The definitions of congruence and similarity in terms of transformations are an interesting and challenging component of a modern high school geometry curriculum. In this section of the lesson, I develop the definition of similarity and use it to explain why in similar triangles corresponding pairs of angles must be congruent and corresponding side lengths must be proportional.
I project Defining Similarity so that students can see it. To begin, I only have the definition of congruence showing. My students should know this definition well as they have seen it many times. Next I ask the students to predict what the analogous definition of similarity would be. I give students one minute to think about their prediction privately, then I have the students discuss their predictions with their A-B partners. My students usually predict that the similarity definition will include something about dilations. Once I can tell that practically all students have made and discussed their predictions, I show the definition of similarity on the projector screen. I emphasize that the definition says "rigid transformations AND/OR dilations" and point out how this reinforces the fact that congruent figures are also similar.
Next I spend time talking about the implications this has for triangle similarity. This is an important line of reasoning for students to understand. So in addition to the statement that is on the sheet of notes, I break it down further to show underlying logic in the statement. My discussion goes like this:
In similar fashion, I lay out the logical statements related to segment lengths of similar triangles:
After explaining this, I give student time to rehearse the following in their A-B pairs:
In this section of the lesson, we'll be establishing that AA is sufficient to determine that triangles are similar. I start this section by giving students AA Similarity Criterion.
I wrote this resource to be self-explanatory and interactive, but I still guide my students through it. One of my overarching goals for the course if for my students to have the sense that they are doing what mathematicians would do. So in this section of the lesson, we start by creating our own corollary to the definition of similarity. Our corollary: Two figures are similar if and only if one can be made congruent to the other through only dilation. I explain to my students that we have already done lots of work on congruence in terms of transformations and that we will now be able to leverage that work to help with our current task.
The first bit of reasoning for students to understand is that two triangles are similar to each other if one can be made congruent to the other through a dilation. Why? If the triangles can be made congruent, then our work with congruence tells us that one can be mapped onto the other through a sequence of rigid transformations. So if one can be made congruent to the other by a dilation, then it stands to reason that it can be mapped onto the other through dilation and rigid transformation.
The first part of AA Similarity Criterion develops this idea. I choose students randomly to read a chunk of text at a time. I stop at key points (usually marked "Discussion" or "Why?" on the handout) to have students pair-share. I also restate, clarify, and elaborate on what is in the text as I feel the need arise.
Once I feel that my students have adequately understood the reasoning on the first page of the handout, we move on to the second page. The second page asks students to summarize the reasoning involved in the specific case we studied and then to explain generally why AA is a sufficient similarity criterion. I have my students work on this part of the handout independently, switch papers with their A-B partner for review and then make revisions. Students keep the handouts at this point because they may need it as a reference as they complete tasks in the next two sections.
In this section we explain why SAS is a sufficient criterion. The reasoning is similar to the reasoning we used for AA, so I treat this section also as a reteaching and check for understanding for the AA argument. To begin this section, I hand out SAS Similarity Criterion.
The first part of the handout aims to distinguish between SAS congruence and SAS similarity criteria. After reading these two criteria as a class, I ask students to individually think about how the two criteria differ. Then I have the students discuss their responses with their A-B partners. Finally, I have a 2 or 3 students share their responses with the class.
Next, students are asked to create a diagram with two non-congruent triangles that satisfy SAS. to make sure students get this step correct, I have them compare their sketches with at least two other students, and then I select four our five correct examples to show under the document camera.
Finally, I have the students work independently to answer the final prompt. If students appear stuck, I advise them to think about the argument we made for AA and how this one is similar. I also check to see if they know the corollary we established as it is foundational for the argument they are about to make.
When students have finished writing, I have them switch papers with their A-B partners, review, and revise.
Time permitting, I will call on a few students to read their responses while placing them under the document camera.
In this section, students will be working independently to define the SSS similarity criterion and explain why it is sufficient to determine triangle similarity. Successful completion will rely on what students were supposed to have learned in the previous two sections so this is a good check for understanding.
I allow students to look at the handouts from the two previous sections as they complete SSS Congruence Criterion. If I have class time to devote to this portion of the lesson, students will write in class and I will select students to share their arguments. Otherwise, this will be a take-home assignment.