.20 stones in groups of 4
20 stones make 5 groups of 4.

You can count the groups with hops. Hop 4 each time from 0. Stop at 20. Count the hops.

A line from 0 to 20. Five hops of 4 land on 4, 8, 12, 16 and 20.481216200
Five hops of 4 reach 20. So 20 ÷ 4 = 5.

Worked example

How many groups of 4 are in 20?

  1. Count in 4s until you reach 20: 4, 8, 12, 16, 20.
  2. You said 5 numbers. So there are 5 groups.
  3. Check by times: 5 × 4 = 20.

Answer: 5 groups

Sharing asks how many go in each group. Grouping asks how many groups. Both use the ÷ sign.

See it three ways

With things

Put 12 stones in a pile. Deal them out, one at a time, to 3 people.

As a picture

Twelve stones in 3 equal groups of 4, labelled 4 stones each.4 stones each

In symbols

12 ÷ 3 = 4. Each person gets 4.

Try it

Share, then group, 12 beans

You need: 12 beans, or 12 small stones, 3 pieces of paper.

  1. Put 12 beans in a pile. Put the 3 pieces of paper in a row.
  2. Sharing: deal the beans, one at a time, onto the papers. Count one paper.
  3. Now push all 12 beans back together.
  4. Grouping: make groups of 3 beans. Count how many groups you make.

What you will see: Sharing gives 4 beans on each paper. Grouping gives 4 groups of 3. The number 4 comes up both times.

If you cannot try it, this is what happens.
Twelve beans in 3 equal groups. Each group is labelled 4.444
Two pictures of 12 beans. The top picture shows 4 groups of 3 beans, labelled Shared by 4: 3 each. The bottom picture shows 3 groups of 4 beans, labelled Groups of 4: 3 groups.Shared by 4: 3 eachGroups of 4: 3 groups
Both ways give 3. The 3 means beans each in one, and groups in the other.

A wrong idea many people have

Sharing and grouping give different answers.

Why it feels right: The two questions sound different. One asks how many in each, and one asks how many groups.

The test that breaks it: Take 12 beans. Share them between 4 people and count one share. Then make groups of 4 and count the groups.

What is true: Both give 3. It is 12 ÷ 4 = 3 either way. The 3 means something different each time.

Key words

sharing
dealing things out equally, one at a time.
grouping
finding how many groups of one size you can make.
divide
to split into equal groups. The sign is ÷.

Back to your guess

The answer is B: 4 beans, because 4 + 4 + 4 = 12.

Give one bean to each friend in turn, until none are left. Each friend gets 4. We write 12 ÷ 3 = 4.

Check yourself

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

  1. Share 15 stones between 3 friends. How many stones does each friend get?

    Tap your answer.

    1. Not this one. You took 3 away. Sharing means each friend gets the same. Deal them out to see. Try another.

    2. Not this one. The 3 is the number of friends. It is not the number each one gets. Try another.

    3. Yes, that is right. 15 ÷ 3 = 5. Check: 5 × 3 = 15.

    See the answer

    5 stones

    15 ÷ 3 = 5. Check: 5 × 3 = 15.

    • If you chose A: You took 3 away. Sharing means each friend gets the same. Deal them out to see.
    • If you chose B: The 3 is the number of friends. It is not the number each one gets.

  2. Tariq has 28 stones. He puts them in groups of 4. How many groups does he make?

    See the answer

    7 groups

    Count in 4s: 4, 8, 12, 16, 20, 24, 28. That is 7 numbers. Check: 7 × 4 = 28.

  3. From an earlier chapter

    Mei shares 40 beans between 10 friends. How many beans does each friend get?

    See the answer

    4 beans

    Use the 10 times table. 4 × 10 = 40, so 40 ÷ 10 = 4.

  4. Share 20 beans between 5 friends. Then put 20 beans in groups of 5. Do you get the same answer?

    Tap your answer.

    1. Yes, that is right. 20 ÷ 5 = 4. Each friend gets 4 beans, and there are 4 groups of 5.

    2. Not this one. Count in 5s up to 20: 5, 10, 15, 20. That is 4 groups, not 5. Try another.

    3. Not this one. Deal 20 beans to 5 friends. Each gets 4, not 5. Count the groups of 5 as well: 4. Try another.

    See the answer

    Yes. Both give 4.

    20 ÷ 5 = 4. Each friend gets 4 beans, and there are 4 groups of 5.

    • If you chose B: Count in 5s up to 20: 5, 10, 15, 20. That is 4 groups, not 5.
    • If you chose C: Deal 20 beans to 5 friends. Each gets 4, not 5. Count the groups of 5 as well: 4.

  5. Write the sum for 18 stones shared between 3 friends. Then find the answer.

    See the answer

    18 ÷ 3 = 6

    Each friend gets 6 stones. Check: 6 × 3 = 18.

  6. Kofi has 24 stones. He puts them in groups of 6. How many groups?

    Tap your answer.

    1. Not this one. The 6 is the size of each group. You are asked how many groups there are. Try another.

    2. Not this one. You took 6 away. Grouping counts how many 6s fit in 24. Try another.

    3. Yes, that is right. Count in 6s: 6, 12, 18, 24. That is 4 numbers. Check: 4 × 6 = 24.

    See the answer

    4 groups

    Count in 6s: 6, 12, 18, 24. That is 4 numbers. Check: 4 × 6 = 24.

    • If you chose A: The 6 is the size of each group. You are asked how many groups there are.
    • If you chose B: You took 6 away. Grouping counts how many 6s fit in 24.

  7. Three friends share this pile. Which sum shows the sharing?

    A pile of 18 stones in 2 rows of 9, labelled pile of stones.pile of stones
    Tap your answer.

    1. Not this one. You took 3 away. Sharing deals the pile out equally, so you divide. Try another.

    2. Yes, that is right. Deal 18 stones to 3 friends. Each friend gets 6. Check: 6 × 3 = 18.

    3. Not this one. That is true, but it is a times sum. Sharing is written with ÷. Try another.

    See the answer

    18 ÷ 3 = 6

    Deal 18 stones to 3 friends. Each friend gets 6. Check: 6 × 3 = 18.

    • If you chose A: You took 3 away. Sharing deals the pile out equally, so you divide.
    • If you chose C: That is true, but it is a times sum. Sharing is written with ÷.

  8. Amina shares 12 beans between 3 friends. 12 ÷ 3 = 4. What does the 4 tell you?

    Tap your answer.

