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.
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.