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Symmetry

What Is Symmetry?

- Fundamental organizing principle in nature and

art - Preserves distances, angles, sizes and shapes

Symmetry of the Alphabet

- Sort the letters of the alphabet into groups

according to their symmetries - Divide letters into two categories
- symmetrical
- not symmetrical

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

Four Types of Symmetry in a Plane

- Rotation
- Translation
- Reflection
- Glide Reflection

Rotation

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

Translation

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

Reflection

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

Glide Reflection

- Involves more than one step
- Combination of a reflection and a translation

along the direction of the mirror line

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?

Symmetry In The Real World

- Plants and animals exhibit many forms of symmetry

M.C. Escher

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

Butterflies

Fish and Boats

Lizards

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

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

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.

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

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)

Results

90o rotation

reflection

FIELD TRIP!

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

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

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

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