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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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For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

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