Chapter 1 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

M2.3.1 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

  • Write 4 groups of 3 as 4 × 3
  • Show 4 × 3 as rows and columns of dots

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.

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.