Humans & Transformations
Emily DiMaulo-Milk
Emma Halecky
Troy Karanfilian
AJ Perlowin
Matt Monaghan
Table of Contents
Rotation
Translation
Reflection
Tessellations
Dilations
Emily DiMaulo-Milk
ROTATIONS
Rotations
A transformation in which a figure is turned
about a fixed point
Center of rotation
The fixed point around which the preimage
rotates. Can be inside, outside, or on the figure.
Angle of rotation
Rays drawn from the center of rotation to a
point and its image form the angle at which the
figure rotated
Rotations are expressed in degrees, and the
amount of degrees a figure rotates is the angle
of rotation.
180º
Rotational Symmetry
When a figure can be mapped onto itself by a
clockwise rotation of 180º or less
180º
Rotational Symmetry
All regular figures have rotational symmetry,
but a figure can still have rotational symmetry
even if it is not regular, just look at lines of
symmetry. To find the angle of rotation needed
to achieve rotational symmetry…
- Regular figures- Find the measure of one
interior angle
- Irregular figures- Look at the lines of
symmetry, and divide the numbers of sides a
figure has by the number of lines of
symmetry. This works as a general rule and
should be used with general logic.
Let’s Try It!
Does the shape have rotational symmetry?
Yes
Yes
No
Rotate a Shape
Step 1.
Draw a line connecting the original point to the
center of rotation
Rotate a Shape
Step 2.
Measure how many degrees you want to rotate
the figure with a protractor
Rotate a Shape
Step 3.
Measure the distance between the original
point and the center of rotation, then draw the
new point the same distance from the center of
rotation at the proper distance. Repeat with
each point
Let’s Try It!
Using a coordinate plane, draw triangle ABC.
A(1,9)
B(5,8)
C(8,8)
Rotate the figure 50º clockwise.
Regular Rotations on a Coordinate
Plane
Just use these equations-Counterclockwise
R - (x,y) = (-y, x)
R - (x,y) = (-x, -y)
R - (x,y) = (y, -x)
90
180
270
Clockwise
R - (x,y) = (y, -x)
R - (x,y) = (-x, -y)
R - (x,y) = (-y, x)
90
180
270
Let’s Try It!
Without drawing the shape, rotate figure
ABCDE 180º counterclockwise.
A(1,0)
B(3,8)
C(4,9)
D(-7,9)
E(-8,0)
Answers
A’(-1,0)
B’(-3,-8)
C’(-4,-9)
D’(7,-9)
E’(8,0)
Rotations In the Human Body
• When you turn your head from left to right,
that’s approximately 120º of rotation
• You can roll your eyes 360º
• You can rotate your entire body by spinning
in a circle
Let’s play a game!
Simon Spins
This game is a lot like “Simon Says”
The leader calls out a motion or body part, and
you only do the motion or touch the body part if
it is a rotation in your body or has the potential
to rotate. If it is not, stay still! Any motion will
make you get out.
Interesting Fact
A 2008 study found that Simon Says actually
has a psychological benefit and helps young
children learn to suppress impulsiveness.
Troy Karanfilian
TRANSLATIONS
Vocabulary
Translation- A transformation that maps every 2 points P and Q in the plane to points P’ and Q’
so that the following properties are true:
1) PP’ = QQ’
As you can see in Figure 1, the distances between the original points and the new points are
the same.
2) PP’ is parallel to QQ’ OR PP’ and QQ’ are collinear
As you can see in Figure 2, the old points and new points create line segments. The slopes of
these lines are the same, meaning that they are parallel
Also, in Figure 3, the line segments still have the same slope, but they are collinear. However, it
is still a translation since the segments can be either parallel OR collinear.
Fig 1:
Fig 2:
Fig 3:
Vocabulary cont.
Vector- A quantity that has both direction and magnitude (NOTE:
When a vector is written as vector PQ, P is the initial point and Q
is the end point)
Component Form (or Vector Form)- combines the horizontal and
vertical components (If a point is translated 4 units to the left and 3
units up, it would be written as <-4,3>)
Matrix- an array of numbers in brackets that represent points; each
column represents a point, the first row represents the xcoordinates, and the second row represents the y-coordinates
Ex (Fig 4):
Entry- A slot in a matrix
Ex (Fig. 5):
Describing a Translation
Let’s say that “a” represents the units a point is
translated on the x-axis and “b” represents the
units a point is translated on the y-axis.
