Converse of the Pythagorean Theorem and Special Right Triangles Investigation

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Students will be able to apply the Pythagorean Theorem to derive shortcuts for 30-60-90 triangles and 45-45-90 triangles.

Big Idea

By using linguini noodles, students will see how the Pythagorean Theorem and its converse work. Then, in a small group investigation, students will discover the special right triangles shortcuts and justify their reasoning.

Is the Converse of the Pythagorean Theorem True?

15 minutes

Since students have seen the Pythagorean Theorem in middle school, I must activate their prior knowledge in new ways.  One strategy that I like to use is to work as a class to write the Converse of the Pythagorean Theorem.

Once the Converse is written, I assign each group a set of side lengths to test, checking that they satisfy the Pythagorean Formula, a^2+b^2=c^2.  We further test the converse by building models. Each group builds a triangle using linguine noodles, then examines the triangle for its properties (Is it a right triangle?, MP5, MP6). 

Once all of the groups have constructed and analyzed a triangle, we take a quick gallery walk. I ask each group to check the other groups’ work.

  1. Were they accurate?
  2. Are you convinced it is a right triangle?

At this point of the year, I would be surprised if we found non-right triangle. But, the incentive of looking for errors motivates students to think deeply about the Converse. 

Debrief and Notes

10 minutes

Once the Gallery Walk loses momentum (after 3-4 minutes), I plan to debrief the big ideas of the lesson. As usual I will have my students take notes in their notetakers as I question students or answer their questions. 


Special Right Triangles Investigation

20 minutes

With the focus of the lesson shared, I will now use a powerpoint to introduce the Special Right Triangles Investigation to students.  In this investigation, each group of students will get a set of 45-45-90 triangles (later, 30-60-90 triangles after they check in with the teacher) for which they will determine the lengths of all sides, look for patterns, and generalize their findings (MP7). 

In the past, I have found it essential to emphasize the fact that the 45-45-90 and 30-60-90 triangles come from squares and equilateral triangles, respectively.  I reiterate to students that they must justify why the special right triangles rules work for all 30-60-90 triangles and 45-45-90 triangles—an algebraic argument or similar figures argument would suffice (for example, all squares and equilateral triangles are similar and all 30-60-90 triangles and all 45-45-90 triangles are similar by AA~).    

As students work, I circulate the room to check in with groups who are ready to check in or to clarify my expectations around justifying why the rules work.  As I look at groups’ work, I identify 1-2 groups who will share their findings with the class during our debrief discussion.


5 minutes

I debrief the Special Triangles Investigation with my students by first having groups present their findings.  I try to ensure that there will be an algebraic proof that is shared, as well as a similar figures argument.  We then expand on the notes in our notetakers. As time allows, I like to give my students additional practice problems to work. Learning how often one can use the special right triangles ratios is an important outcome of this lesson.

Group Quiz: Special Right Triangles

20 minutes

I want to hold my students accountable to the learning from the lesson, so I will close things out today with an assessment. I plan to give my students a Special Right Triangles Group Quiz to assess their understanding of the special right triangles shortcuts.