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.