Definition of Similarity and Similar Triangles

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SWBAT apply the definition of similarity to identify measurements in similar figures. SWBAT to argue that HL is a “similarity shortcut” for right triangles.

Big Idea

Students' prior knowledge of transformations and congruence helps them make connections as they learn to use similarity shortcuts.

Launch Similarity

15 minutes
I launch my unit on Similarity and Trigonometry by having students re-evaluate their work on Pre-Assessment. I use this activity to begin our development of a working definition of "what it means for polygons to be similar."  
In small groups, students will discuss their ideas from the Pre-Assessment using the following protocol:
  1. Starting with the Facilitator, students will share out counter-clockwise
  2. The Recorder/Reporter will document:
    1. The shape that is the exception in each set
    2. What they noticed about the corresponding sides and angles for each figure in the set
When we debrief in a whole-class discussion, I will first call on Recorder/Reporters to share out (1) and (2) for each set of polygons. My goal is to ground the discussion in my students' prior knowledge and previous experiences with transformations and congruence, that is, two 2-D figures are congruent if the second can be obtained by the first by a sequence of translations, reflections, and rotations.  As I call on Recorder/Reporters, I want to highlight the definition of dilation as it begins to come out in the discussion, that is, a dilation is a transformation that produces an image by enlarging or shrinking the original figure (pre-image) by a scale factor.  
Since this is our first lesson on similarity, I want students to understand that two 2-D figures are similar if the second can be obtained by the first by a sequence of translations, reflections, rotations, and dilations. To provide a visual context for this explanation, I plan to use a document camera to demonstrate that a dilation preserves an angle while scaling a length by using a set of triangles forming a Pythagorean Triple (on graph paper).

Practice: Similar Polygons

15 minutes

A number of big ideas were introduced during the Launch section of this lesson. To give my students some time to think about what we've covered so far I will give them some familiar problems to work on. I plan to have students, in pairs, practice applying the definition of similarity to solve for missing angles and side lengths. After about 10 minutes of work time, I will pass out answer keys so that students can make corrections and ask clarifying questions.  



Classwork: Make Sense of Similarity in Triangles

15 minutes
In this Triangle Similarity Shortcuts Classwork activity, my students extend their understanding of triangle congruence shortcuts to consider whether there are triangle similarity shortcuts. My students will also consider whether HL~ can guarantee triangles are similar, comparing HL to SSA, which they know does not guarantee triangles to be congruent.
For this Activity I ask students to work in pairs and to be prepared to report out their conclusions to the other pair in their table group (MP3).  My goal is for the group of four to come to consensus about their ideas around triangle similarity shortcuts. I let the class know that I expect table groups to be able to defend their conclusions with convincing evidence (words, examples, diagrams, constructions, etc.)  

Whole Class Discussion and Debrief

8 minutes
I pass out a Graphic Organizer for students to use to take notes on the ideas that emerge during the whole-class debrief discussion, making it clear to students that I will collect and look at their notes.

Something I Heard that Reinforced My Current Understanding of Similarity

Something I Heard that Challenged My Understanding of Similarity





To get this debrief rolling I will select 1-2 student volunteers to share out their group's conclusions. Each reporter will explain their groups' ideas while projecting their own work using the document camera.  
Next, I will select 1-2 students in the class to reflect on the notes they took during these presentations. Again, using the document camera, I will have them share out what they heard that either reinforced or challenged their understanding of triangle similarity.


Check for Understanding

5 minutes

Because we've done a lot of partner work during this initial similarity lesson, I want to provide my students with an opportunity to share their own thinking.  On the back of the Graphic Organizer students used during the whole-class discussion, I have provided a quick assessment. The assessment checks students' understanding (see page 2) by asking them to determine whether the triangles are similar. I make clear to students that I want them to take their time and use precise academic and geometric vocabulary to explain what they know and how they know it (MP6). In order to get a sense of the connectedness or depth of a student's understanding, I ask the students to show how they know in more than one way. Asking students to provide multiple explanations will give me greater insight into their thinking and understanding. 


Homework: Are the Triangles Similar?

2 minutes

I want students to apply what they learned in this lesson, which is why I give them this homework assignment.  In Are the Triangles Similar?, students determine whether given pairs triangles are similar and justify how they know by showing work and writing similarity statements.