My plan for the whiteboard session was based on an introduction to functions. I had decided that I wanted the students to represent a situation using a table, graph, function rule and description. Previously they had practiced the individual parts, and now I wanted them to put it all together.
I randomly arranged them into groups of three (thanks Fawn!), and they got organized into the groups with their boards in the vertical position (thanks Alex ).
I gave them a situation. I told them that the first situation had to do with the grocery store. They were to call out input values (independent variable values) and I would give them the output (dependent variable values).
The results were something like the following:
them: 3, me: 3.75
them: 16, me: 20
them: 50, me: 62.5
them: zero!, me: zero!
them: 1, me: 1.25
After this it suddenly became quiet as the students realized that they had enough information in front of them to begin to figure out the function. The also asked: "Can we begin working on the graph?" Of course!
What was interesting to see at the end of this first session was the choice of written descriptions. One group chose the cost per Granny Smith apple, another chose the cost per piece of beef jerky. Another group said it was a 25% tax per item purchased. None of these examples were in the textbook...
I took the opportunity at this point to introduce them to new notation for a function rule f(x)=1.25x instead of what they had been using which was y=1.25x
For the second round, after I had the students erase the boards and re-randomized the groups of three, the situation had to do with public transportation.
The results for the input/output went like this:
them: 0, me: 47.25
them: 1, me: 47.25 (them: wait a minute.....)
them: ok, how about 3 miss? me: 47.25
them: 2, me: 47.25
them: 10, me: 47.25
Again they got straight to work creating the table of values, the graph, writing the rule and writing the description.
And then I was blown away.
As I wandered looking at the work I noticed that many of their representations of functions looked a little non-traditional to the part of my brain which is the math-is-traditionally-represented-in-this-way-always part.
I was seeing stuff like:
f(x) = 47.25 +0x
and, f(x) = 47.25x/x
I came to my senses and realized that the students were merely representing the function in a way that made sense to them.
When questioned, each group who had used one of these types of strategies explained that they were trying to find a way to "get rid of" the variable x.
Miraculous!
