Radian Measure

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Objective

Students will be able to use similarity to define radian measure as the constant of proportionality between arc length and the radius.

Big Idea

Time invested in understanding similarity and in the π project pays off when students see the common sense behind the idea of radian measure.

Opener: Finding the Constant of Proportionality

10 minutes

In order to review the term constant of proportionality, which is part of the learning target we're studying now, I begin today's class with two classically straight-forward direct variation word problems.  My hope is that these problems will look familiar to kids, and that they might say things like, "hey, alright, this is easy!" or "wait, what are we studying today?"

If I notice that the scaffold is necessary, while students are working on this opener, I'll put the values from each problem on the board in two column tables.

As students finish, I either elicit through conversation or just show on from arc length to radians Slide 3 the equations for each of these direct variations.  I explicitly note the value of the constant of proportionality in each equation, then on Slide 4, I point out that phrase in learning target Circles 3.

Mini-Lesson: Developing a Definition for Radian Measure

55 minutes

Students look at the arc length chart they completed during the previous class.  I instruct students to select an arc length column, and to determine the constant of proportionality between arc length and the radius (from arc length to radians Slide 6 and determine k notes).  In groups, students complete this task for all columns of the arc length chart, then they collect their results in the chart of Slide 7.  The big reveal happens on Slide 8, when we simply change the heading titles for this chart, and students see that they have already converted degrees to radians!

Of course, it's not exactly clear to all students what just happened, and I expect students to now wonder "Wait, so what is a radian?" -- this is the title of Slide 8.  In my experience, when students first learn about radians, their greatest conceptual leap is to understand that radians are a unit of angle measurement.  I compare the relationship between degrees and radians to the relationship between inches and centimeters or pounds and kilograms.

Once that leap is made, I note say that we know there "are 360 degrees in a circle," so "how many radians are in a circle?"  From there, you can use any method that makes sense to help students use proportions to make these conversions.  I prefer to set students straight to practice and to begin constructing their own processes, so I set them to task on a Delta Math assignment.

Practice / Homework: Delta Math

  • Click here for an introduction to Delta Math
  • Trigonometry - Intro to Radians
  • Converting Radians to Degrees
  • Convert Degrees to Radians
  • Radians, Radius and Arc Length
  • Radians on the Clock

I set the assignment so students have to get 5 right in a row on each of the above. Students are given time to start on this assignment in class.  As students work and see the solutions to these exercises, I check in with each group to make sure that they're grasping converstion from 360 degrees to 2π radians and therefore from 180 degrees to π radians.

Check-In Quiz

10 minutes

The four questions on Check In Quiz 1 will help me understand what students understand so far. The first three are computational exercises.  I'm most interested in seeing what students give me for #4 - this will show whether or not they have a picture for how big "one radian" is, and whether or not they have the concept of the intercepted arc matching the length of the radius.  I'll use the results of this quiz to target individual students in the next class, where we'll be reviewing Unit 2.