Introduction to Similar Figures

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Objective

Students will be able to understand the relationships between the sides and the angles of similar figures.

Big Idea

What are dilations good for? Let's look at similar figures!

Linking Dilations to Similar Figures

10 minutes

I begin by asking the students to plot a quadrilateral on their graph paper and to record the coordinates of the vertices.  I then ask them to think back to our unit on Transformations, and to perform a dilation that I specify on their quadrilateral.  Then, I will ask the class to share information about the results of their dilation of the quadrilateral:

  1. Are their pre-image and image congruent? Why or why not?
  2. What remained invariant in their figures?  Why?  What changed and how?

The change in the lengths of sides is easy for the students to see and comprehend.  The invariance of the angles is not so obvious to many of the students, however, and needs to be emphasized.  Often students expect, for example, that if the sides double, the angles will double, too.

It is at this point, I introduce the term similar to describe their figures (image and preimage).

Working with Similar Polygons: What Conclusions Can We Make?

33 minutes

Students are given Similar Polygons.  I lead them through the notes, giving them time to work in their groups to complete each section of problems.  I then allow the groups to compare answers with their neighbors, we discuss any differences in their answers, and move on to the next section of problems. 

This series of problems presents students with a summary of the concepts. It requires them to complete numerical problems involving proportions. Finally, it asks students to reflect and to make observations based on their solutions. For example:

  • How is scale factor related to perimeters of similar figures?
  • Are all quadrilaterals similar?  If not, are there quadrilaterals that are always similar?  

The section of "Always, Sometimes, Never" questions is always tough for the majority of my students, and will require a good deal of class discussion, examples, and counterexamples.

 

Lesson End + Homework

2 minutes

For Homework, the students are asked to complete the final three problems in the Similar Polygons notes (see Page 4).  I remind them to refer back to the class notes if they struggle with the homework problems. It is my hope that these problems will provide enough practice to keep the concepts fresh in students' minds so they are ready for the next day’s lesson.

To help me assess their understanding of this lesson, I give a Ticket out the Door, asking the students to briefly explain the differences between similar and congruent figures in their own words.