Chapter 6 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

M2.4.6 Chapter 6

Same amount, different names

Is 2/4 more than 1/2, since the numbers are bigger?

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

  • Show that two fractions name the same amount
  • Find a fraction equal to 1/2, 1/3 or 3/4
  • Explain why 2/4 is not more than 1/2

Cut a bread into 2 equal halves. Then cut each half into 2 equal pieces. Now you have 4 pieces in all.

You still have the same bread. It has the same size. Only the cuts are new. One half is now 2 of the 4 pieces.

Fractions that name the same amount are equal fractions. We write 1/2 = 2/4.

Draw it first

Cut each part in two

  1. Three bars of the same whole width, stacked. The top bar is cut into 2 parts with 1 shaded, labelled 1/2. The middle bar is cut into 4 parts with 2 shaded, labelled 2/4. The bottom bar is cut into 8 parts with 4 shaded, labelled 4/8. Every shaded part ends at the same place.12

    1 Draw a bar. Cut it in 2 equal parts. Shade 1 part. That is 1/2.

  2. Three bars of the same whole width, stacked. The top bar is cut into 2 parts with 1 shaded, labelled 1/2. The middle bar is cut into 4 parts with 2 shaded, labelled 2/4. The bottom bar is cut into 8 parts with 4 shaded, labelled 4/8. Every shaded part ends at the same place.1224

    2 Draw a bar the same length. Cut it in 4. Shade 2 parts. That is 2/4.

  3. Three bars of the same whole width, stacked. The top bar is cut into 2 parts with 1 shaded, labelled 1/2. The middle bar is cut into 4 parts with 2 shaded, labelled 2/4. The bottom bar is cut into 8 parts with 4 shaded, labelled 4/8. Every shaded part ends at the same place.122448

    3 Draw one more bar the same length. Cut it in 8. Shade 4 parts. That is 4/8.

Bars with the same length have the same whole. Look down the bars. Each shaded part ends in the same place.

Two bars of the same whole width. The top bar is cut into 3 parts with 1 shaded, labelled 1/3. The bottom bar is cut into 6 parts with 2 shaded, labelled 2/6. The shaded parts end in the same place.1326
1/3 = 2/6. Cut each third in 2, and 1 part becomes 2 parts.
Two number lines of the same length, from 0 to 1. The top line is cut into halves. The bottom line is cut into quarters. A point on each line is at the same place, at 1/2 on the top line and at 2/4 on the bottom line.012101424341
On equal lines, 1/2 and 2/4 are in the same place.

See it three ways

With things

Take two breads of the same size. Cut one in 2 and one in 4. Put 1 piece of the first by 2 pieces of the second.

As a picture

Two circles of the same size. The left circle is cut into 2 equal parts, with one part shaded, labelled one half. The right circle is cut into 4 equal parts, with two parts shaded, labelled two quarters. An equals sign sits between them.one half=two quarters

In symbols

1/2 = 2/4 = 4/8. Each time, the top and the bottom both double.

Worked example

Make a fraction equal to 2/3 that has 6 parts.

  1. Cut each third in 2. The 3 parts become 6 parts.
  2. Each shaded part is cut in 2 too. The 2 shaded parts become 4.
  3. The bottom doubled, from 3 to 6. The top doubled, from 2 to 4.

Answer: 2/3 = 4/6

Try it

Two strips, one amount

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

  1. Fold the first strip in half. Open it. Shade one half.
  2. Fold the second strip in half, then in half again. Open it.
  3. Shade 2 of the 4 parts on the second strip.
  4. Put the strips side by side, with the left ends together.

What you will see: Both shaded parts end in the same place. So 1/2 and 2/4 are the same amount.

If you cannot try it, this is what happens.
Two strips of the same length side by side. The top strip has 2 parts with 1 shaded, labelled one half. The bottom strip has 4 parts with 2 shaded, labelled two quarters. The shaded parts have the same length.one halftwo quarters

A wrong idea many people have

2/4 is more than 1/2, because 2 and 4 are bigger numbers.

Why it feels right: With whole numbers, a bigger number means more.

The test that breaks it: Make the two strips from the Try it. Lay them side by side. The shaded parts end in the same place. Neither sticks out.

What is true: Cutting parts smaller gives more parts. The amount does not change. 2/4 is the same amount as 1/2.

Key words

equal fractions
fractions that name the same amount.
same whole
bars or lines that have the same length.

Back to your guess

The answer is B: No. They are the same amount.

Two quarters is the same amount as one half. Bigger numbers do not mean more. The parts are smaller, and there are more of them.

Check yourself

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

  1. Which fraction is equal to 1/2?

    Tap your answer.

    1. Yes, that is right. Cut each half in 2. The bottom doubles from 2 to 4. The top doubles from 1 to 2.

    2. Not this one. One part of 4 is less than a half. A half is 2 parts of 4. Try another.

    3. Not this one. Adding 1 to both numbers changes the amount. Double both numbers. Try another.

    See the answer

    2/4

    Cut each half in 2. The bottom doubles from 2 to 4. The top doubles from 1 to 2.

    • If you chose B: One part of 4 is less than a half. A half is 2 parts of 4.
    • If you chose C: Adding 1 to both numbers changes the amount. Double both numbers.

