You will need copies of the Measuring Radians handout. You will also need two pieces of string per team (one between 40cm and 70cm and the other approximately 4 times as long), a piece of chalk and a meter stick for each team. I put the string and chalk in a baggie so that each team has the correct combination of string lengths. It is much more interesting if the shorter strings are NOT all the same length so that each team gets different data.
Team Work (15-20 min): This activity requires each team of students to have an area at least three meters square, so if your room does not accommodate that, I suggest moving to a carpeted hallway or a paved area outside.
For this activity I have students work in teams of 2-3, but you can use as many as 4 per group. If you have more than 2 students per team, I suggest assigning specific tasks to each student to encourage participation and accountability. Possible tasks include recorder, measurer, timekeeper, materials manager, or reporter. Give each team member a copy of the handout entitled “Measuring radians”. I allow approximately 2 minutes for the teams to read through the handout and develop their plan for successfully completing the activity. (MP1, MP4) When the teams have completed reading the handout I ask if there are any questions then have the materials manager collect the tools for his/her team. At this point I either move the class to the area we will be working, or designate areas within my classroom for each team and assign them a team number for completing the class data chart. If we are working somewhere other than my classroom I tell students that they will be recording their data to the board when we return to class.
If we remain in the classroom, while teams are drawing their circles and beginning to make their radii measurements I draw a chart on the board as shown in the resource labeled “Class data”. If we leave the classroom I either post the chart before we leave or have it ready to project when we return.
There are several possible areas of confusion for the students as they work through this activity. Some will struggle with creating the circle either through general carelessness or because they didn’t read the activity and so don’t understand what they are trying to do. Step 4 is a good indicator of how accurate each team has been, because the three measurements should be very close if it is a well-drawn circle. (MP6) Step 4 is also a place where some teams will gripe that they already know the radius because that’s what the string and chalk were for. My response to this is to ask them what value they gain by checking their accuracy. At step 7 some teams may ask what to do with the chalk and can be referred to step 4…I try not to repeat written directions unless there is definite confusion. Because we are a very small school some students have become accustomed to receiving what amounts to personal instruction whenever they are the least bit confused or frustrated. What I mean is that they have learned (and trained some of the teachers who have been here for a while) that if they don’t pay attention to directions or read the instructions, they can get that same information simply by asking the teachers. Some of my colleagues find it easier to give this kind of assistance than to require kids to be active learners, but I find it much less frustrating in the long run to encourage responsibility from the first in my classes. Students may ask how they are to use the radius string to measure the circumference (step 8). Rather than give an explicit answer I ask them to brainstorm ways to accomplish the task and assure them that there is more than one correct way. (MP1, MP2)
When all teams have posted their data on the board I remind managers to return the materials and also remind all students that they need to have the data copied on their own paper. I ask each student to individually examine the class data and write three statements about the data on their lab sheet in the space for “Statements”. (MP2) I allow 3-5 minutes for this, and encourage students to think of previous compare and contrast activities we’ve done. I then ask students to share their statements with their teammates and select two that they feel are the most interesting or important about radians. I have one student from each team share these two statements with the class (I select the teams in random order to reduce arguments about who has to/gets to go first.) If none of the statements bring out the idea that radians don’t change as the circumference and radius change then I take the class there through leading questions. Otherwise, we note that the number of radians is almost the same for all the circles and I ask them if they know the name for the number of radians in a circle. If no one responds, I ask how we calculate the circumference of a circle and the discussion usually gets to “pi” or 2π. I state that pi is number like 4 or -25 or 360, can be used to count how many of something there are and can be used with all the operations we use with other numbers. I continue to give and ask for examples until it appears that most students are comfortable with pi as a number in and of itself. (MP6) (There are always a couple of students who want to use 3.14 for pi and don’t accept that it is a unique number in its own right. I just keep trying to make a connection these students can accept, like √2. Sometimes making the connection to a mole from chemistry helps.)
Individual Work (10-15 min): You will need copies of the Degrees vs Radians handout. For this activity students work individually, which gives me a chance to see which ones are struggling with radians. I begin this activity by asking if anyone can make a connection between what we did at the beginning of class and the activity we just completed about radians. This usually generates discussion about circles and measuring circles in degrees and radians. Give out the handout entitled “Degrees vs Radians” and ask each student to label the point on the circle that is both 0 degrees 0 radians. (I also project this handout on my whiteboard so we can add things to it as a class. Check to see that everyone has the zero points correctly marked before continuing. I take note of which kids didn’t get it so that I can help them as needed through the rest of this activity.) I ask for other possible labels for this point and have students mark them on the board. I tell the students to mark as many points on their circle as they can, including degrees and radians for each point. (MP1, MP2) You can have students add labels to the board, but be careful that some students aren’t just coasting, waiting to add points as they see them on the board. The learning stretch here is to make a connection between the known degree measures and the relatively new radian measures. I have included a partially completed key to give an idea of what my students generally come up with. I encourage students to continue individually until they have at least eight different points marked, then have them work in teams to find at least sixteen points. Some teams will struggle with the fractions for radians. I give them hints about the measures in the first quadrant and let them work from there. to add measures from other quadrants.
You will need copies of the Homework Problems handout for this section. When I feel that all of my students are ready to apply radians in finding arc lengths, I give them this push: I remind them of their confusion at the beginning of class when asked to find the measure of an arc of a circle with a given radius and angle measure. Then I ask them to think of a way to use radians to find the measure of an arc, given the radius and angle measure. (MP1, MP2) I have students pair-share their ideas then try them with the following problem, which I write on the board: (angle measure = π/4, radius = 12m) Some students will convert radians to degrees because that’s what they’re comfortable with, but generally when they see how much easier it is to just multiply π/4 x 12, they become more interested in working with radians. I put one more problem on the board for the class, because it uses a real-life application that my mostly rural students can relate to. (A sprinkler system rotates 3π/4 rads and has a range of 35 meters. What is the arc covered by the sprinkler?) I give my students a minute or two to set up the problem, then go over it with the class. I assign the Homework Problems handout for homework. I usually just have this copied on the back of their Degrees vs Radians paper.