    1. Yes, that is right. 12 beans dealt out to 3 friends gives 4 each. 4 + 4 + 4 = 12.

    2. Not this one. The friends are 3. The 4 is the number of beans each friend gets. Try another.

    3. Not this one. All 12 beans are dealt out, so none are left over. 4 + 4 + 4 = 12. Try another.

    See the answer

    Each friend gets 4 beans.

    12 beans dealt out to 3 friends gives 4 each. 4 + 4 + 4 = 12.

    • If you chose B: The friends are 3. The 4 is the number of beans each friend gets.
    • If you chose C: All 12 beans are dealt out, so none are left over. 4 + 4 + 4 = 12.

  9. Which sum helps you find 24 ÷ 4?

    Tap your answer.

    1. Not this one. That multiplies, so it makes a bigger number. Divide asks what times 4 makes 24. Try another.

    2. Not this one. You took 4 away once. Divide asks how many 4s make 24. Try another.

    3. Yes, that is right. Division and times go together. 6 × 4 = 24, so 24 ÷ 4 = 6.

    See the answer

    6 × 4 = 24

    Division and times go together. 6 × 4 = 24, so 24 ÷ 4 = 6.

    • If you chose A: That multiplies, so it makes a bigger number. Divide asks what times 4 makes 24.
    • If you chose B: You took 4 away once. Divide asks how many 4s make 24.

Keep it

Copy this into your exercise book.

Sharing: deal out one at a time. 12 ÷ 3 = 4 means 4 each. Grouping: how many groups of 4 are in 20? Count in 4s. That is 5.

Draw: 12 dots in 3 groups of 4. Write 12 ÷ 3 = 4. Then draw 5 hops of 4 from 0 to 20.

M2.3.4 Chapter 4

The 3, 4 and 8 times tables

I know my 4 times table. How can that help me with the 8 times table?

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

You can count in 3s and in 4s. Do it as you did in 5s.

The 3 times table counts 3, 6, 9, 12 and on. The 4 times table counts 4, 8, 12, 16 and on.

A line from 0 to 18. Six hops of 3 land on 3, 6, 9, 12, 15 and 18. Each hop is labelled +3.+3+3+3+3+3+30369121518
Six hops of 3 land on 18. So 6 × 3 = 18.

The 4 times table is the 2 times table doubled. To double is to make twice as many. The 8 times table is the 4 times table doubled.

Two arrays of dots. The left array has 2 rows of 5 dots, labelled 2 × 5 = 10. The right array has 4 rows of 5 dots, labelled 4 × 5 = 20.2 × 5 = 104 × 5 = 20
Twice as many rows gives twice as many dots. 10 doubled is 20.

Draw it first

Double 4 × 3 to get 8 × 3

  1. A first array of 4 rows of 3 dots, labelled 4 × 3 = 12. A second array of 4 rows of 3 dots below it, labelled 12 again. Together they make 8 rows of 3, labelled 12 + 12 = 24 and 8 × 3 = 24.4 × 3 = 12

    1 Draw 4 rows of 3 dots. That is 4 × 3 = 12.

  2. A first array of 4 rows of 3 dots, labelled 4 × 3 = 12. A second array of 4 rows of 3 dots below it, labelled 12 again. Together they make 8 rows of 3, labelled 12 + 12 = 24 and 8 × 3 = 24.4 × 3 = 1212 again

    2 Draw 4 more rows of 3 dots below. That is 12 again.

  3. A first array of 4 rows of 3 dots, labelled 4 × 3 = 12. A second array of 4 rows of 3 dots below it, labelled 12 again. Together they make 8 rows of 3, labelled 12 + 12 = 24 and 8 × 3 = 24.12 + 12 = 248 × 3 = 24

    3 Now there are 8 rows of 3. Add: 12 + 12 = 24.

8 rows is double 4 rows. So 8 × 3 is double 4 × 3. The 8 times table is the 4 times table doubled.

Two rows of answers for 1 to 10 groups. Row 4s reads 4, 8, 12, 16, 20, 24, 28, 32, 36, 40. Row 8s reads 8, 16, 24, 32, 40, 48, 56, 64, 72, 80. Each answer in 8s is double the answer above it.4s4812162024283236408s8162432404856647280
Each answer in the 8s is double the answer above it.

Worked example

Find 8 × 7. Use the 4 times table.

  1. Find 4 × 7 first. Count in 4s seven times to get 28.
  2. 8 is double 4, so double the answer.
  3. Double 28 is 28 + 28, which is 56.

Answer: 8 × 7 = 56

Now look at the last digits of the 3s, 4s and 8s. They make patterns.

The last digit of each answer. Row 3s: 3, 6, 9, 2, 5, 8, 1, 4, 7, 0. Row 4s: 4, 8, 2, 6, 0, then again 4, 8, 2, 6, 0. Row 8s: 8, 6, 4, 2, 0, then again 8, 6, 4, 2, 0.3s36925814704s48260482608s8642086420
The 3s use every digit. The 4s and 8s use only even digits.

The 4s go 4, 8, 2, 6, 0. The 8s go down in twos: 8, 6, 4, 2, 0. Both repeat.

See it three ways

With things

Put 18 stones in 6 rows. Put 3 stones in each row. Count in 3s.

As a picture

Eighteen stones in an array of 6 rows with 3 stones in each row, labelled 6 rows of 3.6 rows of 3

In symbols

6 × 3 = 18. Count: 3, 6, 9, 12, 15, 18.

Try it

Double an array of beans

You need: 32 beans, or 32 small stones, a flat table.

  1. Make an array of 4 rows with 4 beans in each row.
  2. Count the beans. That is 4 × 4.
  3. Make a second array exactly the same, under the first.
  4. Count all the beans. Is it double?

What you will see: The first array has 16 beans. Both arrays make 8 rows of 4, which is 32. So 8 × 4 = 32.

If you cannot try it, this is what happens.
Thirty-two beans in an array of 8 rows with 4 beans in each row, labelled 8 rows of 4.8 rows of 4

A wrong idea many people have

8 × 7 is 54.

Why it feels right: 54 sounds right. It is a real answer in the tables, because 6 × 9 = 54.

The test that breaks it: Double the 4 times table. 4 × 7 is 28. Now double 28.

What is true: 28 + 28 = 56. So 8 × 7 = 56. The number 54 belongs to 6 × 9.

Key words

double
to make twice as many. Double 4 is 8.

Back to your guess

The answer is B: Double each answer.

8 is double 4. So 8 groups are double 4 groups. 4 × 6 = 24, so 8 × 6 = 24 + 24 = 48.

