Chapter 4 of 8

All 8 chapters
  1. 1Fair shares: equal parts, readunderstood
  2. 2One part: halves, quarters, thirds, fifths, readunderstood
  3. 3More than one part, readunderstood
  4. 4Fractions of a group, readunderstood
  5. 5The number line, past 1, readunderstood
  6. 6Same amount, different names, readunderstood
  7. 7Which is bigger, readunderstood
  8. 8Adding and taking away parts, readunderstood

M2.4.4 Chapter 4

Fractions of a group

I have 12 beans. How many beans is 1/3 of them?

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

  • Find a fraction of a group of things
  • Make equal groups, then take some of them
  • Explain why 2/3 of 12 is 8

So far the whole was one thing, like a bread. A whole can be a group of things too.

A group is a number of things together. 12 beans are a group. We can take a fraction of them.

Take 1/3 of 12 beans. The bottom number is 3. Make 3 equal groups. Equal groups have the same number of things.

Draw it first

Find one third of 12 beans

  1. Twelve beans in a loose set, labelled 12 beans. Below, the same beans in 3 equal groups of 4. Then the first group is filled in, labelled one third of 12 is 4.12 beans

    1 Start with 12 beans. That is the whole group.

  2. Twelve beans in a loose set, labelled 12 beans. Below, the same beans in 3 equal groups of 4. Then the first group is filled in, labelled one third of 12 is 4.12 beans

    2 Make 3 equal groups. Each group has 4 beans.

  3. Twelve beans in a loose set, labelled 12 beans. Below, the same beans in 3 equal groups of 4. Then the first group is filled in, labelled one third of 12 is 4.12 beansone third of 12 is 4

    3 Take 1 group. 1/3 of 12 is 4.

The bottom number makes the groups. The top number says how many groups to take.

For 2/3 of 12, make 3 equal groups. Then take 2 groups.

Worked example

Find 2/3 of 12 beans.

  1. The bottom number is 3. Make 3 equal groups.
  2. 12 beans in 3 equal groups is 4 in each group.
  3. The top number is 2. Take 2 groups.
  4. 2 groups of 4 is 4 + 4 = 8.

Answer: 8 beans

The same 12 beans can make 4 equal groups. Then each group is smaller. It holds fewer beans.

Two sets of 12 beans. The top set is in 3 equal groups of 4, with one group filled in, labelled one third of 12 is 4. The bottom set is in 4 equal groups of 3, with one group filled in, labelled one quarter of 12 is 3.one third of 12 is 4one quarter of 12 is 3
More groups means fewer beans in each group.

See it three ways

With things

Put 8 stones in a pile. Share them into 4 equal groups. Take 3 groups.

As a picture

Eight stones in 4 equal groups of 2. Three groups are filled in and one group is not.

In symbols

8 ÷ 4 = 2. 2 + 2 + 2 = 6. So 3/4 of 8 is 6.

Try it

Share beans into groups

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

  1. Count out 12 beans.
  2. Share them one at a time into 3 groups. Stop when none are left.
  3. Count one group.
  4. Now push 2 groups together. Count them.

What you will see: Each group has 4. One group is 1/3 of 12, which is 4. Two groups are 2/3 of 12, which is 8.

If you cannot try it, this is what happens.
Twelve beans in 3 equal groups. Each group is labelled 4. Two of the groups are filled in.444

A wrong idea many people have

1/3 of 12 is 3, because of the 3.

Why it feels right: The 3 is the only number you see in the fraction. It looks like the answer.

The test that breaks it: Count out 12 beans. Make 3 equal groups. Count the beans in one group.

What is true: The 3 says how many groups to make. Each group has 4 beans, so 1/3 of 12 is 4.

Key words

group
a number of things together.
equal groups
groups that all have the same number of things.

Back to your guess

The answer is B: 4 beans.

Make 3 equal groups of the 12 beans. Each group has 4 beans. So 1/3 of 12 is 4.

Check yourself

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

  1. Look at the picture. How many stones are in 1/5 of 10 stones?

    Ten stones in 5 equal groups of 2. One group is filled in.
    Tap your answer.

    1. Yes, that is right. 10 stones in 5 equal groups is 2 in each group. 1/5 of 10 is one group.

    2. Not this one. The 5 tells how many groups to make. It does not tell how many stones. Try another.

    3. Not this one. You counted the stones not filled in. That is 4/5 of 10. 1/5 is 1 group. Try another.

    See the answer

    2 stones

    10 stones in 5 equal groups is 2 in each group. 1/5 of 10 is one group.

    • If you chose B: The 5 tells how many groups to make. It does not tell how many stones.
    • If you chose C: You counted the stones not filled in. That is 4/5 of 10. 1/5 is 1 group.

  2. What is 1/4 of 12 beans?

    See the answer

    3 beans

    12 beans in 4 equal groups is 3 in each group. One group is 3 beans.

