Finding Equivalent Fractions Visually - Third Grade Mathematics

In mathematics, different numbers can sometimes name the exact same amount. Fractions that look different because they have different numerators and denominators, but describe the exact same quantity or location on a number line, are called equivalent fractions. In this chapter, you will discover equivalent fractions using fraction strips, area models, and number lines, unlocking the secrets of mathematical equality.

What Does "Equivalent" Mean?

The word "equivalent" comes from words meaning "equal value." Equivalent fractions are different names for the very same amount.

Think of money:
- Having 1 one-dollar bill is worth the exact same amount as having 2 half-dollar coins, or 4 quarters!

- Even though the coins look different and you have different numbers of pieces, the total value is completely identical.
In fractions: 1/2 = 2/4 = 4/8!

Visual Area Model of 1/2, 2/4, and 4/8:

One Whole:
+---------------------------------------------------------------+

1/2 (One out of two parts shaded):
+-------------------------------+-------------------------------+
|/////////////// SHADED ////////|            Unshaded           |
+-------------------------------+-------------------------------+

2/4 (Two out of four parts shaded):
+---------------+---------------+---------------+---------------+
|//// SHADED ///|//// SHADED ///|    Unshaded   |    Unshaded   |
+---------------+---------------+---------------+---------------+

4/8 (Four out of eight parts shaded):
+-------+-------+-------+-------+-------+-------+-------+-------+
|// S //|// S //|// S //|// S //|   U   |   U   |   U   |   U   |
+-------+-------+-------+-------+-------+-------+-------+-------+

Notice how the shaded space takes up the exact same length in all three bars! 1/2, 2/4, and 4/8 describe the identical amount of space. They are equivalent fractions.

Finding Equivalent Fractions Using Fraction Strips

Fraction strips align fraction families directly underneath each other so your eyes can compare them immediately.

+---------------------------------------------------------------+
|                           1 Whole                             |
+-------------------------------+-------------------------------+
|              1/2              |              1/2              |
+---------------+---------------+---------------+---------------+
|      1/4      |      1/4      |      1/4      |      1/4      |
+-------+-------+-------+-------+-------+-------+-------+-------+
|  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |
+-------+-------+-------+-------+-------+-------+-------+-------+

Look down the center line of the chart:
- 1/2 aligns perfectly with 2/4.

- 2/4 aligns perfectly with 4/8.
Look at 3/4 on the chart:
- 3/4 aligns with 6 of the 1/8 pieces: 3/4 = 6/8!


Equivalent Fractions on Number Lines

When two fractions share the exact same location on a number line, they are equivalent.

Number line for Halves:
0                              1/2                              1
+-------------------------------+-------------------------------+

Number line for Fourths:
0              1/4             2/4             3/4              1
+---------------+---------------+---------------+---------------+

Number line for Sixths:
0        1/6      2/6      3/6      4/6      5/6        1
+---------+--------+--------+--------+--------+---------+

Look vertically at the halfway mark:
- 1/2, 2/4, and 3/6 sit at the exact same location on the number line!
Therefore: 1/2 = 2/4 = 3/6.

Also look at 1/3 and 2/6:
- 1/3 sits at the exact same position as 2/6!
Therefore: 1/3 = 2/6.

The Rule Behind Equivalent Fractions

Do you see the pattern?
- 1/2 -> multiply top and bottom by 2 -> (1 x 2)/(2 x 2) = 2/4!

- 1/2 -> multiply top and bottom by 3 -> (1 x 3)/(2 x 3) = 3/6!

- 1/2 -> multiply top and bottom by 4 -> (1 x 4)/(2 x 4) = 4/8!


When you multiply (or divide) both the numerator and the denominator by the exact same non-zero number, you are cutting each existing slice into smaller pieces without changing the total amount of shaded space.

Chapter Practice Exercises

  1. Use visual models to find the missing numerator: a. 1/2 = /6 b. 1/3 = /6 c. 2/4 = /8 d. 3/4 = /8
  2. Which of the following fractions is equivalent to 1/2? a. 2/3 b. 3/6 c. 4/6 d. 5/8
  3. Draw or describe two rectangles of the same size. Shade 2/3 of the first rectangle and 4/6 of the second rectangle. Explain why they are equivalent.
  4. A recipe calls for 1/2 cup of honey. You only have a 1/4 cup measuring scoop. How many scoops of honey must you use to equal 1/2 cup?
  5. True or False: 2/6 and 1/3 are equivalent fractions. Explain using number lines or visual strips.

Solutions and Step-by-Step Answers

  1. Missing numerators: a. 1/2 = 3/6 (half of 6 is 3). b. 1/3 = 2/6 (each third splits into two sixths). c. 2/4 = 4/8 (doubling top and bottom). d. 3/4 = 6/8 (three fourths equals six eighths).
  2. 3/6 is equivalent to 1/2 because 3 is half of 6. (Option b).
  3. First rectangle: Divide into 3 equal columns and shade 2 columns (2/3). Second rectangle: Divide into 6 equal columns and shade 4 columns (4/6). Both shaded regions take up the exact same physical space of the whole bar, showing that 2/3 = 4/6.
  4. Since 1/2 is equivalent to 2/4, you need 2 scoops of 1/4 cup to get 1/2 cup of honey.
  5. True. On a number line from 0 to 1, 1/3 and 2/6 land on the exact same point because dividing a line into 3 parts and taking 1 hop is the same distance as dividing into 6 parts and taking 2 hops.