Check yourself

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

  1. 4 × 6 = 24. Use it to find 8 × 6.

    Tap your answer.

    1. Yes, that is right. 8 × 6 is double 4 × 6. 24 + 24 = 48.

    2. Not this one. You added 4. Eight groups is double four groups, so double 24. Try another.

    3. Not this one. You took half. 8 is more than 4, so the answer is more than 24. Try another.

    See the answer

    48

    8 × 6 is double 4 × 6. 24 + 24 = 48.

    • If you chose B: You added 4. Eight groups is double four groups, so double 24.
    • If you chose C: You took half. 8 is more than 4, so the answer is more than 24.

  2. Count in 3s to find 3 × 9.

    See the answer

    27

    Count in 3s nine times. You say 3, 6, 9 and so on, up to 27.

  3. Write the 4 times table, from 1 × 4 to 10 × 4.

    See the answer

    4, 8, 12, 16, 20, 24, 28, 32, 36, 40

    Count in 4s ten times. Each answer is 4 more than the last.

  4. From an earlier chapter

    Amina shares 24 stones between 8 friends. How many stones does each friend get?

    Tap your answer.

    1. Yes, that is right. 8 × 3 = 24, so 24 ÷ 8 = 3.

    2. Not this one. You took 8 away. Sharing means dealing out equally. Think: 8 times what is 24? Go to M2.3.3. Try another.

    3. Not this one. The 8 is the number of friends. Each friend gets fewer than 8. Go to M2.3.3. Try another.

    See the answer

    3 stones

    8 × 3 = 24, so 24 ÷ 8 = 3.

    • If you chose B: You took 8 away. Sharing means dealing out equally. Think: 8 times what is 24? Go to M2.3.3.
    • If you chose C: The 8 is the number of friends. Each friend gets fewer than 8. Go to M2.3.3.

  5. What is 8 × 7?

    Tap your answer.

    1. Not this one. 54 is 6 × 9. Double 4 × 7 instead: 28 + 28. Try another.

    2. Yes, that is right. 4 × 7 = 28. Double it: 28 + 28 = 56.

    3. Not this one. 64 is 8 × 8. You need 8 sevens, not 8 eights. Try another.

    See the answer

    56

    4 × 7 = 28. Double it: 28 + 28 = 56.

    • If you chose A: 54 is 6 × 9. Double 4 × 7 instead: 28 + 28.
    • If you chose C: 64 is 8 × 8. You need 8 sevens, not 8 eights.

  6. Ravi counts in 3s: 3, 6, 9, 12. Which number comes next?

    Tap your answer.

    1. Not this one. You added 1. Counting in 3s adds 3 each time. Try another.

    2. Not this one. You added 4. That is counting in 4s. Counting in 3s adds 3 each time. Try another.

    3. Yes, that is right. Each number is 3 more than the last. 12 + 3 = 15. So 5 × 3 = 15.

    See the answer

    15

    Each number is 3 more than the last. 12 + 3 = 15. So 5 × 3 = 15.

    • If you chose A: You added 1. Counting in 3s adds 3 each time.
    • If you chose B: You added 4. That is counting in 4s. Counting in 3s adds 3 each time.

  7. Kofi has 6 stools. Each stool has 4 legs. How many legs are there?

    Tap your answer.

    1. Yes, that is right. Count in 4s six times: 4, 8, 12, 16, 20, 24. So 6 × 4 = 24.

    2. Not this one. You added 6 and 4. Times means 6 groups of 4. Go to M2.3.1. Try another.

    3. Not this one. That is 5 groups of 4. Count one more 4 to reach 6 groups. Try another.

    See the answer

    24

    Count in 4s six times: 4, 8, 12, 16, 20, 24. So 6 × 4 = 24.

    • If you chose B: You added 6 and 4. Times means 6 groups of 4. Go to M2.3.1.
    • If you chose C: That is 5 groups of 4. Count one more 4 to reach 6 groups.

  8. Think of 8, 16, 24, 32 and 40. Which list shows the last digit of each?

    Tap your answer.

    1. Not this one. Those are the first digits. The last digit is on the right: 16 ends in 6. Try another.

    2. Yes, that is right. The last digit of 8 is 8. Of 16 it is 6. Of 24 it is 4. Of 32 it is 2. Of 40 it is 0.

    3. Not this one. The last digit changes each time. 16 ends in 6, and 24 ends in 4. Try another.

    See the answer

    8, 6, 4, 2, 0

    The last digit of 8 is 8. Of 16 it is 6. Of 24 it is 4. Of 32 it is 2. Of 40 it is 0.

    • If you chose A: Those are the first digits. The last digit is on the right: 16 ends in 6.
    • If you chose C: The last digit changes each time. 16 ends in 6, and 24 ends in 4.

  9. Look at the two arrays. Together they show 8 × 5. What is 8 × 5?

    Two arrays of dots, each with 4 rows of 5 dots, labelled first array and second array.first arraysecond array
    Tap your answer.

    1. Not this one. That is one array. 8 × 5 is both arrays together. Try another.

    2. Not this one. You added 8 and 5. Times means 8 groups of 5. Go to M2.3.1. Try another.

    3. Yes, that is right. Each array is 4 × 5 = 20. Double it: 20 + 20 = 40. So 8 × 5 = 40.

    See the answer

    40

    Each array is 4 × 5 = 20. Double it: 20 + 20 = 40. So 8 × 5 = 40.

    • If you chose A: That is one array. 8 × 5 is both arrays together.
    • If you chose B: You added 8 and 5. Times means 8 groups of 5. Go to M2.3.1.

Keep it

Copy this into your exercise book.

Count in 3s and 4s. The 4 times table is double the 2 times table. The 8 times table is double the 4 times table. 8 × 7 = 28 + 28 = 56.

Draw: Draw 4 rows of 3 dots, then 4 more rows below. Write 4 × 3 = 12 and 8 × 3 = 24.

M2.3.5 Chapter 5

The 6, 7 and 9 times tables

I can never remember 6 × 7 or 9 × 8. Is there a trick?

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

A times table lists the answers for one number. You know the 2, 3, 4, 5 and 10 tables. Now we build the last three from them.

Start with the sixes. A six is two threes. So 6 × 4 is double 3 × 4. To double is to make twice as much.

3 × 4 = 12. Double 12 is 24. So 6 × 4 = 24.

Now the nines. Nine is one less than ten. So 9 × 7 is 10 rows of 7, less one row.

