Lesson 12 of 15
Objective: SWBAT use conjectures about intersections to justify a claim. Students will understand how conjectures related to intersections are applied in the construction of a bisector.
The Warm Up prompt for this lesson asks whether, in the construction of a (segment) bisector, 4 arcs must all be constructed with the same radius. The goal of the question is to get students to help each other recall the method of constructing a bisector. I expect them to answer that, yes, all 4 arcs must be constructed with the same radius, since that is the way I taught them to construct a bisector.
After reviewing the teams' answers to the warm-up problem and praising them for recalling the procedure they were taught, I reveal that the question is a setup. Displaying the Agenda and Learning Targets, I tell the class that the goal of today's lesson of is to use conjectures about the intersections of points, lines, and planes to reason about the properties of a geometric figure. To start, we will analyze the construction of a bisector to understand why it works...and see that there is room to deviate from the method as we learned it.
Analyzing a Construction
In this section, I lead the class in an analysis of the construction of a bisector, using the line of reasoning demonstrated in the video (MP3) and summarized here:
Referring to the slide I point out...
- That to be a point on the bisector of A and B, we must locate point C so that it is equidistant from those points. In other words, distance AC must equal distance BC. We can make this happen by constructing a pair of arcs, centered at A and B, using the same compass setting.
- Likewise, to be a point on the bisector of A and B, point D must be located so that it is also equidistant from A and B. That means that distance AD must equal distance BD. It is not necessary, however, that AD=AC or that BD=BC. (We learned to construct it that way simply because it is convenient.) We can test this conclusion by deliberately changing our compass setting before constructing two more arcs to locate point D.
- As long as points C and D are both points on the bisector of A and B, we can be sure that the bisector is line CD. Why? Because one and only one line exists between any two points. Since there is only one bisector of A and B, this line must be it.
This argument seems a little funny to students at first. Speaking from personal experience, I remember finding arguments like this to be strange when I first learned geometric proofs. With repeated exposure, however, the conventions of formal proof become more familiar, and students learn to judge what is considered a viable argument. The purpose of this demonstration and the activity that follows is to give students experience early in the course, so that they can internalize the ways of thinking that mathematicians use to make their arguments (MP3).
Of course, this argument makes a few big assumptions: that a bisector exists, and that it is unique. We will return to those points when we use transformations to prove the properties of a bisector in a later lesson.
To check whether we have reasoned correctly, I ask every student to get out construction tools and try the alternate method of constructing a bisector.
I display the instructions and ask students to get out their Guided Notes on Intersections. In the Activity that follows, students work together to answer questions and determine which conjecture about intersections provides the best support for thier answer.
This activity follows the Rally Coach routine, because I want students to critique each other's work. Does the chosen conjecture really apply in this situation (MP3)?
The Lesson Close asks students to make a connection: Why, when using geometric notation, do we name a line using two points, but name a plane using three points?
This activity follows our Team Size-Up routine.
For homework, I assign problems #38-40 of Homework Set 2. Problems #38-39 ask students to use conjectures about intersections to justify claims, similar to the problems they completed in class. Problem #40 previews the next lesson by asking students to describe the intersection of a relatively simple polyhedron with a plane.