Coordinate Notation Format(x,y) ---> (x+a,y+b)
Component Form<a,b>
Describing a TranslationExamples
Let’s say that triangle ABC is moved to the left 4 units and up 3
units
Fig. 6:
Coordinate notation- ex:
(x,y)-->(x-4, y+3)
Component Form<-4, 3>
• Let’s say that triangle DEF is moved to the right 5 units and
down 2 units
Fig. 7:
• Coordinate notation- ex:
• (x,y)-->(x+5, y-2)
• Component Form• <5, -2>
Graph an Image on a Coordinate
Plane Given a Preimage and a Rule
Take each point and apply the rule to them. Then connect the
dots.
Ex. Triangle GHI is translated (x,y)-->(x+2, y-3)
G(4,2)
H(3,1)
Fig. 8:
I(-1,-2)
G’(4+2, 2-3)
H’(3+2, 1-3)
I’(-1+2, -2-3)
G’(6,-1)
H’(5,-2)
I’(5,-2)
Write a Rule
Using Coordinate Notation:
You would count how many units are added to the point
across the x-axis (a) and how many units are added to
the point across the y-axis (b). Then you would plug it
into this format:
(x,y)-->(x+a, y+b)
Using Component Form:
You would count how many units are added to the point
across the x-axis (a) and how many units are added to
the point across the y-axis (b). Then you would plug it
into this format:
<a,b>
Find the Coordinates of the Preimage given
the Coordinates of the Image and the Rule
If you have the image and you are looking for
the preimage, you would take “a” (amount of
units that are added to the point across the xaxis) and “b” (amount of units that are added to
the point across the y-axis) and multiply each
of the by -1. You would take the solutions and
make a new rule. Then, you would apply the
new rule to the points of the image. Then, you
get the preimage.
Finding the Preimage- Example
Triangle KJL is translated (x,y)-->(x+2, y-3)
2(-1) = -2
-3(-1) = 3
(x,y)-->(x-2, y+3)
Fig. 9:
K’(4,2)
J’(3,1)
L’(-1,-2)
K(4-2, 2+3)
J(3-2, 1+3)
L(-1-2, -2+3)
K(2,5)
J(1,4)
L(-3,1)
Use Matrices to Find the
Coordinates of a Translation Image
Let’s say that the rule <4,-3> is applied to triangle MNO
M(1,2)
N(-3,2)
O(-5,-1)
You would put the points into a matrix
Ex. (fig. 10):
Then you would put the rule into a matrix
Ex. (fig. 11):
Once you know the matrices, add them.
Ex:
+
Continued on Next Slide
Matrix Example cont.
To actually add the two matrices, you would add the
corresponding entries (so the top left entries would be added
together)
Ex. (fig. 12):
-----------------------Fig. 13:
Then, you would convert that matrix back into coordinates and
label them
M’(5,-1)
N’(1,-1)
O’(-1,-4)
Translations in the Human Body
One translation in the human body is a
dislocated shoulder:
The humerus translates down past the scapula
causing pain in the arm.
Translations in the Human Body
cont.
Another translation in the human body is
chewed up food traveling through the
esophagus:
The Bolus (or chewed food) is translating down
through the esophagus
Games!
• Matrix 2048! http://scratch.mit.edu/projects/21659646/
(Login: Period3; Password: mathp3)
• Translation Rule Memory Game!http://scratch.mit.edu/projects/21697690/
(Login: Period3; Password: mathp3)
Translation Crossword Puzzle!
AJ Perlowin
REFLECTIONS
Reflection
An object can be reflected in a mirror line or axis of
reflection to produce an image of the object.
For example,
Each point in the image must be the same distance from
the axis of reflection as the corresponding point of the
original object.
Reflecting shapes
If we reflect the quadrilateral ABCD in a mirror line we label
the image quadrilateral A’B’C’D’.
A’
A
B’
B
Pre-image
Image
C’
C
D
D’
Axis of reflection
This transformation is isometric.
