Radians Day 1 of 2
Lesson 4 of 19
Objective: Students will be able to use radians to measure angles.
Warm up and Homework review
I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Ups- Radians, which asks students to identify which quadrants have the same cosine.
I also use this time to correct and record the previous day's Homework.
History of Degrees
This is a historical introduction to our lesson on radians. This introduction is based off of information obtained here. This goal is for students to see why we use a method to measure angles other than degrees. I highly recommend reviewing the information in the article before teaching this lesson. Any time that history or other connections are included in a lesson, the teacher has a lot of power over whether this will go over well or not. I find that the students will connect better to the lesson if I use some flare and enthusiasm. This is a story and goes better when told as such. These slides are just a guide I use to keep the flow, I tell the story and not just read the slides.
The next portion of this lesson is a guided investigation that uses the physical representation of a radian and connects it to π.
My personal preference, when it comes to investigations, is to do a whole class guided investigation rather than having them do it individually or in pairs. I find that I can keep the pacing appropriate and we can stop together to talk about the key points. If you and your students do well with independent investigations, please feel free to put the next slides into a sheet for that purpose. Please watch this short video on Guided Investigations.
I pass out a circle to each student and a ruler, protractor, pipe cleaner and scissors to each pair of students. There should be a variation in the sizes of the circles and the centers should be marked. It wouldn't be a good use of time in an already full lesson to have the students draw their circles and cut them out. The best scaffolding for this lesson is to model the activity yourself at the front of the room.
All the instructions are prepared in the PowerPoint. The students measure the radius of their circle with a length of pipe cleaner. They then lay the pipe cleaner around the circle, marking that length as they go. Next, they measure their pipe cleaner in cm and use that to find the percent of the remaining portion (which should be around 0.28).
Some scaffolding may be necessary for some students when it comes to finding percent. I walk around the classroom and give individual help. If too many are struggling, I may address the class and ask if someone can explain how to find the percent.
I have them identify to total number of radians, including the percentage, as a decimal. I then write their numbers on the whiteboard. They will be somewhere around 6.3ish. There will obviously be some variation in the numbers. This is an excellent time to discuss Math Practice 6. Why didn’t we all get the same answers? What could we have done to make our solutions more accurate?
Before pulling up the 2π slide, I see if any students can figure this out by discussing it in pairs and then as a class.
The final portion of this investigation has them measure the angle that cuts of the arc that is one radian. They will get an angle around 57o. To many this will seem like a crazy number to be a unit, so I remind them that degrees are the arbitrary ones and not radians.
The Unit Circle
I pass out the Radian Measure and the Unit Circle sheet. The students will fill in the radian and degree measures today and put in the coordinates and discuss the trig ratios in the next lesson.
The final goal is to give the students the opportunity to figure out a method for finding radians from angles. The question I asked is "What if I wanted to know how many radians are equal to 140o?". If time is short, this can be done with a class discussion, otherwise, I have them work as pairs and then ask for volunteers to share out. I am careful not to just give them the formula but build it so the it makes sense to them why it works. This is an excellent opportunity for students to write out an explanation (Math Practice 3). For example, they could say that you find what percent the degrees are to 180 and then multiply that by π. Expect that there will be a number of students that will end up just plugging numbers in but they should have the opportunity to understand why it works.
Next the students convert 55o, 110o, and 440o into radians to practice the skill. Notice that these are multiples of each other. Some students may catch on to this and use it to find the radian measure (Math Practice 7). After all three problems have been completed, I see if I can get someone to bring it up by asking a leading question like "Did anybody notice anything interesting about these three problems?". The better students can connect the fractional parts of a circle to radians, the better off they are going to be.
The final portion of this lesson asks students to figure out how to convert radians to degrees. This may prove more challenging and require more scaffolding. I may ask the students guiding questions like "How would you undo what you did in the last type of problem?" (Math Practice 2) Again, I have them write a quick explanation of the process rather than writing a straight formula. There are a few practice problems to finish off this lesson.