This is a sequel to my last post (Seeing is Believing?), addressing the same issue from a different point of view.

Fractions, of course, are difficult. When teaching 4th and 5th grade in the 1970’s I struggled with this, and came up with a powerful learning tool: fraction rectangles. The idea is that it is much easier to think about (say) 2/3 and 4/7 when looking at a 3 by 7 rectangle:

This representation allows us to readily compare those fractions, or add them, or subtract one from the other. It does not accomplish miracles, but it provides multipurpose visual support for teacher-led student reflection, discussion, and generalization. Best of all, it is always available to students if they have access to grid paper, and actually any sort of paper. (They can quickly make a good enough grid if the numbers are sufficiently small.) They do not need a teacher, or manipulatives, or a computer to have access to this tool:

I discussed this in great detail in 2013 on my website: Fraction Arithmetic on Grid Paper. Last year, I added videos to that presentation, as I’m told that nowadays “people expect that”. (The text and videos are addressed to the teacher.) The rest of this post will make more sense if you take some time to visit that page.

Unfortunately, the fraction rectangles approach is turning out to be a hard sell. For reasons I cannot fathom, math educators I respect fail to be impressed, perhaps because the approach is unfamiliar. If there are robust arguments against this tool, I’d love to hear them. Not that I would be convinced: I have a few years of experience using this in the classroom, and I know it’s useful. One objection I’ve heard is that what’s going on here is not visually obvious, as it requires students to count the shaded squares. True! But counting is not a sin. It is something in which almost all students have a solid grounding. And having to count can lead to useful and important shortcuts: skip-counting, and “length times width”. The point of the fraction rectangles representation is not to make things easy or obvious. It is to make things understandable, and that requires students to engage.

Just to be clear: I do not claim this can do everything. While it’s a great foundation to think about common denominators, it does nothing for *least* common denominators. Nor does it help with reducing fractions to lowest terms. I propose this as a complement to other representations, not an alternative to them. The number line and other linear models remain essential in understanding that fractions are numbers and in comparing them visually. The pie representation helps in making connections with angles and with time (on an analog clock). The role of fraction rectangles is to expand that repertoire with an intuitive way to tackle comparison and the four operations. Effective teaching of important topics requires the use and articulation of multiple representations.

A couple of weeks ago, I made a bare-bones GeoGebra applet as a support for this approach: Fraction Rectangles. (I used it to make the first figure in this post.) I was tempted to make the applet more “powerful yet easy to use” by automating the process. Perhaps have the students enter the fractions, and have the rectangles automatically adjust to the right dimensions. Or have the shaded areas automatically described as fractions with the common denominator. Or have sum and difference automatically displayed. I could do any of these things, but doing them would undermine the basic pedagogical concept here: this approach is intended to empower the student, not highlight my GeoGebra prowess. I decided to facilitate the drawing of the figures, and not go beyond that. There is no royal road to fractions, and there is no shortcut to understanding: students must engage intellectually, not watch passively. (This also means that merely watching the teacher demonstrate fraction rectangles is unlikely to have much of an impact. The whole point is to have *students* use the tool, which is what would make possible a fruitful discussion of the underlying mathematics.)

That said, there is a place for fancier fraction applets and I’m sure you can find many good ones on the web. I may even create one some day. My advice is to use those late in the game, when students have developed enough understanding to be able to dissect step by step what those applets do.