I base all of the trigonometric functions/ratios in the unit circle. We use this activity as an anchor for our future discussions about all sorts of things: the graph of sine and cosine, how we create triangles within the circle, and even sine being the distance from the x-axis and cosine the distance from the y-axis.

I overheard multiple conversations between students who noticed the rate of change both sped up and slowed down. They were convinced they were doing it wrong. That's the kind of dissonance I'm looking for!

I overheard multiple conversations between students who noticed the rate of change both sped up and slowed down. They were convinced they were doing it wrong. That's the kind of dissonance I'm looking for!

Here is a complete write up of the lesson.

Here is the only setup:

- This diagram shows a circle with one equilateral triangle inside and one equilateral triangle outside.
- Find the exact ratio of the areas of the two triangles.
**Be convincing in your explanation.**

After spending 30 minutes or so working on this problem in groups, we ran out of time. Meh. We planned on starting class the next day talking about our ideas for solution methods. Normally, with two minutes left of class, they line up at the door like cattle. Some of the students weren't done with the problem yet.

Here's an activity we do at the beginning of our unit on linearity: cup stacking. I took this idea hook line and sinker from Dan Meyer and Andrew Stadel. Rather than do this as a 3 ACT lesson, I use it as a chance to get students out of their seats and talking to one another.

We begin by giving 5 styrofoam cups to every pair of students. Then I show a picture of me getting my height measured at my doctor's office (in cm). They have to figure out how many cups tall I am. We also almost always need to come to an agreement about how to measure the cups. Quick setup and clean up and almost complete engagement. You can't ask for much more than that.

]]>We begin by giving 5 styrofoam cups to every pair of students. Then I show a picture of me getting my height measured at my doctor's office (in cm). They have to figure out how many cups tall I am. We also almost always need to come to an agreement about how to measure the cups. Quick setup and clean up and almost complete engagement. You can't ask for much more than that.