Similarity Problem Solving and Proofs

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Objective

Students will be able to apply the definition of similarity to determine whether polygons, triangles in particular, are similar and solve for sides and angles when possible.

Big Idea

Students will deepen their understanding of triangle similarity by solving similarity problems and writing proofs.

Warm-Up: Similar Polygons

10 minutes

In the Similar Polygons Warm-Up, I want my students to have the opportunity to apply their developing ideas about similarity to classify figures as always, sometimes, or never similar.  Like most warm-ups, I ask students to work individually before sharing out in their group to ensure that all students have had an opportunity to think on their own.  As students share out in their groups, I circulate the room, taking note of the diagrams students have drawn to get a sense of their arguments.

When we debrief the warm-up, I call on student volunteers who are willing to share out and defend their answers.  In small group and whole-class discussions, I remind students to use language like, “I agree with _____ because…” or “I disagree with ____ because…” to ensure that we bring in multiple voices to the discussion with a focus on constructing viable arguments and critiquing each other’s reasoning (MP3).  One of my goals in this discussion is for students to be able to generalize their thinking to discover something new; for example, students will be able to explain why any two regular polygons with the same number of sides will be similar, using precise mathematical vocabulary.

Notes: Similar Polygons and Triangle Similarity Shortcuts

10 minutes

As typical for our class, we take notes in our notetakers to formalize our understanding.  In our notes, we differentiate between similar and congruent figures, using the idea of transformations to ground our discussion.  Since we have only discussed dilations informally up until this point, I take time to talk about the notion of a scale factor, emphasizing that dilations scale lengths while preserving angle measures.  In terms of the triangle similarity shortcuts, we use SSS~, SAS~ and AA~.  I make a point of asking the class why we do not have to include HL~ in our shortcuts, which encourages them to look for and make connections between HL~ and SAS~ as well as SSS~.

Classwork: Solve and Prove Similar Polygons

50 minutes

Now that I have presented several key ideas around similarity, I want to give students an opportunity to apply their understanding by solving problems and writing proofs.  In this particular classwork, I like having students work in pairs to try to increase students’ math talk. To keep the conversation growing, I will ask pairs to then compare their thinking as a group of four with their table partners. 

When groups of four have finished comparing their work, they may call me over to check their work against an answer key:

To debrief the classwork, I go over a few key problems, like #4, which requires students to first establish that the triangles are similar by AA~ before solving for side lengths, as well as any of the proofs students would like to discuss.

CFU: Are the triangles similar?

5 minutes

At this point in the unit, I want to make sure to check each individual student’s understanding. In this check for understanding, my students must first determine if the triangles are similar. Then they need to justify how they know before solving for m, one of the side lengths.  When I assess students’ work, I will look for a correct triangle similarity statement, for example Triangle DTA~Triangle RAK by AA~, as well as a correct solution for the value of m, to show that students understand the proportional relationship between corresponding sides.