Transformations - River Dell Regional School District

Report
Transformations
By Yena Shin, Nicola Licata, Mona Moshet
& Andi Deng
Table of Contents
O Tessellations
O Reflections by Mona Moshet
O Dilations by Nicola
O Rotation by Yena Shin
O Translation by Andi Deng
O Bibliography
Tessellations
Vocabulary
• Tessellation- a repeating pattern of figures that
Completely covers a plane without any gaps or overlaps
• Edge- the intersection between two bordering tiles
• Vertex- the intersection of three or more bordering tiles
• Regular tessellation- when a tessellation uses only one type of
regular polygon to fill up a plane
• Semi-regular tessellation- when a tessellation uses more than
one type of regular polygon to fill up a plane
• Translational symmetry-when a translation maps that tessellation
onto itself
Vocabulary (continued)
• Glide reflectional symmetry-when a glide reflection maps the
tessellation onto itself
• Reflection or line symmetry-when a figure is reflected across the axis
& the image is the same as the original
• Rotational symmetry-when a rotation of 180 degrees or less is
performed on a tessellation and the resulting image is the same as
the original image
• Point symmetry-when a tessellation rotates 180 degrees and the
image is the same
Determining which polygons
to tessellate or not
• To see which regular polygons can tessellate or not you use the
following formula to find the polygon’s angle measure
• a=180(n-2)/n
• If the angle measure is a factor of 360, then that polygon can
tessellate
• For example, if you wanted to
see if a regular pentagon, like the one
• at the right, could tessellate, you use
the formula to find that each inner
angle is 108 degrees
• Then you would divide 360 By 108
360/108=3 1/3, therefore a regular pentagon cannot tessellate
Tessellations Activity
• Johnson wants to tile the floor of his house.
However he wants to semi-tessellate the tiles
with 3 different shapes. Can you help Johnson
pick which shapes he needs?
Reflections
By Mona Moshet
Vocabulary
Line of
symmetry
•
Reflecting Figures & Finding the Line
of
Reflection
Rules for reflecting figures:
Finding the Minimum Distance
Activity
• Take the paper that you have and bring the top
left tip to the right side of the paper till a triangle
is made.
• Then fold the small rectangle under the triangle.
You now have a square!
• Fold the square in different ways to find how
many lines of symmetry the square has
• How many lines of symmetry does a square
have?
• This proves that a regular polygon’s number of
lines of symmetry is equal to it’s number of sides
Dilations
Nicola Licata
What is a Dilation?
• A dilation is a transformation that reduces or
enlarges a polygon by a given scale factor
around a given center point.
• When a figure is dilated the new figure will
always be similar to the original figure
Center
Point
There is a
scale factor
of 2 which
means that
the new figure
is 2 times larger
The Scale Factor
• The scale factor is the amount by which the
image grows or shrinks
• To find the scale factor of two given shapes
simply find two corresponding sides or points
and put the new shape’s side over the original
shape’s corresponding side.
15
Here the scale factor is
3 because 15 is the new
distance
and 5 is the old
distance. 15/5 = 3
5
Reductions and Enlargements
• If the scale factor of your dilation is greater
than 1 than the dilation is an enlargement.
• If your scale factor is greater than 0 but less
than 1 than the dilation is a reduction
This is an enlargement because the scale
factor is 5.
20
100
Activity
• Johnny wants to make a scale model of the
clock tower big ben in London. If he wants to
make it 1/100 of the actual height how tall will
the scale model be?
316 ft.
Rotation
Key Vocabulary
Rotation
A transformation where a figure is turned around a center of rotation.
Isometry
A transformation where the figure stays congruent
Center of Rotation
A fixed point anywhere that the figure rotates about. The center of
rotation can be anywhere. It can be inside or outside the figure.
Angle of Rotation
The angle created by rays drawn from the center of rotation to a point
and its image.
Key Concepts
Theorem
Line K and line M intersect at point P. Then a reflection in line K
and the line M is a rotation about point P.
The angle of rotation is
double the angle formed
by K and M.
Since the angle formed
by lines Kand M is 70°,
the angle of rotation is
140°.
Key Concepts Cont.
Rotational Symmetry
A figure in the plane has rotational symmetry if the figure
can be mapped onto itself by clockwise rotation of 180
degrees or less.
