Introducing Quadratics

Quadratics were the first non-linear functions my students ever encountered. From reading different blogs I had the idea that it would be cool for the non-linearity of the relationship to be a surprise that they would discover and therefore take to heart by exploring the relationship between two quadratically related variables. This plan was also influenced by my own experience in 9th grade of being utterly confounded by seeing the teacher graph a parabola for the first time. I was convinced it was wrong. When you graph a function, it has to be straight! This can’t be right.

As for the two quadratically related variables, I figured it would be good to work with a physical shape, so I went for the base-length of a rectangle with constant perimeter and the area of that shape.

The collective student response I anticipated:

“Gee, I wonder how the base length of a rectangle with constant perimeter is related to the rectangle’s area. This is a neat-o activity. Okay, now I’ll plot the ordered pairs I found and…wait a minute! Something’s weird here. My data points don’t lie in a straight line! How can this be? This is like nothing I’ve seen before! Well, since I’m working with a physical shape with which I’m quite familiar, and have proven for myself that these two variables are related in this curved way, I must accept the reality and strength of this kind of relationship. I wonder what other variables are related this way. Wowzers, this is fascinating!”

The actual collective student response:

“Why are we doing geometry? What’s the point of this? What do I do with the values in the table? How do I graph them? And then I connect the dots? Okay.”

Not even the slightest hint of interest that they had graphed a curve instead of a line.

Problems:

1) The apparent aimlessness and disconnectedness of the rectangle activity. Blame this on my thirst for surprising them with the quadratic relationship. I should have introduced the activity with something like, “These two values are related quadratically. This activity will help you explore what that means.” I hadn’t wanted to taint their discovery process with my teacher-y information, but I ended up robbing them of the sense of purpose that they needed.

2) Many students lacked automaticity for graphing a table of values. A problem indeed.

3) Many students showed no meaningful comprehension of the concepts of mathematical relationships and functions (related to problem 2). While certainly not everyone will be intrigued by the same mathematical discoveries that I was, I’m sure some of the lack of interest in the curve was caused by the meaninglessness of even line graphs to these students. If a graphed line communicates no information to you, a graphed curve is hardly different.

Improvements for next time:

1) Give an intro to the rectangle activity. Okey dokes.

2) Provide physical rectangles that allow them to adjust and measure the side lengths; allow for non-whole number side lengths. (This time through, the rectangle was only hypothetical, and they were dealing with whole numbers exclusively.)

3) Consider spending time practicing reading graphs before teaching quadratics, and moving the review of domain and range up from where it currently is (right before teaching inverse trig functions). Although a more realistic expectation would be to review domain and range before quadratics and before inverse trig functions.

4) Prep self for the potential dismay of the pre-calc teacher that my students have covered so little material because I was forever adding preparatory and enrichment activities to the material we did cover. Dismay or no, I’m convinced that it’s better for these students to understand some things than to have seen many things and understand none.

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