Chapter 8 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

M2.4.8 Chapter 8

Adding and taking away parts

My friend says 1/4 + 1/4 is 2/8. Is that right?

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

  • Add fractions that have the same bottom number
  • Take away fractions that have the same bottom number
  • Show adding and taking away as hops on a number line

Two quarters of a bread make half the bread. Let us see why.

In this chapter the bottom numbers are always the same. Parts with the same bottom number are like parts.

You can count like parts as you count beans. 1 quarter and 2 quarters make 3 quarters.

Draw it first

Add 1/4 and 2/4 with hops

  1. A number line from 0 to 1 cut into quarters. Step 1 draws the line. Step 2 draws a hop of 1/4 from 0. Step 3 draws a second hop of 2/4 and a point where you land, at 3/4.01424341

    1 Draw a number line from 0 to 1. Cut it into quarters.

  2. A number line from 0 to 1 cut into quarters. Step 1 draws the line. Step 2 draws a hop of 1/4 from 0. Step 3 draws a second hop of 2/4 and a point where you land, at 3/4.014243410142434114

    2 Start at 0. Make a hop of 1/4 to the right.

  3. A number line from 0 to 1 cut into quarters. Step 1 draws the line. Step 2 draws a hop of 1/4 from 0. Step 3 draws a second hop of 2/4 and a point where you land, at 3/4.014243410142434114014243411424

    3 Hop 2/4 more. You land on 3/4. So 1/4 + 2/4 = 3/4.

The hops are all quarters. So the bottom number stays 4. Only the tops are added.

Worked example

Add 1/4 + 2/4.

  1. The bottom numbers are both 4. The parts are the same size.
  2. Add the top numbers. 1 + 2 = 3.
  3. Keep the bottom number. It is still 4.

Answer: 3/4

To take away, hop to the left. Start at 5/6 and take away 2/6.

A number line from 0 to 1 cut into sixths. A hop to the left, labelled 2/6, goes from the point at 5/6 to the point at 3/6.01626364656126
Hop left by 2/6. You land on 3/6.

Worked example

Take away 5/6 − 2/6.

  1. The bottom numbers are both 6.
  2. Take away the top numbers. 5 − 2 = 3.
  3. Keep the bottom number. It is still 6.

Answer: 3/6. That is the same amount as 1/2.

Sometimes you hop past 1. Then the top is bigger than the bottom. For example, 3/4 + 2/4 = 5/4.

Now back to the question. Is 1/4 + 1/4 the same as 2/8? Look at the bars.

Two bars of the same whole width. The top bar is cut into 4 parts with 2 shaded, labelled 2/4. The bottom bar is cut into 8 parts with 2 shaded, labelled 2/8. The shaded part of 2/4 is twice as long.2428
1/4 + 1/4 = 2/4. Two quarters cover half the bar. Two eighths cover only a quarter.

See it three ways

With things

Cut a bread in 4 equal pieces. Eat 1 piece. Then eat 2 more. You ate 3 pieces of 4.

As a picture

Three circles of the same size, each cut into 4 equal parts. The first has 1 part shaded, labelled 1/4. The second has 2 parts shaded, labelled 2/4. The third has 3 parts shaded, labelled 3/4. A plus sign sits after the first and an equals sign after the second.1/4+2/4=3/4

In symbols

1/4 + 2/4 = 3/4. Add the tops. Keep the bottom.

Try it

Add on a paper strip

You need: a strip of paper, a pencil.

  1. Fold the strip in half. Fold it in half again. Open it. It has 4 equal parts.
  2. Shade 1 part.
  3. Shade 2 more parts, next to the first.
  4. Count the shaded parts.

What you will see: You shaded 3 of the 4 parts. So 1/4 + 2/4 = 3/4. Every part is a quarter, so the bottom stays 4.

If you cannot try it, this is what happens.
A strip cut into 4 equal parts with 3 parts shaded, labelled three quarters.three quarters

A wrong idea many people have

1/4 + 1/4 = 2/8.

Why it feels right: You add the tops, 1 + 1. Then you add the bottoms, 4 + 4. Adding everything seems fair.

The test that breaks it: Cut a bread in 4 equal parts. Eat 1 part, then 1 more. You ate half the bread. Now cut a bread in 8. Two of those pieces are only a quarter.

What is true: The parts stay quarters, so the bottom stays 4. 1/4 + 1/4 = 2/4. That is 1/2.

Key words

like parts
parts that have the same bottom number.
hop
a jump along the number line.

Back to your guess

The answer is B: 2/4. Add the tops and keep the bottom.

Two quarters of a bread are half the bread. That is 2/4. And 2/8 is only a quarter. Add the tops and keep the bottom.

Check yourself

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

  1. Add. 2/7 + 3/7

    Tap your answer.