Reflecting shapes
If we draw a line from any point on the object to its image
the line forms a perpendicular bisector to the mirror line.
A’
A
B’
B
Pre-image
Image
C’
C
D
D’
Axis of reflection
Reflection on a coordinate grid
y
A’(–2, 6) 7
B’(–7, 3)
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
C’(–4, –1) –2
–3
–4
–5
–6
–7
A(2, 6)
B(7, 3)
1 2 3 4 5 6 7 x
C(4, –1)
The vertices of a
triangle lie on the
points A(2, 6), B(7, 3)
and C(4, –1). Reflect
the triangle in the
y-axis and label each
point on the image.
Notice that this
reflection follows
the rule
(x,y)  (-x,y)
This is true for all
reflections over the
y-axis
Reflection on a coordinate grid
y
A(–4, 6)
D(–5, 3)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
D’(–5, –3)
–2
–3
–4
–5
–6
–7
A’(–4, –6)
B(4, 5)
C(2, –2)
1 2 3 4 5 6 7 x
C’(2, –2)
B’(4, –5)
The vertices of a
quadrilateral lie on
the points A(–4, 6),
B(4, 5), C(2, –2) and
D(–5, 3). Reflect the
quadrilateral in the
x-axis and label each
point on the image.
Notice that this
reflection follows
the rule
(x,y)  (x,-y)
This is true for all
reflections over the
x-axis
Reflection on a coordinate grid
B’(–1, 7)
C’(–6, 2)
y
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
x=y
A’(4, 4)
A(4, 4)
1 2 3 4 5 6 7 x
B(7, –1)
C(2, –6)
The vertices of a
triangle lie on the
points A(4, 4), B(7, –1)
and C(2, –6). Reflect
the triangle in the line
y = x and label each
point on the image.
Notice that this
reflection follows
the rule
(x,y)  (-x,-y)
This is true for all
reflections over the
line y = x.
Finding the axis of reflection
To find the axis of reflection all you have to do is use the
pre-image and the image and find the point of perpendicular
bisection.
A
A’
B’
B
Pre-image
Image
C’
C
D
D’
Axis of reflection
Lines of Symmetry
A line of symmetry is the line that a shape reflects on to its
self. There can be many lines of symmetry in a shape or
none all depending on the shape.
Lines of Symmetry In Triangles
How many lines of symmetry does each triangle have?
Notice that the
isosceles triangle
has one line of
symmetry and the
equilateral triangle
has two while the
scalene triangle has
none. This is true
for all triangles.
Lines of Symmetry
How many lines of symmetry does each quadrilateral have?
Notice that the
square has four
lines of symmetry,
the rhombus and
rectangle both have
two while the kite
and isosceles
trapezoid both have
one. This is true for
all of the
corresponding
quadrilaterals.
Lines of Symmetry
How many lines of symmetry does each regular polygon
have?
Notice that all the
regular polygons
have the same
number of lines of
symmetry as they
have sides. This is
true for all regular
polygons.
Minimum Distance
Minimum Distance is the point on a line that is the shortest
distance between two points that the path intersects on a
line. This is normally represented as point C.
A
B
Line
C
Minimum Distance
To find the minimum distance you have to first reflect point A
over the line to get A’. Then you have to draw a strait line
between A’ and B. Lastly you draw in point C where the line
intersects with your connecting line segment.
A
B
Reflection 
Line
C
   Connecting Line
A’
Reflections In Real Life
Word Scramble
immuinm dtaiencs is the point on a line that is
the shortest distance between two points that
the path intersects on a line.
An object can be reflected in a rmoirr niel or
ixsa fo felocitern to produce an image of the
object.
This transformation is tmiisceor.
A ilen fo yerymtms is the line that a shape
reflects on to its self. There can be many in a
shape or none all depending on the shape.
Matt Monaghan
TESSELLATIONS
Vocabulary
Tessellation- A tessellation of a flat surface is
the tiling of a plane using one or more
geometric shapes with no gaps or overlaps.
Tessellations are like a puzzle.
Tessellations in the Human Body
All the cells fit together like a puzzle and
tessellate.
