Matrices

Overview

The purpose of this lesson is to have students apply matrix operations using a geometric approach.


Objectives

Students will be able to use matrix multiplication to perform reflections.

Students will discover the matrices used to rotate a triangle around the axis.

Standards Addressed

NCTM Principles and Standards

Students will be able to:

• develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases

• develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices

• understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices

• draw and construct representations of two- and three-dimensional geometric objects using a variety of tools

• select, apply, and translate among mathematical representations to solve problems

• understand how mathematical ideas interconnect and build on one another to produce a coherent whole

• develop and evaluate mathematical arguments and proofs

Georgia Performance Standards

Students will be able to:

• Make and investigate mathematical conjectures

• Organize and consolidate their mathematical thinking through communication

• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

• Select, apply, and translate among mathematical representations to solve problems

  
Background Knowledge

Students should be able to multiply matrices.

Students should be able to solve for unknown matrices using inverse matrices.

Students should be familiar with Geometer Sketchpad (GSP).


Materials Needed

Geometer Sketchpad
Procedures


Time Allotment

One 90 minute class period
                    or
Two 50 minute class periods


Rationale

Matrix algebra is often perceived by students as a very abstract and meaningless topic.   Many aspects of matrix algebra, especially matrix multiplication, are taught emphasizing procedures.  During our student teaching we found few real life problems using matrix multiplication.  Inverse matrix applications are often restricted to solving systems of equations.  We hope this lesson will provide motivation for learning both matrix multiplication and inverse matrix.  Students will be able to geometrically interpret the impact of matrix multiplication rather than viewing the result in row-cloumn form.  We hope to encourage students to think critically about what they are learning in their linear algebra units and to apply their knowledge in a unique way.  

We  chose to use Geometer  Sketchpad, rather than another program such as Graphing Calculator, because we wanted to emphasize the shape and direction of the rotation, rather than an equation.  Geometer Sketchpad allows students to create circles and triangles without equations.  It also allows students to rotate objects, in our case a triangle, about the origin easily.  Students are able to make predictions based on past observations, and then test those predictions using the rotation feature. 

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Activity