EMBRACING TRANSFORMATIONAL
GEOMETRY IN CCSS-MATHEMATICS
Jim Short
[email protected]
Presentation at Palm Springs
11/1/13
Introductions
Take a minute to think about, and then be ready to
share:
 Name
 School
 District
 Something you are doing to implement CCSS-M
 One thing you hope to learn today
3
Workshop Goals


Briefly explore the Geometry sequence in CCSS-M
Deepen understanding of transformational
geometry and its role in mathematics
 In
the CCSS-M
 In mathematics in general

Engage in hands-on classroom activities relating to
transformational geometry
 Special
thanks to Sherry Fraser and IMP
 Special thanks also to CMP and the CaCCSS-M
Resources
4
Workshop Norms
1. Bring and assume best intentions.
2. Step up, step back.
3. Be respectful, and solutions oriented.
4. Turn off (or mute) electronic devices.
ATP Administrator Training Module 1 – MS/HS Math
Transformation Geometry



What is a transformation?
In Geometry: An action on a geometric figure that
results in a change of position and/or size and or
shape
Two major types
Affine – straight lines are preserved (e.g. Reflection)
 Projective – straight lines are not preserved (e.g. map of the
world)


School mathematics focuses on a sub-group of affine
transformations: the Euclidean transformations
Flow of Transformational Geometry

Ideas of transformational geometry are developed
over time, infused in multiple ways
Develop Understanding of
Attributes of Shapes
Develop Understanding of
Coordinate Plane
Develop Understanding of
Effect of Transformations on
Figures
Develop Understanding of
Transformations as Functions
on the Plane/Space
Develop Understanding of
Functions

Transformations are a big mathematical idea,
importance enhanced by technology
Geometry Standards Progression



Share the standards with your group. Take turns
reading the content standards given
Analyze the depth and complexity of the standards
Order the standards across the Progression from K
– High School
Geometric Transformations In
CCSS-Mathematics






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Begins with moving shapes around
Builds on developing properties of shapes
Develops an understanding of dynamic geometry
Provides a connection between Geometry and Algebra
through the co-ordinate plane
Provides a more intuitive and mathematically precise
definition of congruence and similarity
Lays the foundation for projections and transformations
in space – video animation
Lays the foundation for Linear Algebra in college – a
central topic in both pure and applied mathematics
Golden Oldies: Constructions



“Drawing Triangles with a Ruler and Protractor”
(p. 125-126)
Which of the math practice standards are being
developed?
How can this activity be used to prepare students
for transformations?
More With Constructions




Please read through “What Makes a Triangle?” on
p. 134-135
Please do p. 136, “Tricky Triangles”
How can we use constructions to prepare students
for a definition of congruence that uses
transformations as the underlying notion?
What, if any, is the benefit of using constructions to
motivate the development of geometric reasoning?
Physical Movement in Geometry



Each person needs to complete #1 on p. 148
Each group will then complete #2 for one of the 5
parts of #1.
What are the related constructions, and how do we
ensure that students see the connections?
Transformations



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In any transformation, some things change, some
things stay constant
What changes?
What stays constant?
What are the defining characteristics of each type
of transformation?
 Reflection
 Rotation
 Translation
 Dilation
Reflection
Is This A Reflection?
Is This A Reflection?
Reflection




Do “Reflection Challenges” on p. 168 either using
paper and pencil, or using Geometer’s Sketchpad
(or Geogebra or other dynamic geometry system)
What is changed, what is left constant, by a
reflection?
What is gained by having students use technology?
What is lost by having students use technology?
..\..\..\Desktop\Algebra in Motion\Geometric
Transformations (reflect, translate, rotate, dilate
objects).gsp
Rotations

Do activity “Rotations”
 Patty

Do “Rotation with Coordinates” p. 177
 What

are students connecting in this activity?
Look at “Sloping Sides” on p. 178.
 What

paper might be helpful for this activity
are students investigating and discovering?
..\..\..\Desktop\Algebra in Motion\Geometric
Transformations (reflect, translate, rotate, dilate
objects).gsp
Translations



Look at “Isometric Transformation 3: Translation” (p.
180)
Do “Translation Investigations” p. 183
..\..\..\Desktop\Algebra in Motion\Geometric
Transformations (reflect, translate, rotate, dilate
objects).gsp
Dilations

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Do “Introduction to Dilations”
Look at p. 189, “Dilation with Rubber Bands”
Now do “Enlarging on a Copy Machine” (p. 191192)
“Dilation Investigations” – read over and think
about p. 193
..\..\..\Desktop\Algebra in Motion\Geometric
Transformations (reflect, translate, rotate, dilate
objects).gsp
Euclidean Transformations

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What changed and what remained the same in the
four Euclidean transformations?
Complete “Properties of Euclidean Transformations”
How do we now define congruent figures?
How do we now define similar figures?