Transformations Through Flags
Table of Contents
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Translations
Rotations
Dilations
Reflections
Tessellations
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David A.
David C.
Matt
Dom
Ethan
Translations
David Angione
Summary
• A translation is when you slide a figure on a
coordinate grid without turning or flipping the figure.
Vocabulary
• Vector- A quantity that has both direction and
magnitude, or size.
• Initial Point- The starting point of the vector.
• Terminal Point- The ending point of the vector.
• Component Form- A format in which to describe a
vector that combines the horizontal and vertical
components.
• Vector Form- A format in which to describe a vector
by putting the change in the x-axis on the left, and
the change in the y-axis on the right. Needs special
parenthesis. <.
Concepts
• You can write component form by following this
model. (x, y)=(x+a, y+b).
• You can write vector form by putting the change in
the x-axis on the left, and the change in the y-axis on
the right. Needs special parenthesis. <. i.e. Vector <7,
2>.
Mathematical
Examples
• The figure was translated four spaces to the
right, and two spaces up. The figure on the
left is the original shape, and the figure on the
right is the shape after the translation.
Activities
1. Graph a figure with points: A=(-4, 0), B=(-4,
4), C=(0, 0), and D=(0, 4).
2. Fill in the figure with the colors of the
country that you family is from.
3. Translate the figure by using the vector <5, 2>.
4. Graph the new figure and name each point.
5. Fill in the new figure with the country that
you would like to visit.
Real-Life Applications
• A real-life application is when someone raises
a flag to the top of a flagpole, or when they
lower a flag down to half staff.

Key Terms
• Rotation- A transformation where a figure is turned around the
center of rotation as an isometry
• Isometry- the figure is the same before and after the
transformation
• Center of Rotation- A fixed point that can be inside or outside the
shape
• Angle of rotation – the measure of degrees that figure is rotated
about a fixed point
• A rotation of the Japanese flag with the center of rotation outside
of the shape
Rotation About The Origin
• The center of rotation, the origin, is located
at (0,0)
• The equations for rotations about the origin
are
– R90° (x,y) = (-y, x)
– R180° (x,y) = (-x,-y)
– R270° (x,y) = (y,-x)
– R-90° (x,y) = (y,-x)
Rotation About The Origin (continued)
The flag is rotated
180 degrees
about the
origin
Use R-90° (x,y) =
(y,-x)
Rotational Symmetry
• Rotational Symmetry- A figure has rotational
symmetry when the figure can be mapped onto
itself by a clockwise rotation of less than 180
• When you rotate the flag of Switzerland, one of
the two square flags, 90
degrees you will get the same
shape which means the flag
has rotational symmetry
• Click to see
The Angle of Rotation When an Object
is Reflected Over Two Lines
• When you reflect a figure
over two lines that are
not parallel the angle of
rotation is double the
angle between the two
lines
• For example, angle ACB
is 65 degrees so when
the triangle reflects over
the two lines the angle
of reflections is 130
degrees
B
C
65 degrees
A
Real Life Situation
• The flag needs to be
rotated so it can go
on the pole
• How many degrees
counter clock-wise
does the flag need
to rotated about the
center of the flag so
it can be corrected?
Answer
Real Life Situation Answer
• 180 degrees counter clockwise
Rotation Activity
• What is the angle
of rotation between
flag A and flag D?
A
D
B
C
Answer
Rotation Activity (answer)
• 270 degrees
Dilations
Matthew Wechsler
Key Definitions
• Dilation- a transformation in which a polygon
is enlarged or reduced by a given scale factor
around a given center point
• Reduction- 0 < X < 1
• Enlargement- X > 1
Reduction
Enlargement
Matrices
[ ]
293
456
Scale factor: 3
• To get the answer multiply all of the exponents by the
scale factor (click for answer)
• Answer:
[
6 27 9
12 15 18
]
The Flag Situation
• You have a small flag. You want it to be larger
but it has to stay the same shape, what is the
scale factor of the smaller flag to the larger
X
flag? Then find
x
5
30
10
Answer
Answer
• Scale factor =
• X = 15
1
3
Activity
• Get a piece of graph paper
• Draw a rectangle with the points:
(-4, -1)
(-1,-1)
(-4, -4)
(-1, -4)
• Make the scale factor for the new shape 2.5
Answer
• What are the new points?
Answer
(-10, -2.5)
(-2.5, -2.5)
(-10, -10)
(-2.5, -10)
Reflections
By Dominick Gagliostro
Key Definitions
• Reflection- a transformation which uses a line
that acts like a mirror, with an image reflected in
the line.
• Line of Reflection- the line which acts like a
mirror in a reflection
• Line of Symmetry- a line that divides a figure into
two congruent parts, each of which is the mirror
image of the other. When the figure having a line
of symmetry is folded along the line of symmetry,
the two parts should coincide.
Normal Reflections
Reflections over y-axis
Reflections over x-axis
Reflections when X isn’t zero
Minimum distance
Original Points
Description
• To find the minimum
/
distance. First reflect point
A. Next draw a line from A’
to B. Then the point where
that line crosses the x-axis is
the minimum distance.
Minimum Distance continued
Real World Application
• The cemetery wants to put an American flag
in their cemetery for two war veterans that
were recently buried there. They want it to be
the minimum distance between both graves.
Find the minimum distance to help the
cemetery out.
Real World Application
Where do you put the flag?
Answer
What Are Tessellations
• Tessellations are a repeating pattern of figures
that completely covers a plane without any
gaps or overlaps.
Some Vocab
• An edge is the intersection between two
bordering tiles.
• A vertex is the intersection of three or more
bordering tiles.
• A regular tessellation is when a tessellation
uses only one type of regular polygon to fill up
a plane.
• A semi-regular tessellation uses more than
one type of regular polygon to fill up a plane.
Tessellations & Symmetry
• Translational symmetry is when a translation maps the
tessellation onto itself
• Glide Reflectional symmetry is when a glide reflection
maps the tessellation onto itself
– glide reflection is when you reflect then translate an object
• 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
• Reflection or line symmetry: when a figure is reflected
across and axis and the image is the same as the original
• Point symmetry: when a tessellation rotates 180 degrees
and the image is the same
Will It Tessellate
• use the formula for the measure of an angle of a
regular polygon 𝑎 =
180(𝑛−2)
)
𝑛
• substitute a number of sides for n
• if the figure simplifies without a remainder into
360, it will tessellate
• in other words, the product of the expression has
to be a factor of 360
Flag Tessellations
How To Make A Tessellation
Construct Segment AB
Construct Point “C” above AB
Mark the Vector From A B
Translate Point “C” by the Created Vector
Construct the Remaining Sides of the
Parallelogram
Steps 4-7
Construct Irregular Segments
Construct Your New Polygon’s
Interior
Translate the Polygon Interior
Translate the Newly Created Column
Bibliography
•
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•
•
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http://www.globeslcc.com/2013/03/29/the-weekly-reel-how-the-media-can-tug-at-patrioticheartstrings/com-american-flag-pub-dom/
http://nathandahm.com/happy-flag-day/
http://sedalianewsjournal.com/2012/12/12/state-honors-fire-chief-meador/
http://language-assessment-anddevelopment.pusd.schoolfusion.us/modules/groups/group_pages.phtml?gid=942978&nid=69489
http://pantbeer.com/to-clubs-pubs.html
http://www.viecoballoons.com/flpinwheels.htm
Links
• http://content1.riverdell.org/access/web?id=10718736173
731352748
• http://sevendesktop.com/wpcontent/uploads/2013/02/England-flag-art-background.jpg
• http://www.flagwallpapers.com/bulkupload/flagwallpapers/Canada/canada
-flag-wallpaper.jpg
• http://flagwallpapers.com/bulkupload/flagwallpapers/Greece/greekflag.jpg
• http://www.orlandoflagcenter.com/US_FLAGS.htm
http://stationary.prissed.com/images/flag-backgroundpatriotic.jpg
Bibliography
• http://www.mapsofworld.com/flags/japan-flag.html
• http://www.worldatlas.com/webimage/flags/countrys/mideast/tur
key.htm
• http://www.millersmotorcycles.com
• http://axismonday.blogspot.com/2008/12/simple-flags.html
• http://en.wikipedia.org/wiki/File:Flag_of_Canada.svgn
• http://www.britannica.com/hispanic_heritage/article9093933?flagHistory=y
• http://mathworld.wolfram.com/Rotation.html
• http://www.smscs.com/picview.php?title=flags+wallpaper&photo=
http%3A%2F%2Fwww.university500.com%2Fwpcontent%2Fuploads%2F2012%2F02%2F1288-map-of-flags-of-theworld-wallpaper-wallchan-1024x768.jpg
Pictures
• http://www2.waketech.edu/blogs/esl_mlbk/fi
les/2012/10/us_flag.png
• http://www.all-flags-world.com/countryflag/Italy/flag-italy-XL.jpg