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Presentation of Unit Lesson Plans on Geometric

Transformations

- EDUC 366
- June 27, 2007

Unit on Geometric Transformations

- Lesson 1 Reflections
- Lesson 2 Translations
- Lesson 3 Rotations

Reflections

Who is this man? Who is this woman? What are

the words in the red box? How are they all

connected? Click here to find out!

S.W.B.A.T.

- Identify properties of reflection transformations
- Identify and locate reflection images of figures
- Use a coordinate system to formulate general

mappings for reflections. - Construct reflections on the Cartesian coordinate

system

Reflections are the first examples of

transformations that we will be studying in this

unit. A transformation is a change in position,

shape, or size of a figure.

Did you take a look in the mirror this morning?

You were actually the pre-image, or the original

figure. The view you caught in the mirror was

your image, or the figure after a transformation.

Leonardo da Vincis writings were the pre-image.

In order to read them, you must hold them up to a

mirror, then read the image.

????????????

What are some types of reflections? Well, Im

glad you asked.

There are reflections along a vertical line,

Reflections along a horizontal line,

And reflections along a diagonal line!

Have some practice with the characteristics of

reflections! Click here to experiment with an

activity.

Now that youve had a chance to work with some

reflections, lets discuss what you saw.

A transformation maps the pre-image onto its

image. Take a look at the example below.

The above statement would be read Triangle FUN

maps to Triangle F prime, U prime, N prime.

Notice that prime notation is used to identify

points on the image.

Is congruent to ? For a

review of congruency, click here.

If the original figure and its image are

congruent, then the transformation is an

isometry. Are reflections isometries?

Have you ever noticed how ambulances have reverse

printing on their fronts? Why do you think that

is?

A reflection reverses the orientation of a

figure. The image will always appear reversed

from the pre-image.

Remember..

Reflections are flips!!!

Based on what weve learned so far, can anyone

predict what the two main properties of

reflections are?

Have you given it a try? Check your predictions

here.

Just to make sure its crystal clear Is the

figure on the right side of the mirror line a

reflection of the figure on the left side of the

mirror line? Yes or no

Is the figure on the right side of the mirror

line a reflection of the figure on the left side

of the mirror line? Yes or no

Activity

Now that youve learned more about

transformations and reflections, show what you

know! Complete this activity on reflections of

polygons, including the six questions on the

activity page.

Lets relate our knowledge of reflections to what

we know about the coordinate system.

Suppose we have coordinate points A(3, 1), B(4,

2), and C(3, 5). If we reflected these points

over the x-axis, what would be the coordinates of

their images, A, B, and C? Can you find a

pattern and make a generalization for the mapping

of any general point (x, y) to its image?

If you need help, click here for the solution.

Lets try another one. Use the same points as

above with A(3, 1), B(4, 2), and C(3, 5). If we

reflected these points over the y-axis, what

would be the coordinates of their images, A, B,

and C? Can you find a pattern and make a

generalization for the mapping of any general

point (x, y) to its image?

Give it your best shot, then check here for the

solution.

Activity

Test your skills on mirror lines and coordinates

of reflections in the following online worksheet.

Click on the shaded boxes at the top of the page

to advance to the next problem. The problem

youre working on currently will be displayed by

an x in the shaded box. Good luck!

Lets wrap it up

Who can give me an example of a reflection with a

pre-image and image? Why are you certain it is

a reflection? What properties does it

have? Given a figure, how can you map a

reflection on the Cartesian coordinate system?

Something to think about.

According to the Wall Street Journal, most drugs

are made up of two versions of the same molecule.

One version is called an R-isomer, the other

version is called an S-isomer, and they are

mirror images of each other with different

healing properties. In an effort to produce

drugs with fewer side effects, researchers have

learned to produce pure batches of these isomers.

They run tests to see which version has the

least amount of side effects, then produce the

drug from this batch. For example, the R-isomer

of Albuterol treats asthma, but its mirror

version (the S-isomer) has been shown to increase

the risk of future attacks. Pretty neat, huh?

