Chapter 7 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

M2.4.7 Chapter 7

Which is bigger

Which is more, 1/4 or 1/8, since 8 is bigger than 4?

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

  • Compare two fractions with the same bottom number
  • Compare two fractions with the same top number
  • Explain why 1/8 is less than 1/4

To compare two fractions is to find which one is more. Or to find that they are the same.

Always draw bars with the same whole. Then the bars have the same length, and the picture is fair.

Draw it first

Compare 1/4 and 1/8

  1. Two bars of the same whole width. The top bar is cut into 4 equal parts with 1 shaded, labelled 1/4. The bottom bar is cut into 8 equal parts with 1 shaded, labelled 1/8. The shaded part of 1/8 is shorter than the shaded part of 1/4.

    1 Draw two bars of the same length. Each bar is one whole.

  2. Two bars of the same whole width. The top bar is cut into 4 equal parts with 1 shaded, labelled 1/4. The bottom bar is cut into 8 equal parts with 1 shaded, labelled 1/8. The shaded part of 1/8 is shorter than the shaded part of 1/4.

    2 Cut one bar in 4 equal parts. Cut the other in 8 equal parts.

  3. Two bars of the same whole width. The top bar is cut into 4 equal parts with 1 shaded, labelled 1/4. The bottom bar is cut into 8 equal parts with 1 shaded, labelled 1/8. The shaded part of 1/8 is shorter than the shaded part of 1/4.1418

    3 Shade one part in each bar. The 1/8 part is shorter than the 1/4 part.

So 1/8 is less than 1/4. More parts in the whole means smaller parts.

First rule. The same bottom number means parts of the same size. More parts is more.

Two bars of the same whole width, both cut into 8 equal parts. The top bar has 3 parts shaded, labelled 3/8. The bottom bar has 5 parts shaded, labelled 5/8. The bottom bar has more shaded.3858
The parts are the same size. 5 parts are more than 3 parts.
A number line from 0 to 1 cut into eighths. A point named less is at 3/8. A point named more is at 5/8, further to the right.0182838485868781lessmore
On the number line, further right means more.

Second rule. When the top numbers are the same, look at the bottom numbers.

Two bars of the same whole width. The top bar is cut into 4 parts with 3 shaded, labelled 3/4. The bottom bar is cut into 8 parts with 3 shaded, labelled 3/8. The shaded part of 3/4 is longer.3438
Both show 3 parts. The parts of 3/4 are bigger.

A bigger bottom number means smaller parts. So 3/8 is less than 3/4.

See it three ways

With things

Take two breads of the same size. Cut one in 4 and one in 8. Take 1 piece from each.

As a picture

Two circles of the same size. The left circle is cut into 4 equal parts with 1 shaded, labelled one quarter. The right circle is cut into 8 equal parts with 1 shaded, labelled one eighth. A more than sign sits between them, open to the quarter.one quarter>one eighth

In symbols

1/4 > 1/8. The open side of the sign faces the bigger one.

Try it

Fold two strips

You need: two strips of paper of the same length, a pencil.

  1. Fold the first strip in half. Fold it in half again. Open it. It has 4 parts.
  2. Fold the second strip in half 3 times. Open it. It has 8 parts.
  3. Shade one part on each strip.
  4. Put the strips side by side, with the left ends together.

What you will see: The shaded part of the second strip is half as long. So 1/8 is less than 1/4.

If you cannot try it, this is what happens.
Two strips of the same length side by side. The top strip has 4 parts with 1 shaded, labelled one quarter. The bottom strip has 8 parts with 1 shaded, labelled one eighth. The shaded part of the bottom strip is half as long.one quarterone eighth

A wrong idea many people have

1/8 is more than 1/4, because 8 is bigger than 4.

Why it feels right: With whole numbers, 8 is more than 4. The same rule seems to work here.

The test that breaks it: Share one bread between 4 people. Share another bread of the same size between 8 people. Ask who gets the bigger piece.

What is true: The bottom number is how many parts the whole is cut into. More parts means smaller parts. So 1/8 is less than 1/4.

Key words

compare
to find which is more, or that they are the same.
same whole
bars or lines that have the same length.

Back to your guess

The answer is B: 1/4, because its parts are bigger.

The bottom number tells how many parts the whole is cut into. More parts means smaller parts. So 1/4 is more than 1/8.

Check yourself

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

  1. Which is more, 3/8 or 5/8?

    Tap your answer.

    1. Not this one. The parts are the same size. 5 parts is more than 3 parts. Go to M2.4.3. Try another.

    2. Yes, that is right. Both are eighths, so the parts are the same size. 5 of them are more than 3 of them.

    See the answer

    5/8

    Both are eighths, so the parts are the same size. 5 of them are more than 3 of them.

    • If you chose A: The parts are the same size. 5 parts is more than 3 parts. Go to M2.4.3.

  2. Which is less, 1/3 or 1/6?

    Tap your answer.

    1. Not this one. You may think 6 is bigger, so 1/6 is more. More parts means smaller parts. Go to M2.4.2. Try another.

    2. Yes, that is right. Sixths are smaller parts than thirds. So 1/6 is less.

    See the answer

    1/6

    Sixths are smaller parts than thirds. So 1/6 is less.

    • If you chose A: You may think 6 is bigger, so 1/6 is more. More parts means smaller parts. Go to M2.4.2.

