Special Right Triangles

As their “Do Now” activity, students are asked to get out the previous day’s assignment, Pythagorean Theorem practice problems (which they were supposed to complete for homework) and to compare their answers with those of the other students in their group.  I then ask if there are questions on any of the problems. If so, I ask for volunteers come to the board to answer questions or explain any issues.

For an Extension I introduce the students to the formula that generates whole numbered triples:

{a,b,c} = {x^2 - y^2  , 2xy  ,  x^2 + y^2}

For example, if a student chose x = 6 and y = 1, he or she would generate a 35, 12, 37 right triangle. 

I allow the students time to play with this formula (they may use calculators to aid in the calculations) and present several questions for them to consider as they play:

• Can you find any new Pythagorean Triples to add to our list?  
• Do you think that ALL Pythagorean triples can be found in this way? 
• Can x and y be any values? (Are there any restrictions on x and y?)

For whatever reason, I have found that students feel empowered when they know the common Pythagorean triples.  Throughout the remainder of the course, I often hear students make remarks like, “Of course AB is 12. It’s just a 5,12,13 triangle.”  To provide practice in recognizing the triples, therefore, I hand out the Pythagorean Triples problems worksheet and ask the groups to work together to fill in the missing sides of the diagrams.  I remind them that they have created lists of triples to which they can refer, and tell them that no calculators or work are allowed.  This last stipulation – no work allowed? – always produces puzzled looks from some of the students, but they tend to catch on quickly and gain confidence as they begin to recognize the triples.

In the Triples problems I incorporate a number of quadrilateral problems.  I will do this throughout the remainder of the right triangles unit, as it provides variety (all the problems aren’t just right triangles!) and, more importantly, I think it really sets the stage for the upcoming unit on quadrilaterals.  In doing so, I find that the quadrilateral unit requires very little time, relative to many of the other units, because the students have already become so familiar with a lot of the diagrams and problems.