Geometry is EVERYWHERE!

Lesson 2 Translations Back to Main Page

Translations

Click on the image below to watch the video of OK

Gos Here It Goes Again on YouTube.

How would you describe the motion of the band

members that makes this video unique?

S.W.B.A.T.

- Locate translation images of figures
- Represent translations using vectors
- Represent translations using matrix addition
- Model translation images on the Cartesian

coordinate system - Formulate a translation based on coordinates of

pre-images and images

Activity

Lets start with an exercise. Click here to

complete ten questions that will help us

determine the characteristics of our next type of

transformation. Please do the five green

questions and the five purple questions by

clicking on the colored squares at the top of the

page.

Take a look at triangle FUN and triangle FUN

above. Does the transformation

seem to be an isometry? Think

about it, then click here for the answer.

Does the transformation change the orientation?

Give it some thought and click here.

The picture above is an example of a

transformation called a translation. A

translation is a transformation that moves points

the same distance and in the same direction.

Every point on the blue triangle was moved ten

units to the right to produce the image of the

red triangle. To put it another way.

Translations are SLIDES!

Activity

Try some translations out for yourself! Take a

few minutes to complete these three activities

(use the arrows above the questions to move from

one activity to the next), then well move on to

describing the distance and direction of

translations.

The orange lines in the diagram above represent

vectors. A vector expresses the distance and

direction of a translation. The vectors in the

triangle diagram are , , and .

Be careful -vectors are not the same as rays!

Although the notation may look identical,

vectors have a fixed length. They do not go on

forever like rays.

Like rays, vectors have initial points. In this

case the initial points are F, U, and N. But

unlike rays, vectors also have terminal points.

In this case the terminal points are F, U, and

N.

Vectors can be expressed in ordered pair

notation ltx,ygt. The x-value is the horizontal

change between the initial and terminal points,

and the y-value is the vertical change between

the initial and terminal points. In the example

above, the blue triangle is translated ten units

to the right and five units down.

The value for vector is lt10,-5gt. What is

the value for vector ? What is the value

for vector ? Think it over and explain your

answer, then check it here.

Youre probably asking How do I represent a

translation? Thats an excellent question!

Mathematically, this translation can be

expressed in three different ways

The first way

(x,y) (x10, y-5)

On coordinate axes, x will become x10, and y

will become y-5.

The second way

T

The 10 means add 10 to all x-coordinates, and

the -5 means subtract 5 from all y-coordinates.

And the third way

Sometimes a translation is just represented as a

vector of a certain length and direction.

This vector moves all x-coordinates 10 units to

the right and all y-coordinates 5 units down and

would have the notation lt10, -5gt.

The coordinates of FUN are F(-3, 7), U(-5.5,

4.5), and N(-3, 1). If, under the translation, x

becomes x10 (we add 10 to all x-coordinates),

and y becomes y-5 (we subtract 5 from all

y-coordinates), then what are the coordinates of

F, U, and N? Click to check your answer step

by step.

Activity

Lets practice vectors and translations using the

information we just learned. Complete the

problems on the interactive worksheet to test

your knowledge.

Addition and subtraction on individual

coordinates is only one way to translate figures

in the coordinate plane. We can also use

addition and subtraction through matrices.

Consider the following example

We have rectangle SWIM. On our coordinate

system, where would the image of SWIM be under

the translation lt-11, 4gt?

First step Set up a matrix with the

x-coordinates of the vertices in the first row

and the y-coordinates of the vertices in the

second row.

Vertices of Pre-Image in Matrix Form

S W I M

x-coordinate

y-coordinate

Next step Set up a translation matrix for the

translation lt-11, 4gt.

Translation Matrix

Last step Add these two matrices together using

matrix addition. For an algebra review of matrix

addition, click here.

The matrix sum is the image, where the first row

contains the x-coordinates of the image vertices

and the second row contains the y-coordinates.

Vertices of Image in Matrix Form

S W I M

Exercise

Now you try one! Figure YOU has vertices (-7,

-3), (-9, -7), and (-5, -7). Find the image of

this figure under the translation lt14, 2gt by

using matrices. After you have sketched out the

figure and its image, check your answer here.

Show what you know and exercise your brain! Try

this interactive activity on vectors and

translations.

Lets Wrap It Up

What are the different ways can we locate an

image under a translation if we are given the

pre-image? What is a vector and how is it used

to represent a translation? Using matrices, what

are the steps we take to find an image if we are

given a translation? If you are given the

coordinates of a point, how do you find the

coordinates of its image under a translation?

Worksheet

Please print out this worksheet, complete the

questions, and turn it in to me when completed.

If accommodations are necessary, please choose

the appropriate version. Worksheet Worksheet

Visual Impairment Worksheet Autism

Did you know..

Translations are a critical part of animation.

Your favorite animated movies wouldnt look the

same without it. But animation isnt only used

for entertainment. Its used in control systems

and flight simulators for pilot training, and

also in scientific research. It can help

surgeons practice without putting a real life in

danger. Automobile companies use it for making

3D models of cars. Its used by architects to

make model houses so their clients can take

virtual tours. And its critical for your video

games. Pretty neat, huh? Geometry is

EVERYWHERE!

Moving on!

Lesson 3 Rotations Lesson 1

Reflections Back to Main Page

Rotations

Can anyone tell me the name of the game pictured

above? What is the object of the game? How is

it played? What happens if you let the pieces

fall without changing them? What are some

strategies? For anyone who is not familiar with

the game, lets take a look at an example.

S. W. B. A. T.

- Identify properties of rotation transformations
- Identify and locate rotation images of figures
- Construct a rotation of a figure and formulate a

general mapping on the Cartesian coordinate

system - Determine whether rotations have symmetry

Rotations are the final examples of

transformations that we will be studying in this

unit. To rotate an object means to turn it

around or to spin it. Every rotation has a

rotocenter, or center of rotation, and an angle.

In the example below, the pre-image (red R) was

rotated around a rotocenter (which also happens

to be the center of the circle) for 90 degrees in

order to produce the image (blue R).

In each of these pair examples, the image on the

right is a rotation of the pre-image on the left.

Activity

Have some practice with the characteristics of

rotations! Click here to experiment with an

activity. Please complete all three

sub-activities listed in the right-hand frame of

the activity page and make sure your answer all

of the questions!

The direction of a rotation can be either

clockwise

or counterclockwise.

Think about your locker combination. You need to

turn it clockwise to a certain number, then

counterclockwise to another number, then

clockwise again to the final number before the

lock will open. You use rotations every day!

Positive angles are typically measured

counterclockwise. If a rotation is clockwise, it

is usually a negative angle.

To work with rotations, you need to be able to

recognize angles of certain sizes and understand

the basic workings of a unit circle.

Remember that a unit circle moves in a

counterclockwise direction!

Activity

Try this activity. Click on the rotation

button at the top of the page when it loads.

Then click on the buttons to experiment with

triangles under ¼ turn rotations clockwise and

counterclockwise, and also ½ turn rotations.

Notice the x and y coordinates of the images

under each transformation.

The rotocenter may be located in three different

areas in a rotation. The center of rotation

could be inside the pre-image, as in the

following picture. Arrow A was rotated 120

degrees about Point C to produce Arrow A

The center of rotation could also be outside of

the pre-image, as shown in the picture below.

Arrow A was rotated 120 degrees about Point C to

produce Arrow A.

As another option, the center of rotation could

be on the pre-image, as shown in this picture.

Arrow A was rotated 120 degrees about Point C to

produce Arrow A.

Under a rotation, does each point move the same

distance? Why or why not?

Is Arrow A congruent to Arrow A? For a review

of congruency, click here.

