Understanding Congruence, Similarity, and Symmetry Using Transformations
and Interactive Figures:
Composing Transformations
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Composing Transformations |
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Rotations; translations,
or slides; and reflections, or flips, are geometric transformations that
change an object's position or orientation but not its shape or size.
The interactive figures in this four-part example allow a user to manipulate
a shape and observe its behavior under a particular transformation or
composition of transformations. In the first part, Visualizing Transformations,
one can choose a transformation and apply it to a shape to observe the
resulting image. In the next part, Identifying Unknown Transformations,
the user is challenged to identify the transformation that has been used.
In Composing Reflections, users can examine the result of reflecting a
shape successively through two different lines. And in this part, Composing
Transformations, the users are challenged to compose equivalent transformations
in two different ways. Activities like these allow students to deepen
their understanding of congruence, similarity, and reflection, and they
also contribute to the study of transformations, as described in the Geometry
Standard.
Task
Use the icons from
the upper left panel to choose up to three transformations to be successively
applied to the red shape. The black shape shows the resulting image. Use
the icons from the lower left panel similarly. The blue shape shows the
results from successively transforming the red shape using the lower left
panel. Consider the red shape in the interactive figure below. Drag it
and observe the behavior of its image after one or more successive transformations
are applied using the top left panel. Choose a different shape and observe
the behavior of its image under the same transformation or composition.
Change the shape of the red square or red triangle by dragging it from
an edge or vertex while pressing the Control key. Change their orientation
by dragging them from a corner. Which single transformation or composition
of transformations, if any, would have the same effect on the original
figure? Use the lower left panel to test your conjecture.
[How
to Use the Interactive Figure]
[Stand-alone
applet]
Discussion
Using
dynamic geometry software, teachers can ask students to consider what
happens when transformations are composed. Teachers can then ask students
to make conjectures about which single transformation, if any, would
have the same effect on the original figure as the composition does.
A challenging assessment of students' understanding of transformations
can be given if two congruent shapes are given and students are asked
to specify a transformation or composition of transformations that will
map one to the other.
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Take
Time to Reflect
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What
new insights into transformations can students gain as they
work on activities like this?
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What
are specific ways in which middle-grades students can identify
each of the transformations or composition of transformations
that would have the same effect on the original figure as
the composition does?
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What
are some ways in which teachers can assess students' understanding
of transformations?
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Composing Transformations |
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