Geometry
Transformational Geometry
Code (Kelly will fill this in)
Course: Geometry
Unit #5: Transformational Geometry
CCSS Curriculum
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Geometry
Transformational Geometry
Code (Kelly will fill this in)
Overarching Question:
What impact does each type of transformation (reflection, rotation, translation, and dilation) have
on the location, size, and orientation of geometric objects?
Previous Unit:
Quadrilaterals and Other
Polygons
This Unit:
Questions to Focus Assessment and Instruction:
1. What is the result of reflecting an object consecutively over two lines?
2. Does the result of performing two consecutive reflections depend on
whether the two mirror lines are parallel, or intersecting? Why?
3. How can the distance of the pre-image from two parallel mirror lines
be used to determine the location of the image after reflecting over
both lines?
4. When reflecting a pre-image over two intersecting lines, what impact
does the angle at which the lines intersect have on the location of the
image?
Key Concepts:
Isometry
Preimage
Translation
Image
Reflection
CCSS Curriculum
Next Unit:
Transformational Geometry
Right Triangle Trigonometry
Intellectual Processes (Standards for Mathematical
Practice):
Model with mathematics: Use transformations to
model mathematical phenomena
Use appropriate tools strategically: Apply and adapt
a variety of appropriate strategies using
transformations to solve problems
Rotation
Composition of two or more
transformations
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Geometry
Transformational Geometry
Code (Kelly will fill this in)
Lesson Abstract
This lesson begins with having students recall previous knowledge about flips, turns, and slides
that they experienced in seventh and eighth grade. Lesson 1 focuses on characteristics of
reflections and how reflecting over a variety of lines changes a pre-image. These can be explored
in the coordinate plane with reflections over the x-axis, y-axis, other vertical or horizontal lines, and
the lines y=x and y = -x. Included in this exploration would be the effect reflections play on
coordinates, and could include developing coordinate rules for different lines of reflections. Lesson
3 moves to exploring reflections over two lines. Students will explore the composition of reflections
over two parallel lines and over two intersecting lines. Conjectures should be made about the
impact of distances between the two parallel lines and the angle formed by the intersecting lines.
Students should then mathematically defend their conjectures.
Common Core Standards
Geometry-Congruence (G-CO)____________________________________________________
Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
2. Represent transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs and give other
points as outputs. Compare transformations that preserve distance and angle to those that do
not (e.g., translation versus horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure
using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms
of rigid motions to decide if they are congruent.
CCSS Curriculum
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Geometry
Transformational Geometry
Code (Kelly will fill this in)
Instructional Resources: Computers with Java installed or Miras and rulers, graph paper
Sequence of Lesson Activities
Lesson Title: Symmetries II, (http://illuminations.nctm.org/LessonDetail.aspx?ID=U139)
Selecting and Setting up a Mathematical Task:

By the end of this lesson
what do you want your
students to understand,
know, and be able to do?







In what ways does the task
build on student’s previous
knowledge?

What questions will you ask
to help students access
their prior knowledge?
Students will articulate their reasoning and judge the reasoning of others.
Use various representations to help understand the effects of simple
transformations and their compositions.
Understand and represent translations, reflections, and rotations of objects
in the plane by using sketches, and coordinates
Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs.
Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given
figure onto another.
Develop definitions of rotations, reflections, and translations in terms of
angles, perpendicular lines, parallel lines, and line segments.

Students have experience with flips, turns, and slides in seventh and eighth
grade. The language of transformations becomes more formal and
students are asked to reach conclusions about combining more than one
transformation on an object.

Think about using flips, turns, and slides with an object. What impact does
each of these operations have on the size and shape of an object? On the
location of an object?
Are there any rules associated with coordinates of a shape that give
information about how the coordinates of the pre-image are related to the
coordinates of the image?
What effect does performing two reflections consecutively have on the
original image?


Launch:

How will you introduce
students to the activity so
as to provide access to all
students while maintaining
the cognitive demands of
the task?
CCSS Curriculum

Begin by giving students a piece of graph paper with a coordinate axes
drawn on it. Select coordinates of a point and have students reflect the
point over the x- and y- axes. They can use a Mira, fold on the reflections
lines or use other logic to determine the new point. Have students discuss
how they got the new point and ask for the coordinates of the new point.
What is the relationship between the coordinates of the pre-image point and
its image? Have students discuss how they can determine the location of an
image point if they are not reflecting over the axes.

