### x - River Dell Regional School District

```Transformati
ons in
Carnivals
By Hannah Lin, Joseph Choi,
Kaitlyn Choi and Kevin Walker
Welcome!
• Rotations by Hannah Lin……………… 3 to 22
• Reflections by Joe Choi……….23 to 34
• Translations by Kaitlyn Choi……..35 to 53
• Tessellations by Kaitlyn Choi and Hannah Lin……..54 to 65
• Dilations by Kevin Walker………….66 to 77
• Bibliography………………………………78 to 82
tion
s
By Hannah Lin
What is a rotation?
• A rotation is a transformation in which a figure is turned about a fixed
point.
Key Vocabulary Terms
• The fixed point of a rotation is
called the center of rotation.
• Rays drawn from the center of
rotation to a point and its image
form an angle called the angle of
rotation.
A figure in the plane has rotational symmetry if the figure can be
mapped onto itself by a clockwise rotation of 180 ° or less.
Some polygons that have rotational symmetry: square, equilateral
triangle, regular pentagon and rectangle. The equilateral triangle can
be rotated 120 degrees and 180 degrees and map onto itself.
Examples of shapes that have rotational symmetry:
Practice Problems
Determine whether each polygon has rotational symmetry. If yes,
describe the rotations that map the polygon onto itself.
1)
2)
3)
4)
Equilateral triangle
Kite
Regular hexagon
Square
1)
2)
3)
4)
Yes, 120° and 180°
No.
Yes, 60°, 120° and 180°
Yes, 90° and 180°
Rotation Theorems
• Theorem 7.2 Rotation Theorem: A rotation is an isometry. (original
shape and after shape are congruent)
• Theorem 7.3: If lines k and m intersect at point P, then a reflection in k
followed by a reflection in m is a rotation about point P. The angle of
rotation is 2x°, where x° is the measure of the acute or right angle
formed by k and m.
Theorem 7.3
If the angle between line l and m is 32
degrees x, then the angle of rotation
about O would be 2x, 64 degrees.
The rotation is a “double reflection”
of shape XYZ: one reflection across
line l, and one reflection across line
m, which is equal to one rotation
around point O.
*not drawn to scale
To find the new coordinates of an image after being rotated a certain
degree clockwise or counterclockwise, you would use these equations:
R90°(x, y)= (-y, x) {counterclockwise}
R180°(x, y)= (-x, -y)
R270°(x, y)= (y, -x)
R-90°(x, y)= (y, -x) {clockwise}
Do it Yourself!
Find the new coordinates of the vertices of the image after a clockwise
rotation of 90 degrees around the origin, using the equations from the
previous slide:
13) A’(1,3) B’(1,5) C’(4,6) D’(4,2)
Rotating and Drawing the New
Figure
Say we want to rotate this carnival ticket 60 degrees counter-clockwise around point P.
When you rotate a figure, you need to do it vertex by vertex. First, draw a light line from P to your
The line is about 2.6 cm long. Remember that.
Next, line up the protractor onto the newly drawn line as you see below. Make sure your angle is
going in the counter-clockwise direction.
Mark 60 degrees and take away the protractor like so.
Then, using a ruler, line up P with the marked angle line. Remember the distance between A
and P? It’s 2.6 cm. Keeping the ruler lined up with P and the marked angle line, mark A` at
2.6 cm so the distance from P to A` is also 2.6 cm. Do the same process for points B, C and D
and you will get your new shape, rotated 60 degrees counterclockwise around point P!
*not to scale on computer, make sure you do it on paper
Do it Yourself!
Rotate figure WXYZ 120 degrees
counter-clockwise around
Point P.
Challenge:
Rotate figure WXYZ 50 degrees
counter-clockwise around point Z.
120 Degrees
50 Degrees (Challenge)
Real-Life Application
After a long day at the carnival, you
decide to end the day by riding a
peaceful Ferris wheel. One rotation of
the wheel takes 20 minutes.
If you want to find the measure of the
angle between each seat, you would
divide 360 by the number of pieces the
wheel is divided into. In this case, it
would be 8. So the measure of the
angle between each seat is 45 degrees.