Draw it first

Find 9 × 7 from 10 × 7

  1. Ten rows of 7 dots. Step 1 labels it 10 × 7 = 70. Step 2 fills in the top row and labels it take away 7. Step 3 labels what is left 70 − 7 = 63 and 9 × 7 = 63.10 × 7 = 70

    1 Draw 10 rows of 7 dots. That is 10 × 7 = 70.

  2. Ten rows of 7 dots. Step 1 labels it 10 × 7 = 70. Step 2 fills in the top row and labels it take away 7. Step 3 labels what is left 70 − 7 = 63 and 9 × 7 = 63.10 × 7 = 70take away 7

    2 Fill in one whole row. This is the row to take away. It has 7 dots.

  3. Ten rows of 7 dots. Step 1 labels it 10 × 7 = 70. Step 2 fills in the top row and labels it take away 7. Step 3 labels what is left 70 − 7 = 63 and 9 × 7 = 63.10 × 7 = 70take away 770 − 7 = 639 × 7 = 63

    3 Count the empty rows. There are 9. So 9 × 7 = 70 − 7 = 63.

The rule for nines: work out 10 × the number. Then take away the number.

Worked example

Work out 9 × 6.

  1. 10 × 6 = 60.
  2. Take away one 6. 60 − 6 = 54.

Answer: 9 × 6 = 54

The sevens are a five and a two. For 7 × 8, do 5 × 8 and 2 × 8. Then add.

See it first with 7 rows of 4 dots. Fill in 5 rows. The other 2 rows are empty.

Seven rows of 4 dots. The top five rows are filled in and labelled 5 × 4 = 20. The bottom two rows are empty and labelled 2 × 4 = 8.5 × 4 = 202 × 4 = 8
A seven is a five and a two. 7 × 4 = 20 + 8 = 28.

Worked example

Work out 7 × 8.

  1. 5 × 8 = 40.
  2. 2 × 8 = 16.
  3. 40 + 16 = 56.

Answer: 7 × 8 = 56

Three rows of answers for 1 to 10 groups. Row 6s reads 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. Row 7s reads 7, 14, 21, 28, 35, 42, 49, 56, 63, 70. Row 9s reads 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.6s61218243036424854607s71421283542495663709s9182736455463728190
The three tables. Say each row out loud.

See it three ways

With things

Put 4 stones in each of 6 rows. Cover 3 rows with your hand. Count 12. Double it.

As a picture

Six rows of 4 stones. The top three rows are filled in and labelled 3 × 4 = 12. The bottom three rows are empty and labelled 3 × 4 = 12.3 × 4 = 123 × 4 = 12

In symbols

6 × 4 = 3 × 4 + 3 × 4 = 12 + 12 = 24.

Try it

Look for a pattern in the nines

You need: a pencil, a sheet of paper, or an old page.

  1. Write 9 × 2 = 18 and 9 × 3 = 27. Then 9 × 4 = 36 and 9 × 5 = 45.
  2. Beside each answer, add its two digits.
  3. Carry on to 9 × 10 = 90.

What you will see: The digits of every answer add to 9. The tens digit goes up by 1. The ones digit goes down by 1. You can use this to check a nine.

If you cannot try it, this is what happens.
Four lines. On the left, 9 × 2 = 18, 9 × 3 = 27, 9 × 4 = 36 and 9 × 5 = 45. On the right, the digits added: 1 + 8 = 9, 2 + 7 = 9, 3 + 6 = 9 and 4 + 5 = 9.9 × 2 = 181 + 8 = 99 × 3 = 272 + 7 = 99 × 4 = 363 + 6 = 99 × 5 = 454 + 5 = 9

A wrong idea many people have

9 × 7 is 10 × 7 take away 9. So it is 61.

Why it feels right: The nine is right there in 9 × 7. It seems the number to take away.

The test that breaks it: Make 10 rows of 7 dots. Take away one row. Count what is left. You get 63, not 61. One row is 7 dots, not 9.

What is true: Take away one row. One row is the other number. 9 × 7 = 70 − 7 = 63.

Key words

double
to make twice as much.
times table
the answers when you count in equal groups of one number.

Back to your guess

The answer is B: Yes. Each hard fact is built from a fact I know.

Sixes are double the threes. Nines are tens with one group taken away. Sevens are fives plus twos. You already know those tables.

Check yourself

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

  1. What is 6 × 8? Double 3 × 8 to find out.

    Tap your answer.

    1. Yes, that is right. 3 × 8 = 24. Double 24 is 48. So 6 × 8 = 48.

    2. Not this one. You may have 6 × 7 in mind. Double 3 × 8 = 24 is 48. Try another.

    3. Not this one. You added 6 + 8. Times means 6 equal groups of 8. Go to M2.3.1. Try another.

    See the answer

    48

    3 × 8 = 24. Double 24 is 48. So 6 × 8 = 48.

    • If you chose B: You may have 6 × 7 in mind. Double 3 × 8 = 24 is 48.
    • If you chose C: You added 6 + 8. Times means 6 equal groups of 8. Go to M2.3.1.

  2. Use 10 × 6 to work out 9 × 6.

    See the answer

    54

    10 × 6 = 60. Take away one 6. 60 − 6 = 54.

  3. Mei puts 8 stones in each of 7 bags. How many stones is that?

    See the answer

    56 stones

    7 bags of 8 is 7 × 8. That is 5 × 8 plus 2 × 8. 40 + 16 = 56.

  4. Kofi says 9 × 4 = 32. Is he right?

    Tap your answer.

    1. Not this one. That is the eights. 9 × 4 is one more group of 4 than that. Try another.

    2. Yes, that is right. 10 × 4 = 40. Take away one 4. 40 − 4 = 36.

    3. Not this one. You took away 9 from 40. Take away one group of 4, not 9. Try another.

    See the answer

    No. 9 × 4 = 36.

    10 × 4 = 40. Take away one 4. 40 − 4 = 36.

    • If you chose A: That is the eights. 9 × 4 is one more group of 4 than that.
    • If you chose C: You took away 9 from 40. Take away one group of 4, not 9.

  5. From an earlier chapter

    Use 4 × 6 = 24 to find 8 × 6.

    See the answer

    48

    Eight is double four. Double 24 is 48. So 8 × 6 = 48.

  6. Look at the dots. Take away the shaded row. How many dots are left?

    Ten rows of 5 dots with the top row shaded, labelled 10 rows of 5.10 rows of 5
    Tap your answer.

