Chapter 5 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

M2.4.5 Chapter 5

The number line, past 1

I saw 5/4 on a page. Can a fraction be more than 1?

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 on a number line
  • Tell if a fraction is before 1, at 1, or past 1
  • Count steps past 1 to place a fraction such as 5/4

A number line is a straight line with numbers on it. Each number has its own place.

Fractions are numbers too. They have places on the line, between the whole numbers.

Draw it first

Find 5/4 on a number line

  1. A number line from 0 to 2. Step 1 marks 0, 1 and 2. Step 2 cuts each whole into 4 equal steps. Step 3 puts a point at 5/4, which is past 1. The point is named 5/4.012

    1 Draw a line. Mark 0, 1 and 2 with equal gaps.

  2. A number line from 0 to 2. Step 1 marks 0, 1 and 2. Step 2 cuts each whole into 4 equal steps. Step 3 puts a point at 5/4, which is past 1. The point is named 5/4.012014243415464742

    2 Cut each whole into 4 equal steps. Each step is 1/4.

  3. A number line from 0 to 2. Step 1 marks 0, 1 and 2. Step 2 cuts each whole into 4 equal steps. Step 3 puts a point at 5/4, which is past 1. The point is named 5/4.0120142434154647420142434154647425/4

    3 Count 5 steps from 0. You land on 5/4. It is past 1.

Four steps of 1/4 reach 1. So 4/4 is the same place as 1.

A number line from 0 to 2, cut into halves. Point A is at 1/2, before 1. Point B is at 3/2, past 1.0121322AB
A is 1/2. It is before 1. B is 3/2. It is past 1.

Look at the top and bottom numbers. Is the top bigger than the bottom? Then the fraction is past 1.

See it three ways

With things

Take two strings, each 1 metre long. Cut each one into 4 equal pieces. Lay 5 pieces end to end.

As a picture

A number line from 0 to 2 in quarters. One hop starts at 0 and lands past 1. The hop is labelled 5/4.01424341546474254

In symbols

5/4 = 4/4 + 1/4. That is 1 and one more quarter.

Try it

Walk past 1 with paper strips

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

  1. Fold one strip in half. Fold it in half again. Open it.
  2. Do the same with the other strip. Each strip has 4 equal parts.
  3. Lay the strips end to end. Write 0 at the left end.
  4. Count 4 parts from 0. Where the first strip ends, write 1.
  5. Count one more part. Write 5/4 at the end of that part.

What you will see: The fifth part is on the second strip. So 5/4 is past 1.

If you cannot try it, this is what happens.
Two strips of the same length laid end to end, each cut into 4 equal parts. The first part of the second strip is shaded. Below them is a number line from 0 to 2 with a point at 5/4.012

A wrong idea many people have

A fraction is always less than 1.

Why it feels right: Most fractions we cut are pieces of one whole.

The test that breaks it: Cut two breads of the same size, each into 4 equal pieces. Take 5 pieces. You hold more than one whole bread.

What is true: A fraction can be less than 1. It can equal 1. It can be more than 1. If the top is bigger than the bottom, it is past 1.

Key words

number line
a line where every number has its own place.
steps
the equal parts of a whole, on a number line.

Back to your guess

The answer is B: Yes. Count more parts than one whole has. You go past 1.

A fraction is a number. It has a place on the number line. Five quarters sits after 1.

Check yourself

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

  1. Count the steps from 0. Which point is at 5/4?

    A number line from 0 to 2 with each whole cut into 4 equal steps. Point A is 3 steps from 0. Point B is 5 steps from 0. Point C is 7 steps from 0.012ABC
    Tap your answer.

    1. Not this one. A is 3 steps from 0. That is 3/4, before 1. Try another.

    2. Yes, that is right. B is 5 steps from 0, and each step is 1/4. So B is 5/4.

    3. Not this one. C is 7 steps from 0. That is 7/4, past 1 but too far. Try another.

    See the answer

    Point B

    B is 5 steps from 0, and each step is 1/4. So B is 5/4.

    • If you chose A: A is 3 steps from 0. That is 3/4, before 1.
    • If you chose C: C is 7 steps from 0. That is 7/4, past 1 but too far.

  2. Is 6/4 before 1, at 1, or past 1?

    Tap your answer.

    1. Not this one. You may think a fraction is always small. Here the top, 6, is bigger than the bottom, 4. Try another.

    2. Not this one. Only 4/4 is at 1. 6/4 is two more steps along. Try another.

    3. Yes, that is right. 4 steps of 1/4 reach 1. 6/4 has 6 steps, so it is past 1.

    See the answer

    Past 1

    4 steps of 1/4 reach 1. 6/4 has 6 steps, so it is past 1.

    • If you chose A: You may think a fraction is always small. Here the top, 6, is bigger than the bottom, 4.
    • If you chose B: Only 4/4 is at 1. 6/4 is two more steps along.

