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.
One of the major jumps in trigonometry during Algebra 2 is the addition of angles outside 0o to 90o. This is a major conceptual shift that challenges many students. Once they have the concept of reference angle, the next major hurdle is to identify the sign of the trig ratio. I have found the students need a lot of experience dealing with this to master it.
This warm up is a small step in that direction. The goal is that the students remember that cosine (on a unit circle) is the x coordinate as this can REALLY help them intuitively figure out the sign rather than rote memorizing. The key here is that both girls are right given this information. Some of my students like Warm Up 1 figured this out. This student 2 figured it out but didn’t connect that both were true. I would say around 30% of my students gave me entirely incorrect answer, like 3. Talking about this warm up as a class was the key to getting that 30% on board.
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.
I got a lot of good feedback from this activity. The students really liked the physical connection and other teachers at my school who also used this lesson said that it will well with their students.
I do have a couple of pointers to keep this lesson running smoothly. Accuracy in measurement was a big issue. If the students weren’t super careful when measuring the radius (which happened more than I would have liked), their measurements were really off. Also, they wanted to put the pipe cleaner around the outside edge of the circle. This also made the measurements incorrect. It is better to measure a bit on the inside than around the outside. A fellow teacher used string and had good luck with it. I chose pipe cleaner due to its ability to hold its shape but both tools can be used successfully. As we went to each step, I would hold up student work as a model. This really seemed to help keep everyone on track. When we went to measure the 0.28 radians (Student Picture), the student's percentages were really spread. I wrote down each pairs’ percentage on the white board so everyone could see the range and then gave them the actual measurement. This lead to a great discussion on accuracy and human error.
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 Unit Circle* handout is the piece of this and future lessons that puts everything together into a coherent package. Today, my students started by adding on the radian measures. This is really important as it helped them see the division of a half circle in terms of radians. I found it helpful to state that π/2 is the same as ½π. While this should be obvious, so many of my students are still frightened by and struggle with fractions. I continued to use this way of talking about radians both in this lesson and in the other lessons as it seemed to reinforce this idea with my students. As we went into the bottom half of the unit circle, I again talked about the other way to say the radians. For example, 3π/2 is the same as 1 1/2 π. They didn’t write this down but it seemed to help them make sense of the idea of radians.
*Please note that this is a completed unit circle. The highlighted portion was added in the lesson on the Unit Circle.
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.
I ended up having to split this section between today’s lesson and the next one. Rather than give my students the formulas π/180 and 180/π, I worked on getting the students to make sense of and come up with a method for figuring how to convert between degrees and radians. Many of them were to be able to take degrees and convert them to Radians It made sense that we had to find what percent of 180o the given angle was and this was the radians. This is where we got to on the first day.
Going from radians to degrees the next day proved to be a bit more challenging. When we looked at it in terms of 2π/3 is 2/3 of 180o, this seemed to help a bit. I think if I change anything, I will extend this with a few more practice problems and insist that they write down clearer notes. Several students did not write clear enough notes and then had nothing to refer back to towards the end of the chapter.
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
Today's Exit Ticket asks the students to find the radian measure given degrees.