Real-life application continued
If you want to find out the degree of rotation to
get to any seat, multiply 45° by the number of
“leaps” you take to get to that seat. For
example, what is the degree of rotation to get
from seat 1 to 5(clockwise)? You are going to
take 4 leaps, so 45 * 4= 180 degrees.
Challenge: How many minutes would it take to
get to the top of the Ferris wheel, seat 4, from
seat 1? (clockwise) {refer back to the
description}
Write a proportion.
(45 * 3= 135)
Reflections in
Carnivals
By Joseph Choi
What is a Reflection?
• A reflection is a transformation
which uses a line that acts like a
mirror, with an image reflected in
the line
• A shape is reflected over the line of
reflection, which acts like a mirror
for the shape
• Theorem 7.1 Reflection Theorem –
A reflection is an isometry
Key Vocabulary
• Line of Reflection –
The line that a shape
is reflected over in a
transformation
• Line of Symmetry –
The line that goes
through a shape and
allows the shape to
be mapped out onto
itself
Equations
Reflecting Over Lines:
• rx-axis (x , y) = (x , -y)
• ry-axis (x , y) = (-x , y)
• ry=x (x , y) = (y , x)
• ry=-x (x , y) = (-y, -x)
Point Reflection
• R180 (x , y) = (-x , -y)
Reflections are
Everywhere
Whenever there is a symmetric figure in real life, such as a Ferris
wheel, that figure can be created by reflecting half of the figure
on the line of symmetry
Reflection: True or
False
True or False
True or False
True or False
Minimum Distance
The tent is on fire and the clown must go to the well to put it out!
The shortest distance to go to the well and then the tent would be
the minimum distance.
Minimum Distance
In order to find this we would first draw a line at the well (point C)
parallel to the ground. Then, we would reflect the clown (point A) over
the line that we drew earlier. Then in order to find the distance, we
would connect point A’ to the tent (point B).
Minimum Distance
So to recap…
Step 1: Draw a parallel line to
the ground from point C.
Step 2: Reflect point A over
the line you drew.
Step 3: Connect point A’ to
point B
Step 4: Measure the line from
point A’ to point B
Try it Yourself
Point C
Solutions
Minimum Distance:
√65
Cotton Candy Reflections:
1) False
2) False
3) False
Fun in Carnivals
Pin the Tail on the correctly reflected point
Translati
ons
By Kaitlyn Choi
What is a translation?
• A translation is a transformation that maps two points, A and B, in the plane
to points A’ and B’, so that the following properties are true:
1. AA’ is congruent to BB’
2. AA’ is parallel to BB’, or AA’ and BB’ are collinear.
Examples:
Key Vocabulary Words
Vector – a quantity that has both direction and magnitude, or size.
Initial point – the starting point
Terminal point – ending point
terminal point
Initial point
Isometry – a transformation that has congruent pre-images and images
Key Vocabulary Words cont.
Component form – combines the horizontal and vertical components
of a vector
To find the component form of a vector, you can use this equation: < x2 – x1, y2 – y1 >
OR you can count the number of units it takes to reach your terminal point, both
horizontally and vertically.
Ex. 1
< x2 – x1, y2 – y1 > = < 4 – 1, 5 – 1 >  < 3 , 4 >
You can also see that it was moved three units to
the right, and four units up. So the component
form of the vector would be < 3 , 4 >.
Translation Theorems
Theorem 7.4: A translation is an isometry.
Theorem 7.5: If lines x and y are parallel, then a reflection in line x
followed by a reflection in line y is a translation. If B” is the image of B,
then the following is true:
1. BB” is perpendicular to x and y.
2. BB” = 2d, where d is the distance between x and y.
Theorem 7.5
Keep in mind that:
- Lines x and y are parallel.
- The images shown from each axis over is a reflection.
x
y
A
A’
B
B’
A”
As you can see here, there was a translation taken place
from AB to A”B” after two reflections. Also, BB” is
perpendicular to lines x and y.
B”
Remember that the distance between lines x and y is d,
and the distance between B and B” is 2d.
So for example, if d was four, the length of BB” would have
to equal eight.