  3. From an earlier chapter

    Which fraction is equal to 1/3?

    Tap your answer.

    1. Yes, that is right. Cut each third in 2. The bottom doubles from 3 to 6. The top doubles from 1 to 2.

    2. Not this one. Cut each third in 2. The shaded third becomes 2 parts, not 1. Go to M2.4.6. Try another.

    3. Not this one. Changing only the top changes the amount. Double both numbers. Go to M2.4.6. Try another.

    See the answer

    2/6

    Cut each third in 2. The bottom doubles from 3 to 6. The top doubles from 1 to 2.

    • If you chose B: Cut each third in 2. The shaded third becomes 2 parts, not 1. Go to M2.4.6.
    • If you chose C: Changing only the top changes the amount. Double both numbers. Go to M2.4.6.

  4. Put these in order, from least to most: 1/2, 1/8, 1/4.

    See the answer

    1/8, 1/4, 1/2

    All three have the top 1. The bigger the bottom number, the smaller the part. So 1/8 is the least and 1/2 is the most.

  5. Kofi eats 3/4 of a bread. Lucía eats 3/8 of a bread of the same size. Who ate more?

    Tap your answer.

    1. Yes, that is right. Both ate 3 parts. A quarter is bigger than an eighth, so Kofi ate more.

    2. Not this one. Each has 3 parts. But eighths are smaller than quarters. Go to M2.4.2. Try another.

    3. Not this one. Both have 3 parts, but the parts are not the same size. Go to M2.4.2. Try another.

    See the answer

    Kofi

    Both ate 3 parts. A quarter is bigger than an eighth, so Kofi ate more.

    • If you chose B: Each has 3 parts. But eighths are smaller than quarters. Go to M2.4.2.
    • If you chose C: Both have 3 parts, but the parts are not the same size. Go to M2.4.2.

  6. Ravi has 5/8 of a bread. Amina has 3/8 of a bread of the same size. Who has more?

    Tap your answer.

    1. Yes, that is right. Both are eighths, so the parts are the same size. 5 of them are more than 3.

    2. Not this one. Bigger number, smaller part is a rule for the bottom number. Here both bottoms are 8, so the top decides. 5 parts are more than 3. Try another.

    3. Not this one. Both are eighths. But Ravi has 5 of them and Amina has 3. Try another.

    See the answer

    Ravi

    Both are eighths, so the parts are the same size. 5 of them are more than 3.

    • If you chose B: Bigger number, smaller part is a rule for the bottom number. Here both bottoms are 8, so the top decides. 5 parts are more than 3.
    • If you chose C: Both are eighths. But Ravi has 5 of them and Amina has 3.

  7. Which is more, 3/5 or 3/10?

    Tap your answer.

    1. Not this one. Tenths are smaller than fifths. 3 small parts are less than 3 bigger parts. Try another.

    2. Not this one. Both have 3 parts, but the parts are not the same size. Fifths are bigger than tenths. Try another.

    3. Yes, that is right. Both have 3 parts. Fifths are bigger parts than tenths, so 3/5 is more.

    See the answer

    3/5

    Both have 3 parts. Fifths are bigger parts than tenths, so 3/5 is more.

    • If you chose A: Tenths are smaller than fifths. 3 small parts are less than 3 bigger parts.
    • If you chose B: Both have 3 parts, but the parts are not the same size. Fifths are bigger than tenths.

  8. Mei says 1/8 is more than 1/4. Her reason is that 8 is more than 4. Is she right?

    Tap your answer.

    1. Not this one. That works for whole numbers. In a fraction, the bottom number counts the parts. More parts are smaller. Try another.

    2. Yes, that is right. Eighths are smaller parts than quarters. So 1/8 is less than 1/4.

    3. Not this one. The top is the same, but the bottom numbers are not. An eighth is a smaller part than a quarter. Try another.

    See the answer

    No. More parts in the whole means smaller parts.

    Eighths are smaller parts than quarters. So 1/8 is less than 1/4.

    • If you chose A: That works for whole numbers. In a fraction, the bottom number counts the parts. More parts are smaller.
    • If you chose C: The top is the same, but the bottom numbers are not. An eighth is a smaller part than a quarter.

  9. Put 3/4, 3/5 and 3/8 in order, least to most. Which list is right?

    Tap your answer.

    1. Yes, that is right. All three have 3 parts. The bigger the bottom number, the smaller the part. So 3/8 is the least.

    2. Not this one. That puts the smallest bottom number first. Eighths are the smallest parts, so 3/8 is the least. Try another.

    3. Not this one. Each has 3 parts, but the parts are not the same size. Quarters, fifths and eighths differ. Try another.

    See the answer

    3/8, 3/5, 3/4

    All three have 3 parts. The bigger the bottom number, the smaller the part. So 3/8 is the least.

    • If you chose B: That puts the smallest bottom number first. Eighths are the smallest parts, so 3/8 is the least.
    • If you chose C: Each has 3 parts, but the parts are not the same size. Quarters, fifths and eighths differ.

Keep it

Copy this into your exercise book.

Compare fractions on bars of the same whole. Same bottom number: more parts is more. Same top number: a bigger bottom number means smaller parts. So 1/8 is less than 1/4.

Draw: Two bars of the same length. Cut one in 4 and one in 8. Shade one part in each.

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.