Representing Fractions on a Number Line - Third Grade Mathematics

A fraction is not just an illustration of pizza or a slice of pie; a fraction is a true mathematical number with an exact address on the number line. When you place fractions between 0 and 1, you see that numbers do not jump straight from zero to one—there is a whole universe of numbers living in between. In this chapter, you will learn how to partition a number line into equal intervals and locate any fraction with absolute precision.

The Number Line from 0 to 1

To represent fractions on a number line, we focus on the distance between the whole numbers 0 and 1. This distance represents ONE whole unit.

0                                                               1
+---------------------------------------------------------------+
<------------------------ One Whole Unit ----------------------->

The denominator of your fraction tells you how many equal intervals (hops) you must divide that distance into. The numerator tells you how many intervals to travel from 0 to reach your fraction's location.

Partitioning into Equal Intervals

Partitioning means cutting the distance between 0 and 1 into equal parts.

Partitioning into Halves (Denominator = 2):
0                              1/2                              1
+-------------------------------+-------------------------------+
<--------- Interval 1 ---------><--------- Interval 2 --------->

Partitioning into Fourths (Denominator = 4):
0              1/4             2/4             3/4              1
+---------------+---------------+---------------+---------------+
<-- Int. 1 ----><-- Int. 2 ----><-- Int. 3 ----><-- Int. 4 ---->

Crucial Insight: Count the spaces (intervals), NOT just the tick marks!
- When you divide a whole into 4 equal parts, you make 3 interior tick marks, creating 4 equal spaces.

- Always check that every space between consecutive tick marks is the exact same width!


Step-by-Step Method to Plot Any Fraction

Follow these four steps to locate any fraction on a number line:
1. Draw a straight line and mark 0 on the left and 1 on the right.

2. Look at the denominator: divide the segment between 0 and 1 into that many equal segments.

3. Label each tick mark starting from 0:
- 0 is equal to 0/d. - The first tick mark is 1/d. - The second tick mark is 2/d. - Continue until the whole number 1, which is d/d.
4. Place a bold point on the tick mark matching the numerator.


Example: Locating 3/8 on a Number Line

  1. Start with 0 and 1.
  2. Denominator is 8: divide into 8 equal parts (halve the line, halve each half to get fourths, halve each fourth to get eighths).
  3. Label tick marks: 0/8, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8.
  4. Count 3 hops from 0 and place a dot at 3/8!
0      1/8    2/8    3/8    4/8    5/8    6/8    7/8    8/8 (1)
+------+------+------+------+------+------+------+------+
                      *
                    (3/8)

Fractions Beyond 1 on a Number Line

Fractions do not stop at 1! You can continue the exact same equal intervals past 1 and toward 2:

0       1/3     2/3     3/3(1)  4/3     5/3     6/3(2)
+-------+-------+-------+-------+-------+-------+

Notice that:
- 3/3 is exactly equal to 1.

- 4/3 is one hop past 1 (also known as 1 and 1/3).

- 6/3 is exactly equal to 2 (because 6 ÷ 3 = 2).
Seeing fractions on a number line makes it clear that fractions are real numbers that live alongside whole numbers.

Chapter Practice Exercises

  1. Draw or describe how you would partition a number line between 0 and 1 to show: a. Thirds b. Sixths c. Fourths
  2. What fraction names the point located at each position on a number line partitioned into 6 equal parts: a. 1 hop from 0 b. 4 hops from 0 c. 6 hops from 0
  3. Identify the fraction marked by the point X on this number line: 0 -------- + -------- X -------- + -------- 1 (There are 4 equal segments).
  4. On a number line divided into eighths, how many hops from 0 is the fraction 5/8?
  5. A frog starts at 0 and hops 1/4 of a unit on each jump. a. Where does the frog land after 2 jumps? b. How many jumps does it take to reach 1?

Solutions and Step-by-Step Answers

  1. Partitioning: a. Thirds: Divide the space between 0 and 1 into 3 equal lengths by placing 2 evenly spaced tick marks. b. Sixths: Divide into 3 equal parts first, then divide each part in half to create 6 equal intervals (5 interior tick marks). c. Fourths: Halve the line between 0 and 1, then halve each half to create 4 equal intervals (3 interior tick marks).
  2. Locations on sixths line: a. 1 hop from 0 = 1/6. b. 4 hops from 0 = 4/6. c. 6 hops from 0 = 6/6 (which equals 1).
  3. The number line is partitioned into 4 equal segments (fourths). Point X is at the second tick mark after 0, which represents 2/4 (or 1/2).
  4. The fraction 5/8 is exactly 5 hops from 0 on a number line partitioned into eighths.
  5. Frog hops: a. After 2 jumps of 1/4 each, the frog lands at 2/4 (which is at the halfway mark, 1/2). b. It takes 4 jumps of 1/4 to reach 4/4, which is 1 whole unit.