Lesson 4: Creating the Unit Circle
Standards Addressed
MCC9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Activity Description
In this lesson students will begin to make connections between the special right triangles and the unit circle. They will use the knowledge that they have gained about right triangles and reference angles to explore the unit circle using an online applet. Students will explore the unit circle in increments of 45 and then 30 degrees. Students must select either the 45 or 30 degree increment and then a slider will appear which they can move to shade the represented angles in on the circle. The applet can be found in the subsection of this page labeled "Applet Activity."
Once students have established the connections between the special right triangles and the unit circle they will use an online applet to make connections between the sine and cosine ratios and the coordinates on the unit circle. The QR code can be scanned using a QR reader on a mobile device to access the applet. This applet is a basic exploration applet with minimal directions.
The students will need access to the online applets, the applet activity handout, pencil, and a calculator. This applet allows students to make connections to the unit circle using prior learning from lessons 2 and 3 of this unit. Once a majority of the class has finished their exploration, we recommend that the teacher engages in a discussion with the class about their explorations and the relationships dicovered.
Once students have established the connections between the special right triangles and the unit circle they will use an online applet to make connections between the sine and cosine ratios and the coordinates on the unit circle. The QR code can be scanned using a QR reader on a mobile device to access the applet. This applet is a basic exploration applet with minimal directions.
The students will need access to the online applets, the applet activity handout, pencil, and a calculator. This applet allows students to make connections to the unit circle using prior learning from lessons 2 and 3 of this unit. Once a majority of the class has finished their exploration, we recommend that the teacher engages in a discussion with the class about their explorations and the relationships dicovered.
Activity Rationale
This applet provides an interactive way for students to make connections to the unit circle, special right triangles and reference angles. It allows students to review and solidify previous information while simultaneously developing new concepts. Students are able to explore and find patterns within the unit circle using their current understanding of trigonometric concepts. The applet provides the constructions of the triangles thereby allowing students more time for exploration.
Teacher Guidelines
The primary role of the teacher in this activity is that of a facilitator. The directions for how to use this technology are included on the applet and are easy to follow; therefore, the teacher should only have to tell students how to access the applet and should not have to explain how the applet works. However, the teacher may want to tell students that they need to select an angle increment for the slider to appear. There are also additional questions to guide exploration through the patterns of the square roots. This applet provides angle measures in degrees and radians so the teacher can instruct the students to look at the degree measure but may want to revisit this applet in Lesson 5. There should be no technical issues when students are engaging with the applet. TThe following are possible questions to pose to students that are not included in the activity, but they could provide deeper thinking and understanding:
Also, some students may have trouble remembering the coordinates of the Unit Circle. If they are having trouble doing so the instructor can direct students attention to the following animation that describes a trick to remember the first quadrant of the Unit Circle. The animation is included here:
- If you did not have this applet, how can you find the x and y coordinated using special right triangles?
- How can you use your knowledge of reference angles and the quadrants to find the coordinates for the rest of the unit circle given the coordinates from the square root pattern
Also, some students may have trouble remembering the coordinates of the Unit Circle. If they are having trouble doing so the instructor can direct students attention to the following animation that describes a trick to remember the first quadrant of the Unit Circle. The animation is included here:
Lastly, students responses should be sequenced as they arrive from the applet activity.
Assessment
Students will construct unit circles using the information they learned in these activities. The unit circles should contain the 30, 45, 60 and degree angles and their reference angles as well as their coordinates. They will need a protractor, compass, ruler and grid paper OR they can use paper plates. Color pencils and/or markers may be useful.