Integrating
Technology into
Mathematics (6-12)
MELT 2015
Appalachian State University
Kayla Chandler
DAY 3
Agenda
Time
Activity
8:30 – 10:00
• Exploring Properties using GSP
10:00 – 10:15
Break
10:15 – 11:45
• Transformations
11:45 – 1:00
Lunch
1:00 – 2:30
• Transformations
2:30 – 2:45
Break
2:45 – 4:15
• Lesson Planning Work Session
4:15 – 4:30
Wrap up
SESSION 1
Exploring Properties with GSP
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Open the ExploringProperties.gsp file and
identify the properties of the five pentagons.
Record the properties in the chart provided
and generate a definition for each figure.
Questions to Consider
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Reflect on your thinking processes when
generating your definitions. What properties did
you consider important or not important? Did
you create definitions that did not work? How did
you know if a definition “worked” or not?
Compare your definitions with the definitions
created by others. Consider similarities,
differences, strengths, and weaknesses in the
different definitions.
Questions to Consider


Describe the benefits and drawbacks of allowing
students to interact with a constructed figure in a
DGE to generate their own definitions versus a
teacher providing a formal definition to students.
When students are generating their own
definitions, describe how a teacher can bring
these different ideas together so that the class is
eventually working from a single definition.
Partitional vs. Hierarchical
Definitions
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Partitional definition: considers concepts
disjointly from one another; exclusive
definition
Hierarchical definition: defined as a subset of
a more general concept; inclusive definition
Questions to Consider

Consider the following definitions of a trapezoid:
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Definition 1: A trapezoid is a quadrilateral with exactly one pair of parallel
sides.
Definition 2: A trapezoid is a quadrilateral with at least one pair of
parallel sides.
How would you categorize each definition, partitional or
hierarchical? Why?
Why might students prefer partitional definitions to
hierarchical definitions?
When students are generating their own definitions,
describe how a teacher can bring these different ideas
together so the class is eventually working from a single
definition.
Questions to Consider


Look at your properties for the unknown
pentagons. Determine which properties are
“inherited” from other properties. Use this
information to create a hierarchical classification
of these pentagons.
Describe how you could apply this same
approach you experienced to have students
investigate properties of quadrilaterals and
generate the accompanying sketch.
SESSIONS 2 & 3
Geometric Transformations


When you hear this term, what comes to
mind?
What geometric transformations might you
use to describe the dancers?


http://www.youtube.com/watch?v=tyZeGOsR9IA
http://tumteresources.wikispaces.com/file/view/Da
ncing+stick+figure.avi/338311984/Dancing stick
figure.avi
Translations


Create a stick figure in GSP.
Translate using polar coordinates

5 cm, 0 degrees
Questions to Consider

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What is true about the size and shape of the
two stick figures?
Use the Arrow tool to drag a point on the leg
of the original stick figure (pre-image). Try
dragging a point on the arm of this stick
figure. What else changes when you perform
this dragging? Why?
Questions to Consider

Measure the distance between the endpoint
of the left leg of the original stick figure and
the corresponding point on the left leg of the
new stick figure. Measure several other
distances for pre-image/image pairs of points.
What do you notice? Explain.
Questions to Consider


Translate your stick figure to create an entire
chorus line of dancers that are equidistance
apart. Describe how you chose to do this.
A student in a high school class asks what
the 5 and 0 represent (the values we typed
into the polar coordinate dialog box to
perform the translation). How do you
respond?
Translations
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Open a new sketch and create a new stick
figure.
Construct a line segment and label it AB.
Translate by marked vector.
Questions to Consider
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Translate the second stick figure to have
three stick figures on the screen. What do
you notice about the size and shape of the
three stick figures?
How is the location of the stick figures related
to the translation vector?
Questions to Consider


Drag the line segment representing the
translation vector so that the tail corresponds
with a point on the original stick figure. Where
is the head of the vector located? Why?
Use the segment tool to create a segment
that joins a point from the original stick figure
to its corresponding image point. What do
you notice about this segment and the
vector?
Questions to Consider
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Measure several distances of preimage/image points as you did on your
previous sketch. How do these distances
relate to one another? How do they relate to
the vector AB?
What do you think will happen when you
translate a line segment? What about a line?
What do you think will happen when you
translate an angle?
Questions to Consider
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What do you think will happen when you
translate two lines that are parallel to each
other?
You have had the opportunity to perform
several different translations. Based on your
experiences, describe a translation. Be sure
to explain what stays the same and what
changes.
Questions to Consider

