Teaching Equivalent Fractions: Making Equivalent Fractions Concrete

Using models to make the concept of fractional amounts more concrete is fairly standard in most math classrooms.

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But too often, teachers stray away from concrete models when teaching more complex fraction concepts. We teach shortcuts that rely on abstract ideas and leave students unclear about why the shortcut works.

Modeling Equivalency

This model shows the same whole and the same amount of the model shaded. The only difference is that each of the original parts has been divided into two parts. Students who are familiar with fraction models could identify the fraction 2/6.

When the two models are compared to each other, there is opportunity for discussion around whether the fractions are the same, or similar, or different, ultimately leading to the idea that two different fractions can represent the same amount and be equivalent.

To reinforce the concept of the same amount of the whole being shaded, the division of the original parts into smaller parts should be shown in different ways:

And with different numbers of divisions.

Taking it to the Next Level

Let’s compare two of the models that we know are equivalent:

1 out of 3		2 out of 6
1 part shaded/3 equal parts		2 parts shaded/6 equal parts
1/3		2/6

What can we see from the comparison:

The two fraction models represent the same amount
Each of the original parts was turned into 2 parts when we added the dotted line
The numerator in 2/6 is two times the numerator in 1/3 (1 x 2 = 2)
The denominator in 2/6 is two times the denominator in 1/3 (3 x 2 = 6)

This work will eventually lead to the understanding that if the numerator and denominator of any fraction are each multiplied by the same value, the fractions represent the same amount and are equivalent.
13/22=26/44