The Warm-Up prompt for the lesson asks students to explain the difference between a line and a bisector. I display the prompt using the slideshow for the lesson. The lesson opener follows our Team Warm-up routine.
The team answers give me a chance to check for understanding. Have my students learned to go beyond descriptions that begin with "looks like" and specify the properties of these objects? For more on this topic, see my reflection that accompanies this section.
Following the warm-up, I display the Agenda and Learning Targets. Today, students will have a chance to show how well they understand the properties of lines, circles, and bisectors. The challenge today is to apply those properties in a creative way in order to find a solution to a problem that many high-tech companies are working on today. (MP4, MP5)
The goal of this activity is for students to apply the properties of bisectors to locate the center of a circle, given two points on its circumference.
I distribute the Activity and display it using a document camera and overhead camera.
I spend about 5 minutes introducing the problem and making sure that students understand the facts of the situation they are given and what they have to find. I may ask students to read the set up and the problem. Through questioning, I make sure that students understand that points A and B are equidistant from the source of the signal (since signal strength is proportional to distance from the source). Similarly, points C and D are equidistant from the source, though not necessarily the same distant as points A and B.
Displaying the instructions, I then ask students to work together to find the location of the signal source on the map, using their knowledge of the properties of lines, circles, and bisectors (MP1, MP4).
As students are working, I am looking for signs of progress that the rest of the class can build upon, giving hints, and answering clarifying questions. After about 5 minutes, I call the class together in order to guide students toward the solution.
Students who understand the properties of bisectors and circles may arrive at the solution by a process of abstract reasoning:
In this case, I ask those students to share their work and explain their thinking.
Alternatively, I use the line of reasoning shown in the video. In this approach, students consider the different circles that could intersect points A and B and notice that the centers all lie on a line: in fact, they lie on the line which is the bisector of points A and B (MP7).
This activity uses the Rally Coach format.
Students are free to choose and two pairs of points on the circumference of the circle. They may wonder if the choice of points matters at all. In fact, the intersection of the resulting bisectors locates the center of the circle more precisely if the points are spaced so that the bisectors intersect at a larger angle: the closer to perpendicular the better. (MP6)
Displaying the Lesson Close prompt, I ask students to explain the difference between two examples of geometric notation: AB and "bar AB". Students often miss the distinction until it is pointed out to them (MP6). This activity follows the Team Size-Up routine, which is basically like the team warm-up we used to begin the lesson.
Homework Set 2 problem #31 provides practice in locating the center of a circle using construction tools. In Problem #32, students must apply their knowledge of geometric vocabulary and notation to follow instructions. The resulting construction is a square inscribed in a circle. Problem #33 gives additional practice in vocabulary and geometric notation.