Having tried Ratio Triplets with many classes, we’ve been quite pleased at the way the task helps draw attention to a number of important mathematical ideas, and how the task promotes mathematical discourse in class. Here are a few final thoughts and questions we’d like to share:
► Why do so many students seem to have an easier time relating to money-per-ounce than to ounces-per-money? Especially on Version 1 and Version 3, where it explicitly asks if Alisha is correct (in using the inverse ratio as Mark), we find a high number of students who simply say “No”. That is, from our perspective, many students seem to think money-per-ounces can tell a person which is the better buy, while ounces-per-money offers no useful information.
► How can we better assess student understanding of proportional reasoning? Many of the classes where we have done Ratio Triplets have already had lessons in finding unit rates, setting up and solving proportions, and many of the typical exercises one might expect for proportional reasoning. Yet, while a teacher might expect something such as Version 1 to pose no difficulty, it was precisely the interpretation of results on that Version that seemed to pose the most difficulty! We wondered how often students might seem to be able to “find answers” without an ability to “explain answers”, which made us wonder about how to assess student understanding.
► What can a teacher expect from trying something like Ratio Triplets in class? One key ingredient for assessment depends on the ability of the student to communicate their reasoning, and Ratio Triplets certainly acts as a catalyst for student discourse. Perhaps practice in articulating their thinking verbally can transfer into their written explanations. Also, something noticed in common among students’ reflections about their experience with Ratio Triplets was how they came to an awareness of multiple solution strategies. As one student put it, the task “impacted my understanding by showing more than one way I can find an answer to that type of problem.”
Yet, beyond “that type of problem” exemplified by our Ratio Triplet task, we believe it can be useful to try this idea with many types of problems. That is, we encourage teachers to try different versions of problems with their classes, let them discuss their ideas, and pull out the relevant mathematics from the discourse that ensues. In doing so, the aim is the fostering of student reasoning and communication of mathematics, a wonderful goal for all grades.