Frieze Patterns
• Frieze Pattern- A pattern that extends itself to the left and right
in such a way that the pattern can be mapped onto itself by a
horizontal translation.
• Classification of Frieze Patterns
• T - Translation
• TR - Translation and 180
• TG - Translation and horizontal glide reflection
• TV - Translation and vertical line reflection
• THG - Translation, horizontal line reflection and horizontal glide
reflection
• TRVG - Translation, 180 rotation, horizontal line reflection,
vertical line reflection, and horizontal glide reflection
• TRHVG - Translation, 180 rotation, horizontal line reflection,
vertical line reflection, and horizontal glide reflection
Tessellations
Can you tell what each of these are?
Emma Halecky
DILATIONS
Dilations
• Dilations- a type of transformation with center
C and scale factor k, that maps every point P
on a plane to a point P’ so that the following
two properties are true:
• •If P is not the center point C, then the image
point P’ lies on line CP. The scale factor k is
a positive number such that k=CP’/CP and k
is not equal to 1
• •If P is the center point C, then P=P’
• Equation: Dk(x,y)= (kx,ky)
Scalar Multipication
Dilations in the Human Body
Dilations happen very often
in your eyes. Your pupils
dilate due to changes in the
light and even changes in
emotion. Your pupils would
dilate if you just came out of
a dark room into bright
sunlight. They would also
dilate if you get scared,
excited, or see someone you
like!
Practice Problems
1) John goes to the movies on a
sunny day. Outside the theater his
pupils are are 0.25 cm wide. When
he goes inside his eyes dilate by a
scale factor of 2. How wide are his
pupils inside the theater?
Answers:
1) 0.5cm
2) 1/3
2) Kelly goes to the eye doctor. Her
pupils are originally 6mm. When the
doctor shines the light in her eyes
they dilate to 2mm. What is the
scale factor of the dilation?
Final Game
Frankenstein's Art Project
This is like a color by number, except it’s color
by transformation!
Color all…
Rotations- green
Translations- purple
Reflection- yellow
Tessellation- blue (only if there is no other
transformation)
Dilation- red
2 or more transformations- orange
Game Board for Frankenstine’s art
Project
Save the photo to
word and print if
you’d like to play.
Interesting Fact
Frankenstein was actually the scientist, not the
monster.
Bibliography
• http://www.carnagill-school.ik.org/img/man-turnhead.gif
• http://www.msnpro.com/emoticons/bestemoticons/roll-eyes.gif
• http://upload.wikimedia.org/wikipedia/commons/a/
a7/Frankenstein's_monster_(Boris_Karloff).jpg
• http://en.wikipedia.org/wiki/Peristalsis
• https://www.mathway.com/
• http://coolmath.com/algebra/24-matrices/01whats-a-matrix-01.htm
• http://coolmath.com/algebra/24-matrices/02adding-subtracting--01.htm
Bibliography (cont.)
• http://forum.woodenboat.com/showthread.ph
p?112363-Voronoi-Diagrams-in-Nature
• http://langfordmath.com/ECEMath/Geometry/
FriezePatternPractice.html
• http://www.fun-stuff-to-do.com/geometricshapes-worksheets.html
• http://educationportal.com/academy/lesson/dilation-in-mathdefinition-meaning-quiz.html#lesson
• http://image.tutorvista.com/cms/images/38/dil
ation-graph.JPG
Bibliography (Cont.)
• http://www.mathwarehouse.com/transformati
ons/dilations/images/picture-of-dilation-inmath2.png
• http://www.mathwarehouse.com/algebra/matr
ix/images/matrix-multiplication/scalarmultiplication.png
• http://drjoannabuckley.files.wordpress.com/2
011/05/single-eye.jpg
• http://www.nature.com/eye/journal/v24/n6/im
ages/eye2009275f3.jpg
Bibliography (cont.)
• http://www.mathopenref.com/axis.html
• http://www.healthylifestylesliving.com/enlight
en-the-soul/law-of-attraction/autosuggestionand-the-person-in-the-mirror/
• http://therightathome.com/the-two-houseswith-identical-planning
• http://outlandishobservations.blogspot.com/2
013/09/friday-fun-facts-9272013.html