Converse of the Pythagorean Theorem and Special Right Triangles Investigation

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. 

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. 

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.

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.

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.