To introduce this lesson I hand each student a Missing a Leg Launch Entrance Slip. The task asks students to perform an error analysis for each triangle problem, then provide some written feedback. I plan to project the Missing a Leg Launch Slides one at time. I will ask each student to analyze the work on their own. Afterward, they can discuss their thinking with a shoulder partner, if they wish, before completing their slips. I conclude my instructions by asking students to provide feedback and make appropriate corrections, like a teacher would when assessing student work.
Teacher's Note: Error analysis is a potent learning strategy, more so when students are asked to provide feedback and make appropriate corrections. It´s also a good formative assessment tool.
Once the students have their written responses there will be some time for them to share their work with the class.
Some students prefer using c2 – a2 = b2,, for example, when finding the length of one of the legs of the triangle, and a2 + b2 = c2 when finding the hypotenuse. I always let students know that they can use the route they feel more comfortable with.
At this point I make sure the objective of the lesson, finding leg lengths of right triangles, is clear by both stating it and writing it on the board.
Up to this point, I've only used Pythagorean triples in the problems we've covered in this unit. Today I will talk to the students about the fact these these measurements are common, but there are many real world situations that will require use to work with different numbers, some of which are irrational. As we discuss the different types of numbers that we will be working with in coming days, it is a good time to discuss the issue of place value and measurement accuracy.
Next, I hand each student my Snail Slip Pythagorean Radicals (the figure resembles a snail. I ask students to the use Pythagorean theorem to find the missing sides. This work requires close attention to detail as I walk around the room. I expect some students may still be struggling with handling radicals. This is one reason I want students to do this exercise on their own.
I also include an exercise on Square Roots that I give to these students and ask that they complete it in class if time permits, or do it for homework.
For this activity I print the Missing a Leg Application Problems and make four cards for each group using large index cards or strong construction paper. This allows me to more easily reuse the cards. The tasks are best done in small groups, preferably no larger than 3 students. When forming groups I try to take into account mastery levels, preferring that students work in relatively homogeneous groups.
I place a set of 4 cards on each group table. I let each group solve them problems in whatever order they choose. Since there are no Pythagorean triples involved, I ask students to round their answers to the nearest hundredth in each case.
As they begin, I expect student partners to discuss and solve those problems they feel are easiest to them. I expect at least two problems should be done with all work shown in notebooks before the end of the class. More advanced groups will probably get all four problems done.
As they work I will motivate students to strive to fully comprehend the problems they are working on, rather than trying to work quickly. Most of my students will find Problems 3 and 4 challenging.
Here are the answers to the Application Problems.
I close the lesson calling on a volunteer to go up and explain one of the problems completed by all groups. Any problem not completed should be tried at home and discussed in class the next day. I want to see different approaches to the same problem displayed on the board for all students to see as well.