Equation
To translate a figure, use this equation: Ta,b(x, y) = (x + a, y + b)
A and b are constants. A moves horizontally while b moves vertically.
The number of units you move to the right or left equals a, and the
number of units you move up or down equals b. Next, x and y are the
coordinate points of the pre-image.
This is all explained in coordinate notation. The equation for this is:
(x + a, y + b)
Plug in to solve!
Translate each line one by one.
Example
Let’s start off with an easy example.
Let’s go piece by piece and start off with just one line–just one segment of
a triangle.
Refer to the graph to the left.
What is the image after the translation (x + 5, y – 2)?
To translate this line, move five units to the right and two units down.
Matrices
A matrix is another way to find the image of a translated figure, given coordinate notation or a
vector in component form. It is a rectangular arrangement of numbers in rows and columns.
Matrix addition is what we use to find the coordinates of an image.
So first, set up the translation rule in the matrix. If the coordinate notation is (x + 2, y + 4), or if the
vector/component form is < 2 , 4 >, insert it like this:
[2 2 2 2]
[4 4 4 4]
[ -3 -4 1 0 ]
[ 0 -2 -1 -3 ]
The amount of times you must insert this is the amount of vertices you have. So if you have
four vertices, like in this case, you’d have to write it four times.
Next, take the coordinates of your pre-image: (-2, 0) (-2, -4) (1, -1) (0, -3)
And write it like this.
number of the first matrix to the first number of the second matrix, and so on.
[2 2 2 2]
[4 4 4 4]
+
[ -3 -4 1 0 ]
[ 0 -2 -1 -3 ]
=
[ -1 -2 3 2 ]
[ 4 2 3 1]
So the new coordinates of the image
Would be:
(-1, 4) (-2, 2) (3, 3) (2, 1)
Do It Yourself!
Describe each translation with coordinate notation.
1.
8 units to the right and 3 units down
2.
5 units to the left and 4 units up
Find the coordinates of the image with the given
picture and information.
Refer back to the picture again and find the
coordinates of the image using matrices.
1. (x + 1, y – 4)
1. (x – 6, y + 7)
2. (x + 5, y – 2)
2. (x – 1, y – 3)
Describe each translation with coordinate notation.
1.
8 units to the right and 3 units down (x + 8, y – 3)
2.
5 units to the left and 4 units up (x – 5, y + 4)
Find the coordinates of the image with the given
picture and information.
1. (x + 1, y – 4)
P: (0, -3) Q: (3, 0)
R: (7, -1) S: (3, -5)
2. (x + 5, y – 2)
P: (4, -1) Q: (7, 2)
R: (11, 1) S: (7, -3)
Refer back to the picture again and find the
coordinates of the image using matrices.
1. (x – 6, y + 7)
P: (-7, 8) Q: (-4, 11)
R: (0, 10) S: (-4, 6)
2. (x – 1, y – 3)
P: (-2, -2) Q: (1, 1)
R: (5, 0) S: (1, -4)
Translate the Ball
At carnivals, there are lots of fun games that require throwing or translating a ball.
Describe the translation [in component form AND coordinate notation] we can use to get the ball to
hit the target.
(2, 4)
(-5, 2)
At carnivals, there are lots of fun games that require throwing or translating a ball.
Describe the translation [in component form AND coordinate notation] we can use to get the ball to
hit the target.
(2, 4)
(-5, 2)
Component form: <7, 2>
Coordinate notation: (x + 7, y + 2)
Word Search
Complete the word search.
Componentform
Coordinatenotation
Image
Initialpoint
Isometry
Preimage
Terminalpoint
Transformation
Translation
Vector
Real Life Application
D (6,8)
.
Emily is at a carnival. She is getting
on an hot air balloon.
The trip is supposed to go from
Point A to Point B, taking a stop,
then stop at point D. However,
because of the wind, it goes off
course to point C.
.
.
B (5,3)
A (0,0)
.
C (7, 5)
Real Life Application Cont.
1. Write the component form
of the vectors shown in the
image.
D (6,8)
.
2. Write the component form
of the vector that describes
the path the balloon can
take to arrive at its terminal
point.
