Transformations with Matrices

Mr. J Jay Yohe

Categories Details
Standard(s)
  • 2.8.11.I -- Use matrices to organize and manipulate data, including matrix addition, subtraction, multiplication and scalar multiplication.
  • 2.9.11.H -- Construct a geometric figure and its image using various transformations.
  • 2.9.11.J -- Analyze figures in terms of the kinds of symmetries they have.
Level Grades 10-12 (Algebra II--Geometry Strand)
Materials
  • Worksheet (attached)
  • Graphing Calculators (TI-83)
  • Graph Paper
  • Rulers
  • Reference:
    • "Advanced Algebra" --Holt, Rinehart and Winston, Inc.
    • "Contemporary Mathematics in Context -- A Unified Approach (Course 2 Prt. A)" Everyday Learning Corp.
    • "Matrices and Transformations--A Technological Approach" by Tom Evitts
Assessment
  • Exploration #1: Given a polygon on a coordinate plane, students will be asked to provide the matrix coordinates of the polygon. Students will give several different transformations that will reflect and slide a polygon. Students will also describe matrices that will enlarge and reduce the polygon. Students will find the product of each transformation matrix and matrix [A] to find the coordinates of the transformed figures and develop matrix models for symmetry reflections.
  • Exploration #2: Students will rotate polygons (in a clockwise and counterclockwise direction) by developing matrix models.
  • Exploration #3: Students will form transformational polygons given pictures of the transformed figures in the coordinate plane. Students will write matrix representations for these transformations.
  • Exploration #4: Students will perform written transformations by finding the matrix or matrices necessary to rotate, slide, size or reflect each of the coordinates of the original polygon.
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Objectives: Students will use matrices to represent and transform geometric objects on a coordinate axis.

Intro/Review

Polygons: are made up of straight lines and can be described by the vertex points that make up the figures.
Coordinates: of the vertices are placed in a matrix. The x-coordinates will be placed in the first row of the matrix and the y-coordinates will be placed in the second row of the matrix.

Mathematical Meaning of Symmetry:

  • Two points P and Q are symmetric with respect to a line l if l is the perpendicular bisector of segment PQ
    Seg Line
  • Two points P and Q are symmetric with respect to a point M if M is the midpoint of segment PQ.
    Mid Point

Symmetry/Translation

Y-Axis Sym  X-Axis Sym

x=y Sym  y=-x Sym

Origin Sym

Transformation

Begin:

Create polygon A,B,C,D,E,F,G (illustrated below) on graph paper, using the coordinates listed below (the top row of [A] holds the x-coordinates and the bottom row of [A] holds the y-coordinates). Also place Matrix A in your TI-83 calculator:

Arrow PolygonPolygon

                       Matrix A

Exploration #1 (Activities 1-4)

Activity #1

  1. Perform the matrix multiplication below (Matrix A is the original polygon Matrix):
    Matrix Mult
  2. Graph the resulting polygon by plotting the vertices of [B][A]

    Graph
  3. Compare the original polygon with the new polygon. Are they congruent? Describe how they are alike and how they differ. How do A and A' relate to each other? What does product Matrix B accomplish? The two matrices are a reflection over what axis?
  4. Observe the method used to determine Matrix B below. How can this model be used to find matrix B for other types of symmetry? Explain!

    Y-Axis Matrix Model

  5. Store the product of Matrix B * Matrix A into Matrix C. Then find the product of Matrix B with Matrix C (refer below). Describe the result? What happens and why?
    Matrix Mult

Activity #2

  1. Form the Matrix Product Illustrated below (Matrix A is the original polygon):
    Matrix Mult

  2. Graph the resulting polygon and label the points A'', B'', etc.
  3. Compare the original polygon with the new polygon. Are they congruent? Describe how they are alike and how they differ. How do A and A'' relate to each other? What does product Matrix B accomplish? The two matrices are a reflection over what axis?
  4. Again, observe the model used to calculate Matrix B. Find Matrix B in order to reflect polygon A,B,C,D,E,F,G over the line y=x, over the line y=-x and over the origin!

    X-Axis Matrix Model

  5. If you did everything correctly, you should now have the polygon in three of the four quadrants of the coordinate axis. Can you find a series of matrix multiplications to transform the original polygon to the missing quadrant? Can you find one matrix multiplication to transform the original polygon to the missing quadrant?
  6. Once you find a series of matrix multiplications to transform the polygon to the missing quadrant (or by a single matrix multiplication), draw the polygon in this quadrant. All four reflected polygons should now form a symmetrical image. Describe the type(s) of symmetry of the resulting figure!

Activity #3

  1. Get a new piece of graph paper and graph the original polygon A,B,C,D,E,F,G.
  2. Add the Matrix B (below) to your original Matrix. Graph the resulting Matrix.
    Addition Matrix
  3. How does this graph relate to and differ from the reflection of polygon A,B,C,D,E,F,G in Activity 1.2?
  4. What kind of transformation results from the addition of Matrix B to Matrix A?
  5. Can you generalize about transformations involving the addition of matrices based on this exploration?

Activity #4

  1. Compute the product [B][A] where [A] is the original polygon matrix and [B] is the matrix below:
    Matrix B
  2. Graph the new polygon (use the same graph paper that you used in Activity #3).
  3. Find and test matrices which accomplish the following:
    • enlarge matrix A 125%
    • reduce matrix A 75%
    • reduce the matrix 50% and slide it down 10 units (graph this one also)
  4. In your own words, explain how to enlarge and reduce an object (polygon) represented by a matrix. Write a general matrix model to accomplish this task (refer to the matrix models in activity 1 and 2).

Click here to go to Exploration #2 -- Rotation