Last week's class began with an activity in proportional reasoning. A scenario was given in which there was a child and a giant and each had measured themselves with their own hands. The giant then measured the child with his hands, and we were asked to find out how many child hands tall the giant was. If that is a little confusing, I am sorry! The idea was that we could use own knowledge of each of their heights in their own hands to find an equivalent fraction.
I approached this problem by finding the relative size of each hand. The child was six child hands tall, or four giant hands, therefore we could say that 6/4 was the size ratio between the two hands. Using my knowledge that one giant hand was 6/4 or 3/2 or 1.5 child hands, I could say the giant was then 6 x 1.5 child hands tall, or 9. The class was asked to demonstrate this proportional reasoning using the document camera, something I had never used before, so I wanted to try it out. Before I could however, I had the opportunity to see a peer's strategy for finding the giant's height, which gave me a new idea for solving the problem.
Rather than equating the size of each hand, which some may find confusing, I decided to demonstrate the problem by equating their heights instead. Working in giant hands, we knew the child was four tall and the giant 6 tall. This indicated that the giant was 6/4 or 1.5 times as tall as the child. I thought that this was a much easier way to visualize and explain the problem than my initial approach! Using this ratio, we then knew the child was 6 child hands tall, and the giant was 6/4 that value, which works out to 36/4 = 9. The latter approach is nicer in that we set up our solution by equating the height of each individual. As our height does not change, we can quickly see that there exists a proportional relationship between them that is static. Equating hands is a perfectly reasonable approach, but as discussed in class a student may find the idea of measuring in hands somewhat unclear, leading to confusion when trying to carry forward this ratio to find a solution.
The activities for the class also addressed proportional reasoning. The first of such dealt with Hallowe'en candy taken from one student and placed in a bag with other candy. The candy of the student was distinct (this was not clear on the handout, but explained to each group) and therefore with each handful taken from the bag we were able to estimate the proportional relationship between the student's candy and the rest in the bag. This was very similar to the exercise used for the grade 6 EQAO problem solving task, except had fewer variables (two kinds of candy vs. three colours of gumballs) which I liked. The activity was a great way to investigate proportional reasoning, and could be further extended to later topics such as data management and probability.
The second activity consisted of a series of diagrams, each having two distinct sets of objects. The student was asked to indicate which set had more of a certain element. The instructions for this were intentionally vague, which spoke to the devil's advocate in me as the task could be approached in two ways. The first was to take each situation literally by simply counting which group had more, whereas the second approach was to represent each element as a fraction of the set and compare the fractions. Each approach resulted in different answers, which could be a valuable exercise for students to explore.
The last activity dealt with two families eating pizza and the number of pizzas each was to buy. Each family purchased their pizza based on a proportion to the number of people, and the total amount for each case was considered. The last step of this activity had the two characters eating a total of two pizzas at different rates, and the number of slices each person ate was to be found. I liked this addition to the worksheet as it not only dealt with proportional reasoning, but with rates as well. I have found that students in my practicum have difficulty relating problems to applications, and their poor understanding of unit rates is generally responsible for this.