If the original figure and its image are

congruent, then the transformation is an

isometry. Are rotations isometries?

Does a rotation change orientation?

So, what are the two properties of a rotation?

So, what are the two properties of a rotation?

1) A rotation is an isometry. 2) A rotation does

not change orientation.

Lets work through some problems together.

This is regular pentagon PENTA and it has been

divided into five congruent triangles.

What is the image of E under a 72 degree rotation

about X?

Have you thought about it? Click for an

explanation.

What is the rotation that maps E to N? Check your

answer here.

What is the image of any point under a 360 degree

rotation? Are you ready for the answer?

What do you notice about regular pentagon PENTA

each time it is rotated 72 degrees? Does it look

different? What do YOU think? Heres what Ive

noticed. See if its different from your ideas.

Activity

Lets practice what youve learned about rotation

symmetry. Please link to this activity.

Complete question B1, parts A and B, then check

your answers.

Lets relate our new knowledge about rotations to

what we know about the coordinate

system. Suppose we have points A(3, 1), B(4, 2),

C(3, 5). If we rotated these points 90 degrees

counterclockwise, what would be the coordinates

of their images, A, B, and C? Can you find a

pattern and make a generalization for the mapping

of any general point (x, y) to its image?

If you need help, click here for the solution.

Lets try another one. Use the same points as

above with A(3, 1), B(4, 2), and C(3, 5). If we

rotated these points 180 degrees, what would be

the coordinates of their images, A, B, and C?

Can you find a pattern and make a generalization

for the mapping of any general point (x, y) to

its image?

Give it your best shot, then check here for the

solution.

Activity

Now try this matching game. You will be

presented with 16 squares. Eight squares will

have a description, such as Rotate 180 degrees.

The other eight squares will have a mapping,

such as (x, y) (-x, -y). You need to click

on the two squares that match each description to

its correct mapping.

Activity

Now that you are a master of rotations, test your

skills in the following online worksheet. Click

on the shaded boxes at the top of the page to

advance to the next problem. The problem youre

working on currently will be displayed by an x

in the shaded box. Please complete the five

brown questions and the five red questions. Good

luck!

Lets wrap it up

What are the properties of a rotation? What is

an example of a rotation where the center is

inside the pre-image? What is an example of a

rotation where the center is outside of the

pre-image? What is rotation symmetry?

Extra Activity

Try a fun game called Pentominoes. Its said to

be part of the inspiration for the game Tetris.

Choose a 10 x 6 grid, and rotate and move the

pieces to fill the entire grid. Good luck!

Something to think about.

Why do clocks run clockwise?

Before clocks were invented, sundials were used

to tell time. In the northern hemisphere, the

sun's shadow rotated in the direction we now call

clockwise. The clock hands were built to mimic

the natural movement of the sun. However, if

clocks had been invented in the southern

hemisphere, clockwise would have been in the

opposite direction. Pretty neat, huh? Geometry

is EVERYWHERE!

Lesson 1 Reflections Lesson 2

Translations Back to Main Page

This man is Leonardo da Vinci.

This woman is Mona Lisa.

This is the sentence I love geometry. written

from right to left. How are they all connected?

Leonardo da Vinci created both of them! In his

notebooks, da Vinci wrote using "mirror writing,"

writing that went backwards from right to left,

instead of from left to right. In order to read

his writing normally, you must place a mirror

beside the writing and read the reversed image in

the mirror. No one knows why he wrote this way,

but some possibilities have been suggested

- He was trying to make it harder for people to

read his notes and steal his ideas. - He was hiding his scientific ideas from the

powerful Roman Catholic Church, whose teachings

sometimes disagreed with what Leonardo

observed. - Leonardo wrote with his left hand. Writing left

handed from left to right was messy because the

ink just put down would smear as his hand moved

across it. Leonardo chose to write in reverse

because it prevented smudging. - Try it out for yourself!

Back to lesson

ABSOLUTELY!

The pre-images and images are exactly the same

size and shape, only reversed.

Back to lesson

- The two main properties of reflections are
- A reflection is an isometry (The figure and

its image are congruent). - A reflection reverses orientation (The image

appears backwards). - Did you get it? Nice job!

Back to lesson

Nice try, but no. The image on the right does

not have a reversed orientation. Try again!

back to lesson

Nice job! The figure on the right is not a

reflection because the orientation of the

pre-image is not reversed.

Back to lesson

Nice try, but no. The transformation is not an

isometry. The figure on the right is not

congruent to the figure on the left. Its

dimensions are larger. Try again!

Back to lesson

Very nice! The transformation is not an

isometry. The figure on the right of the mirror

line has reverse orientation, but it is not

congruent to the figure on the left. Its

dimensions are much larger. Way to go!

Back to lesson

The first thing we need to do is graph the points

we are given.

We need to reflect the points over the x-axis

(remember, its the horizontal one). This means

Point A would keep the same x-coordinate. Its

y-coordinate would be the same distance away from

the x-axis, but below the axis.

So the coordinates for Point A would be (3, -1).

If we follow the same logic, the coordinates for

Point B would be (4, -2) and the coordinates for

Point C would be (3, -5).

Now lets take a look at the pre-image and image

coordinates A (3, 1) A (3, -1) B (4, 2)

B (4, -2) C (3, 5) C (3, -5) What

generalization can we make about coordinates that

are reflected over the x-axis?

Well, it looks like the x-coordinate stays the

same, but the y-coordinate becomes negative.

So, the mapping becomes.. (x, y) (x,

-y)

Back to lesson

The first thing we need to do is graph the points

we are given.

We need to reflect the points over the y-axis

(remember, its the vertical one). This means

Point A would keep the same y-coordinate. Its

x-coordinate would be the same distance away from

the y-axis, but left of the axis.

So the coordinates for Point A would be (-3,

1). If we follow the same logic, the coordinates

for Point B would be (-4, 2) and the coordinates

for Point C would be (-3, 5).

Now lets take a look at the pre-image and image

coordinates A (3, 1) A (-3, 1) B (4, 2)

B (-4, 2) C (3, 5) C (-3, 5) What

generalization can we make about coordinates that

are reflected over the y-axis?

Well, it looks like the y-coordinate stays the

same, but the x-coordinate becomes negative. So,

the mapping becomes.. (x, y) (-x, y)

Back to lesson

Sure it is! Remember from Lesson 1, an isometry

is a transformation in which the original figure

and its image are congruent. These two triangles

have sides of the same length and the same

angles.

Back to lesson

No, the transformation does not change the

orientation. The order of the vertices in the

pre-image is counterclockwise (FUN), and so is

the order of the vertices in the image (FUN).

Back to lesson

Since translation is an isometry, the original

figure and its image are congruent. Therefore,

each point of the image will be an equal distance

from the original figure. So the value of

vectors will all be lt10, -5gt.

Back to lesson

First well calculate the coordinates of F.

The x-coordinate of F is -3. We want to add 10

to all x-coordinates, so the x-coordinate of F

is (-3 10), or 7. The y-coordinate of F is 7.

We want to subtract 5 from all y-coordinates, so

the y-coordinate of F is (7 5), or 2. Thus

the coordinates of F are (7,2).

Now well calculate the coordinates of U.

The x-coordinate of U is -5.5. We want to add 10

to all x-coordinates, so the x-coordinate of U

is (-5.5 10), or 4.5. The y-coordinate of U

is 4.5. We want to subtract 5 from all

y-coordinates, so the y-coordinate of U is (4.5

5), or -0.5. Thus the coordinates of U are

(4.5,-0.5).

Finally well calculate the coordinates of N.

The x-coordinate of N is -3. We want to add 10

to all x-coordinates, so the x-coordinate of N

is (-3 10), or 7. The y-coordinate of N is 1.

We want to subtract 5 from all y-coordinates, so

the y-coordinate of N is (1 5), or -4. Thus

the coordinates of N are (7,-4).

