##
* *Reflection: Developing a Conceptual Understanding
SAS Area Formula - Section 1: Developing the formula

Teaching decision-making is one of my overarching goals in the geometry course. In this lesson, students should continually be in decision-making mode. For example, they should be deciding:

- which base is most efficient to use for a particular triangle (regardless of its orientation),
- which height corresponds to a particular base,
- whether to use Pythagorean Theorem or trigonometry to find the height,
- whether it is even possible to find the height,
- under what circumstances we should apply the formula
- whether trigonometric ratios apply to all triangles

In order to get students to ask themselves these questions, I have to model this decision-making process by thinking out loud and being explicit about all decisions I make in the problem-solving process and my reasons for making them.

*Decision Making*

*Developing a Conceptual Understanding: Decision Making*

# SAS Area Formula

Lesson 4 of 6

## Objective: SWBAT to derive and apply the formula for finding the area of a triangle given two side lengths and the included angle measure.

#### Developing the formula

*35 min*

In this section students will be working on Developing the SAS Triangle Area Formula. I created the problems on the handout to force my students to think strategically (as opposed to blindly applying formulas). All of the problems require students to find the area of a triangle, but each is different in terms of what is given and how students should approach the problem.

Students should continually be in decision-making mode. For example, they should be deciding:

- which base is most efficient to use for a particular triangle (regardless of its orientation),
- which height corresponds to a particular base,
- whether to use Pythagorean Theorem or trigonometry to find the height,
- whether it is even possible to find the height,
- under what circumstances we should apply the formula
- whether trigonometric ratios apply to all triangles

So I begin by asking students to find the answer for #1. This is a straightforward application of the triangle area formula so I simply have my students find the answer quickly on their own and then compare answers with a neighbor.

Problem #2 emphasizes the importance of choosing the base that makes the solution most efficient. After giving students time to attempt the problem, I explain that many students tend to think they have to use *BC* as the base because it is on the bottom. However, since this is a right triangle, it is best to use one of the legs as a base and then the other height (which is perpendicular to the leg/base) must be the corresponding height. This is an important understanding for students so I take time to make sure I'm clear about it.

Problem #3 is the first real problem on the handout. Again, after giving students time to solve the problem on their own, I give a thorough explanation of the thought process involved in solving the problem. I explain the following:

- we cannot assume triangle ABC is a right triangle.
- the only full side length we have is
*AC*, which is 9; we should use this as the base. - if
*AC*is the base, then*BD (*the altitude length) is the height; that's why segment*BD*is drawn. - We need to know
*BD*, which is a leg in right triangle ABD, - Since we have two side lengths in triangle ABD, we can use Pythagorean theorem to find
*BD.* - Having base
*AC*and corresponding height*BD*, we simply apply the triangle area formula.

As I am explaining #3, I am conscious of what students will be asked to do in problem #4, so I try to front load the concepts and strategies they'll need for #4.

In problem #4, students are asked to find the area of a triangle again given a base length and no height. I give them some time to start on the problem. As I am doing this, I walk around trying to see if students have drawn the altitude. If they haven't, I ask "What base have you chosen?..What would the corresponding height be for that base?" If students are stuck on how to find the height, I have them label the diagram with the given information if they haven't, and then I say something like, "In the last problem, we used Pythagorean Theorem to find the height; based on what's given here, how might we find the height?" Since this problem is at the heart of the lesson, I allow time for at least 90% of students to finish (on their own or through collaboration). When they are finished, If I can, I find an exemplary work sample from a student who I feel will give a cogent and thorough explanation to the class. Otherwise, I provide the demonstration and explanation.

Next students work on problem #5, in which they are asked to derive a formula for the general case. I have students work in pairs on #5. I give them enough time to think about the problem and come up with a solution. I walk around as pairs are working. If there is a pair that seems to be genuinely stuck, I will try to identify what has them stuck and get them past it through questioning without giving the answer away. For example, I might say, what is the general formula for the area of a triangle? What does the "b" in the formula mean? What are you using for the "b"? What does the "h" mean? What will you be using for the "h"? Or I might say, how is this problem similar/different with respect to #4? How might you use some of the strategies we used in #4?

Because students have seen (from me or from peer exemplars) what I expect from a high quality presentation, I explain to them that I will be choosing students at random to present their answers to #5. Therefore I advise them to make sure their work is legible and well-organized. And I tell them to be prepared to explain every step of their work and reasoning.

When the students have had enough time to finish, I call on one or two students, at random, to come to the document camera to present their work.

Finally, before working on #6, we read the scenario together as a class. Then I ask students in their A-B pairs, to play the roles of Carl and Carla and basically paraphrase (without looking at the paper) what each one is saying. Then I have them work together (outside of their roles) to decide who they think is correct. I tell them not to write anything until they have agreed with each other first.

When the students have decided, I walk around and identify groups who have sided with Carl and those who have sided with Carla. First I take a poll, by show of hands, to see how many side with Carl and how many with Carla. Then I call on representatives of each side to present their rationale. One student presents the Carl argument. Then another student (without directly rebutting what student one has said) presents the Carla argument. Then I repeat the poll to see how many students have changed their minds based on what the other side has said. If neither has changed their mind, I have the students critique the other side's argument more directly, then we repeat the poll. Finally, I explain why Carla is correct and write a formal statement that summarizes what we've learned:

Whenever we have two side lengths of a triangle and the measure of the ** included** angle, we can find the area of the triangle by multiplying 1/2 times the product of the side lengths times the sine of the included angle measure.

**A = (1/2)(ab)(sin(C))**

#### Resources

*expand content*

#### Practice

*20 min*

In this section, students will be working on SAS Area Practice. The problems assess students' ability to decide when and how to apply the formula we derived in the last section.

Before I have students begin working, I give a quick mini-lesson on the convention used for naming vertices and sides of triangles. I explain that a, b, and C in the formula A=(1/2)(ab)(sinC) are not inherently special letters. These can be rearranged or even replaced by other letters.

That said, If A, B, and C are the vertices of a triangle, then the side lengths opposite those vertices are a, b, and c respectively. I emphasize that vertices are ** by convention** named using upper case letters and the sides opposite the vertices are named using the same letter, but lower case. To make sure students get this, I as them to draw a diagram of triangle PBJ and label all vertices and side lengths according to the convention just described.

Then I have them express the area of the triangle three different ways:

A = (1/2)(pb)(sinJ)

A = (1/2)(jb)(sinP)

A = (1/2)(jp)(sinB)

Once we have this convention squared away, students get to work on the problems. After 15 minutes or so, I collect the work and analyze it to see how well students have understood the lesson.

#### Resources

*expand content*

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras