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Matrix/Spreadsheet/Transformation Lesson

Introduction

This is a carefully guided, student centered inquiry/investigation lesson. For details on key components of various instructional models see Strategies for Teachers by Eggen and Kauchak, 1996. Students will use this lesson to investigate the connections between matrix multiplication and geometric transformations. In an ideal situation, the teacher will

a.) Create the spreadsheet worksheet necessary for this lesson

b.) Have the spreadsheet worksheet set up on an LCD or similar projection device

  1. Have the spreadsheet worksheet set up on indivual computers for students (I prefer groups of two for this activity)
The spreadsheet is easy to set up. Directions are included in this lesson. Student will use the spreadsheet to answer specific questions on the worksheets, designed to teach students how matrix multiplication is related to geometric transformations. Students can make conjectures, enter numbers into the spreadsheet and get instantaneous feedback from the computer. Is is a simple yet powerful way for students to gain an understanding of how math is used in real world applications such as computer graphics.
 
 
 
 
 
 

Common Curriculum Goals

Mathematics

Algebraic Relationships

Patterns, Functions and Operations

  1. Create, extend and reproduce patterns, using a variety of materials.
  2. Use Patterns and functions to describe (model) problems.
  3. Recognize and use patterns, functions and algebraic operations to solve problems.
  4. Use a variety of methods and tools to solve problems.
Representations of Mathematical Relationships
  1. Describe patterns and other relationships using tables, graphs and open sentences.
  2. Observe, analyze and explain relationships.
Geometry

Concepts and Properties

  1. Identify, describe, draw, compare and classify physical models of geometric figures.
  2. Construct two- and three- dimensional models using a variety of materials and tools.
  3. Make and test conjectures about geomtric shapes and their properties, incorporating technology where appropriate.
  4. Describe analyze and reason about the properties of two- and three- dimensional figures.
  1. Relationships
  1. Relate geometric ideas to measurement and number sense.
  2. Find and analyze relationships among geometric figures using transformations (e.g., reflections, translations, rotations, dilations).
  3. Prove solutions using geometric relationships, spatial reasoning and coordinate geometry.
Problem Solving

Interpret Reasonableness

  1. Generalize solutions and strategies to new problem situations
  2. Review and verify solutions to prove their accuracy and reasonableness.
Technology

Technological Knowledge

  1. Understand that technology can be used to solve problems and meet needs.
  2. Understand the relationship between technology and other disciplines.
Technological Applications
  1. Use a variety of technological systems.
  2. Adapt technological concepts and processes to biological, informational and physical systems to form technologies and solve practical problems.

    3.  
     
     
CIM / Grade 10 Benchmarks Met

Mathematics

Algebraic Relationships

Patterns, Functions and Operations

  1. Model situations and solve problems using linear and nonlinear functions and inequalities.
  2. Use recursive relationships and/or matrices to represent and solve problems.
Representations of Mathematical Relationships
  1. Represent and analyze discrete structures and continuous functions using tables, graphs, matrices, generalizations and equations.
  2. Solve equations using symbolic, graphic and numeric strategies.
  3. Translate among numeric, symbolic and graphic representations of functions.
Geometry

Concepts and Properties

  1. Interpret, draw and describe two- and three- dimensional objects
  2. Represent and solve problems applying geometric models.
  1. Relationships
  1. Using given assumptions, justify or generalize relationships, between properties of figures
  2. Explore, deduce or prove charecteristics of figures, using transformations, coordinates and/or geometric properties.
Problem Solving

Interpret Reasonableness

  1. Accurately compute and/or apply models to solve problems.
  2. Review the work and support the reasonableness of the results.
Technology

Technological Knowledge

  1. No state benchmarks are established.
Technological Applications
  1. No state benchmarks are established.

    2.  
Instructional Objectives
  • Students will be able to multiply an N*2 matrix by a 2*2 matrix.
  • Students will be able to use a matrix to represent a 2 dimensional figure by labeling vertices.
  • Students will be able to graph a two dimensional figure given an appropriate matrix.
  • Students will be able to use a spreadsheet to graphically represent a two dimensional matrix.
  • Students will be able to identify and name geometric transformations.
  • Students will be able to apply geometric transformations to a given figure.
  • Students will be able to describe algebraic functions using concepts of geometric transformations.
  • Students will name on real world application of matrix transformations.
  • Students will be able to predict an image given a preimage and a matrix transformation.
  • Students will be able to name the dimensions of a matrix.
  • Students will define and classify: preimage, domain, image, range, function, mapping and transformation.
  • Students will be able to give a preimage given an image and a transformation.
  • Students will give the appropriate transformation matrix given an image and a preimage.
  • Students will identify fixed points in the transformation.

    •  
The Spreadsheet Students Will Use

 
 

The Student Inquiry Lesson Worksheets


 
 

Homework/ Worksheets