In the previous lesson, students were introduced to the process of proving similar triangles using the Angle-Angle Similarity Postulate (e.g., Homework Proof). In this lesson, I ask the students to apply this postulate, as well as their prior knowledge of geometry, to different sets of triangle diagrams, in order to determine if the triangles are similar.
When the students have seated themselves in the clusters of four, I begin a brief discussion with the students, asking them to recall the similarity postulate that was introduced the day before. My questions to the students include:
What does it mean when we say triangles are similar?
I will be listening to make certain that the students understand that the sides of the triangles are in the same ratio, while the angles are congruent. Often students are under the mistaken impression that the angles will be in the same ratio as well.
For triangles, what is the minimum that we need to prove triangle similarity?
Here I will listen for and direct the conversation toward the understanding that if two angles of one triangle are congruent to two angles of another triangle, the third angles must be congruent, and that, therefore, only two angles in each triangle are required.
Thinking back to congruent triangles, what are some of the ways in which we have proven angles congruent thus far in Geometry?
I hope for such answers as vertical angles, the isosceles triangle theorems, and angles formed by parallel lines. I list the students’ answers on a side board and will revisit this list at the end of the class to see if any additions can be made to it.
At this point, I announce that students will be working together in groups for the next half hour or so. I distribute the handout entitled Triangle Diagrams. I call the students’ attention to a set of directions I have written on the smart board:
I make it clear that the entire group should be working together and discussing each diagram, one diagram at a time (MP3). I want to eliminate the idea that a group can split the problems up, assigning two per person, for example (which reduces the opportunity for MP3).
I also let the students know that each packet in each group has been numbered from 1 to 4, and that I will use these numbers to call on students in the Discussion phase of the lesson. My intent is to make sure that each student in each group is knowledgeable of each of the problems and is prepared to discuss any of them with his or her classmates when called upon (MP1).
Once the students begin to work on the problems, I circulate the room, making sure that everyone is on task. I do not intervene in group discussions unless it is clear that all work for that group has come to a halt. If it is the case that the entire group does not know how to begin a problem, I offer leading questions, hoping to steer them in a productive direction. If it is the case that students in a group disagree or have differing approaches or opinions, however, I encourage them to note all approaches on their papers and to offer these in the discussion phase.
With a few minutes left to go in the period, I direct the students’ attention back to the board on which they had previously offered methods for finding congruent angles. I ask if any of the groups encountered each of them in today’s lesson.
I then ask them to look back at their packets to see if there were any concepts they used that were not listed on the board. If so, we add them to the list. By the end of class a typical list will include: