We have been deep into a unit on ratio, rate and proportions. The students have been studying proportions in tables, graphs and equations. One specific application of unit rates is scale factor. I spent a day in class introducing the concept of scale as a rate, that compares a real thing to a model of the real thing. The students were having quite a bit of problems with understanding the concept and relating it to our work on rates and unit rates.
In response to their struggle, I gathered up a collection of toys and models:
A key chain of the Eiffel Tower,
a teeny tiny Statue of Liberty,
a small stuffed tiger,
a Lionel train,
a matchbox car,
a replica of the Sears Tower,
a toy soup can and a real soup can,
a Barbie doll
I had iPads out on the tables, as well as rulers and calculators. I held up the tiny stuffed tiger, and asked the students to find pictures of a real tiger on the internet. I had them take a few minutes to discuss with the person next to them whether they thought the stuffed tiger was made proportionally.
Most of the students could agree that the stuffed tiger was NOT made proportionally, but we spend a fair amount of time trying to determine how we knew it was not proportional to a real tiger. Many students made comments like “the tail is too long” or “the legs are too short”. And, of course, my response went something like “what do you mean by ‘too long’ or ‘too short’. We struggled a bit with these ideas, until finally one of the students made their intention clear, finally, by comparing the tail to the length of the body. Thus we finally hit upon the notion that just as ratios are comparisons, scale was a comparison between a real thing and a model (or a toy). One measurement, (the length of the tail) set up one ratio. In order to determine whether it was proportional, the students needed two separate measurements.
I distributed the rest of the toys/models and the students went to work. They enjoyed the activity. As I walked around to assist, I saw students engaged – one was measuring the toy, and the other was searching on the internet for real measurements of these models.
I then asked the students if they thought that the Barbie doll was made to proportion. They were in agreement that she was not, if you compared her waist to her hips, for example. I then asked them if they thought that Barbie’s legs were too long. They all decided that they were. I put up the measures of an “average”woman’s height and “inseam” (taken from clothing Vogue Patterns, based on a woman’s size 12). I then asked the students to measure the Barbie doll height, and then leg length, but only to the heel, not to the toe (Barbie stands on tiptoe!!). Surprisingly, Barbie’s legs are proportional to her height as compared to an average woman!