  3. What is 2/3 of 9 stones?

    See the answer

    6 stones

    9 stones in 3 equal groups is 3 in each group. Two groups is 3 + 3 = 6.

  4. From an earlier chapter

    A bar has 5 equal parts. 4 parts are shaded. What fraction is shaded?

    A bar cut into 5 equal parts. The first 4 parts are shaded.
    Tap your answer.

    1. Yes, that is right. The bar has 5 equal parts. 4 are shaded. So the fraction is 4/5.

    2. Not this one. That is one part. Here 4 parts are shaded, so the top number is 4. Go to M2.4.3. Try another.

    3. Not this one. The bottom number is all the equal parts. That is 5. Go to M2.4.3. Try another.

    See the answer

    4/5

    The bar has 5 equal parts. 4 are shaded. So the fraction is 4/5.

    • If you chose B: That is one part. Here 4 parts are shaded, so the top number is 4. Go to M2.4.3.
    • If you chose C: The bottom number is all the equal parts. That is 5. Go to M2.4.3.

  5. Mei has 12 beans. She gives 3/4 of them to Kofi. How many beans does Kofi get? How many does Mei keep?

    See the answer

    Kofi gets 9 beans. Mei keeps 3 beans.

    12 beans in 4 equal groups is 3 in each group. Kofi takes 3 groups, which is 9. Mei keeps 1 group, which is 3.

  6. Amina finds 2/3 of 12 beans. Which working is right?

    Tap your answer.

    1. Not this one. The bottom number, 3, makes the groups. The top number says how many to take. Try another.

    2. Yes, that is right. 12 beans in 3 equal groups is 4 in each group. 2 groups is 4 + 4 = 8.

    3. Not this one. That is 1/3 of 12. The top number is 2, so take 2 groups. Try another.

    See the answer

    Make 3 equal groups of 4. Take 2 groups. That makes 8 beans.

    12 beans in 3 equal groups is 4 in each group. 2 groups is 4 + 4 = 8.

    • If you chose A: The bottom number, 3, makes the groups. The top number says how many to take.
    • If you chose C: That is 1/3 of 12. The top number is 2, so take 2 groups.

  7. Hana has 10 beans, as in the picture. What is 3/5 of them?

    10 beans in two rows. The label is 10 beans.10 beans
    Tap your answer.

    1. Yes, that is right. 10 beans in 5 equal groups is 2 in each group. 3 groups is 2 + 2 + 2 = 6.

    2. Not this one. That is one group, 1/5 of 10. The top number is 3, so take 3 groups. Try another.

    3. Not this one. You took 5 away from 10. The bottom number says how many equal groups to make. Try another.

    See the answer

    6 beans

    10 beans in 5 equal groups is 2 in each group. 3 groups is 2 + 2 + 2 = 6.

    • If you chose B: That is one group, 1/5 of 10. The top number is 3, so take 3 groups.
    • If you chose C: You took 5 away from 10. The bottom number says how many equal groups to make.

  8. Tariq finds 3/4 of 8 stones. How many equal groups does he make first?

    8 stones in two rows. The label is 8 stones.8 stones
    Tap your answer.

    1. Yes, that is right. The bottom number is 4. So Tariq makes 4 equal groups, with 2 stones in each.

    2. Not this one. The top number says how many groups to take. The bottom number says how many to make. Try another.

    3. Not this one. There are 2 stones in each group. But Tariq makes 4 groups. Try another.

    See the answer

    4 groups

    The bottom number is 4. So Tariq makes 4 equal groups, with 2 stones in each.

    • If you chose B: The top number says how many groups to take. The bottom number says how many to make.
    • If you chose C: There are 2 stones in each group. But Tariq makes 4 groups.

  9. Priya has 6 stones in equal groups, as in the picture. She takes 1 group. What fraction of the stones is that?

    6 stones in 3 equal groups of 2. The first group is filled in. Each group is labelled 2 stones.2 stones2 stones2 stones
    Tap your answer.

    1. Not this one. You used the 2 stones in a group. The bottom number counts the groups, which is 3. Try another.

    2. Yes, that is right. The 6 stones are in 3 equal groups. Priya takes 1 group, so she takes 1/3.

    3. Not this one. That is one stone out of 6. Priya took a whole group of stones. Try another.

    See the answer

    1/3

    The 6 stones are in 3 equal groups. Priya takes 1 group, so she takes 1/3.

    • If you chose A: You used the 2 stones in a group. The bottom number counts the groups, which is 3.
    • If you chose C: That is one stone out of 6. Priya took a whole group of stones.

Keep it

Copy this into your exercise book.

The bottom number says how many equal groups to make. The top number says how many groups to take. 2/3 of 12 is 8.

Draw: 12 dots in 3 groups of 4. Fill in 2 groups. Write 4 + 4 = 8.

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 (790 KB). It opens on any phone.