Chapter 3 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

M2.3.3 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

  • Share 12 things equally and write 12 ÷ 3 = 4
  • Find how many groups of 4 are in 20

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.

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.