I give daily quizzes with a variety of purposes. And my lab exercises in AP Physics often involve creating a curved graph with direct data collection, followed by linearizing that graph.

Historically, I've struggled getting students to understand graph linearization. Only a few students have truly understood how to figure out which variables go on which axes, and what the slope of the graph means. Most of the class has needed multiple consultations with me and with their friends to get each experiment done; and more often than not they haven't been able to reproduce their analysis later on. Graph linearization is abstract and difficult.

This year, I made graph linearization a common topic for daily quizzes. I started simply:

I make a graph of the net force experienced by
an object on the vertical axis, and the acceleration experienced by the object
on the horizontal axis. What is the
physical meaning of the slope of the graph?

I teach that we solve the relevant equation for the vertical axis... then using the equation

*y=mx+b*, identify the*y*and*x*variables. What's left is represented by the slope of the line. In this case, the relevant equation is*F=ma*. The vertical axis is*F*, the horizontal axis is*a*, so the slope is the cart's mass.
The biggest misconception is to deal with units not with variables in an equation. Someone will get the answer right by saying "the units are N/(N/kg), which is kilograms. That's mass." Well, that sometimes works. It sometimes is too difficult to mess with (i.e. for those who don't recognize alternate forms of units for N or m/s/s). And it is very often wrong.

In the lab, a student releases cars from rest on an incline, and
measures the distance they travel on the incline. The relevant equation is

*x*=*v*_{o}*t*+ ½*at*^{2}, with*v*_{o}= 0. The student keeps the time of travel constant while changing the cart’s acceleration. He graphs the distance traveled on the vertical axis, and the acceleration on the horizontal. What is the meaning of this graph’s slope?
Now the vertical axis is distance

*x*, and the horizontal axis is*a*. That leaves the slope as (1/2)*t*^{2}. The student doing a unit analysis might get the*t*^{2}part, but he certainly won't get the factor of 1/2.
Things get even more complicated when I ask students to figure out for themselves what to graph. But I'm still using daily quizzes to get them to practice -- primarily because I can do one every day or two, and give them instant and brief feedback on their answers.

Last week I did the standard period-vs.-mass-of-a-spring experiment. I have students collect period vs. mass data, then they linearize such that the slope of their graph allows determination of the spring constant. When they're all done, I use my five-second spring constant measurement method to check each group's result.

Try this quiz. It asks directly what a graph of period vs. mass for a spring looks like. (Learning to sketch the shape of a graph is a different skill that I'm also working on through daily quizzes.) Next, it asks for a possible linearization and the meaning of the slope.

Not only do we go over and grade this quiz for immediate feedback, we go straight into the lab to

*do*the experiment. I've primed my students' brains to know what to expect from the experiment. Then when it's time to linearize, there's much less fussing than in previous years. We just discussed the linearization, and for a grade, even. Everyone paid careful attention (because they care deeply what grade they get on a quiz). That doesn't mean everyone interprets their graph perfectly... but we're five stepping stones ahead of where we were in previous years, even though I've done fewer experiments in this style. Quizzes work!
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