If you are having issues understanding the students thinking ask them. Tell students if they can not show at least 4 steps per problem, maybe some don’t have that many, they must in a sentence tell their thoughts on how they solved. You will get a lot of I don’ know I just guessed which will tell you a great deal as well.

Jim Doherty

My guess is that the notion of the defining features of a linear function versus a quadratic function is not firmly in place in any way. What we are looking at is an attempt at a linear function with the x coefficient being the change in x. The y intercept is correct but there is an awful lot of confusion in this one problem. Given that it is identified as a mastery question it is appropriate not to include any prompt, but I wonder what the student might have done if asked to write a function of the form y = ax^2 + bx + c or y = a(x -h)^2 + k

mpershan

I agree with much of what Jim said. The student has only seen (I’d bet) equation-writing problems in a linear context, and so his knowledge about writing equations is associated with only linear equations. Seeing two points and asking for an equation immediately triggers his “linear functions” knowledge.

What I’m wondering is whether mistakes such as this one point to a need for setting up different categories of functions for students before studying any of them in depth. Like, maybe students should be working through different tables and completing them, and then categorizing them before any of them are given a name. Would that be enough to link the proper equation writing skills to the proper function?

Or maybe an activity like this would be better placed after students study linear functions for the first time?

This reminds me of another mistake (that I’ll try to hunt down and post) where nearly every single student in my classroom thought that quadratic functions had a slope.

Jim Doherty

I’d say that an even more interesting problem down the line would be to discuss a variety of functions that all might contain the two given points. I think that this could accomplish a great deal in the way of teasing out mathematical thinking, of tying together equation knowledge with graphing knowledge, set the table for interesting math later on (such as multiple trig functions that look radically different yet are all identical), but clearly that would be WAY down the line here.