    1. Not this one. You took away 9. The shaded row has 5 dots, so take away 5. Try another.

    2. Not this one. You took away one dot. The shaded row has 5 dots, so take away 5. Try another.

    3. Yes, that is right. 10 rows of 5 is 50. Take away one row of 5. 50 − 5 = 45. That is 9 × 5.

    See the answer

    45

    10 rows of 5 is 50. Take away one row of 5. 50 − 5 = 45. That is 9 × 5.

    • If you chose A: You took away 9. The shaded row has 5 dots, so take away 5.
    • If you chose B: You took away one dot. The shaded row has 5 dots, so take away 5.

  7. What is 7 × 6? Use 5 × 6 and 2 × 6.

    Tap your answer.

    1. Not this one. You added 7 and 6. Times means 7 groups of 6. Go to M2.3.1. Try another.

    2. Yes, that is right. 5 × 6 = 30 and 2 × 6 = 12. 30 + 12 = 42.

    3. Not this one. That is only 5 × 6. Add 2 × 6 = 12 as well. Try another.

    See the answer

    42

    5 × 6 = 30 and 2 × 6 = 12. 30 + 12 = 42.

    • If you chose A: You added 7 and 6. Times means 7 groups of 6. Go to M2.3.1.
    • If you chose C: That is only 5 × 6. Add 2 × 6 = 12 as well.

  8. Hana knows 3 × 9 = 27. Which sum gives 6 × 9?

    Tap your answer.

    1. Not this one. You added the 6. Six is double three, so double 27. Try another.

    2. Not this one. That is 4 × 9, one more group of 9. Six groups is double three groups. Try another.

    3. Yes, that is right. Six is double three. Double 27 is 27 + 27 = 54. So 6 × 9 = 54.

    See the answer

    27 + 27 = 54

    Six is double three. Double 27 is 27 + 27 = 54. So 6 × 9 = 54.

    • If you chose A: You added the 6. Six is double three, so double 27.
    • If you chose B: That is 4 × 9, one more group of 9. Six groups is double three groups.

Keep it

Copy this into your exercise book.

Sixes are double threes: 6 × 4 = 12 + 12. Nines are tens less one group: 9 × 7 = 70 − 7. Sevens are fives plus twos: 7 × 8 = 40 + 16.

Draw: Ten rows of 7 dots. Fill in one row. Write 70 − 7 = 63.

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

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.

M2.3.7 Chapter 7

Remainders

I have 17 stones. I make groups of 5. What about the extra stones?

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

Some numbers do not share out evenly. A few things are left over.

Take 17 stones. Make groups of 5. You can make 3 full groups. That uses 15 stones.

The 2 stones left over are the remainder. We write 17 ÷ 5 = 3, remainder 2.

Draw it first

Share 17 stones into groups of 5

  1. Seventeen stones. Step 1 shows them loose, labelled 17 stones. Step 2 puts 15 of them in 3 groups of 5 and labels the groups 3 groups of 5. Step 3 labels the 2 stones left over remainder 2.17 stones

    1 Put out 17 stones.

  2. Seventeen stones. Step 1 shows them loose, labelled 17 stones. Step 2 puts 15 of them in 3 groups of 5 and labels the groups 3 groups of 5. Step 3 labels the 2 stones left over remainder 2.3 groups of 5

    2 Make groups of 5. You can make 3 full groups. Two stones are in no group.

  3. Seventeen stones. Step 1 shows them loose, labelled 17 stones. Step 2 puts 15 of them in 3 groups of 5 and labels the groups 3 groups of 5. Step 3 labels the 2 stones left over remainder 2.3 groups of 5remainder 2

    3 The 2 stones left over are the remainder.

You can always check. Groups times size, plus the remainder, gives the start. 3 × 5 + 2 = 17.

The remainder is always smaller than the group size. Here 2 is smaller than 5. If it were 5, you could make one more group.

Two pictures of 20 stones. The top picture shows 3 groups of 5 and 5 stones apart, labelled 3 groups of 5, 5 left over. The bottom picture shows 4 groups of 5, labelled 5 left over makes a 4th group.3 groups of 5, 5 left over5 left over makes a 4th group
A remainder of 5 is not a remainder. Those 5 stones make one more group.

Worked example

Work out 23 ÷ 4.

  1. Count in 4s: 4, 8, 12, 16, 20. The next one, 24, is too big.
  2. So 5 groups of 4 fit. 5 × 4 = 20.
  3. What is left? 23 − 20 = 3.

Answer: 5 groups, remainder 3

In a story, you must decide what to do with the remainder. Look at two stories.

Story one. 17 children must ride in cars. A car holds 5. How many cars?

17 ÷ 5 = 3, remainder 2. Two children still need a seat. They need another car. So 4 cars. This is to round up. You go up to the next whole number.

Seventeen dots for children. Three groups of 5 are labelled 3 full cars. A fourth group of 2 dots is labelled car 4.3 full carscar 4
The 2 children left over need a car. Round up to 4 cars.

Story two. 17 eggs go in boxes. A box holds 5. How many full boxes?

Again 17 ÷ 5 = 3, remainder 2. Two eggs do not fill a box. So 3 full boxes. This is to round down. You drop the remainder.

Seventeen stones stand for the eggs. Three rings of 5 stones, one ring for each box, are labelled 3 full boxes. Two stones outside the rings are labelled left over.3 full boxesleft over
The 2 eggs left over do not fill a box. Round down to 3 boxes.

See it three ways

With things

Put out 17 stones. Take away groups of 5 until you cannot take another.

As a picture

Three groups of 5 stones. Two more stones sit apart from the groups, labelled remainder 2.remainder 2

In symbols

17 ÷ 5 = 3, remainder 2. Check: 3 × 5 + 2 = 17.

Try it

Share 14 stones two ways

You need: 14 stones, beans or sticks.

  1. Put out 14 stones.
  2. Make groups of 4. Count the full groups. Count the stones left over.
  3. Put them all back. Now make groups of 7.
  4. Count the full groups. Count the stones left over.

What you will see: Groups of 4 leave 2 stones. Groups of 7 leave none. The remainder is 0 when a number shares out exactly.

If you cannot try it, this is what happens.
Two pictures of 14 stones. The first has 3 groups of 4 and 2 stones apart, labelled 3 groups of 4, remainder 2. The second has 2 groups of 7 and nothing apart, labelled 2 groups of 7, remainder 0.3 groups of 4, remainder 22 groups of 7, remainder 0

A wrong idea many people have

The remainder is always dropped. 17 children in cars of 5 need 3 cars.

