We did a great activity to introduce Sine and Cosine from the Unit Circle. This is a standard activity from the high schools my GK12 students work with but none of our undergrads had seen it.

The activity... you start with the unit circle and measure the arc-length using a piece of string. That string is laid down as the independent variable (x-axis). You then measure either the cosine or sine values that correspond with the given arc-length on the unit circle. So if you're measuring cosine you start with angle 0 and measure the perpendicular length from the y-axis to the circle at angle 0 and you cut that length in spaghetti and lay it down on the graph.

By physically having to move the spaghetti from the unit-circle graph to the cosine plot the students gain a better intuition for where these values come from as a function. Also it results in great discussion about the fact that the x-axis (as they call it) is now labeled theta and the y-axis is now measuring the x-values from the unit circle which is used to create the cosine function. It sounds confusing now but the hands-on interaction and group work really keep it interactive and engaging.

When I repeated this activity with my GK12 partnership teams it lead to even more great discussion about the conventions we use in our classrooms. I require all my students to add arrows to the end of their plots of continuous functions. This is not a mathematical convention that is necessarily used in advanced mathematics but by requiring the use of arrows, labeling the axis with the correct variable and function names, as well as adding the correct spacing for the axis I find that I am able to see more of my students conceptual misunderstandings.

Before, without arrows, many students believed that f(x)= x^2 only went from [-2,2] on the range [0,4] because that's all they've ever seen of that plot. Requiring the arrows on these plots (especially the trig functions) has helped clear some of these misunderstandings. Another idea that was presented was to have students write a sentence or two on the edge of their plot to indicate what happens as the function approaches infinity, I like this idea and will be trying it out next semester.