Understanding Fractions as Equal Parts

Fractions is often formally introduced in 3rd Grade. When this topic is introduced, it usually starts with naming fractions (for example, parts of a pie), or fraction of  a set (for example, 1/4 of 20 items). From our experience, students often skipped over the very fundamental concept of fraction – EQUAL PARTS. Without a strong visual understanding of fraction as equal parts, fraction often gets reduced to number operations.

The Common Core addresses this, but under a different heading – Geometry. In the 2nd grade standard 2.G.3, students are required to:

Partition circles and rectangles into two, three, or four equal shares,
describe the shares using the words halves, thirds, half of, a third of,
etc., and describe the whole as two halves, three thirds, four fourths.
Recognize that equal shares of identical wholes need not have the
same shape.

In this blog post, we describe the progression that teachers and parents can take to lay the foundation for fractions, i.e. understanding fractions as equal parts through visual representations using geometric shapes.

Recognizing Equal Parts

The first step to introducing the concept is to identify division of a standard geometric shape. The easiest shape to work with is a right-oriented square, which allows for different combinations of sub-divisions to illustrate equal parts. For example, the following figures all show equal part division of a standard square.

Fractions as equal parts - right square examples
Fractions as equal parts – right square examples

2nd grade students may not be able to see the third example as equal parts since they are of different shapes. Teachers and parents can show, using cutting and rearranging, that although of different shapes, the parts are in fact of equal area. This is also a good place to introduce the notion of congruence (same size and same shape). Students should understand that equal parts means equal area and that parts need not be congruent to be equal.

Different Orientations

Continuing with shapes and equal parts, students can be asked to divide a rotated square into equal parts. Kids usually love this challenge, and can spend a considerable amount of time working on this (answers at the end of the blog)!

For example, divide the following figure into 3 equal parts.

Exercise to divide a rotated square into equal parts
Exercise to divide a rotated square into equal parts

Non Examples

It would also be useful to introduce non-examples. For example, students should be able to recognize that the following figures are not divided into equal parts.

Non examples - examples of non-equal partitions
Non examples – examples of non-equal partitions

Completing Figures With Equal Parts

When students are comfortable recognizing equal part division of geometric shapes, we can then ask them to complete figures to show the number of equal parts required.

For example, how many lines do you need to draw to divide the circle into 4 equal parts? How many lines do you  need to draw to divide the square into 8 equal parts?

Completing figures to find number of equal parts
Completing figures to find number of equal parts

For advanced students, we can introduce shapes other than squares and circles, such as triangles and hexagons, and ask them to divide into equal parts.

Dividing advanced shapes
Dividing advanced shapes

We can also give students a hint, by drawing a line to show them where to start (answers at the end of the blog).

For example, divide the following triangle into 3 equal parts.

Divide triangle with hint
Divide triangle with hint

Naming Proper Fractions

Once students understand the concept of equal parts, naming proper fractions becomes much easier. Students need to understand fractions naming conventions where the number of shaded equal parts is represented by the numerator, while the total number of equal parts is the denominator. For more advanced students, we can combine the equal parts concept and naming fractions in one question: What fraction of the circle below is shaded?

Naming proper fractions
Naming proper fractions

Conclusion

Introducing fractions as equal parts using visual representation is a great way to lay the foundation for fractions in 3rd grade. The steps described above can be incorporated into the 2nd grade curriculum, with the understanding of how it is linked to the topic of fractions in 3rd grade.

Answers:

Different Orientations: Divide the following figure into 3 equal parts.

Rotated square example
Rotated square example

Completing Figures: Divide the following triangle into 3 equal parts.

Triangle example
Triangle example

This is the first part of a series of blog posts on Fractions:


More Fraction Resources

For more fraction resources, refer to our main fractions page.

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