Chapter 1 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

M2.4.1 Chapter 1

Fair shares: equal parts

My brother cut the bread in two. Is his piece really a half?

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

  • Tell if a shape is cut into equal parts
  • Share a group of things fairly between 2, 3 or 4 people
  • Explain why two pieces of different sizes are not halves

Two friends share one flat bread. They cut it into two pieces. One piece is big. One piece is small. Is that fair?

A share is fair when every piece is the same size. We call these pieces equal parts.

Draw it first

Cut a strip into two equal parts

  1. A strip that is one whole. A fold line cuts it into two equal parts. One part is shaded and labelled one half.one whole

    1 Draw a long strip. The whole strip is one whole.

  2. A strip that is one whole. A fold line cuts it into two equal parts. One part is shaded and labelled one half.one whole

    2 Fold it so the two ends meet. Draw a line on the fold.

  3. A strip that is one whole. A fold line cuts it into two equal parts. One part is shaded and labelled one half.one wholeone half

    3 Shade one part. It is one half of the strip.

Two bars. The top bar has two parts of the same size, so they are halves. The bottom bar has a big part and a small part, so they are not halves.halvesnot halves
Both bars have two parts. Only the top bar has two equal parts.

See it three ways

With things

Put 6 beans in a line. Give one to each of 2 people in turn. Stop when none are left.

As a picture

Six beans in two groups of three. One group is filled in.

In symbols

6 ÷ 2 = 3. Each person gets 3.

Try it

Fold a fair share

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

  1. Fold the paper so that its edges meet.
  2. Press the fold flat, then open it.
  3. Draw along the fold. Now fold it again and look.

What you will see: One half lies exactly on top of the other. Nothing sticks out. That is how you know the two parts are equal.

If you cannot try it, this is what happens.
A sheet of paper with a line down the fold, making two equal halves of the same size.two equal halves

A wrong idea many people have

Any two pieces are two halves.

Why it feels right: People say "cut it in half" for any cut into two pieces.

The test that breaks it: Cut a bread into one big piece and one small piece. Ask your friend to pick. Nobody picks the small one.

What is true: Halves are always equal. Two pieces of different sizes are just two pieces.

Key words

whole
all of one thing, before it is cut.
equal parts
parts that are all the same size.
half
one of two equal parts of a whole.

Back to your guess

The answer is B: Only if the two pieces are the same size.

Two pieces are halves only when they are the same size. A cut down the middle is one way, not the only way. The sizes are what count.

Check yourself

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

  1. Which bar is cut into halves?

    Two bars, each 240 long. Bar A has two parts, 72 and 168 long. Bar B has two parts, 120 and 120 long.AB
    Tap your answer.

    1. Not this one. You may be counting pieces. Bar A has two pieces, but they are not the same size. Try another.

    2. Yes, that is right. B has two parts of the same size. A has a small part and a big part.

    See the answer

    Bar B

    B has two parts of the same size. A has a small part and a big part.

    • If you chose A: You may be counting pieces. Bar A has two pieces, but they are not the same size.

  2. Share 12 stones fairly between 3 friends. How many stones does each friend get?

    See the answer

    4 stones each

    12 stones in 3 equal groups is 4 in each group. 4 + 4 + 4 = 12.

  3. Ravi has 10 beans. He gives 6 to his sister and keeps 4. Is that a fair share?

    Tap your answer.

    1. Not this one. Both having some is not enough. A fair share means the same number each. Try another.

    2. Yes, that is right. 10 beans shared by 2 is 5 each. 6 and 4 are not the same.

    See the answer

    No. A fair share is 5 each.

    10 beans shared by 2 is 5 each. 6 and 4 are not the same.

    • If you chose A: Both having some is not enough. A fair share means the same number each.

  4. Draw a square. Cut it into 4 equal parts. Then find a second way to do it.

    See the answer

    One way: a line across and a line down, through the middle. Another way: three lines from top to bottom. They make 4 strips of the same width.

