Coterminal and Reference Angles

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SWBAT describe the measure of angles as rotations in the coordinate plane.

Big Idea

What happens when we have angles more than 2*pi or 360 degrees?

Bell work

5 minutes

Today I use the student homework assignment to develop reference angles. I ask my students to take out the homework activity.  Then, I have them discuss the following two questions as they review the homework with each other:

  • Why did I group the angles together in tables in this manner?
  • How is the first angle in each table related to the other angles in the table?

As students discuss I listen for comments, such as "the denominators are the same." Or if you multiply 30 by 5 you get 150. After 2-3 minutes of discussion, I expect we will be able to quickly fill in the tables. Then, we will return to the questions posed at the start of class. I will ask my students to share their ideas. As they volunteer them, I will scribe them on the board for students to see and discuss further.

Reference Angles

15 minutes

After discussing the questions from the Bell Work, I will have my students label a coordinate plane and plot the angles in the first table. I will ask my students to label the lines with both the degree and radian measure. I may need to remind my students to label the terminal side of the angle. This is a common confusion of students that should be addressed multiple times. I will ask questions such as, "When we label this line 45 degrees what are we meaning?"

After placing the angles on the coordinate plane, I will share with students the idea that I could plot a point on the terminal side line, add a vertical line, and make a right triangle so that the x-axis is one side of a right triangle and the hypotenuse is a segment on the terminal side. Then, I'll ask:

What can you say about all the triangles made like this?

I want my students to observe that it is possible to make a number of similar triangles, because the triangles all share the acute angle at the origin. Then, the students finish labeling the coordinate plane by using the their angles from their homework page

If you draw a vertical line to the x-axis from the pi/3 angles around the coordinate plane what can you say about the triangles? What about the pi/6 angles?

I tell students that the angle we have identified is called a reference angle. I like to have students write a definition in their own words. I will give them a prompt like, "Thinking about what we just did how would you define a reference angle to someone that missed class today?" This is a good opportunity to use a think-pair-share protocol to develop a class explanation.

Next, we will test the class explanation of reference angles to work on some problems. I wrote the problems as fill in the blank. I think that this will help my students to recognize the process they used as they found the reference angle. I also give students a table with several angles that are multiples of the special angles. Students work to find the reference angles and explain how they found the angle.

I allow students 3-4 minutes to work on these calculations. As they work I move around the room asking questions such as:

  • In which quadrant is the angle?
  • According to our class explanation, where do you draw the right triangle?

Common errors include forgetting to go to the nearest x-axis when in Quadrant IV. For example, my students often subtract from pi instead of 2*pi. I am also on the lookout for students who are struggling with how to draw the given angle. When I come across these challenges, I ask questions such as:

  • Where is the x -axis with respect to that angle?
  • About how much has the angle rotated from the horizontal?

By looking over student work I am able to both correct errors and make an informal assessment of what my students currently understand. I also find students with fluent processes or interesting methods that can share work with the class.

Reference angles that are multiples of a unit angle.

5 minutes

I begin working with special angles because  students will need to use these angles the most through the rest of the year. I don't want students to think that the numerator is always just pi on a reference angle. So I now give the student the angle 3pi/5 to find a reference angle.

When students are struggling to find the result I ask questions such as:

  • What quadrant is the terminal side of the angle?
  • Where could I draw the right triangle?
  • How can I find the distance from the terminal side to the x-axis?

After working several problems I put a black xy coordinate plane on the board. Then, I ask:

If you were going to find the a reference angle in quadrant II how would you do this?

I looking to see how quickly my students recognize that in this case we work with the 180-the angle or take pi-angle. I plan to repeat this line of questioning with each quadrant. I do Quadrant I last, considering a rotation of more than 2pi. At this point, the question helps transition to the next section of this lesson.

Coterminal Angles

10 minutes

Another term I introduce today is coterminal angles. I start by asking my students to find the reference angle of a 370 degree angle. I use an angle in degrees because students understand degrees better than radians, immediately recognizing that we have gone past a full rotation.

I plan to let my students consider this question for a minute, giving them the opportunity to think about whether or not this is a complicated example or the start of something new. After about a minute I begin asking students, "Where will the terminal side of the angle would be located?"I expect that some of my students may think you cannot graph a 370 degree angle because it is more than 360 degrees. If so, I will ask, "If I have a spinner can it rotate more than 360 degrees when I spin the pointer? How can I measure the amount of rotation from a spin using an angle measure?" 

Once students consider a real situation, they often quickly realize where to place the 370 degree angle.  One or more students will usually say it is once around and 10 more degrees, so it is in Quadrant I.  Once we establish its location, I will ask my students to find the reference angle. Then, I will say, "Is it okay to have more than one angle that has the same initial side and terminal side? What do you think?" Giving students some ownership over the idea of coterminal angles is helpful. They need to buy into the idea of a an infinite number of possible angles, a process that begins with believing that two is possible and acceptable.

Once we think about and discuss (if necessary) the idea of two different angles sharing the same initial and terminal sides, I will ask, "How many angles could have the x-axis as its initial side and have a terminal side the same as 370 degrees (or 10 degrees)?" As a class we will work towards an understanding of the idea that an infinite number of coterminal angles are possible.

An important addition to this conversation is the idea that we can we can rotate in both a clockwise and a counterclockwise direction. I don't have a specific plan for introducing this. Sometimes a student brings it up, sometimes I raise it when I think that there is enough understanding to consider an example without the conversation breaking down.  

Overall, this is a time for completing several examples  with angles in both radians and degrees to develop understanding and fluency (MP6, MP8). As we work, I will eventually share a definition of coterminal angles with the class.  And, I will start to ask students to volunteer other angles, both positive and negative, that are coterminal with the examples we just completed.


5 minutes

Today students have an Exit Slip. I will ask my students to find a reference angle and 2 coterminals for 13pi/7 on a small sheet of paper and turn it in to me as they leave. I will use the Exit Slip to informally assess students' understanding of today's discussion and think forward to upcoming lessons.