Radians: Just Another Way

Before: 

Review the degree measures of the special angles on the Unit circle.

Students should be paired and each student will be given a circle/paper plate (Note: If using basic paper plates the ridges should be flattened out as much as possible. This can be done by running the plate across the edge of their desk or table.) Students should fold the paper plate or circle in half and then fold the half in half creating 4 equal sections. Use a pen or marker to draw over the fold lines to create the x- and y-axis. Students should label 0°. Have students tell you the number of degrees in a circle (360°). Have them label 360°. Now have them wrap the strip of paper around the outside edge of the paper plate (example). Have one student hold the paper in place while the other student wraps the strip. Cut the strip of paper so that it is exactly the distance from 0° to 360° or the exact circumference. Since this strip represents one full rotation around the circle we are going to define this length as 360°. (As a time saver have the students cut the second strip of paper into the same length but lay it aside for later.) Now we will label the special angles on the circle in degree measures by dividing the strip into equal sections. First, fold the strip of paper in half. Ask students to determine the length of the folded strip (360/2 = 180). Place the strip at 0 and measure around the outside of the circle. The end of the strip should be halfway around the circle at 180°. Now fold the strip in half again. Ask students to determine the length of this strip (180/2 = 90). When you unfold the strip, you should have 4 equal sections each measuring 90°. Place the strip at 0 and wrap it around the plate marking 90°, 180° and 270°.  Finally, refold the strip into the 90° lengths and fold it in half one more time. Ask students to determine the length of the folded strip (90/2 = 45°). Again unfold and observe 8 equal sections (360/8 = 45°). Wrap the strip starting at 0 and label 45°, 135°, 225°, and 315°. Now lay this strip aside and pick up the second strip. Ask the students to identify the length of this strip. (360° if they cut it the same length of the first strip.) Students will be asked to fold the strip into 3 equal sections (example). Ask students to determine the length of the folded strip (360/3 = 120°). Now have them fold the strip in half and identify the length of the strip (120/2 = 60°). Ask them to fold the strip in half one more time. Unfold the strip and note the 12 equal sections (360/12 = 30°) Finally have the students start at 0° and wrap the strip (counterclockwise) labeling the angles 30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°.   Students should now have a unit circle with degree measures labeled.  

During:

Next, tell students that we will be finding a different angle measurement for these special angles. Share the definition of a radian with students. Have the students add the definition to the Radian Notes Page. Show students the radian demonstration. Set the radius to 3 and either set the length to 3 or move the slider so that the length is 3. Explain to students that the length represents the arc length. Next, move the slider to unwrap the arc length and check the box beside "show radius scale." The students should be able to see that the arc length is the same as the radius. Finally, check the box beside "show angle measure" so students see the measure is equal to one radian. Stress to students that a radian is an arc length that is equal to the radius of the circle. Have students draw a picture to illustrate the definition on their notes page. Use the short strip of paper to measure the radius of the circle and have students determine how many radians are in the circumference (they should find there is just a little more than 6 approximately 6.28 or 2π). Add this result to their Radian Notes Page. You could return to the radian demonstration and drag the slider for length to the end (complete rotation around the circle). When you unwrap this the students will be able to see that it is more than 6 radians. To confirm this, ask students to find the circumference of the circle using the formula for circumference (C=2πr, r=1 so C=2π). Label 0 radians and 2π radians on the circle/paper plate. Pick up the strip of paper that was folded into 8 equal sections. Ask the students to find the measures of the angles associated with these folds using radians (2π/8 = π/4). Add to notes page. Label these on their circle/paper plate. (Be sure that students reduce fractions π/4, 2π/4=π/2, 3π/4, 4π/4=π, 5π/4, 6π/4=3π/2, 7π/4.) Next, pick up the other long strip folded into 12 equal sections. Ask students to find the radian measures of the angles associated with each fold (2π/12=π/6). Add notes to notes page. Label these on their circle/paper plate. (Be sure that students reduce fractions π/6, 2π/6=π/3, 4π/6=2π/3, 5π/6, 7π/6,8π/6=4π/3, 10π/6=5π/3, 11π/6.) Students should now have a complete unit circle with both degree and radian measures. Have students work together to complete the section of the Radian Notes Page to develop conversion rules. They should find that each radian has 180/π degrees and each degree has π/180 radians. Once each pair has found the conversion rules demonstrate to students how to convert a degree to a radian measure and how to convert a radian measure to a degree measure.  

Assign the Practice Problems Worksheet.  Allow students to work in pairs to complete these.

After:

Have students complete a fist list. Trace their hand and for each finger on their hand, they must list one fact about radians. Some possible answers include:  There are 2π radians in a circle, to convert from degrees to radians multiply by π /180, to convert from radians to degrees multiply by 180/π, a radian is an arc length that is equal to the radius of the circle, radians are units of angle measurement, and 180°=π  radians.  In the palm of the hand have them convert 5π/8 radians to degrees. This fist list is their ticket out the door.