Have students access the applet Reflection of a Point found in Lesson 1,
Describing Reflections, and answer the questions #6-9 provided with the
applet individually if there are enough computers. Otherwise pair up
students making sure they switch off who has control of the computer.
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Geometry
Transformational Geometry

What will be heard that
indicates that the students
understood what the task is
asking them to do?
Code (Kelly will fill this in)

Continuing in Lesson 1, using the applet Describing Reflections, have
students try #10-12 Select students to demonstrate to the class the result of
changing the reflection line or points that determine the shape in response
to these three questions.

Have each student explore #13-15. If students are sharing computers, have
one student change the shape, and the second student draw what they
expect the reflection to look like. Use the Show Reflection tab to verify the
correctness of the image. Switch places and have the other students do the
same.

Students will discuss how they are able to determine the location of the
image over a reflection line.

There should be some discussion about the line of reflection being the
perpendicular bisector of the segment joining the pre-image point to its
image. Or perhaps there could be comments about measuring points on
the pre-image the same distance on either side of the reflection line at a 90
degree angle.
Supporting Student’s Exploration of the Task:
 The exploration for students now turns to Lesson 3, Reflections Across Two
Mirror Lines. Provided for student use are two applets: Composition of
Reflections Across Parallel Mirror Lines and Composition of Reflections
Across Intersecting Mirror Lines.

What questions will be
asked to focus students’
thinking on the key
mathematics ideas?

What questions will be
asked to assess student’s
understanding of key
mathematics ideas?

What questions will be
asked to encourage all
students to share their
thinking with others or to
assess their understanding
of their peer’s ideas?

How will you extend the
task to provide additional
challenge?

What is the relationship between the line of reflection and points on the preimage and image of an object?

What happens when an object is reflected over the same line twice?

What happens when an object is reflected over two parallel lines?

What is the relationship between the distance between the two parallel lines
of reflection and the location of the image compared to the pre-image?
Explain mathematically why this is true. How were you able to determine
this relationship?

When an object is reflected over two intersecting lines, what happens to the
resulting image?

How can you determine the angle that the pre-image rotates through after
reflecting an object over intersecting lines? Why does this angle make
sense?

To extend the task, have students use coordinates to represent an object in
the coordinate plane. Using these coordinates and two vertical or horizontal
parallel lines, find the coordinates of the object’s image. To make this more
challenging, make the distance between the lines a variable distance such
as h or k, and describe the coordinates using these variables.
Sharing and Discussing the Task:
CCSS Curriculum
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Geometry
Transformational Geometry


Code (Kelly will fill this in)
What specific questions will
be asked so that all
students will:

What happens when a shape is reflected twice across the same mirror line?
Does it matter where the mirror line is in relationship to the object? Will this
relationship always hold for two reflections across parallel lines?
o
Make sense of the
mathematical ideas
that you wanted them
to learn?

What one transformation would be the same as two reflections across
intersecting lines? Will this relationship always hold true?

Summarize the result of composing two reflections across parallel lines.
o
Expand on, debate,
and question the
solutions being
shared?

Summarize what happens when a shape is reflected across two intersecting
lines.

o
Make connections
between the different
strategies that are
presented?
Have students share their strategies about how they determined the
relationships that occur when objects are reflected consecutively over two
parallel or intersecting lines.

If students have used coordinate rules in their exploration, have students
share their results and defend their ideas to their classmates.
o
Look for patterns?
o
Begin to form
generalizations?
What will be seen or heard
that indicates all students
understand the
mathematical ideas you
intended them to learn?
Formative Assessment:
For Lesson 1, students should be asked to draw an object and a line of reflection (or mirror line), and then draw the
object that results after reflecting over the mirror line. To check progress each students could be given the same
object and line to reflect over and compare answers with one another. The students could share how they were able to
produce the reflection. This could be done using a Mira, a ruler, graph paper, paper folding and tracing, or a computer
applet. Another assessment could include giving coordinates that determine a pre-image and the equation of a line of
reflection, and have students give the coordinates of the image.
For Lesson 3, students should be asked to draw the resulting image when a pre-image is reflected over two lines, both
in the case of parallel and intersecting lines. They should also be able to give the distance from the pre-image to the
image, when using parallel lines, and the angle the pre-image moves for intersecting lines.
CCSS Curriculum
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