Using Matrices to Find the Area of a Triangle

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Objective

SWBAT use the determinant to find the area of triangles and verify that 3 points are collinear.

Big Idea

When the determinant of the matrix used to find area of a triangle is zero the 3 points are collinear.

Bell Work

5 minutes

Students start today explaining in their own words how to use the Determinant of a matrix to verify the matrix has an inverse. This bell questions brings students back to yesterday's lesson

Area of a triangles

25 minutes

After reviewing that a matrix has an inverse only if it is a a square matrix and the determinant is not zero, I explain to the class that matrices are a great tool in mathematics and we are going to see some uses today.

I put up how to find area of triangle on the board. Students read and note any questions they have about the directions. A main question will be how to find the Determinant of a 3X3 matrix. This will be done with the TI-84. I give students instructions for finding the determinant using their calculators. After the instructions I also have a practice problems so that students can test the instructions and get help on using their calculators. I give students a few minutes to practice and then put the correct answer on the board.

To help students understand why we will use the calculator to find the Determinant of a 3X3, I demonstrate the process for finding the Determinant of a 3X3 matrix. Once students see the process they are more than ready to use the calculator. I ask "Why did we find the 2X2 by hand?" Students realize that it is quicker to find the Determinant of a 2X2 by hand since you need to input the matrix which is time consuming.

Students are given some practice problems. Since these problems are mainly contextual I give the students all the problems with room to do the work. This will give students notes to use later.

I work with the class to find complete the first question. We refer to the book rule as we place the points in the matrix. I ask:

  • Why does the book have the plus/minus in front of the 1/2?
  • Can the Determinant of a matrix be negative?
  • Can the area of a figure be negative?

As we discuss this issue students realize that we are taking the absolute value of the Determinant. I ask "What is the notation for absolute value? Could this notation be confused with the notation of determinate?" Student begin to see that when we are using notation that could be confusing books will use alternate notation such as plus/minus. to be clear.

After working the first problem students work in their groups to do the other examples.

Collinear points

10 minutes

After students have worked for about 10 minutes I have students share their answers with the class. If students have not finished the problems I make sure the students have done the last problem.

When students do the last problem the answer comes out 0. I ask:

  • The answer came out zero, what does that mean?
  • Can a triangle have an area of zero?
  • If a triangle cannot exist what does this tell us about the points?

Students start to draw sketches of the points, I plot the points using Desmos graphing application. Students say the points lie on a line. "What is the term we use when 3 or more points are on a line?" A student usually says collinear.

Students are asked to create rule about how to determine if 3 points are collinear using matrices. I have one group share their rule and the other groups alter the rule until the class feels they have a good rule.

Closure

5 minutes

As class ends I want students to extend their thinking about finding area of triangles. I ask students to work in pairs to determine how finding the area of triangles could be used to find the area of other polygons.

Each pair turns in an exit slip with ideas.