Lesson 10 of 15
Objective: SWBAT describe the intersection of two geometric figures. Students will understand the meaning of an intersection.
I open the lesson by introducing students to a way of thinking about the edges of a polygon that will be new for most: as the intersections of adjacent faces. To guide students to this realization, I give a warmup problem that takes the form of a riddle:
A tetrahedron has four faces, each with three sides, yet the total number of edges is 6 rather than 12. Explain.
I ask students to discuss this question and come up with an explanation as a team. I am looking for answers that show students realize that adjacent faces share an edge, so each edge counts as a side of two faces.
I then say, "We have been studying intersections. What does this problem have to do with intersections?" I follow this up with a few more questions, cold calling on students, to ensure that the idea gets across. For example:
- What is the intersection of face ABC and face ACD?
- An intersection is a part that belongs to both figures. What part of triangle ABC is segment AC?
- What is the intersection of edges AB, AC, and AD?
- What part of these segments do they share?
I stress that an intersection is the part of two geometric figures that they share, not just "where they meet." I ask students to answer precisely, identifying objects with geometric notation wherever it applies (MP6, MP7).
Displaying the Agenda and Learning Targets, I remind the class that we have been learning to describe the intersections of geometric objects. Today we will investigate intersections further and learn to think of intersections precisely. This will pay off, because the concept of an intersection is fundamental to many areas of mathematics.
As I explain in the video, the purpose of this activity is to help students recognize the different ways in which geometric figures can intersect. I distribute physical models: representations of lines, rays, circles, and other geometric objects printed on plastic transparencies. Students can use these manipulatives to help them visualize the parts of the objects which "overlap". The discs are meant to represent planes. I cut a narrow slit in each disc, so that two discs can be made to intersect at right angles by fitting together the two slits.
I display the instructions as students get to work. This activity uses the Rally Coach format, so that students can support each other as well as check each other's answers for accuracy and completeness.
As students work, I circulate. I check in with each team, focusing on how they have answered the Key Question that accompanies each pair of problems. It is not necessary for students to think of every way a pair of objects can intersect, especially at the beginning of the exercise. The sequencing of the problems, together with the "leading" key questions, is intended to help students expand their concept of an intersection as they work. I help by making suggestions and asking questions.
My goal is for every team to complete the first four problems, focusing on the following learnings:
- The intersection of two objects can be more than a point. An intersection can consist of figures which are connected, or figures which are disjoint (such as a pair of distinct points).
- It is possible for the intersection of two objects to be one of the objects. In this case, that object is a part of the other object.
- It is possible for two objects to have an intersection that is "nothing". There are names for special cases: parallel lines and (can students guess it?) parallel planes.
- It is possible for the intersection of two objects to be the objects themselves. In this case, all parts of the two objects "overlap" precisely. When one object can be superimposed on the other, we say that they coincide in all points. Of course, this is the definition of congruence, which will be stressed throughout the course.
Since most of my students have probably learned that congruent means "same shape, same size," I am not expecting to have a discussion on the meaning of congruence during this lesson. I do want to lay some groundwork, however, so I make sure that what students learn today is consistent with the meaning of congruence that they will study later.
For example, the question will probably come up: since two lines, say, can intersect at different points depending on how they are positioned, does that count as more than one way in which they can intersect? For the purposes of this exercise, my answer is that all those points of intersection are the same, so students should identify a point as only one way that two lines can intersect. On the other hand, two rays can intersect to form line segments of different lengths, which would count as separate ways of intersecting. The rule I am applying is whether the two intersections are congruent, not whether they are distinct.
I distribute the Guided Notes on Intersections and lead the class in completing the Big Ideas and the terms on the front side. We will return to the notes at the end of the next lesson. Some of the features of the notes are highlighted in this video. More on how I use Guided Notes can be found in my strategies folder.
For homework, I assign problems #32-34 of Homework Set 2. These problems all give students practice in describing the intersections of geometric figures. Problem #34 is important, because it gives students an opportunity to describe the intersections of parts of a polyhedron. We completed a problem like this following the warmup, but we did it as a class rather than in pairs with peer support.