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

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.

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.

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.