Why it feels right: The sum 17 ÷ 5 gives 3. So 3 looks like the answer.

The test that breaks it: Put 17 stones in 3 groups of 5. Two stones are in no group. In the story, two children would have no seat.

What is true: Read what the story asks. If the rest still need a place, round up. If only full groups count, round down.

Key words

remainder
what is left over when things do not share out evenly.
round up
to go up to the next whole number.
round down
to drop the remainder and keep the whole number.

Back to your guess

The answer is B: There are 3 groups of 5, and 2 stones left over.

Three groups of 5 use 15 stones. Two are left over. That is the remainder. A story then tells you what to do with it.

Check yourself

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

  1. Put 20 stones in groups of 6. How many stones are left over?

    Tap your answer.

    1. Yes, that is right. 3 groups of 6 use 18 stones. 20 − 18 = 2 left over.

    2. Not this one. That is the number of full groups. 3 groups of 6 use 18 stones. Now take 18 from 20. Try another.

    3. Not this one. You took away one group of 6. Keep making groups until no full group is left. Try another.

    See the answer

    2

    3 groups of 6 use 18 stones. 20 − 18 = 2 left over.

    • If you chose B: That is the number of full groups. 3 groups of 6 use 18 stones. Now take 18 from 20.
    • If you chose C: You took away one group of 6. Keep making groups until no full group is left.

  2. Work out 29 ÷ 5. Say the groups and the remainder.

    See the answer

    5 groups, remainder 4

    5 × 5 = 25. 29 − 25 = 4. The remainder is smaller than 5, so it is right.

  3. Lucía takes 23 children across a river. A boat holds 6. How many boats?

    Tap your answer.

    1. Yes, that is right. 23 ÷ 6 = 3, remainder 5. Five children still need a boat. Round up to 4.

    2. Not this one. You dropped the remainder. 5 children would have no boat. Round up. Try another.

    3. Not this one. Four boats hold 24 children. That is enough for 23. Try another.

    See the answer

    4 boats

    23 ÷ 6 = 3, remainder 5. Five children still need a boat. Round up to 4.

    • If you chose B: You dropped the remainder. 5 children would have no boat. Round up.
    • If you chose C: Four boats hold 24 children. That is enough for 23.

  4. Joseph has 23 sticks. A bundle needs 6 sticks. How many full bundles can he make?

    See the answer

    3 bundles

    23 ÷ 6 = 3, remainder 5. Five sticks cannot make a full bundle. Round down to 3.

  5. From an earlier chapter

    Share 24 beans fairly between 4 friends. How many beans does each friend get?

    See the answer

    6 beans each

    24 ÷ 4 = 6, because 4 × 6 = 24. Nothing is left over.

  6. Stones go in groups of 5. Which number can be a remainder?

    Tap your answer.

    1. Not this one. With 5 left over you could make one more group. A remainder is smaller than the group. Try another.

    2. Yes, that is right. The remainder is always smaller than the group size. Only 3 is smaller than 5.

    3. Not this one. With 7 left over you could make one more group of 5. A remainder is smaller than 5. Try another.

    See the answer

    3

    The remainder is always smaller than the group size. Only 3 is smaller than 5.

    • If you chose A: With 5 left over you could make one more group. A remainder is smaller than the group.
    • If you chose C: With 7 left over you could make one more group of 5. A remainder is smaller than 5.

  7. Sipho has 31 beans. A pot needs 7 beans. How many full pots can he fill?

    Tap your answer.

    1. Not this one. You rounded up. A 5th pot would hold only 3 beans. It is not full. Try another.

    2. Not this one. The 3 is the remainder, the beans left over. Count the full pots instead. Try another.

    3. Yes, that is right. 4 × 7 = 28. 31 − 28 = 3 beans are left. Only full pots count, so round down to 4.

    See the answer

    4 pots

    4 × 7 = 28. 31 − 28 = 3 beans are left. Only full pots count, so round down to 4.

    • If you chose A: You rounded up. A 5th pot would hold only 3 beans. It is not full.
    • If you chose B: The 3 is the remainder, the beans left over. Count the full pots instead.

  8. A boat holds 4 people. How many boats do these people need?

    Dots for people waiting to cross a river, in two rows. The label is people waiting.people waiting
    Tap your answer.

    1. Not this one. You dropped the remainder. 3 people would be left on the bank. Round up. Try another.

    2. Not this one. The 4 is how many one boat holds. Count how many boats everyone needs. Try another.

    3. Yes, that is right. 11 ÷ 4 = 2, remainder 3. Three people still need a boat. Round up to 3.

    See the answer

    3 boats

    11 ÷ 4 = 2, remainder 3. Three people still need a boat. Round up to 3.

    • If you chose A: You dropped the remainder. 3 people would be left on the bank. Round up.
    • If you chose B: The 4 is how many one boat holds. Count how many boats everyone needs.

  9. Which number makes groups of 4 with nothing left over?

    Tap your answer.

    1. Not this one. 14 is even, but 3 groups of 4 use 12. Two are left over. Try another.

    2. Not this one. 34 ends in 4. It is not in the 4 times table. 8 groups of 4 use 32. Two are left over. Try another.

    3. Yes, that is right. 5 × 4 = 20. So 5 groups of 4 use all 20. None are left over.

    See the answer

    20

    5 × 4 = 20. So 5 groups of 4 use all 20. None are left over.

    • If you chose A: 14 is even, but 3 groups of 4 use 12. Two are left over.
    • If you chose B: 34 ends in 4. It is not in the 4 times table. 8 groups of 4 use 32. Two are left over.

Keep it

Copy this into your exercise book.

The remainder is what is left over. Groups times size, plus remainder, gives the start. 17 ÷ 5 = 3, remainder 2. In a story, round up if the rest need a place.

Draw: 17 stones. Ring 3 groups of 5. Show the 2 left over. Write 3, remainder 2.

M2.3.8 Chapter 8

Times and divide with bigger numbers

What happens to 35 when I times it by 10?

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

Big numbers use the same ideas as small ones. First, times 10.

The place value of a digit is what it is worth. It depends on where the digit stands.

35 is 3 tens and 5 ones. Ten times as much is 3 hundreds and 5 tens.

