Explainer: Fractions on a Number Line

In this explainer, we will learn how to divide a number line between 0 and 1 into equal parts and label each part with a fraction over the same denominator.

We will divide the number line into equal parts because a (unit) fraction of something is always one of a number of equal parts.

We will see how to do this with halves, thirds, and quarters. We will start by thinking about halves.

First, check your understanding of these useful mathematical words.

Definition: Useful Language

Numerator and Denominator of a Fraction

The Whole and Equal Parts

Unit Fraction: It is a fraction where the numerator is 1, for example, 12, 13, and 14.

How to Represent One-Half on a Number Line

We know that one-half is one of two equal parts.

We will see how to show 12 on a number line.

Step One

To show one-half on a number line, we have to divide the distance between 0 and 1 into two equal parts, because the denominator (the number on the bottom of the fraction) is 2.

Step Two

Now we can label the fractions on the number line. The denominator always tells us how many equal parts there are. Since we divided the line into 2 equal parts, the denominator is 2.

Step Three

The numerator (the top number in the fraction) counts the parts from zero to where we are. When we are at zero, we have not gone through any of the parts, so the numerator is 0. When we are at the line in the middle, we are at the endpoint of the first part. So, the numerator of the fraction is 1. The fraction at the end, at 1, shows the whole. There are two equal parts between 0 and 1, so the numerator is 2.

So, if we divide the distance between 0 and 1 into two equal parts and draw an arrow from zero to the end of the first part, we have shown one of two equal parts, which is 12.

We can use the same steps to show other unit fractions on a number line. Read on to see how to show one-third and one-quarter on a number line.

Example 1: Representing One-Third on a Number Line

Represent one-third on the number line.

Answer

If we divide the distance between 0 and 1 into three equal parts, this will allow us to show thirds on the number line.

The number of equal parts tells us the denominator of the fraction, which is 3.

Then, we count the number of parts from zero to tell us the numerator of the fraction.

We can label zero-thirds (which is equal to 0), one-third, two-thirds, and three-thirds (which is equal to the whole, or 1) on the number line.

So, if we divide the distance between 0 and 1 into three equal parts and draw an arrow from zero to the end of the first part, we have shown one of three equal parts, which is 13.

Example 2: Showing One-Quarter on a Number Line

Show one-quarter on this number line.

Answer

To show quarters, we need to divide the distance between 0 and 1 into four equal parts because the denominator of 14 is 4.

Then, we can show one-quarter as being at the endpoint of the first part.

Now that we have seen how to find unit fractions on a number line, we will look at some more questions.

Example 3: Showing One-Fifth on a Number Line

Show 15 on this number line.

Answer

To show one-fifth, or 15, we look at the denominator to tell us the number of equal parts we need to divide the line into.

We would divide the distance between 0 and 1 (the whole) into 5 equal parts.

Then, we look at the numerator to tell us how many parts we have to include. The numerator is 1. This means that one-fifth is the endpoint of the first part, the part that starts at 0.

Example 4: Showing Unit Fractions on Number Lines

Label the fractions on this number line. Draw an arrow to show the unit fraction.

Hint: Count the number of equal parts to find the denominator.

Answer

The number line is divided into ten equal parts, so it shows tenths. If we draw an arrow from zero to the end of the first part, then we have shown one-tenth, or 110.

We have seen that a unit fraction (a fraction whose numerator is 1) can be shown on a number line by dividing it into equal parts and finding the endpoint of the first part.

We can show other fractions on a number line as well.

How to Show Multiple Quarters on a Number Line

Think about dividing the distance between 0 and 1 into 4 equal parts. We have already seen that one-quarter is at the endpoint of the first part.

The length of each part is one-quarter, or 14, of the length from 0 to 1.

If we draw an arrow from zero to the end of the second part, then the length of the arrow is one-quarter plus one-quarter which is two-quarters. So, this arrow shows 24.

If we draw an arrow from zero to the end of the third part, then the length of the arrow is 14+14+14, which is 34.

Example 5: Fractions on a Number line

What fractions do these arrows represent?

Answer

  1. The number line has been divided into 5 equal parts. So, the denominator of the fraction is 5. The number line shows fifths.
    To find the numerator, count the number of parts from zero to the end of the arrow. There is 1 part.
    So, the arrow shows one-fifth, or 15.
  2. The number line has been divided into 6 equal parts. So, the denominator of the fraction is 6. The number line shows sixths.
    To find the numerator, count the number of parts from zero to the end of the arrow. There are 5 parts.
    So, the arrow shows five-sixths, or 56.
  3. The number line has been divided into 10 equal parts. So, the denominator of the fraction is 10. The number line shows sixths.
    To find the numerator, count the number of parts from zero to the end of the arrow. There are 7 parts.
    The arrow shows seven-tenths, or 710.

Some of the lines on a number line can represent more than one fraction. When two fractions are the same distance from zero on the number line, they are called equivalent. Let’s see how to simplify two quarters by using what we have learned about fractions on number lines.

How to Simplify Two Quarters

If we divide the interval between 0 and 1 into 2 equal parts and draw an orange arrow from 0 to the end of the first part, then the length of the orange arrow is 12.

Also, if we divide the interval between 0 and 1 into 4 equal parts and draw a blue arrow from 0 to the end of the second part, then the length of the blue arrow is 24.

Notice that the length of the blue arrow is the same as the length of the orange arrow. This means that 12 and 24 are the same distance from 0. We call one-half and two-quarters equivalent fractions.

So, we can write 12 instead of 24. This is called simplifying the fraction, because we make the numbers smaller. We simplify 24 by dividing the numerator and denominator by the same number.

Key Points

  • The numerator and denominator of a fraction are explained as follows:
    The whole and equal parts are explained as follows:
    The unit fraction is a fraction where the numerator is 1, for example, 12, 13, and 14.
  • By dividing the distance between 0 and 1 on a number line into an equal number of parts, we can represent different fractions of a whole.
  • Some fractions such at 24 can be simplified to another fraction. In the case of 24, it can be simplified to 12.

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