UNIT ONE

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UNIT ONE

• OVERVIEW
• OF
• TRANSFORMATIONS

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Goals for Activity

• Understand that a transformation is a function
  that maps the plane onto itself.
• Identify some basic properties of
• isometries and dilations.
• Link the concept of congruency to
• isometry.
• Link the concept of similarity to dilation.

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GO (Warm-up) Materials needed Notebook Paper,
2 Pencil, Color Pencil

• With your pencil, draw the parallelogram shown
  below on a sheet of notebook paper.
• Turn your paper over to the back. Using a
  regular colored pencil, draw over the
  parallelogram on the back of the paper.
• Put your notebook paper on top of the original
  parallelogram. Slide the notebook paper about 2
  inches to the northeast (i.e. slide right and
  then up).
• 4. Trace over the notebook paper parallelogram.
  Pencil markings from the back of the notebook
  paper should transfer to this paper. Use light
  markings to draw the resulting figure. This
  drawing technique is called the two-sided
  transfer technique.

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Which pictures accurately depicts the results of
sliding the patty paper?
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ABOUT TRANSFORMATIONS
A transformation of the plane is a one-to-one mapping (function) of the plane onto itself.
Refer to the warm-up. You were given a
parallelogram. Then, using notebook paper you
performed a slide to locate a new parallelogram.
This action can be thought of as a
transformation. The resulting figure is called
the image of the original figure.
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1. Color the original figure blue. Label it
parallelogram ABCD. 2. Color the image red.
Label it parallelogram ABCD. Be sure to
mark corresponding points with the same letter.
A
D
B
C
A
B
D
C
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DISCUSSION QUESTIONS
Is distance preserved under this
transformation? What does this mean?

YES, For this translation it does not matter
what two points are chosen because the distance
between any two points corresponding points on
the original figure will be the same as the
distance between corresponding points on the
image.
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DISCUSSION QUESTIONS
Is parallelism preserved under this
transformation? What does that mean?
YES, because the lines are parallel in the
original figure and mapped to images that are
parallel lines as well.
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DISCUSSION QUESTIONS
In question two, Is collinearity preserved under
this transformation?
YES, In mathematical terms, the importance of
this statement is that lines are mapped to
lines. Example. Chose three points that lie
on the line in the original figure . Then
identify the corresponding points on the image.
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DISCUSSION QUESTIONS
Is angle measure preserved under this
transformation?
Angle measure is preserved. In other words,
angles are taken to angles of the same measure.
Is betweeness preserved under this
transformation?
Betweeness is preserved. In other words, if B
is between A and C on a line, then B is between
A and C on the images.
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3. Using the chart below, write TRUE or FALSE
for each statement.
Under this transformation TRUE OR FALSE





1. Distance is preserved. In other words, lines
are taken to lines, and line segments to line
segments of the same length.
TRUE
2. Parallelism is preserved. In other words,
parallel lines are taken to parallel lines.
TRUE
3. Angle measure is preserved. In other words,
angles are taken to angles of the same measure.
TRUE
4. Collinearity is preserved. In other words,
if three points lie on the same line, then their
images lie on the same line.
TRUE
5. Betweeness is preserved. In other words, if
B is between A and C on a line, then B is
between A and C on the images.
TRUE
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Introduction to Isometries
We say two figures are congruent if there is a
sequence of isometries of the plane onto itself
that map one figure onto the other. In other
words, congruent shapes have the same shape and
size. Which of the pairs of figures above
appear to be congruent? Write how could you
justify the pairs you pick.
Isometry is a transformation that preserves shape
and size.
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Are images 1,2,3,4, or 5 an isometry?

1. Yes, Why?
2. No, Why?
3. Yes, Why?
4. No, Why?
5. Yes, Why?
6. No, Why?

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Lets Continue
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REVIEW GOALS

• Understand that a transformation is a function
  that maps the plane onto itself.
• Identify some basic properties of
• isometries and dilations.
• Link the concept of congruency to
• isometry.
• Link the concept of similarity to dilation.

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Discussion Question
How is dilation different from similarity?
A dilation is a function. Similarity is a
property of two figures. These are related
because a dilation creates similar figures.