Draw it first

Times 10 moves the digits left

  1. A place value chart with the headings hundreds, tens and ones holds 35. The label × 10 sits under it. A second chart with the same headings holds 350. The last label reads 35 × 10 = 350.hundredstensones35

    1 Write 35 on a chart. The 3 is in tens. The 5 is in ones.

  2. A place value chart with the headings hundreds, tens and ones holds 35. The label × 10 sits under it. A second chart with the same headings holds 350. The last label reads 35 × 10 = 350.hundredstensones35× 10hundredstensones350

    2 Times 10. Each digit moves one place left. The 3 is in hundreds. The 5 is in tens.

  3. A place value chart with the headings hundreds, tens and ones holds 35. The label × 10 sits under it. A second chart with the same headings holds 350. The last label reads 35 × 10 = 350.hundredstensones35× 10hundredstensones35035 × 10 = 350

    3 The ones place is empty, so it holds a 0. 35 × 10 = 350.

Times 100 moves every digit two places left. 35 × 100 = 3500.

A place value chart with the headings thousands, hundreds, tens and ones holds 35. The label × 100 sits under it. A second chart with the same headings holds 3500, with the 3 in thousands and the 5 in hundreds.thousandshundredstensones35× 100thousandshundredstensones3500
Times 100 moves each digit two places left. The two empty places hold 0.

Now a number such as 23 × 4. Split 23 into 20 and 3. To split a number into parts like this is to partition it.

Times each part by 4. Then add. We draw a box for each part.

Draw it first

Work out 23 × 4 in boxes

  1. Two boxes in a row. Step 1 writes 23 = 20 + 3 on top, then 20 over the wide box, 3 over the narrow box, and 4 beside both. Step 2 writes 80 in the wide box and 12 in the narrow box. Step 3 writes 80 + 12 = 92 and 23 × 4 = 92.23 = 20 + 32034

    1 Split 23 into 20 and 3. Draw a box for each. Write 4 beside them.

  2. Two boxes in a row. Step 1 writes 23 = 20 + 3 on top, then 20 over the wide box, 3 over the narrow box, and 4 beside both. Step 2 writes 80 in the wide box and 12 in the narrow box. Step 3 writes 80 + 12 = 92 and 23 × 4 = 92.23 = 20 + 320348012

    2 Times each part by 4. 20 × 4 = 80 and 3 × 4 = 12.

  3. Two boxes in a row. Step 1 writes 23 = 20 + 3 on top, then 20 over the wide box, 3 over the narrow box, and 4 beside both. Step 2 writes 80 in the wide box and 12 in the narrow box. Step 3 writes 80 + 12 = 92 and 23 × 4 = 92.23 = 20 + 32034801280 + 12 = 9223 × 4 = 92

    3 Add the two boxes. 80 + 12 = 92. So 23 × 4 = 92.

Worked example

Work out 36 × 5.

  1. Split 36 into 30 and 6.
  2. 30 × 5 = 150.
  3. 6 × 5 = 30.
  4. Add. 150 + 30 = 180.

Answer: 36 × 5 = 180

Dividing works the same way. Split the big number into parts that share out well.

Take 84 ÷ 4. Split 84 into 80 and 4. Both parts divide by 4.

Draw it first

Work out 84 ÷ 4 with blocks

  1. Base ten blocks. Step 1 shows 84 as 8 tens and 4 ones, labelled 84. Step 2 shares them into 4 equal groups of 2 tens and 1 one, each labelled 21, and the line 4 equal groups. Step 3 writes 80 ÷ 4 = 20, 4 ÷ 4 = 1 and 84 ÷ 4 = 20 + 1 = 21.84

    1 Show 84 as 8 tens and 4 ones.

  2. Base ten blocks. Step 1 shows 84 as 8 tens and 4 ones, labelled 84. Step 2 shares them into 4 equal groups of 2 tens and 1 one, each labelled 21, and the line 4 equal groups. Step 3 writes 80 ÷ 4 = 20, 4 ÷ 4 = 1 and 84 ÷ 4 = 20 + 1 = 21.212121214 equal groups

    2 Share them into 4 equal groups. Each group gets 2 tens and 1 one.

  3. Base ten blocks. Step 1 shows 84 as 8 tens and 4 ones, labelled 84. Step 2 shares them into 4 equal groups of 2 tens and 1 one, each labelled 21, and the line 4 equal groups. Step 3 writes 80 ÷ 4 = 20, 4 ÷ 4 = 1 and 84 ÷ 4 = 20 + 1 = 21.212121214 equal groups80 ÷ 4 = 204 ÷ 4 = 184 ÷ 4 = 20 + 1 = 21

    3 Each group is 21. 80 ÷ 4 = 20 and 4 ÷ 4 = 1. So 84 ÷ 4 = 21.

Worked example

Work out 78 ÷ 3.

  1. Split 78 into 60 and 18. Both divide by 3.
  2. 60 ÷ 3 = 20.
  3. 18 ÷ 3 = 6.
  4. Add. 20 + 6 = 26.

Answer: 78 ÷ 3 = 26

See it three ways

With things

Make 4 piles. Each pile has 2 bundles of ten sticks and 3 single sticks. Count the tens, then the ones.

As a picture

Four groups of base ten blocks, each labelled 23. Each group has 2 tens and 3 ones.23232323

In symbols

23 × 4 = 20 × 4 + 3 × 4. 20 × 4 = 80 and 3 × 4 = 12. 80 + 12 = 92.

Try it

Times 10 on a paper chart

You need: a pencil, a sheet of paper, or an old page.

  1. Draw three columns. Write hundreds, tens and ones over them.
  2. Write 27 in the chart. The 2 is in tens. The 7 is in ones.
  3. Now times 10. Move each digit one column to the left.
  4. Write 0 in the empty ones column. Read the number.

What you will see: You read 270. Check it. Add 27 ten times. You also get 270.

If you cannot try it, this is what happens.
A place value chart with the headings hundreds, tens and ones holds 27. The label × 10 sits under it. A second chart with the same headings holds 270.hundredstensones27× 10hundredstensones270

A wrong idea many people have

23 × 4 = 812, because 2 × 4 = 8 and 3 × 4 = 12.

Why it feels right: You times each digit by 4. It seems fair to write the answers side by side.

The test that breaks it: Add 23 four times. 23 + 23 = 46. 46 + 46 = 92. The answer is 92. It is far less than 812.

What is true: The 2 is 2 tens, not 2 ones. 2 tens × 4 = 80. Then add 3 × 4 = 12. 80 + 12 = 92.

Key words

place value
what a digit is worth, because of the place it stands in.
partition
to split a number into parts, such as 20 and 3. The parts use tables you know.

Back to your guess

The answer is B: Each digit moves one place left. It becomes 350.

Times 10 moves every digit one place to the left. The empty ones place holds a 0. So 35 × 10 = 350.