    1. Yes, that is right. Add the tops. 2 + 3 = 5. Keep the bottom, 7.

    2. Not this one. You added the bottoms too. The parts stay sevenths, so the bottom stays 7. Try another.

    3. Not this one. You may have multiplied the tops. Add them. 2 + 3 = 5. Try another.

    See the answer

    5/7

    Add the tops. 2 + 3 = 5. Keep the bottom, 7.

    • If you chose B: You added the bottoms too. The parts stay sevenths, so the bottom stays 7.
    • If you chose C: You may have multiplied the tops. Add them. 2 + 3 = 5.

  2. Take away. 5/6 − 2/6

    See the answer

    3/6

    5 − 2 = 3. Keep the bottom, 6. The answer 3/6 is also one half.

  3. Ravi hops along this line. He hops 3/4, then 2/4. Where does he land?

    A number line from 0 to 2 with each whole cut into 4 equal steps. A first hop of 3/4 starts at 0. A second hop of 2/4 goes on from there and lands past 1.0123424
    See the answer

    5/4

    3 + 2 = 5, and the bottom stays 4. So he lands on 5/4. That is past 1.

  4. From an earlier chapter

    What is 1/3 of 12 stones?

    See the answer

    4 stones

    Share 12 stones into 3 equal groups. Each group has 4. 4 + 4 + 4 = 12.

  5. A jug has 7/8 litre of water. Ana pours out 3/8 litre. How much is left?

    See the answer

    4/8 litre

    7 − 3 = 4. The bottom stays 8. 4/8 is the same as one half litre.

  6. Leila eats 1/6 of a bread. Then she eats 3/6 more. How much has she eaten in all?

    Tap your answer.

    1. Not this one. You added the bottoms too. All the parts are sixths, so the bottom stays 6. Try another.

    2. Yes, that is right. Add the tops. 1 + 3 = 4. Keep the bottom, 6.

    3. Not this one. That is only the second part she ate. In all means add both parts. 1 + 3 = 4. Try another.

    See the answer

    4/6

    Add the tops. 1 + 3 = 4. Keep the bottom, 6.

    • If you chose A: You added the bottoms too. All the parts are sixths, so the bottom stays 6.
    • If you chose C: That is only the second part she ate. In all means add both parts. 1 + 3 = 4.

  7. Yusuf has 5/8 of a bread. He gives 2/8 away. How much is left?

    Tap your answer.

    1. Not this one. You added. Giving away is taking away, so do 5 − 2. Try another.

    2. Not this one. That is the part Yusuf gave away. The question asks what is left. Try another.

    3. Yes, that is right. Take away the tops. 5 − 2 = 3. Keep the bottom, 8.

    See the answer

    3/8

    Take away the tops. 5 − 2 = 3. Keep the bottom, 8.

    • If you chose A: You added. Giving away is taking away, so do 5 − 2.
    • If you chose B: That is the part Yusuf gave away. The question asks what is left.

  8. Hana starts at 5/6 and hops left by 3/6. Where does she land?

    A number line from 0 to 1 cut into 6 equal steps. The marks 0 and 1 are labelled. A point labelled start is near the right end, one step before 1.01start
    Tap your answer.

    1. Yes, that is right. A hop to the left takes away. 5 − 3 = 2. The bottom stays 6.

    2. Not this one. You hopped to the right. A hop to the left takes away. Try another.

    3. Not this one. That is the size of the hop. Where she lands is 5 − 3 = 2 sixths. Try another.

    See the answer

    2/6

    A hop to the left takes away. 5 − 3 = 2. The bottom stays 6.

    • If you chose B: You hopped to the right. A hop to the left takes away.
    • If you chose C: That is the size of the hop. Where she lands is 5 − 3 = 2 sixths.

  9. Mei hops to the left on the line. Which number sentence shows her hop?

    A number line from 0 to 1 cut into 4 equal steps. A point labelled start is at 3/4. A hop to the left, labelled 2/4, goes from the start point.0124start
    Tap your answer.

    1. Not this one. The hop goes to the left, so it takes away. A hop to the right adds. Try another.

    2. Yes, that is right. Mei starts at 3/4 and hops left 2/4. 3 − 2 = 1. The bottom stays 4.

    3. Not this one. You wrote the size of the hop. 3 − 2 = 1, so she lands on 1/4. Try another.

    See the answer

    3/4 − 2/4 = 1/4

    Mei starts at 3/4 and hops left 2/4. 3 − 2 = 1. The bottom stays 4.

    • If you chose A: The hop goes to the left, so it takes away. A hop to the right adds.
    • If you chose C: You wrote the size of the hop. 3 − 2 = 1, so she lands on 1/4.

Keep it

Copy this into your exercise book.

Same bottom number: add or take away the tops. Keep the bottom. 1/4 + 2/4 = 3/4. Two quarters of a bread make half the bread. 1/4 + 1/4 is 2/4, never 2/8.

Draw: A number line in quarters. Hop 1/4, then hop 2/4, and land on 3/4.

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.