Discovering What it Means to be Similar Triangles Continued

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Objective

SWBAT understand angle relationships with similar triangles, discover correct methods of enlarging similar triangles and then apply this to dilating figures.

Big Idea

Imagine teaching similarity, angle-angle similarity for triangles, and dilations all at once! It's a three for one deal!

Opening Bellringer Activity

20 minutes

Beginning the New Material Lesson

10 minutes

The goal of the lesson today is to connect the observations from yesterday to both similar figures and dilations.  I told students to get out the activity pages and their unit organizer.  We spent a few minutes reviewing the observations we made on day one and used pictures taken with my IPad to remind students of the tangram comparisons we made on the previous day.  Linked below are those images:

 

Tangram Comparison 1 (adding length) 

Tangram Comparison 2 (subtracting length)

Tangram Comparison 3 (multiplying to lengthen)

 

We spent several minutes recording important notes onto the unit organizer across from similarity.  Notes included ideas such as:

  • Similar figures have congruent corresponding angles because angles determine shape. 
  • Side lengths are different but proportional (they remembered this from 7th grade) through multiplication with a common scale factor. 
  • Scale factors means to multiply each side by the same number. 
  • A scale factor greater than one enlarges and a scale factor between 0 and 1 reduces. 

 

While adding these notes to the organizer we also reviewed answers to the two homework questions.

Using New Information to Practice

30 minutes

Pass out page three of the activity and discuss with students the goal of the activity today.  The goal today is to understand how to identify when two figures are similar because you understand the application of a scale factor to dilate triangles correctly.  The leap to connect dilations to similar figures was easily made as most students remembered this from 7th grade and the concept that side lengths are proportional. 

 

In the following tables students will be given the side lengths of two different triangles.  Students need to look between corresponding sides to find a common pattern that holds true for all corresponding sides and then record this pattern.  The follow-up question is, would this pattern create a correct dilation and therefore make the two triangles similar.  Look at the example together and then give students time to work in cooperative groups as you move about the room providing feedback.  The last table is the most difficult as the scale factor is 2.5.  I had two groups look at this scale factor in unique ways to figure it out so I asked both groups to present their thinking to the class during our mini wrap up time.  The leap to connect dilations to similar figures was easily made as most students remembered this from 7th grade and the concept that side lengths are proportional. 

 

After about 10 minutes of group time, call the class together and allow students to present answers to each table.  Once you finish, make important notes to the unit organizer about dilations.  Notes include:

  • Dilations produce similar figures
  • Dilations follow a common scale factor (common multiplier to all side lengths)
  • Dilations do not change angle measures
  • Figures will enlarge or reduce proportionally in all directions
  • Scale factor > 1 enlarges and a scale factor that is between 0 and 1 reduces.  

Standards Addressed

1 minutes

This activity extends from similar triangles towards understanding that dilation is the process of creating similar triangles and dilations involve a scale factor.  The activity does not explicitly say dilation; therefore on day two, it is up to you to make this fact direct with students.  Once this fact is established, the core standard addressed today is 8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

 

The math practice standards used throughout this activity arise from students working in cooperative groups to discuss and record observations (MP3 Construct viable arguments and critique the reasoning of others) that arise from hands-on exploration ( MP5 Use appropriate tools strategically) in an effort to make generalizations about all similar triangles ( MP7 Look for and make use of structure.)