Symmetry

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Symmetry
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What Is Symmetry?

• Fundamental organizing principle in nature and
  art
• Preserves distances, angles, sizes and shapes

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Symmetry of the Alphabet

• Sort the letters of the alphabet into groups
  according to their symmetries
• Divide letters into two categories
• symmetrical
• not symmetrical

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Symmetry of the Alphabet

• Symmetrical A, B, C, D, E, H, I, K, M, N, O, S,
  T, U, V, W, X, Y, Z
• Not Symmetrical F, G, J, L, P, Q, R

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Four Types of Symmetry in a Plane

• Rotation
• Translation
• Reflection
• Glide Reflection

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Rotation

• To rotate an object means to turn it around
• Every rotation must have a center and an angle

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Translation

• Move it without rotating or reflecting it
• Every translation has a direction and a distance

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Reflection

• Produce an objects mirror image
• A reflection must have a mirror line

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Glide Reflection

• Involves more than one step
• Combination of a reflection and a translation
  along the direction of the mirror line

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Group Activity

• Choose a letter (other than R) with no symmetries
• On a piece of paper perform the following tasks
  on the chosen letter
• rotation
• translation
• reflection
• glide reflection

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Questions

• What happens if you do the same transformation
  twice?
• How many combinations of two transformations are
  there?
• What happens if you combine more than two
  transformations?

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Symmetry In The Real World

• Plants and animals exhibit many forms of symmetry

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M.C. Escher

• Dutch graphic artist
• No formal training in math or science
• Used intricate repeating patterns in his artwork

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Butterflies
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Fish and Boats
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Lizards
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What is a matrix?
A. A movie starring Keanu Reeves B. A complicated
maze C. The new Nike sneaker D. A rectangular
array of numbers
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Multiplying Matrices

• How?
• Each matrix is composed of rows (horizontal
  numbers) and columns (vertical rows).
• The two matrices must have dimensions mxn and
  nxp, where the number of rows in matrix A must be
  equal to the number of columns in matrix B.
• Thus the resulting matrix is of the dimensions mxp

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Practice Problems
1. What are the dimensions of the resulting
matrix when a 2 by 2 matrix is multiplied by a 2
by 3 matrix?
Answer 2 by 3 (2 Rs, 3 Cs)
2. Can a matrix having dimensions 3 by 2 have
resulted from the multiplication of two matrices
having dimensions 2 by 2 and 2 by 3?
Answer NO!!! It does not follow the mxn and
nxp formula.
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Matrix Magic
Matrix A
Matrix B
Matrix C
1. Multiply A by C
2. On graph paper, plot C and AxC
3. Repeat using B and C
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The Geometers Sketchpad

• www.keypress.com
• Plot the points (1,1) (4,2) (2,3)
• (a) Reflect the figure across the line from
  (5,-5) to (-5, 5)
• (b) Rotate the figure 90o about the origin
• Find the matrices that give you the two resulting
  figures in (a) and (b)

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Results
90o rotation
reflection


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FIELD TRIP!
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The Curriculum

• Grade 9
• represent problem situations using matrices
• model, solve, and create problems involving the
  matrix operations of addition, subtraction, and
  scalar multiplication
• use mapping notation to represent translations,
  reflections, rotations, and dilatations of
  geometric figures and interpret such notations

• Grade 7
• Draw, describe, and apply translations,
  reflections, and rotations, and their
  combinations, and identify and use the properties
  associated with these transformations

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The Curriculum (continued)

• Grade 10
• analyse graphs or charts of given situations to
  derive specific information
• derive, analyse, and apply procedures for matrix
  multiplication
• solve problems using various analysis
  capabilities (graphic calculators/ software)

• explore the patterns, make and test conjectures
  about the properties of, and symmetry in, 2- and
  3- dimensional figures
• investigate and use line, point, and plane
  symmetry
• apply transformations when solving problems

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The Curriculum (continued again)

• Grade 12
• derive, analyse, and apply algebraic procedures,
  including those involving algebraic expressions,
  and matrices, in problem solving

• solve problems involving relationships using
  graphing technology as well as appropriate
  paper-and-pencil techniques