Begin the lesson by drawing two circles on the board, one clearly larger than the other. Ask students, "Which of these circles has a greater radius?" Then ask, "Which of these is more round?" You may need to phrase this second question in several different ways so that students understand what you are asking. These questions are meant to plant the seeds about ideas of curvature. After students suggest that the smaller circle has greater curvature, you may want to informally define curvature as follows:
Curvature is the amount by which a geometric object deviates from being flat.
To examine the curvature of the two circles that you drew on the board, you may want to cut a small rectangle from the center of a piece of paper, and then reveal just a portion of the circle through the removed rectangle, as shown below. By considering just a piece of the circle, students are better able to see how much more curved the small circle is than the large circle. (In fact, notice that the exposed arc of the larger circle almost looks like a straight line, whereas the exposed portion of the small circle is nearly a semicircle.)
Once students understand the general idea of curvature, distribute the Circle Packing activity sheet to students.
Allow students to complete the activity sheets in groups of 3–4. As students work, circulate throughout the room and offer assistance as necessary. In particular, students may need help with the following:
- Question 8: Students will need to use the equation and substitute three values. The resulting equation will be a quadratic in one variable.
- Question 10: Students often do not see how a circle can be drawn that is to the right of the two circles shown.
- Question 11: A straight line has zero curvature, because it is straight. You may need to ask some leading questions for students to derive this conclusion on their own.
- Questions 12–13: Students will need to use 0 as the value for one of the variables.
To conclude the lesson, gather the students for a whole-class discussion beginning with the Questions for Students below. Then, allow students to demonstrate what they learned by using one of the Assessment Options, or allow students to explore further by considering one of the Extensions.