Back to lesson

Lets take a look at the matrices we needed for

our transformation.

Back to lesson

The distance each point moves depends on how

close it is to the center of rotation.

The front point of Arrow A traveled a farther

distance than a point on the arrow very close to

Point C. So the points farthest from the center

of rotation move the farthest.

Return to lesson

ABSOLUTELY!

The pre-images and images are exactly the same

size and shape, only in different positions.

Rotating them doesnt change size or orientation.

Back to lesson

No, rotation does not change orientation.

Rotation just changes the angle the figure is

viewed. It does not change the way the figure

would be described.

Back to lesson

Well, lets think about it step by step. Without

a protractor, how would we know what a 72 degree

angle would be?

We want to rotate vertex E through one segment of

the pentagon. Doing this would move vertex E to

vertex P. Therefore, P is the image of E under a

72 degree rotation about X.

Back to lesson

Lets look at the info we know

Were rotating through 4 of the angles around X.

Remember, were going counterclockwise!!! If

each angle is 72 degrees, then the angle of

rotation from E to N would be 72 X 4 288

degrees.

Back to lesson

Its the same point! Choose any point to start,

say Point P.

We know there are 72 degrees in each central

angle about X. We want to rotate 360 degrees. How

many 72 degree rotations would that be? 360 72

5

So, we need to make five 72 degree

rotations. First rotation - Point A Second

rotation - Point T Third rotation - Point

N Fourth rotation - Point E The fifth and final

rotation puts the image on Point P.

Back to lesson

Here we have regular pentagon PENTA.

Suppose we rotate PENTA 72 degrees clockwise

about Point X.

Heres the image

It looks the same! What if we rotate it another

72 degrees?

It still looks the same! This is called rotation

symmetry.

Heres the test for rotation symmetry

If you can rotate a figure less than 360 degrees

around a center point, and its image is identical

to its pre-image, then the figure has rotation

symmetry. The center point is called the center

of rotation, and angle used to turn the figure is

the angle of rotation.

In our PENTA example, the center of rotation was

Point X (also the center of the pentagon), and

the angle of rotation was 72 degrees.

Rotational symmetry plays a big part in the study

of crystals and also in design arts. Back to

lesson

The first thing we need to do is graph the points

we are given.

We need to rotate the points counterclockwise 90

degrees around the origin. This would move all

of the points from quadrant 1 into quadrant 2.

Imagine that we took the entire coordinate system

and turned it ¼ turn.

So the coordinates for Point A would be (-1, 3).

If we follow the same logic, the coordinates for

Point B would be (-2, 4) and the coordinates for

Point C would be (-5, 3).

Now lets take a look at the pre-image and image

coordinates A (3, 1) A (-1, 3) B (4, 2)

B (-2, 4) C (3, 5) C (-5, 3) What

generalization can we make about coordinates that

are rotated 90 degrees around the origin?

Well, it looks like the y-coordinate of the

pre-image takes the opposite sign, then the x-

and y-coordinates switch places. So, the

mapping becomes.. (x, y) (-y, x)

Back to lesson

The first thing we need to do is graph the points

we are given.

We need to rotate the points counterclockwise 180

degrees around the origin. This would move all

of the points from quadrant 1 into quadrant 3.

Imagine that we took the entire coordinate system

and turned it ½ turn.

So the coordinates for Point A would be (-3,

-1). If we follow the same logic, the

coordinates for Point B would be (-4, -2) and

the coordinates for Point C would be (-3, -5).

Now lets take a look at the pre-image and image

coordinates A (3, 1) A (-3, -1) B (4, 2)

B (-4, -2) C (3, 5) C (-3, -5) What

generalization can we make about coordinates that

are rotated 180 degrees around the origin?

Well, it looks like the x- and y-coordinates just

take the opposite sign. So, the mapping

becomes.. (x, y) (-x, -y)

Back to lesson

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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!

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