Last weekend, I shared my thoughts about teaching fractions with teachers of grades 3-5 at the Asilomar meeting of the California Math Council. After decades of work in high school, and hundreds of presentations to teachers of grades 7-12, this was a bit of a departure from my normal routine, and somewhat anxiety-provoking. The reason I decided to step out of my comfort zone was that I wanted to share some of the ideas I came up with back in the 1970’s when I was an elementary school teacher.
Pretty much everything I presented can be found on my website’s Fractions home page.
The biggest part of the session was the well-chosen rectangle. For a full explanation of that approach, you should read the text and/or watch the videos here. Long story short: to work with two fractions visually, make rectangles on grid paper. For the dimensions of the rectangles, use the fractions’ denominators, as in this example for 1/4 + 1/6:
The idea is that the whole rectangle represents 1, and the shaded areas show the fractions we are interested in. This makes it easy to add them by counting the “baby squares” (as one session participant called them.) We see that the sum is 10/24. Doing this many times prepares students for a conversation about using a common denominator, and how one could do this without drawing the picture.
One attendee pointed out that this does not show the least common denominator. That is certainly true, and that is one of the reasons that I propose the well-chosen rectangle as a complement to other approaches — not a substitute. Masha Albrecht suggests that one could see the lowest terms version by rethinking the grid:
Now, if we count the dominoes instead of the baby squares, we see that the sum is 5/12. This is probably not an approach a beginner would be able to use, but it is a nice insight from a teacher’s point of view.
Someone pointed out that my examples only involved numbers less than 1. Does this approach work, say, to add 5/4 to 7/6? Masha, once again, had a great response. Just tile your grid with copies of the well-chosen rectangle. Everything still works:
The sum is 58/24. In fact, we don’t even need to draw the additional rectangles:
Obviously, counting is no longer practical! We need to use multiplication facts: on the left we had 5⨉6, and on the right 7⨉4, so 30+28 — this can help us understand what’s going on when working without grid paper.
I ended the session by introducing the Egyptian Fractions challenge. One participant said it would be sure to interest (and slow down) her speed demons. Unfortunately, we didn’t have time to get seriously into it.
— Henri
PS: I made the figures using this applet, but I would expect students to just work on grid paper.
Henri's Math Education Blog 
I’ve always done this with a square subdivided into rectangles rather than a rectangle subdivided into squares, the idea being that it’s more natural to declare a 1×1 square to be the area unit than some arbitrary rectangle. That the subdivisions are rectangles instead of squares doesn’t seem problematic since all you’re doing is counting them, and they’re manifestly of equal size. The 1×1 square picture also gives a nice way of understanding fraction multiplication in terms of an area model. (Subdividing the horizontal unit length into sixths and vertical unit length into fourths results in the unit area being subdivided into 24ths.) I do not have experience as an elementary school teacher, but I have used this with my own elementary-aged kids. It’s difficult to assess how successful it has been.
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I love that idea! It is indeed more natural, as you say, and there is a bit of a leap of faith to accept that “1” can be shown with different dimensions. However it does not seem as easy for students to implement your approach. Upon seeing a question about two fractions, they can use the well-chosen rectangle approach immediately. In contrast, it would not be easy for a beginner to figure out what size the unit square would need to be on the grid paper, and likewise the dimensions of the embedded rectangles and their orientation. Also, the needed well-chosen rectangles are quite manageable for (say) comparing 2/3 and 4/7. But a square with embedded rectangles would huge (21 by 21), and the rectangle grid would have to be drawn by hand, which would take a long time. This is less of a problem if the square is created by the teacher, as the support offered by the grid paper is not needed. But if the kids are to deal with this, they are far better off with the well-chosen rectangle, which is quick to draw on grid paper, and is pre-divided into a grid. Correct me if I’m missing something!
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I wasn’t thinking hard enough about the implementation aspects, and I see that your way is more practical, since grid paper is the obvious material to use. Working one-on-one with the student I would generally draw the square myself, or help the student to draw it, but the grid paper gives them independence, which is desirable. In an applet setting it might be possible to allow the student to select sevenths or fourths or whatever, and have the software make accurate subdivisions.
The thing I like about my approach is that it emphasizes from an early stage the idea of fractions as situated among the whole numbers on the number line–well, separate horizontal and vertical number lines in this case–and that helps with understanding the important idea that fractions are numbers, something that I suspect can get lost in parts-of-a-whole models. But practice with different fraction models helps students develop flexibility, which is good too.
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