Introduction
Posing interesting problems that get students talking about their thinking is a good goal for all ages. A great topic for encouraging mathematical discourse in the middle and high school grades is proportional reasoning. Rather than pose just one problem to the class, however, we wondered what kind of discourse would result if we used three different versions of the same task with our students: What kinds of mathematical thoughts might be uncovered, and how would these thoughts be expressed in small-group discussions? We called the set of all three versions of the task Ratio Triplets, and the essential feature of the task involved thinking about ways of determining which of two products is the better buy: A 64-ounce container of ice cream selling for $6.79 or a 48-ounce container of ice cream selling for $4.69. This article invites you to see and hear how some of our students reasoned on this task as you visit small-group discussions about Ratio Triplets via a series of short video clips (none of which is longer than 80 seconds). In addition to explaining what we did when using the task, we’ll also share our perspectives concerning some interesting examples of student discourse about the mathematics behind the task. So enjoy your exploration of Ratio Triplets, and be sure to try it with your own class!
