Pythagorean Triples and Special Right Triangles
Lesson 3 of 6
Objective: Students will be able to apply the Pythagorean Theorem to solve novel problems.
Circles and Squares
My students find the special right triangles rules useful, but I have found it beneficial to find and create situations for which students cannot simply apply the rules by rote, without thinking about the problem more deeply. One problem that I like is the Circles and Squares task, which features a deceptively simple diagram of a square inscribed in a circle, which has been inscribed in a square, and asks them about the ratio of the areas of the squares.
I ask my students to work on this task individually at first. While students work silently, I circulate the room to see what kinds of strategies students are trying out. For example, some of my students will draw a diagonal through the center and label the radius, which is an example of students looking for and making use of structure (MP8).
Once I think that everyone has started the task, I will ask students to share their ideas with their group. I want everyone to have something to contribute. I'll say something like, "If your strategy wasn't working, please try that so that others can avoid following the same path. Of course, someone may recognize that your strategy will work, if you apply it a little differently."
After the students have the opportunity to share their initial ideas about how to solve the problem, I will ask them to work together to complete the task. For the group work I will encourage students to attend to precision (MP6), especially when communicating newly formed ideas and when trying to make sense of each other’s arguments.
In our next activity my students must use their experience finding the area of a regular polygons and their knowledge of 30-60-90 triangles to find the area of a regular hexagon, given its perimeter. The figure on Polygon Partners Materials is presented in an unusual way. My students always ask about the five blank spaces and arrows pointing to various parts of the hexagon. Which is really what I want them to do. These graphic organizers will eventually help me to differentiate whom students will work with in the following activity.
This next part is a little chaotic. It is worth the effort because it makes differentiating upcoming work easy. I tell students that they must take their paper and circulate the room looking for five partners:
- diagonal partner
- side partner
- vertex partner
- apothem partner
- center partner
Both partners sign each other's papers on the corresponding blanks. Students must have a different partner for each part of the polygon.
Once things have calmed down, I have students return to their seats. In the next section I explain how the main activity will go.
In the Polygon Partners activity my students will rotate through a series of stations solving a problem at each station. Students have the choice of working individually or with others. Students make this decision when they join a station, after having a chance to look at the problem.
When students join a station, they glue the problem to a page of binder paper, then work out the problem underneath. While all of the problems are Pythagorean Theorem application problems, they can be tricky for my students. This is why I want students to have some choice in how they work.
I give students about 45 minutes to visit all of the stations and attempt all problems. Once this time is up, students will check their work on the problems with their diagonal, side, vertex, apothem, and center partners (from the previous section).
After students have finished working on all of the problems, I ask them to check their work and reflect on each problem with their Polygon Partners. On the back Polygon Partners Materials students will see a reflection tool called Polygon_Partners:_Problem_Solving_Strategies. I ask my students to fill out this graphic organizer while working with their partners. As this part of the lesson proceeds, I circulate the room, taking note of potential pairs I can have present their work to the whole class to debrief the activity.
To debrief this activity, I plan to have one pair of students present each problem to the class. After students check their work and reflect, they staple their problem solving reflection tool to their classwork, which I collect at the end of the lesson.
In this homework assignment, students must use the Pythagorean Theorem to solve the given problem. Before students leave the classroom, I go over my expectations, which are for students to show high-quality work and to provide a high-quality explanation of their solution process.