Students often have difficulty reasoning about
vectors. Because representations of vectors
and rays are very similar, students confuse
these two objects. Describe how you could
assist students in understanding the
differences between vectors and rays.
Questions to Consider

How does this introduction of translations in a
dynamic environment, using dancing and
stick figures, compare with how you teach
translations? What are the benefits and
drawbacks of this approach?
Reflections

Partner dances – moves mirror each other
Questions to Consider

Explain how you could use the picture above
to explain to students that a translation would
not be the appropriate transformation to use
to describe the positions of the dancers.
Reflections
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In a new sketch create a stick figure and a
line or line segment AB.
Select AB and choose “Mark Mirror” from the
Transform menu. (Or double click AB.)
Select entire stick figure and select “Reflect”
from the Transform menu.
Questions to Consider
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What do you notice about the size and shape
of the two stick figures?
Drag a point on the leg of the original stick
figure. What happens? Why? Drag other
points of the pre-image or image of the stick
figure.
Predict what will happen if you drag an
endpoint of the mirror line segment (point A
for example). Drag point A. What happens?
Why?
Questions to Consider
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Predict what will happen if you drag point B.
Drag point B. What happens? Why?
Create a segment joining the hands of the
two stick figures as shown below. What do
you notice about this segment and the line of
reflection?
Based on your interactions with the sketch,
provide a definition of a reflection.
Questions to Consider


Describe at least three different properties of
reflections.
In order for students to develop an
appropriate definition of reflection from this
activity, it is imperative they focus in on the
important properties of reflection. How can a
teacher focus students on the relevant
properties and assist students in
understanding which features of the diagram
are not important to consider?
Questions to Consider

A fixed point for a transformation is when the
point is mapped to itself. Did you encounter
any fixed points while you were dragging
points around in the sketch? Describe fixed
points in terms of input and output.
Questions to Consider

Let’s consider fixed points from an algebraic
context. Describe a linear function for which
there are infinite fixed points. Describe one or
more linear functions for which there are zero
fixed points. Describe one or more linear
functions that have exactly one fixed point.
Questions to Consider

Sometimes young students are taught to
think about reflections as flips. What
properties of reflection are highlighted by
thinking about reflections as flips? What
properties of reflection are not made as
explicit when considering reflections as flips?
Rotations
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Dancers have a vertical axis of rotation (3
dimensions).
We will use an aerial view of synchronized
swimmers to think about rotations in 2
dimensions.
Rotations
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Open a new sketch and create a stick figure
to model one of the swimmers in the picture.
Create point C (not on your figure)
Mark C as the center of rotation by selecting
it and choosing “Mark Center” from the
Transform menu (or double clicking it).
Select the entire stick figure and choose
“Rotate” from the transform menu.
Enter 60 degrees and select “Rotate”.
Questions to Consider
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Consider the pre-image and image stick
figures. Create a description of the
relationship between the two stick figures
using a synchronized swimming scenario.
What do you notice about the size and shape
of the two stick figures?
Drag point C and describe what happens to
the two stick figures. Explain.
Questions to Consider
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
Drag a point on one of the stick figures and
describe what happens to the other stick
figure. Does it matter which stick figure you
drag? Explain.
If you constructed a circle with a center at
point C that also passes through a point on
the original stick figure, what other point will
the circle pass through? Use your DGE to
test your hypothesis.
Questions to Consider
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Describe at least three different properties of
rotations.
Are there any fixed points under a rotation?
To perform a rotation, most DGEs require
that you input a particular angle measure.
Describe how this design feature of the
technology may influence student thinking
about rotations.
Questions to Consider

Are there particular angles of rotation that
would be more or less helpful to use with
students? Explain.
Mystery Transformations

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Open the MysteryTransformations.gsp file.
Drag the points to determine what
transformation was used to create each
image point.

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Translation: describe the translation vector
Rotation: identify the center of rotation and the
angle of rotation
Reflection: locate the line of reflection
Questions to Consider
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Describe your strategies for identifying each
of the mystery transformations.
Which transformation was the most difficult to
determine? Why?
SESSION 4
Planning Session

This session will be devoted to helping you
plan your technology lesson.