Represent! Part 2

In my previous post, I shared the first few questions I asked at a recent #ElemMathChat I hosted. Today I’d like to continue talking about using and connecting mathematical representations with a focus on fractions.

Let’s start with this question from the chat:

Before reading on, pick one of the models yourself and analyze it.

• How does it represent 2/3?
• Where is the numerator represented in the model?
• Where is the denominator represented in the model?
• Can you answer these questions with all three models?

First it might help to differentiate the three models. The top left corner is an area model, the top right corner is a set model, and the bottom middle is a number line.

If you look at the area model, you’ll see that the whole rectangle – all of its area – has been partitioned into three equal parts, each with the same area. When we divide a shape or region into three parts with equal area, we actually have a name for each of those parts: thirds. Those thirds are countable. If I count all of the thirds in my area model, I count, “1 third, 2 thirds, 3 thirds.”

Two of them have been shaded orange. So if I count only the parts that are orange, “1 third, 2 thirds,” I can say that 2 thirds, or 2/3, of the whole rectangle is shaded orange.

If you look at the set model, you might think at first that this model is the same as the area model, but this representation actually has some different features from the area model. In the set model, the focus is on the number of objects in the set rather than a specific area. I used circles in the above image, which are 2D and might make you think of area, but I could have just as easily used two yellow pencils and one orange sharpener to represent the fraction 2/3.

I can divide the whole set into three equal groups. Each group contains the same number of objects. When we divide a set of objects into three groups with the same number of objects in each group, we actually have a name for each of those groups: thirds. Those thirds are countable. If I count all of the thirds in my set model, I count, “1 third, 2 thirds, 3 thirds.”

Two of the groups contain only pencils. So if I count only those groups, “1 third, 2 thirds,” I can say that pencils make up 2 thirds, or 2/3, of the objects in this set.

Finally, we have the number line model which several people in the chat said is the most difficult for them to make sense of. While we have a wide amount of flexibility in how we show 2/3 using an area model or set model, the number line is limited by the fact that 2/3 can only be located at one precise location on the number line. It is always located at the same point between 0 and 1.

In this case, our whole is not an area or a set of objects. Rather, the whole is the interval from 0 to 1. That interval can be partitioned into three intervals of equal length. When we divide a unit interval into three intervals of equal length, we actually have a name for each of those intervals: thirds. Those thirds are countable. If you start at 0, you can count the intervals, “1 third, 2 thirds, 3 thirds.”

However, what’s unique about the number line is that we label each of these intervals at the end right before the next interval begins. This is where you’ll see tick marks.

• So 1/3 is located at the tick mark at the end of the first interval after 0.
• 2/3 is located at the tick mark at the end of the second interval after 0, and
• 3/3 is located at the tick mark at the end of the third interval that completes the unit interval. We know we have completed the unit interval because this is the location of the number 1.

This quote sums up what I was aiming for with this discussion of representations of 2/3:

“Helping students understand the meaning of fractions in different contexts builds their understanding of the relevant features of different fraction representations and the relationships between them.” – Julie McNamara and Meghan Shaughnessy, Beyond Pizzas and Pies, p. 117

The bold words are very important to consider when working with students. What is obvious to adults, who presumably learned all of these math concepts years and years ago, is not necessarily obvious to children encountering them for the first time. What children attend to might be correct or it might be way off base. One common problem is that children tend to overgeneralize. A classic example is shared in Beyond Pizzas and Pies. Students were shown a model like this:

They overwhelmingly said 1/3 is shaded. The relevant features to the students were shaded parts (1) and total parts (3). They weren’t attending to the critical feature of equal parts (equal areas).

I’ll close this post with a Which One Doesn’t Belong? challenge that I shared during the #ElemMathChat. (Note: I revised the image of the set model from what was presented during the chat.) As you analyze the four images, think about the relevant features of the area model, set model, and number line; look for relationships between them; and then look for critical differences that prove why one of the models doesn’t belong with the other three.