    Both ways give 4 parts that are all the same size. There are many more ways.

  5. Tariq wants 3 equal parts of a bread. Which bar shows 3 equal parts?

    Three bars, each 240 long and cut into 3 parts. Bar A has parts 36, 84 and 120 long. Bar B has parts 80, 80 and 80 long. Bar C has parts 120, 60 and 60 long.ABC
    Tap your answer.

    1. Not this one. Bar A has 3 parts, but all three are different sizes. Counting the parts is not enough. Try another.

    2. Yes, that is right. Bar B has 3 parts of the same size. In A and C some parts are bigger than others.

    3. Not this one. Two parts match. But the first part is twice as long as one of them. All 3 must match. Try another.

    See the answer

    Bar B

    Bar B has 3 parts of the same size. In A and C some parts are bigger than others.

    • If you chose A: Bar A has 3 parts, but all three are different sizes. Counting the parts is not enough.
    • If you chose C: Two parts match. But the first part is twice as long as one of them. All 3 must match.

  6. Amina shares 12 beans fairly between 4 friends. How many beans does each friend get?

    Tap your answer.

    1. Not this one. There are 4 friends, but that is not the share. Make 4 equal groups of the 12 beans. Each group has 3. Try another.

    2. Not this one. You took 4 away from 12. Sharing between 4 friends means making 4 equal groups. Try another.

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

    See the answer

    3 beans each

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

    • If you chose A: There are 4 friends, but that is not the share. Make 4 equal groups of the 12 beans. Each group has 3.
    • If you chose B: You took 4 away from 12. Sharing between 4 friends means making 4 equal groups.

  7. Sipho has 15 beans for 3 children. Which plan is a fair share?

    Tap your answer.

    1. Yes, that is right. 15 beans in 3 equal groups is 5 in each group. 5 + 5 + 5 = 15.

    2. Not this one. Every child gets some, and 7 + 5 + 3 = 15. But a fair share gives every child the same number. Try another.

    3. Not this one. 3 + 3 + 3 = 9, so 6 beans are left over. A fair share of 15 beans uses all of them. Try another.

    See the answer

    Give 5 beans to each child.

    15 beans in 3 equal groups is 5 in each group. 5 + 5 + 5 = 15.

    • If you chose B: Every child gets some, and 7 + 5 + 3 = 15. But a fair share gives every child the same number.
    • If you chose C: 3 + 3 + 3 = 9, so 6 beans are left over. A fair share of 15 beans uses all of them.

  8. Leila cuts a bread into pieces to share. How can she check that the pieces are equal?

    Tap your answer.

    1. Not this one. Counting tells how many pieces there are, not how big. Try another.

    2. Not this one. Straight cuts can still leave one big piece and one small piece. Try another.

    3. Yes, that is right. Pieces that match exactly are the same size.

    See the answer

    Lay one piece on top of another. They match exactly.

    Pieces that match exactly are the same size.

    • If you chose A: Counting tells how many pieces there are, not how big.
    • If you chose B: Straight cuts can still leave one big piece and one small piece.

  9. Mei cuts a bread into two parts, as in the picture. She says each part is a half. What do you say?

    A bar of bread with the label bread cut in two. It is cut into two parts. One part is small and one part is big.bread cut in two
    Tap your answer.

    1. Not this one. Two parts are not enough. Halves must be the same size. Try another.

    2. Yes, that is right. One part is big and one part is small. Halves are two parts of the same size.

    3. Not this one. Two parts of any size can make a whole bread. Halves must also be the same size. Try another.

    See the answer

    No. The two parts are not the same size.

    One part is big and one part is small. Halves are two parts of the same size.

    • If you chose A: Two parts are not enough. Halves must be the same size.
    • If you chose C: Two parts of any size can make a whole bread. Halves must also be the same size.

Keep it

Copy this into your exercise book.

Equal parts are the same size. A half is one of two equal parts. Two pieces are only halves if they are equal.

Draw: A strip cut into two equal parts. Shade one part and write "one half".

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.