UNDERSTANDING
GEOMETRIC
TRANSFORMATIONS
The Field Guide to Geometric Transformations
and
Geometric Transformation Workouts
Patrick Callahan, CMP
Denver, CO October 7, 2010
Please individually work on GTW#11. After you complete it for yourself, think about how
students would answer the questions.
Learning and Teaching Geometry
Project Overview
• In year four of a 5-year National Science Foundation project
• Developing videocase-based, PD materials
– 1 Foundation Module
– 4 Extension Modules
• Staff: Nanette Seago (PI), Mark Driscoll (Co-PI), Jennifer Jacobs,
Johannah Nikula, Patrick Callahan, Hilda Borko
• Advisory Board:Harold Asturias, Tom Banchoff, Phil Daro, Megan
Franke, Karen Koellner, Glenda Lappan, Hung-Hsi Wu
• Evaluation Team: [Horizon Research, Inc.] Dan Heck, Kristen Malzahn,
Courtney Nelson
Plan for this hour
•
•
•
•
•
A little background
We will introduce the GTWs and Field Guide
Explore sample items
Analyze the design and use of these tools
Look at some video clips of students using the
GTWs
Background
• The focus of the LTG project is on developing a
robust understanding of similarity via a
dynamic definition of similarity (i.e. in terms
of geometric transformations)
• After the first year of the project we
discovered that most students (and many
teachers) did not have experience with
geometric transformations.
Background
To remedy this situation, inspired by work of
Franke and others on Highly Leveraged
Practices like “number talk,” we designed
some tools that could be used without
requiring much time nor be heavily dependent
on curriculum or pacing guides.
These became the Geometric Transformation
Workouts.
Background
Then it became clear that the GTWs themselves
needed more support. So, inspired by the
laminated fish identification guides you can
use while snorkeling, we created the Field
Guide to Geometric Transformations.
Evidence of Understanding
There is no single piece of evidence that can “prove”
understanding. Rather, one builds a case for understanding
via a preponderance of evidence.
The GTWs are a tool for
-eliciting evidence of understanding of geometric transformations
-providing experience and feedback for students to develop a
robust understanding of geometric transformations
-providing opportunities to make visible a variety of types of
evidence of student understanding of geometric
transformations
Types of Evidence of Understanding
Here are some examples for understanding translations (not in
any specific order):
Design Principles For GTWs
These are designed to be “workouts” in the sense that they
are not meant to introduce new concepts, but rather allow
for practice and refinement around the ideas, definitions
and language of geometric transformations.
These workouts are designed for students to work individually
on them for a short time and then have a teacher led
discussion with the class. The questions are often multiple
choice or simple drawings. They are generally
straightforward questions and are not meant to be lengthy
explorations. The following format is suggested:
o Students work individually for 5 minutes on the GTW
o Teacher leads whole class discussion for 5-10 minutes
FAQ: Is it OK to use these in other ways? A: Yes! We are still exploring what works.
Using the GTWs
• The primary goal is to elicit evidence of
understanding and develop robust definitions
and language around geometric
transformations.
• A secondary goal is to provide experiences
working with these transformations
emphasizing the visual and dynamic aspects
of the transformations.
Just determining who got the correct answer, or asking for explanations without
discussion or comment is not the designed use of these tools.
Exploring the GTWs
• Discuss the items from GTW#11 (focus on
evidence of understanding!)
NAEP item examples…
The 2007 8th grade NAEP item below was classified as “Use similarity of right
triangles to solve the problem.”
Does a Dynamic Approach Help?
The 2007 8th grade NAEP item below was classified as “Use similarity of
right triangles to solve the problem.”
Only 1% of students answered this item correctly.
The 1992 12th grade NAEP item below was classified as
“Find the side length given similar triangles.”
Only 24% of high school seniors answered this item correctly.
The 1992 12th grade NAEP item below was classified as “Find the side length given similar
triangles.”
8
A rotation and a
dilation show the
corresponding sides
of the similar
triangles.
5
6
8
5
6
8
x
12.8
Fig ure A
Fig ure B
Only 24% of high school seniors answered this item correctly.
Common Core Standards
Here is an excerpt from the new Common Core Standards for 7th Grade Mathematics:
1. Verify experimentally the fact that a rigid motion (a sequence of rotations, reflections, and translations)
preserves distance and angle, e.g., by using physical models, transparencies, or dynamic geometry
software:
(a) Lines are taken to lines, and line segments to line segments of the same length.
(b) Angles are taken to angles of the same measure.
(c) Parallel lines are taken to parallel lines.
2.
Understand the meaning of congruence: a plane figure is congruent to another if the second can be
obtained from the first by a rigid motion.
3.
Verify experimentally that a dilation with scale factor k preserves lines and angle measure, but takes a line
segment of length L to a line segment of length kL.
4.
Understand the meaning of similarity: a plane figure is similar to another if the second can be obtained
from the first by a similarity transformation (a rigid motion followed by a dilation).
The basic geometric transformations
•
•
•
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Translations
Reflections
Rotations
Dilations
• Congruence
• Similarity
The Field Guide to Geometric
Transformations
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•
•
Look at the Field Guide
Consider the design features
(note: your copies are B&W, color is an important part of the design)
Learn tools, techniques, and activities to elicit, document and
analyze student understanding of the mathematics.
1.
Watch a video clip
2.
BEFORE discussing: each person write down at least one specific
piece of evidence of understanding.
3.
Discuss/analyze the evidence with small group
CLIPS:
Abby:
vid005,
2:30 - end
Will:
vid018,
4:31 - end
Dominick:
vid007,
11:05-12:50, (plus last 10sec)
Looking at some student data
1. Which of these shows a pair of figures equivalent by a rotation of the square?
a
b
c
d
Looking at some student data
1. Which of these shows a pair of figures equivalent by a rotation of the square?
a
2%
b
86%
16%
c
0%
d
More student data
2. Which of these pictures shows two objects equivalent by a reflection in the dotted line?
a
b
c
3. Are these two figures are equivalent by the following nontrivial transformations?
Mark YES or NO for each.
YES or NO: Translation
YES or NO: Reflection
YES or NO: Rotation
YES or NO: Dilation
d
More student data
2. Which of these pictures shows two objects equivalent by a reflection in the dotted line?
27%
6%
55%
2%
a
b
c
3. Are these two figures are equivalent by the following nontrivial transformations?
Mark YES or NO for each.
YES or NO: Translation
YES or NO: Reflection
YES or NO: Rotation
YES or NO: Dilation
29%,
80%,
57%,
29%,
d
71%
20%
43%
69%
More student data
4.
The same trans lation that
translates triangle P to triangle Q
would translate square X to which
square?
a
P
b
X
Q
d
e
c
More student data
4.
The same trans lation that
translates triangle P to triangle Q
would translate square X to which
square?
8%
a
P
b
22%
X
Q
35%
c
d
e
16%
6%
Next steps