  2. This bar has 3 equal parts, with 1 shaded. Cut each part in 2. How many parts are shaded now?

    A bar cut into 3 equal parts, with 1 part shaded, labelled before.before
    See the answer

    2 of 6, which is 2/6

    Each third becomes 2 parts, so the bar has 6 parts. The shaded third becomes 2 parts.

  3. Is 3/6 equal to 1/2?

    Tap your answer.

    1. Yes, that is right. 3 parts is half of 6 parts. So 3/6 is the same amount as 1/2.

    2. Not this one. Bigger numbers do not mean more. Fold a strip in 6 and shade 3. Compare it with a half. Try another.

    See the answer

    Yes. Both name the same amount.

    3 parts is half of 6 parts. So 3/6 is the same amount as 1/2.

    • If you chose B: Bigger numbers do not mean more. Fold a strip in 6 and shade 3. Compare it with a half.

  4. From an earlier chapter

    A bread is cut into 4 equal parts. Mei eats one part. What fraction of the bread is that?

    Tap your answer.

    1. Yes, that is right. The bread has 4 equal parts, so the bottom is 4. She ate 1 part.

    2. Not this one. You may have counted the parts left. The bottom number is all the equal parts. The bread has 4. Go to M2.4.2. Try another.

    3. Not this one. The bottom number is how many equal parts. The top is how many you take. Go to M2.4.2. Try another.

    See the answer

    1/4

    The bread has 4 equal parts, so the bottom is 4. She ate 1 part.

    • If you chose B: You may have counted the parts left. The bottom number is all the equal parts. The bread has 4. Go to M2.4.2.
    • If you chose C: The bottom number is how many equal parts. The top is how many you take. Go to M2.4.2.

  5. Fill the gap. 3/4 = ?/8. What number goes in place of ?

    See the answer

    6

    Cut each of the 4 parts in 2. Now there are 8 parts. Each shaded part becomes 2, so 3 parts become 6.

  6. Look at the bars. Is 2/4 more than 1/2?

    Two bars of the same whole width, stacked. The top bar is cut into 2 parts with 1 shaded, labelled 1/2. The bottom bar is cut into 4 parts with 2 shaded, labelled 2/4.1224
    Tap your answer.

    1. Not this one. Look at the length, not the number of parts. The parts are smaller, so 2 of them match 1 half. Try another.

    2. Yes, that is right. Both shaded parts end in the same place. So they are the same amount.

    3. Not this one. The parts are smaller, but there are more of them. Together they fill the same length. Try another.

    See the answer

    No. They are equal. The shaded parts have the same length.

    Both shaded parts end in the same place. So they are the same amount.

    • If you chose A: Look at the length, not the number of parts. The parts are smaller, so 2 of them match 1 half.
    • If you chose C: The parts are smaller, but there are more of them. Together they fill the same length.

  7. Which fraction is equal to 1/3?

    Tap your answer.

    1. Not this one. That is one part of 6, a smaller amount. When the bottom doubles, the top must double too. Try another.

    2. Not this one. Adding 1 to both numbers changes the amount. 2/4 is the same as 1/2. Try another.

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

    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 A: That is one part of 6, a smaller amount. When the bottom doubles, the top must double too.
    • If you chose B: Adding 1 to both numbers changes the amount. 2/4 is the same as 1/2.

  8. This bar has 4 equal parts, with 3 shaded. Cut each part in 2. What fraction is shaded now?

    A bar cut into 4 equal parts, with 3 parts shaded, labelled before.before
    Tap your answer.

    1. Yes, that is right. The bar has 8 parts now. Each shaded part becomes 2, so 3 shaded parts become 6.

    2. Not this one. You doubled the bottom number but not the top. Each shaded part becomes 2, so 3 shaded parts become 6. Try another.

    3. Not this one. Adding 1 to both numbers changes the amount. When each part is cut in 2, both numbers double. Try another.

    See the answer

    6/8

    The bar has 8 parts now. Each shaded part becomes 2, so 3 shaded parts become 6.

    • If you chose B: You doubled the bottom number but not the top. Each shaded part becomes 2, so 3 shaded parts become 6.
    • If you chose C: Adding 1 to both numbers changes the amount. When each part is cut in 2, both numbers double.

  9. Which pair of fractions names the same amount?

    Tap your answer.

    1. Not this one. Both have 1 part, but 1/8 is a smaller part. They are not the same amount. Try another.

    2. Yes, that is right. Cut each third in 2 and 1/3 becomes 2/6. The amount stays the same.

    3. Not this one. Adding 1 to both numbers changes the amount. 3/4 is more than 2/3. Try another.

    See the answer

    2/6 and 1/3

    Cut each third in 2 and 1/3 becomes 2/6. The amount stays the same.

    • If you chose A: Both have 1 part, but 1/8 is a smaller part. They are not the same amount.
    • If you chose C: Adding 1 to both numbers changes the amount. 3/4 is more than 2/3.

Keep it

Copy this into your exercise book.

Different fractions can name the same amount. 1/2, 2/4 and 4/8 are equal. Cut every part in two. The top and the bottom both double. The amount stays the same.

Draw: Three bars of the same length, cut in 2, 4 and 8. Shade the same amount 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.