3. Write the component form
of the vector that describes
the path the hot air balloon
could take to go straight
from point A to point D.
.
.
B (5,3)
A (0,0)
.
C (7, 5)
Real Life Application Cont.
1. Write the component form of the vectors
shown in the image. <5, 3> ; <2, 2>
D (6,8)
.
2. Write the component form of the vector
that describes the path the balloon can
take to arrive at its terminal point. <-1, 3>
.
3. Write the component form of the vector
that describes the path the hot air balloon
could take to go straight from point A to
point D. <6, 8>
.
B (5,3)
A (0,0)
.
C (7, 5)
TESSELLATIO
NS
By Kaitlyn Choi
What is a tessellation?
• A tessellation is a transformation that forms a pattern with a repeated
shape with no gaps or overlaps.
KEY vocabulary words
• Frieze pattern or border pattern – A pattern that extends to the left and right in such a way that
the pattern can be mapped onto itself by a horizontal translation. Some frieze patterns can be
mapped onto themselves by other transformations as well.
They are classified with different letters.
- Translation: T
- 180 degrees rotation: R
- Reflection in a horizontal line: H
- Reflection in a vertical line: V
- Horizontal Glide Reflection: G
Key vocabulary words cont.
• Glide reflection – a transformation in which every point is mapped onto a point by the following
steps:
1.
A translation maps A onto A’.
2.
A reflection in a line parallel to the direction of the translation maps A’ onto A”.
Here’s a
little
somethi
ng to
out:
Example
Classify the frieze pattern below.
So in order to see what this pattern can be classified as,
Evaluate each transformation.
Was it translated? YES.
Was it rotated 180 degrees? YES.
Was it reflected horizontally? YES.
Was it reflected vertically? YES.
Was there a horizontal glide reflection? YES.
So this is a TRHVG.
How do you know if A SHAPE
tessellates or not?
Use the equation: (n-2)*180 / 360
If it comes out to be a whole number, the shape will tessellate.
For example:
Let’s try a quadrilateral, which has four sides.
(4-2)*180 / 360 = 360 / 360  1
It can be tessellated.
Now, let’s try a pentagon, which has five sides.
(5-2)*180 / 360 = 540 / 360  1.5
It cannot be tessellated.
Do It Yourself!
Classify the following frieze patterns.
1.
3.
2.
4.
Classify the following frieze patterns.
1.
3.
Translation, vertical line reflection
2.
Translation, 180 degrees rotation
4.
Translation, rotation, vertical line
reflection, horizontal glide reflection
Translation, 180 degrees rotation, horizontal line
reflection, vertical line reflection, horizontal glide reflection
Unscramble the words!
_______________
_______________
_______________
_______________
_______________
_______________
_______________
_______________
_______________
Unscramble the words!
KEY>
TESSELLATION
TRANSFORMATION
VERTICAL
FRIEZE
HORIZONTAL
CLASSIFICATION
BORDER
GLIDEREFLECTION
Real Life Application
Have you ever seen eye illusions? In
carnivals, there are fun houses that
are really exciting!
This tessellated optical illusion has a
trick. Stare at the picture and you will
see gray dots appear.
This eye illusion is formed by black
tessellating squares on a white
background which make seeing the gray
dots possible.
Dilations In
a carnival
By: Kevin Walker
Dilation defined
• Dilation- A transformation in which a polygon is enlarged or
reduced by a given factor around a given center point.
Key Vocab
• Scale factor- the factor in which the measurements of a shape are
dilated by
• Reduction- If the Scale Factor is less than 1 but greater than 0
• Enlargement- If the Scale Factor is greater than 1
• Preimage- shape being dilated
• Image- Shape that is a result of a dilation that is similar to the
preimage.
• Center of dilation- A fixed point on a coordinate plane in which all
points are dilated around
Need to know
• All circles dilate because all circles are similar. ( Look below for
example balloons)
• If the scale factor is 1 then the Image and the preimage are identical
The Equation
• Dk(x,y)=(kx,ky)
• This means that in order to find the new, dilated coordinates, you
simply multiply the original coordinates by the scale factor.
Matrices
• In a matrix you organize the points of a vertices in rows and columns,
rows going from left to right and columns going up and down.
