## Teacher Resources: Extension on Problem 3 - Section 3: Extension

# Determinants and Area of a Triangle (Day 2 of 2)

Lesson 4 of 12

## Objective: SWBAT calculate the determinant of 2x2 and 3x3 matrices and apply determinants to find the area of triangles.

## Big Idea: Students struggle to find the area of a variety of triangles; they learn a quick way to find the area using determinants.

*40 minutes*

#### Application

*20 min*

After spending some time on determinants of 3x3 matrices in the prior day's lesson, I now introduce students to the idea of using a determinant to calculate the area of a triangle using pages 2-3 of today's flipchart, Determinants and Area of a Triangle (Day 2 of 2). We will explore the triangles from the warmup, Student Worksheet: Area of a Triangle Determinant Style, as we consider this application of determinants. To begin, we identify and label the coordinates of the triangle in Problem 1. Then, I show students how to enter these values into a 3x3 matrix. I may do this with or without graphing calculators, depending on the class. Now, I pose this question to my students, “How do you think the plus and minus operate on the determinate of the matrix?” I want my students to be thinking about this peculiarity when I reveal to them that this is a way to ensure that area must always be positive. Thus, we will learn to always choose the positive result.

After walking students through the area calculation, I give students time to calculate the area of the triangles in Problems 2 and 3 using the determinant method.

**Teaching Notes**:

- To give students more engagement with mathematical practices challenge them to see if they can figure out why finding the half of the determinant will work. It works out most easily for Problem 1, so I suggest to my students that they use that triangle for their exploration.

*expand content*

#### Extension

*5 min*

I like to demonstrate to my students why it is possible to find the area of a triangle using determinants. If I had a student discover this already in class, I may have him or her present why this works.

This video demonstrates how to use determinant to find the area of a triangle for Question 1: Determinants and Area: Extension on Problem 1. I use this process to demonstrate to some of my classes because Problem 1 works out ‘nicely’. In this case, the triangle is visibly one-half of the area of the rectangle connecting all three points.

If the class is going well, I sometimes choose to use Problem 3 to demonstrate the area calculation using area calculation using determinants. This triangle does not have an area of one-half the surrounding rectangle, so the explanation is more complicated.

*expand content*

#### Closure + Homework

*5 min*

To close out today’s learning, I will model for students how the 3x3 determinant is calculating the area of the triangle from question 1.

For homework this evening I will assign Homework 3: Matrices -Treacherous Triangles.

*expand content*

I like the warm up for this lesson. Â I was planning on something similar but give them problems that would encourage the use of Heron's Formula or A=1/2 bcSinA. Â

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Basic Matrix Operations
- LESSON 2: Matrices in the World Around Us
- LESSON 3: Determinants and Area of a Triangle (Day 1 of 2)
- LESSON 4: Determinants and Area of a Triangle (Day 2 of 2)
- LESSON 5: Inverses of Matrices
- LESSON 6: Applications of Inverses - Decoding
- LESSON 7: Solving Systems of Equations using Inverse Matrix Operations
- LESSON 8: Cramers Rule - Solving Systems of Equations
- LESSON 9: Determining the Matrix for Transformations
- LESSON 10: Matrix Transformations
- LESSON 11: Matrices Test Review
- LESSON 12: Matrices Exam