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​   In the past 20 years, I’ve taught some version of a Pre-Calculus course for most of those years. And the Unit Circle has become MY FAVORITE topic to teach. I wanted to share one of my favorite activities that I built for the Unit Circle unit–coordinates at certain points on the circle. If you […]

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In the past 20 years, I’ve taught some version of a Pre-Calculus course for most of those years. And the Unit Circle has become MY FAVORITE topic to teach. I wanted to share one of my favorite activities that I built for the Unit Circle unit–coordinates at certain points on the circle. If you want to know more about this unit, email me!

Background: Let me first say that this is after we’ve already talked about angles (negative degrees, more than 360 degrees) and radians (as another way to measure angles). You can read more about one of my favorite radian discovery activities in this previous blog post.

Once we’ve had some lessons to talk about measuring angles in radians and degrees, I introduce the unit circle definition: just that it is a circle whose radius is one unit in length. Then I give them a full page unit circle on a grid that has gridlines every 0.1 units.

Step 1: I ask students to share with their partner what they notice and wonder on this circle (for just a minute or two).

**Below is the circle. I zoomed in on the top half so you can see some of the details better. At the bottom of this post is the full circle.

Step 2: I ask groups to work on a set of questions/tasks.

If all the sectors around the circle above are of equal size, how many degrees are in each angle?

Label the angles marked on the Unit Circle (on the lines) in degrees (from 0 to 360).

If all the sectors around the circle above are of equal size, how many radians are in each angle?

Label the angles marked on the Unit Circle (on the lines) in radians (from 0 to 2pi). Simplify all fractions.

 Use the graph paper the circle is on to estimate (to two decimal places) the coordinates at each of the angles drawn. Write the coordinates outside the Unit Circle.

Question #5 is THE KEY question here. The first four parts keep students reviewing angles in degrees and radians, but #5 is taking it forward. And without me having to tell them students will be able to state a LOT of great facts about the Unit Circle, for example:

“I notice that all the coordinates are decimals less than 1”

“I notice that the coordinates at 45 degrees have the same x and y value” (about 0.7)

“I notice that the y-value of 30 degrees seems to be exactly 0.5”

“I notice that the coordinates of 30 and 60 are basically the same but x and y are switched”

“I notice that the coordinates in Quadrant 2 are the exact same numbers but the x is negative”

Basically, students say all the patterns we want them to understand about the coordinates, AND it really sinks in that these coordinates are all small numbers less than 1 (and greater than -1).

Step 3: Is to have students share out so they can hear these noticings from each other. After this activity students tend to have a more concrete grasp of the unit circle, its patterns, and the magnitude of the coordinates.

Now, we can find some of the exact coordinates in Quadrant 1, and students fly through filling out the rest of the circle. What’s more, I NEVER have to again remind them of the patterns in the circle or why a certain sin/cos answer is negative.

*Below is the full circle if you want to use it! If you want this and any of my other Unit Circle resources, just email me and we can connect!

 In the past 20 years, I’ve taught some version of a Pre-Calculus course for most of those years. And the Unit Circle has become MY FAVORITE topic to teach. I wanted to share one of my favorite activities that I… Continue reading →  Read More