Fraction Brainstorming

I worked with a colleague today to look at various approaches to conceptualizing fractions, and adding & subtracting, multiplying and dividing them.

We talked about area model and how to visually represent adding and multiplying fractions, but then we decided to dig a bit deeper and look at using concrete tools like cuisinaire rods to help us.

The plan we came up with is to start with a concept circle with the rods in place and students need to determine how to write down the value of each rod/combo. Using 12/12 as 1 is helpful for this task.

Next we’d do a concept circle with fractions written down, using quarters, thirds, halves and sixths, and the central 12/12 as 1 and have students build the fractions with the rods.

Next we thought that we could build equivalences with the rods, making combos that are the same length, and then writing them down in math. We know that if they are the same length they are equal, and if it’s a combo you can add up the various blocks. You could get at subtracting with an overshoot and return strategy using the rods as well.

After that we started talking about how to best show fraction division. I shared my recent new learning that fraction division can be done straight across just like multiplication can, and if we get a common denominator first then it simplifies beautifully.

e.g. 3/4 divided by 5/7

is the same as

21/28 divided by 20/28

dividing the numerators gets us 21/20 which would be over the quotient of 28/28 (which is 1) so the answer is 21/20. Now this isn’t a way to use visuals or manipulatives, but it sure is cool.

I was recalling the mini feud between Howie Hua and Mr G (videos are on tiktok and instagram, not sure how to link them here) about various fraction division methods.

This is something fun to dive into as a team. So glad colleagues are keen to ask and try and push me and my thinking in new directions!

I’m glad to have spent some quality time with the book rethinking fractions this summer. It has been very helpful.