Check yourself

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

  1. What is 46 × 10?

    Tap your answer.

    1. Yes, that is right. 4 tens and 6 ones become 4 hundreds and 6 tens. The ones place holds 0.

    2. Not this one. You added 10. Times 10 moves every digit one place left. Try another.

    3. Not this one. You wrote 10 after the 46. Move the digits left and put 0 in the ones place. Try another.

    See the answer

    460

    4 tens and 6 ones become 4 hundreds and 6 tens. The ones place holds 0.

    • If you chose B: You added 10. Times 10 moves every digit one place left.
    • If you chose C: You wrote 10 after the 46. Move the digits left and put 0 in the ones place.

  2. Mei has 6 bags. Each bag holds 100 beans. How many beans is that?

    See the answer

    600 beans

    Times 100 moves the 6 two places left, from ones to hundreds. 6 × 100 = 600.

  3. Work out 32 × 3. Split 32 into 30 and 2.

    See the answer

    96

    30 × 3 = 90. 2 × 3 = 6. 90 + 6 = 96.

  4. Work out 68 ÷ 4. Split 68 into 40 and 28.

    See the answer

    17

    40 ÷ 4 = 10. 28 ÷ 4 = 7. 10 + 7 = 17.

  5. From an earlier chapter

    Use 8 × 7 = 56 to answer 56 ÷ 7.

    See the answer

    8

    Ask: 7 times what makes 56? 8 × 7 = 56. So 56 ÷ 7 = 8.

  6. The chart shows 4. What is 4 × 100?

    A place value chart with the headings hundreds, tens and ones holds 4 in the ones place.hundredstensones4
    Tap your answer.

    1. Not this one. That is times 10, one place left. Times 100 moves the 4 two places left. Try another.

    2. Not this one. You wrote 100 after the 4. Move the 4 two places left. Fill the empty places with 0. Try another.

    3. Yes, that is right. Times 100 moves each digit two places left. The 4 moves from ones to hundreds. 4 × 100 = 400.

    See the answer

    400

    Times 100 moves each digit two places left. The 4 moves from ones to hundreds. 4 × 100 = 400.

    • If you chose A: That is times 10, one place left. Times 100 moves the 4 two places left.
    • If you chose B: You wrote 100 after the 4. Move the 4 two places left. Fill the empty places with 0.

  7. Tariq works out 34 × 3. Which answer is right?

    Tap your answer.

    1. Not this one. You wrote 9 and 12 side by side. The 3 is 3 tens: 30 × 3 = 90. Then add 4 × 3 = 12. Try another.

    2. Not this one. You forgot that the 3 is 3 tens. 30 × 3 = 90, not 9. Then add 4 × 3 = 12. Try another.

    3. Yes, that is right. Split 34 into 30 and 4. 30 × 3 = 90. 4 × 3 = 12. 90 + 12 = 102.

    See the answer

    102

    Split 34 into 30 and 4. 30 × 3 = 90. 4 × 3 = 12. 90 + 12 = 102.

    • If you chose A: You wrote 9 and 12 side by side. The 3 is 3 tens: 30 × 3 = 90. Then add 4 × 3 = 12.
    • If you chose B: You forgot that the 3 is 3 tens. 30 × 3 = 90, not 9. Then add 4 × 3 = 12.

  8. Work out 96 ÷ 3. Split 96 into 90 and 6. What is the answer?

    Tap your answer.

    1. Not this one. You wrote 30 and 2 side by side. Add them: 30 + 2. Try another.

    2. Yes, that is right. 90 ÷ 3 = 30. 6 ÷ 3 = 2. 30 + 2 = 32.

    3. Not this one. That is only 90 ÷ 3. Divide the 6 by 3 as well, and add. Try another.

    See the answer

    32

    90 ÷ 3 = 30. 6 ÷ 3 = 2. 30 + 2 = 32.

    • If you chose A: You wrote 30 and 2 side by side. Add them: 30 + 2.
    • If you chose C: That is only 90 ÷ 3. Divide the 6 by 3 as well, and add.

  9. The groups of blocks are all the same. How many are there in all?

    Three groups of base ten blocks, each labelled one group. Each group has 1 ten and 4 ones.one groupone groupone group
    Tap your answer.

    1. Not this one. You wrote 3 and 12 side by side. 3 × 10 = 30 and 3 × 4 = 12. Add them. Try another.

    2. Yes, that is right. Each group is 1 ten and 4 ones, which is 14. 3 × 10 = 30 and 3 × 4 = 12. 30 + 12 = 42.

    3. Not this one. You added 3 and 14. Times means 3 groups of 14. Go to M2.3.1. Try another.

    See the answer

    42

    Each group is 1 ten and 4 ones, which is 14. 3 × 10 = 30 and 3 × 4 = 12. 30 + 12 = 42.

    • If you chose A: You wrote 3 and 12 side by side. 3 × 10 = 30 and 3 × 4 = 12. Add them.
    • If you chose C: You added 3 and 14. Times means 3 groups of 14. Go to M2.3.1.

Keep it

Copy this into your exercise book.

Times 10 moves each digit one place left. Split to multiply: 23 × 4 = 80 + 12 = 92. Split to divide: 84 ÷ 4 = 20 + 1 = 21.

Draw: A box grid. Write 20 and 3 across the top and 4 at the side. Write 80 and 12 inside.

Key words, A to Z

Every key word in this book, what it means, and the chapter that teaches it.

array
things set out in equal rows. M2.3.1
digit
one of the figures 0 to 9. In 35, the last digit is 5. M2.3.2
divide
to split into equal groups. The sign is ÷. M2.3.3
double
to make twice as many. Double 4 is 8. M2.3.4, M2.3.5
equal groups
groups that all have the same number of things. M2.3.1
even
a number that shares into 2 equal groups with none left. M2.3.2
fact family
times and divide facts that use the same three numbers. M2.3.6
grouping
finding how many groups of one size you can make. M2.3.3
partition
to split a number into parts, such as 20 and 3. The parts use tables you know. M2.3.8
place value
what a digit is worth, because of the place it stands in. M2.3.8
remainder
what is left over when things do not share out evenly. M2.3.7
round down
to drop the remainder and keep the whole number. M2.3.7
round up
to go up to the next whole number. M2.3.7
row
a line of things going across. M2.3.1
sharing
dealing things out equally, one at a time. M2.3.3
times
groups of. 4 × 3 is read "4 times 3". M2.3.1
times table
a list of answers for more and more equal groups. M2.3.2, M2.3.5