Angles on the Coordinate Plane (Lesson 6.2)
Unit 6 Day 2
CED Topic(s): 3.2
Understand how to measure angles in standard position on the coordinate plane and their properties.
Understand that a radian is an angle measure with an arc length of one radius.
Label the angles on the unit circle in radians using proportional reasoning (i.e. partitions of semicircles).
Quick Lesson Plan
Today’s lesson is a fun one and also packed full of content! You will need a paper plate for each student, preferably of 3 or 4 different sizes, so each group member has a different size. If you can’t find this, using just a small plate and a large plate is fine. Two group members should use the small plate and two should use the large plate. We suggest using plates that don’t have a big rim. The cheap paper plates actually work best because they are almost completely flat. Students will also need Twizzlers, scissors, and a ruler. Many alternate materials are available. You can print circles onto pieces of paper and have them cut it out and Twizzlers can be replaced by string, floss, cooked spaghetti, anything that can bend and be cut!
Students first set up their paper plate to resemble a coordinate grid and to allow for a discussion about angles in standard position. They then use a Twizzler whose length is the radius to create an angle whose arc length is the Twizzler. Question 7 may seem painfully obvious but note how we are forging a connection between an angle measure and its subtended arc, as this is a key component for defining radians.
In questions 8-11, students use the Twizzler to subdivide the circle, literally counting the number of Twizzlers to reach Point B (half a circle) and back to Point A (the full circle). Using the circumference formula, students will see that the exact number of Twizzlers that will fit is related to π! Note the language used in question 11. We are subdividing the circle into halves, quarters, and sixths. We are not finding the number of Twizzlers that go into 180˚, 90˚, and 60˚. As I say to my AP Calculus students all the time, “degrees are dead to us.”
In questions 12 and 13, students continue working with the Twizzler to define negative angles and coterminal angles.
Is everyone’s Twizzler length the same? Why or why not?
Is the number of Twizzlers needed to get half-way around the circle the same for everyone? Why is that?
What does the formula C=2πr mean?
How many Twizzler arc lengths should fit around a third of a circle? An eighth of a circle?
Is everyone’s Point E in the same spot?
Does going 4 Twizzler lengths around the circle give everyone in your group the same angle? Is the arc made by 4 Twizzler lengths the same length for everyone in your group?
How many more centimeters of Twizzler would I need to go around the circle again?
It’s easy for students to confuse the terms radius and radian. We tell students that a radius is a length measured with a ruler and a radian is a measure for an angle, it is not the actual arc length. Students begin to associate angles with pi in them as being in radians, but it’s important that students realize pi is also a quantity telling you there are approximately 3.14 radii in half a circle. This is another reason why we use an angle measure of 4 radians in question 12, rather than only looking at key points on the unit circle.
You may have noticed that there are no references to degrees in the entire lesson. The AP Precalculus course framework deals with trigonometry concepts only in radians, just as AP Calculus does. The goal is for students to become native radian thinkers, rather than constantly converting between degrees and radians. There are a few times in upcoming Calc Medic lessons where we do choose to reference degrees but this is for the sake of establishing a concept (such as ratios in special right triangles) (Lesson 6.4) or drawing angles on the coordinate plane by hand (Lesson 6.3).
When filling out the angles of the unit circle for the first time, color coding can be very helpful. We encourage you to show the unsimplified form of pi/3 as 2pi/6 so students see that every multiple of π/3 is also a multiple of π/6. In other words, splitting the semicircle into sixths also indicates how to split the semicircles into thirds. Alternately, splitting the circle into twelfths (counting by π/6) also shows how to split the circle into sixths (counting by π/3). In the Check Your Understanding, students will continue to think about equivalent forms of the angles on the circle and thus, which angles are multiples of one another. Make sure students see that each quadrant is not cut equally into thirds.
When filling out the angles on the circle, model how you are counting by π/6, then counting by π/4. Thinking of π/6 as a quantity that can be counted (just like 1 fish, two fish) is a helpful way to go through the unit circle, especially for students that struggle with adding fractions.