Fluency with Pythagorean Triples

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Objective

SWBAT geometrically find the missing leg length of a right triangle.

Big Idea

Students strengthen the geometric concept of Pythagoras' theorem and knowledge of Pythagorean triples.

Launch

10 minutes

I start the lesson by projecting Box with straw task image on the SmartBoard. The figure shows a straw placed in a rectangular box 12 inches tall with a 3 in by 4 in base. I then hand back yesterday's activity worksheet, so my students can refer to it. This task adds a new wrinkle, but it can be solved with the Pythagorean Theorem. My students may note from previous lessons that 3-4-5 and 5-12-13 are common ratios among the side lengths in a right triangle. Once I have returned all of the worksheets I ask:

If the straw fits exactly into the box diagonally from the bottom left front corner to the top right back corner, how long is the straw?

I plan to give my students a few minutes to determine the length of the straw. I allow them, as usual, to speak to elbow partners. Some students may have some difficulty because they have not been given 3 dimensional figures in the this unit. If many students are confused, I may hint that it might be possible to solve the problem with two calculations, rather than one. This simple indication has worked many times, a fact that reminds me to give students more tasks like this one (see my Prepare students for transfer reflection). In this task, one reason why students don't see a solution strategy quickly is that they have not been given enough exercises where they have had to add segments to a given figure.

Most of the time, once students see that the diagonal in the base is 5, they quickly figure that the straw is 13 inches long. Once students have started to arrive at the answer, it is very important that at least one student come up and explain his/her route to the answer. Otherwise, the students might classify the problem as a trick, rather than an important application.  

New Info

5 minutes

After students present solutions to the launch task, I plan to direct the class to their notes. I want to specifically highlight the side lengths of the triangles that we have worked with so far and identify the fact that there are similar figures, so, it is possible to reason quickly using ratios.  I will list the following triples on the board, if possible, as students volunteer them to me:

3-4-5

6-8-10

5-12-13

8-15-17

7-24-25

We have used these all so far. I label these as Pythagorean Triples and I ask the students why they might be given this name. I also say, "They turn up a lot, especially on exams so it´s a good idea to become familiar enough with them to use them to solve problems."

Once students stop volunteering Pythagorean Triples, I will write some of my own, quickly. My students know me well enough to begin to look for a pattern. Being playful with the students helps them to gain a sense that this is a game, but also an easy way to calculate side lengths for some right triangles.

For those who like to be thorough, here's a lengthy list of Pythagorean Triples:

http://www.tsm-resources.com/alists/trip.html

Application

30 minutes

The Missing Leg Activity.docx can be done working together in pairs. I don´t allow the use of calculators for this activity, because I want my students to think about the area relationships and/or Pythagorean Triples as they work. Calculations can be done by hand.

Questions 1 and 2 ask students to figure out the the length of an unknown side using the areas above a hypotenuse and a leg. 

Whether or not my students handle Questions 3 and 4 will show me their current level with respect to application, because the basic geometry is not so evident.In other words, the route to the solution is less obvious. 

Closure

10 minutes

To close this lesson I will have students (or pairs) go up to the SmartBoard to project the problem and their solution. The class will listen to the students explain the solutions. As they do, I will highlight the reasoning that students used to arrive at the answer. I allow a couple minutes for any questions on the part of the class. I ask questions if I feel it necessary for complete comprehension of the process towards the solution, or to check if the partner not doing the talking also understands what was done in the group. 

I make sure to collect all work from the class to assess and return with proper feedback if needed, before our next class.