For the Love of Ratios

I love teaching ratios. I begin with an image like the one below and use the “Stand and Talk” routine by Sara Van Der Werf. I ask the students “What do you Notice? What do you Wonder?” Students always impress me with their thoughts and ideas and after writing down their ideas, we look at what could be compared.

This leads us into how we can model our comparisons using tape diagrams. I then begin to introducing the idea of tape diagrams to represent simple ratios and then compare equivalent ratios. I usually start with simple examples from their wonderings and noticings and the image with congruent and similar triangles is a great way to do this.

However, in the spring of 2020, we went to distance learning and so did our ratio unit. Teaching students ratios through tape diagrams became a bit more challenging with distance learning and so after our introduction I turned to creating videos to help students better understand visualising ratios using tape diagrams. A video gave them the option of rewatching as many times as needed.

Using the traditional example of making lemonade to introduce how tape diagrams allow for visualisation, I attempted my first video using the program ‘Explain Everything’. We try to choose real-life problems which normally work great for ratios, but my first videos relied on problems that were not great in terms of real-world contexts.

My second ratio video dealt with cookies, but in a three-part ratio. I was still trying to introduce the basic concept of tape diagrams to my students. Not the best choice of question, but the students’ feedback was that it helped them to better understand ratios and using visualisation with tape diagrams. Following this with Dan Meyer’s Nana’s Paint Mixup means that students are quickly able to apply their understanding to a problem.

This led to a more complex problem. This next problem, revolving around post-cards, demonstrates how using tape diagrams can make a complex question much more solvable. The goal was to have students consider what stayed the same within the problem and what changed and how to represent this with tape diagrams. Here is my video explaining this problem.

Dan Meyer’s Three Act Tasks are one of my favourite teaching aspects of ratios. There are so many fabulous tasks which really allow students to think and apply mathematics. Super Bear and Bone Collector are two of my favourites. At school, I am lucky enough to teach with a colleague who has more toys for ratios than one can imagine. Having access to three different sizes of a telephone booth, tennis ball, Rubic’s Cube, etc leads to great ratio understandings. Our goal for the unit is to have the students up and moving as much as possible and measuring toys to determine scale factor allows for movement. Even their summative assessment requires students to be up and measuring different parts of their environment.

What great ideas do you have for teaching ratios?