##
* *Reflection: Continuous Assessment
Area of Triangles - Section 2: Finding Area of Triangles

As I was looking at student work I started to notice that many of the students were only using the Law of Sines and Law of Cosines. The students were not seeing that the altitude was making a right triangle with the base. Instead of doing the sin A= opp side/hypotenuse they would use Law of Sines with the 90 degree angle.

I would ask the students to look at the triangle. Isn't that a right triangle? When is the Law of Sines defined? (of course students can use the Law of Sines with right triangles and it can be used to prove the right triangle properties but our definition was that the Law of Sines and the Law of Cosines is for oblique triangles.)

Once I brought the type of triangle to the students attention, I found out they had 2 issues with the right triangle ratios. The first was that they did not remember the ratios. The other issue was that some students were still struggling with what is meant by the opposite and adjacent sides. I even had students try to use the ratios by saying cos 90 = one leg/hypotenuse. To clear this up I ask the students what the opposite side was and how could this side be the opposite side and the hypotenuse at the same time. I also reviewed that the ratios are for acute angles and 90 is not acute.

My other thought about only using Law of Sines and Law of Cosines is that these were the newest formulas learned so the students assumed that this would be what I was practicing. I am hoping the students will learn that when we do a problem solving activity any concepts can be used not just the most recent.

*Continuous Assessment: Not seeing right triangles*

# Area of Triangles

Lesson 10 of 13

## Objective: SWBAT find the area of triangles when given sides and angles of possible triangles.

*45 minutes*

#### Bell Work

*10 min*

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.

#### Resources

*expand content*

#### Finding Area of Triangles

*20 min*

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."

*expand content*

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?"

*expand content*

#### Closure

*5 min*

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.

*expand content*

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Review of Angles
- LESSON 2: Solving Right Triangles
- LESSON 3: Law of Sines Introduction
- LESSON 4: Ambiguous Case Day 1 of 2
- LESSON 5: Ambiguous Case Day 2 of 2
- LESSON 6: Finding the Second Solution
- LESSON 7: Problem Solving with Triangles
- LESSON 8: Law of Cosines Day 1 of 2
- LESSON 9: Law of Cosines Day 2 of 2
- LESSON 10: Area of Triangles
- LESSON 11: Review of Solving Triangles Day 1 of 2
- LESSON 12: Review of Solving Triangles Day 2 of 2
- LESSON 13: Solving Triangles Assessment