Equations:
R90 (X,Y) = (-Y,X)
R180 (x,y) =( -x,-y)
R270 (x,y) =(y,-x)
R-90 (x,y) = (y, -x)
Rotational Symmetry
Does this figure have rotational symmetry? If so,
what is the angle of rotation?
Rotating on a Coordinate Plane
You can rotate the building
by using the equations
from the previous slide.
You are to rotate counter
clock wise unless told by
the problem.
Equations:
R90 (X,Y) = (-Y,X)
R180 (x,y) =( -x,-y)
R270 (x,y) =(y,-x)
R-90 (x,y) = (y, -x)
Real- Life Application
The Pentagon has a rotational symmetry of 72°
When it is rotated at 72°, it maps onto itself.
Save the Leaning Tower of Pisa!
It is 3025 and the leaning tower of Pisa is now leaning too much. Now, the
people decide that there is a need to reconstruct the tower. Find the angle
you need to rotate the leaning tower to make the tower perpendicular to the
ground by using a protractor.
Translation
By Andi Deng
Vocab
• Vector --A quantity having direction as well as
magnitude
– Initial point – starting point of the vector
– Terminal point – ending point of the vector
– Component form/ coordinate vector– horizontal
and vertical values  < a, b >
• Coordinate notation– (x,y)  (x+a, y+b)
Andi’s explanation
• It’s pretty much a thing move to another place.
• See example: A house moved to a park from a city by a
witch. ( not in scale)
Vector component form/ coordinate vector:
(15, 3)
3
15
Coordinate notation– (x,y)  (x+15, y+3)
Relating to the World
y
8
• Mobile house
7
6
5
4
• A moving
house
3
2
1
–8
• Another
moving
house
–7
–6
–5
–4
–3
–2
–1
–1
–2
–3
–4
–5
–6
–7
–8
1
2
3
4
5
6
7
8
x
Vector component form/
coordinate vector: (10, 5)
practice Coordinate notation
– (x,y)  (x+10, y+5)
• Make the house transfer 10 boxes to right and
y
5 boxes up.
8
7
6
5
4
3
2
1
–8
–7
–6
–5
–4
–3
–2
–1
–1
–2
–3
1
2
3
4
5
6
7
8
x
Bibliography
•
•
•
•
•
•
•
•
http://www.mathnstuff.com/gif/7x7not.gif
http://2.bp.blogspot.com/AZAjgubNMhA/TwgsiY86q5I/AAAAAAAADSc/5vIUw095L7k/s1600/taj+mahel+%25285%25
29.jpg
http://frank.itlab.us/france_2005/small_france/aug_02_north_bridge_2129.jpg
http://lh5.ggpht.com/hFHVVeQ1ivFZv98SQyzvkeJcaaZJDHppDqa_zPdMYclne6eCsrMGaXHBNybBGRxzjqNSBPJZZzKNQzQHzLenos=s285-c
http://1.bp.blogspot.com/G6CJXrFzR48/TgvUn_GsxCI/AAAAAAAAAxA/9C6zW9cOKZ8/s1600/Onassis+resevoir+reflectio
ns+600.jpg
http://uploads.static.vosizneias.com/2013/03/Rite-Aid.jpg
http://brentwood.thefuntimesguide.com/images/blogs/new-brentwood-park-concordroad.jpg
http://www.gbrionline.org/wp-content/uploads/2012/09/Dubai-Marina.jpg
Bibliography
• http://www.ysjournal.com/articles/2009/2/7/images/YoungScientistsJ_2009_
2_7_35_57766_u13.jpg
• http://math.slu.edu/prep08/images/prep08/thumb/3/38/Climatron.jpg/300p
x-Climatron.jpg
• http://cdn.indulgy.com/zB/wC/H7/VakilMosqueShabestanweb.jpg
• http://1.bp.blogspot.com/hM6SOjPC0wg/UQzqZTCkPCI/AAAAAAAAALs/1rHEdIUX90M/s1600/YoungScie
ntistsJ_2009_2_7_35_57766_u12.jpg
• http://1howmany.com/files/uploads/images/pentagon-2.jpg
• http://hdwallpaperfreedownload.com/wp-content/gallery/new-y

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