I've found that not all of my students who apply the Pythagorean Equation successfully, necessarily understand everything they do algebraically. For me, this is a problem. For example, in Step 3 below, a student may correctly identify the answer, c = 5, without understanding the need to perform the same operation on both sides to derive the answer.
1) 32 + 42 = c2
2) 9 + 16 = c2
3) 25 = c2
This learning issue is a necessary consideration when teaching the Pythagorean Theorem. The Launch in this lesson targets this problem.
For the Launch I break my class into groups of three. The groupings today are relatively homogeneous with respect to math expertise. I hand each group a copy of In your group Launch. I expect that all groups will handle Tasks 1 and 2. The strongest groups will likely manage all three. For the groups who are successful, I ask them to try to fix the equations in more than one way.
As my students work I´m looking for those using exponents to form equivalent expressions. My experience is that all groups eventually include, among other ideas, taking the square root of 82 in the first task, (√(82) = 8) , and many students include squaring radical 12 in the second task, (√12)2 = 12)
I walk around assessing the groups and looking for the desired answer for at least the first two questions. To end the launch section I call on students to explain why √(82) = 8 and (√12)2 = 12 are equivalent expressions. Finally, I write on the board:
√(x2) = ?
(√x)2 = ?
I'll ask, "How are these expressions related to the Launch tasks?" I hope that we will eventually arrive at the conclusion that these are equivalent expressions, both equal to x. I think that this is a productive place to move on in today's lesson.
Among the common errors made by my students when using the Pythagorean Theorem are:
The questions in the All is number Acitivty 1.docx-1.docx activity help students overcome these issues. First, squares are drawn on the sides of the triangles to establish a visual connection. Then, the triangles will be presented without the squares.
Students will remain in the groups established for the Launch. I ask students to begin their work by discussing each task until they are all prepared to share their thinking with other groups. In the problems I purposely used different variables, instead of a, b, or c, so students can begin to better understand the generality of the relationship. My students almost always ask questions about the lettering, so we often have a whole class discussion of the diagrams and the naming of variables.
Once students get to work, I make sure students grasp the desired concepts in Activity 1 well before they move on to Activity 2. I am moving quickly from group to group, for a quick assessment, starting with the stronger groups.
Activity 2 is a prelude to solving sets of real world problems. I will soon be giving students challenging problems where students must draw a figure modeling the given information, then solve for a missing measurement.
In Activity 2, illustrations are included with the problems, although the triangles are not all shown. I often mention this in the directions I give the class. I'll say something that sounds a little careless like, "You are going to find the length of the hypotenuse of the right triangle." Then, I will say, "Of course, not all of the situations are missing the measurement of the hypotenuse."
As I walk around, I'll first help all of the students understand that the first step is to identify which situation has an unknown hypotenuse. Then, I will focus on making sure that students apply the Pythagorean Theorem correctly.
There are four problem contexts. The idea is not merely to cover the four in one class. Depending on how quickly the students pick up on the theme of modeling and naming the correct parts, I will be satisfied if students complete at least two of the problems. I want to make sure that we have time to do these on the board for the class to discuss. The problems not covered in class serve as good homework problems.