Children have an intuitive understanding of fractions. When you give your son a cookie and ask him to share ½ of it with his sister, he will know exactly what you want him to do. However, when the teacher asks him whether 1/8 is larger than 1/3, he may be stumped. Children tend to fall back on the rules that apply to whole numbers, that 8 is larger than 3, overly generalizing them to fractions and leading to a great many misunderstandings. Thus, the misunderstanding that 1/8 is larger than 1/3. As parents, it is essential that we support our children in developing a meaningful, conceptual understanding of fractions they can apply flexibly into adulthood.
Below are four fundamental understandings that children must acquire about fractions. These understandings are best developed using concrete materials that children can manipulate with their hands and minds. Standard pattern blocks (shown in the Images below) may be a useful tool in developing these key concepts. One of the visual features that makes pattern blocks so effective is they match color to shape, which aids in building children’s mental models of these math concepts.
Fractions represent a relationship to the whole
When we first think about fractions, it is typical to think of them as numbers, such as ½ or ¼. However, to build toward a deeper understanding of fractions, we actually need to introduce fractions as a relationship to the whole, or what we refer to as the unit. Using pattern blocks, we can introduce the yellow hexagon as the unit equal to one whole or 1. Then we can help children to visualize that the red trapezoid is equal to ½ of 1, because 2 red trapezoids come together to equal 1 (Images 1-3). We can repeat this procedure, demonstrating how a blue rhombus represents 1/3 of 1 because 3 blue rhombuses (also known as rhombi) come together to equal 1 yellow hexagon (Images 4-6). Using concrete pattern blocks, children come to more fully understand that a fraction is not simply a number to be memorized, but rather a relationship to the unit.
Fractions should be applied to many different wholes
Next, we can expand our children’s understanding of fractions to include a variety of different representations of units. For instance, in Images 7-9, a green triangle represents 1/6 of the unit, when the unit is equal to a yellow hexagon. However, when we change the unit so that 1 is equal to a red trapezoid, now a green triangle actually represents the fraction 1/3 (Images 10-12). We can again re-set the unit as equaling one blue rhombus. Here, the green triangle changes again, such that it represents ½ of the unit (Images 13- 15). Children come to see that a green triangle does not hold a value in and of itself, but instead its value varies in relationship to the unit. This relational understanding of the unit is essential as children advance in their work with fractions, so they come to understand that a green triangle can represent 1/6, 1/3, and 1/2, depending on how its relationship to the unit has been constructed.
The numerator and the denominator are defined in relationship to the whole
Typically, when children are introduced to the terms numerator and denominator, they are simply told, “the numerator is on the top and the denominator is on the bottom.” However, this process of labeling does not advance their understanding. Instead, children should be guided to discover that the denominator indicates the number of pieces that comprise the unit and the numerator indicates how many of those equal size pieces you have. For instance, the fraction 3/4 can be understood as first breaking a unit into 4 pieces (e.g., the denominator 4) and then using 3 of those equal-size pieces (e.g, the numerator 3).
In another example, ½ can be thought of as having 1 of 2 equal size parts of the unit (Images 16-17). The numerator tells us how many pieces we have. For 1/2, we have 1 of 2 equal size pieces of the whole. The denominator, the number down under the line, tells us how many equal size pieces there are to the whole. For 1/2, the 2 down under the line tells us that there are 2 equal size pieces to the whole.
Equivalent fractions refer to the same part of the whole
Finally, these understandings can be expanded to develop conceptual understanding of a traditionally-procedural topic: equivalent fractions. Rather than simply instructing children to find a common denominator, you can instead help children explore equivalent fractions using pattern blocks. When children discover that different fractional parts cover the same area of the unit, they see for themselves that these fractions are equivalent (Image 18). For instance, in this example children can visualize that ½ covers the same area of the unit as 3/6, thus ½ is equivalent to 3/6.
We encourage all parents to consider new and creative ways to build these four key understandings with their children, using concrete, everyday materials such as chocolate bars, folded paper, or even small coins that illustrate the relationship between the part and the whole.