Angles and Rotations

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SWBAT sketch an angle diagram rotation and define basic terminology regarding angles.

Big Idea

While learning basic angle terminology students are engaged through multiple clicker questions in this introductory lesson to circular functions.

Warm-up: Clicker Questions

3 minutes

As a warm-up for today’s lesson, I want students to recall some basic facts regarding angles. For the first question (page 2 of the Flipchart_Angles and Rotations.pdf) students review the basic fact that there are 90 degrees in a right angle. For the second question (page 3), students will need to recall that there are 360 degrees in a circle.


7 minutes

Pages 4-9 of the flipchart (Copy of Flipchart_Angles and Rotations) present students with the key vocabulary for this unit. I will have my students copy the definitions in their own words into their personal dictionaries.

Resource Note: Note, you can click on the yellow boxes on page 6 of the flipchart to reveal the rotation diagrams below.

During this section, I place an emphasis on drawing angles in standard position and how now angle measurements can exceed 180 degrees and can even be negative. Today, our angles are still only measured in degrees until students explore radians in the next day’s lesson. I think it’s important to give students time to be comfortable with angle rotations in a unit measurement they are very comfortable with before we introduce radians.

Rotations and Reference Angles

25 minutes

At the conclusion of the next section of this lesson students should be comfortable drawing rotation diagrams, finding coterminal angles, and determining reference angles. Using page 10 of the flipchart Copy of Flipchart_Angles and Rotations.pdf, I will model how to create a rotation diagram and determine the quadrant where the terminal ray falls.  Then students can practice this by completing problems 5-8 on page 11 of the flipchart.    

Pages 12-15 of the flipchart have a few checks for understanding built in. These questions will assess students’ knowledge of rotation diagrams and their ability to apply the definition of coterminal angles. You may want to save these for the closure part of the lesson, depending on time.

Using page 16 of the flipchart I will model how to determine the given angle’s reference angle. I will use rotation diagrams and emphasize the definition of a reference angle to help establish one method to find this. After demonstrating how I would find this, I would like to turn it over to students and ask if they can generalize how to find a reference angle. I think this is a great opportunity to address mathematical practice 8: look for and express regularity in repeated reasoning. Student’s ability to conclude that a good generalization can’t really be made is as equally as important as actually being able to generalize. So I would welcome an answer like, “If the angle terminates in the first quadrant that is the reference angle. If it terminates in the second quadrant, we can find the reference angle by subtracting the given angle from 180 degrees. If it terminates in the third quadrant we can find the reference angle by subtracting 180 degrees from the given angle…” But I would also be okay with students stating that it is difficult to generalize. A second bonus to this quick conversation is that is squashes the common misconception amongst my students that we always just subtract from 180 degrees to find the reference angle.

Pages 17-20 of the flipchart have a few checks for understanding built in. These questions will assess students’ knowledge of reference angles.

Closure: Clicker Checks

15 minutes

Pages 12-15 and 17-20 of today’s flipchart are good resource for closure. I will either use these pages during my presentation or at the end of the lesson, depending on pacing and flow.