• In a 2 dimensional shape the top row is the x coordinates and the y is
the bottom.
• The rows represent different points, so in any given quadrilateral
there will be 2 rows and 4 columns. Look below for an example
Scalar Multiplication
• Now lets try dilating the vertices using a scale factor, we will work
with the coordinates used on the previous slide.
• With a scale factor of two, you simply multiply all the coordinates in
the matrix to find your dilated shape.
• This is called scalar multiplication
How to dilate
• There are 2 ways to dilate, either you can measure the distance from
the center of dilation to the vertices and then multiply the distance
using the scale factor to find the new point; or if you are dilating
around the origin you can simply multiply the vertices using the scale
factor like shown on the previous slide.
Now you try!
• Jumbo the Elephant is, well… jumbo! And his little brother, Tiny the
Elephant is, well… tiny! Find out just how much tinier he is!
Try Again!
• Cousin Ed is even
bigger than
Jumbo! Find out
the height and
width of Ed using
the scale factor
Explanation
• Tiny is exactly 1 half the size of jumbo because if you take jumbos
coordinates and multiply them by ½ you get the coordinates of tiny
since they were dilated around the origin
• Ed is 24 feet tall and 16 feet wide, using the scale factor of 4 you
multiply Jumbo’s characteristics to find Ed’s
Real life applications
• In the real world we see dilations
everywhere. We can find dilations
in a carnival through different
drink sizes; small, medium and
large. They are all similar to one
another; however, they are
different sizes, the size differences
between them could be described
using scale factors.
THE END!
Thanks for coming to the Transformation Carnival!
Bibliography-translations
• Key word definitions and several practice problems: McDougal Littell Inc. Geometry Textbook
Images:
• http://studentweb.wilkes.edu/jennifer.werner/FinalProjectWerner/translation%20triangle.jpg
• http://hotmath.com/hotmath_help/topics/transformations/translation.gif
• http://etc.usf.edu/clipart/49300/49310/49310_graph_blank_lg.gif
• http://cache4.assetcache.net/xc/450468697.jpg?v=1&c=IWSAsset&k=2&d=B53F616F4B95E553AA045CD58424C7D1
0B075991A00C9A70E7F79BAA020FB585
Ball.PNG/120px-Circus_Ball.PNG
• http://www.clipartbest.com/cliparts/yTk/egr/yTkegrREc.png
Bibliography-reflections
• http://misstaraleexo.deviantart.com/art/Cotton-Candy-174819102
• http://www.clipartguide.com/_pages/1386-0901-2323-2707.html
• http://funny-pictures.picphotos.net/clown-clip-art-vector-onlineroyalty-free-public-domain/doblelol.com*thumbs*baby-girl-crawlingclip-art-vector-online-royalty-free-funny_4708006791349289.jpg/
• http://www.freepik.com/free-vector/wishing-well-clipart_384788.htm
• http://www.dreamstime.com/stock-images-grey-donkeyimage1681104
Bibliography-rotations
• http://www.binghamton.edu/campus-activities/docs/ferriswheelcarousel.jpg
• Key concepts and some problems: McDougal Littell Inc. Geometry Textbook
• Pictures: http://www.illustrationsof.com/royalty-free-ferris-wheel-clipart-illustration1075355.jpg, http://www.pamsclipart.com/clipart_images/circus_clown_face_0515-1004-17040147_SMU.jpg, http://www.theory.cs.uvic.ca/~cos/venn/gifs/VictoriaColor.gif,
http://dev.physicslab.org/img/062e65af-eca6-4ab5-be56-5e9c2d963a3c.gif,
http://www.doolins.com/images/products/detail/yellowclownrollticket.jpg,
http://stitchontime.com/osc/images/carnival2.gif
• http://www.emathematics.net/transformations.php?def=rot
ational
Bibliography-tessellations
Key word definitions and several practice problems: McDougal Littell
Inc. Geometry Textbook
Images
• http://www.clker.com/cliparts/L/s/x/o/0/k/carnival-a2jaycees3-hi.png
• http://gwydir.demon.co.uk/jo/tess/optical1.gif
Bibliography-dilations