I start today's class with review questions on the board. These problems are designed to help students quickly remember geometric ideas from middle school. By doing this, I enable students to work more accurately with the worksheet. My students usually come in with some misunderstandings, or they are unable to recall necessary information. For example, many forget what the meaning of the term, altitude, and therefore struggle to find the area of a non-right triangle. Another common misconception is that students think the altitude always bisects the opposite side. I think that this is an overgeneralization; it works for enough problems that it might be worth trying. I discuss the scenarios when the altitude bisects the base, to help my students remember that this only occurs if a triangle is isosceles or equilateral. We use an example of a triangle, when the altitude does not bisect the base. I probe by asking, "if an altitude is always inside a triangle? We then consider cases when an altitude might be outside of a triangle.
Now we have had a chance to quickly review some key concepts, I ask, "How can you find the area if you are not given the base and the altitude." My goal is to have students apply the ideas learned so far in this unit to calculate area. The students should use repeated reasoning to see that the length of an altitude can always be found using the same method. Writing this method as an expression, enables the development of a general formula for the area of a triangle:
A =(1/2)(b)(c)sin (A).
To achieve this goal, students are given an activity worksheet to find the area of several triangles. Only one triangle includes the measurement of the altitude and base. I tell the students that this problem is a warm up to the work. This puts the students into the frame of mind that this is not really hard. For students that struggle with reasoning, seeing a problem that is not deep gets them going. The rest of the triangles require students to find the altitude.
The students work in groups to find the area of the triangles. This example of student work shows the process the students used to solve Question 3. Because this activity requires students to use all the ideas we have discussed and learned recently, I work with students who have been absent. I review or reteach the concepts as the students are working to enable the whole class to keep moving forward.
Questions 2-5 enables students to explore the process of finding the altitude. My goal on question 6 is to have the students write a cohesive argument that explains the formula. We have used arguments to prove the Law of Sines and the Law of Cosines, but this is the first time in this unit the students are asked to write their own argument.
After about 10 minutes of work I begin having students share their results. As the students share I focus students on the method the students used to find the altitude. I say, "So you used the sine ratio to find the height or altitude."
After the area calculations for Problems 1-5 are shared, we'll move on to Question 6 as a class. Before discussion, I give students a few minutes to work on the problem. I tell the class to reflect on how how they determined the altitude of the triangle in the first 5 questions. After about 5 minutes, I will have a student share an argument for finding the area of a triangle.
As a class, we'll at the argument and ask questions. Usually, students will state a formula for a different angle, some not understanding that this is a good thing. When this happens, I ask, "Is the process the same in both arguments? Could we have a general formula for area using any angle?"
For an Exit Slip at the end of today's lesson, I ask my students to write an explanation of what was learned today so that a friend who was absent could review their note and catch-up more quickly.