  3. Each whole is cut into 3 equal steps. Point P is the first tick after 1. What fraction is P?

    A number line from 0 to 2 with each whole cut into 3 equal steps. Point P is on the first tick after 1.012P
    See the answer

    4/3

    1 is 3 steps from 0. One more step makes 4 steps. Each step is 1/3, so P is 4/3.

  4. From an earlier chapter

    Amina eats 3 of the 4 equal parts of a bread. What fraction of the bread did she eat?

    Tap your answer.

    1. Yes, that is right. The bread has 4 equal parts, so the bottom is 4. She ate 3, so the top is 3.

    2. Not this one. That is one part. She ate 3 parts. Go to M2.4.3. Try another.

    3. Not this one. The bottom number is how many equal parts the whole has. The whole has 4. Go to M2.4.3. Try another.

    See the answer

    3/4

    The bread has 4 equal parts, so the bottom is 4. She ate 3, so the top is 3.

    • If you chose B: That is one part. She ate 3 parts. Go to M2.4.3.
    • If you chose C: The bottom number is how many equal parts the whole has. The whole has 4. Go to M2.4.3.

  5. Draw a number line from 0 to 2. Cut each whole in half. Mark 3/2.

    See the answer

    Mark 0, 1 and 2. Mark the middle of each whole. Count 3 half steps from 0. You land on 3/2, halfway between 1 and 2.

    Each step is 1/2. Two steps reach 1, and one more step makes 3/2.

  6. What fraction is point Q?

    A number line from 0 to 3 with each whole cut into 2 equal steps. The marks 0, 1, 2 and 3 are labelled. Point Q is on the line between 2 and 3.0123Q
    Tap your answer.

    1. Not this one. The bottom number is how many steps make one whole. That is 2. The top number counts the steps from 0. Try another.

    2. Yes, that is right. Each step is 1/2. Q is 5 steps from 0, so Q is 5/2.

    3. Not this one. You counted the tick at 0 as a step. Count the steps after 0. Try another.

    See the answer

    5/2

    Each step is 1/2. Q is 5 steps from 0, so Q is 5/2.

    • If you chose A: The bottom number is how many steps make one whole. That is 2. The top number counts the steps from 0.
    • If you chose C: You counted the tick at 0 as a step. Count the steps after 0.

  7. Which of these fractions is past 1?

    Tap your answer.

    1. Not this one. 5 steps of 1/8 do not reach 1. You need 8 steps. Try another.

    2. Not this one. 6 steps of 1/6 reach 1 exactly. It is at 1, not past it. Try another.

    3. Yes, that is right. 6 steps of 1/6 reach 1. 9/6 has 9 steps, so it is past 1.

    See the answer

    9/6

    6 steps of 1/6 reach 1. 9/6 has 9 steps, so it is past 1.

    • If you chose A: 5 steps of 1/8 do not reach 1. You need 8 steps.
    • If you chose B: 6 steps of 1/6 reach 1 exactly. It is at 1, not past it.

  8. Kofi counts steps of 1/3 from 0 and lands on 5/3. How many steps does he count?

    Tap your answer.

    1. Yes, that is right. The top number counts the steps from 0. 5/3 is 5 steps of 1/3.

    2. Not this one. There are 3 steps in each whole. Kofi went past 1, so he counted more than 3. Try another.

    3. Not this one. There are 2 steps past 1. Kofi counted every step from 0. Try another.

    See the answer

    5 steps

    The top number counts the steps from 0. 5/3 is 5 steps of 1/3.

    • If you chose B: There are 3 steps in each whole. Kofi went past 1, so he counted more than 3.
    • If you chose C: There are 2 steps past 1. Kofi counted every step from 0.

  9. Priya says 7/4 is past 1. She says the top is bigger than the bottom. Is she right?

    Tap your answer.

    1. Not this one. A fraction can be more than 1. 7 steps of 1/4 go past the 4 steps that reach 1. Try another.

    2. Yes, that is right. 4 steps of 1/4 reach 1. 7/4 has 7 steps, so it is past 1.

    3. Not this one. Only 4 quarters make one whole, 4/4. 7/4 is 3 more steps along, so it is past 1. Try another.

    See the answer

    Yes. The top is bigger than the bottom.

    4 steps of 1/4 reach 1. 7/4 has 7 steps, so it is past 1.

    • If you chose A: A fraction can be more than 1. 7 steps of 1/4 go past the 4 steps that reach 1.
    • If you chose C: Only 4 quarters make one whole, 4/4. 7/4 is 3 more steps along, so it is past 1.

Keep it

Copy this into your exercise book.

A fraction is a number with a place on the number line. Cut each whole into equal steps. Count the steps from 0. Top bigger than bottom? You are past 1.

Draw: A line from 0 to 2. Cut each whole into 4 steps. Mark 5/4, just past 1.

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.