Download dilating figures

DILATING FIGURES
A stretcher is a rubber band that has been cut and then tied, creating a knot in the
middle. A stretcher also has 2 dots, one on each side of its knot and equidistant
from the knot—in this case, the dots are 2 inches from the knot. (See Figure 1.) In
the questions that follow, you will use a stretcher to create a new version of a triangle. The new version of the triangle is called a dilation of the original triangle. Stretchers are tools that can help you create dilations of any figure.
2 inches
2 inches
Figure 1
1. Follow the directions below to create a dilated version of ΔABC on Grid 1
(page 5).
a. Hold your stretcher so one of the dots is on point P.
b. Pull the other end of the stretcher so the knot is on point A of the original
triangle. (Check to make sure the dot is still directly on point P.) Now, make
a mark on the grid where the second dot of the stretcher is located. Label
this new point D. (See Figure 2.)
P
B
A
C
D
Figure 2
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c. Repeat the process you just completed, but this time move the stretcher so
the knot is on point B of the original triangle. Use the second dot of the
stretcher to help you locate point E of the new triangle.
d. Repeat this process one more time, placing the knot on point C to locate the
position of point F of the new triangle.
e. Use a straightedge to connect the 3 vertices (D, E, and F) of your new triangle.
2. Compare the original triangle (ΔABC) to the dilated version of the triangle (ΔDEF).
a. How are ΔDEF and ΔABC alike? (List 2 or 3 mathematical ways in which they
are alike.)
b. How are ΔDEF and ΔABC different? (List 2 or 3 mathematical differences.)
c. On Grid 1, draw line segments to represent the distances from the knot to
each dot for the 3 positions of the stretcher. For example, draw a segment
that connects points P and B, and then another that connects points B and
E. Do this for all 3 vertices of the triangle.
d. Compare segment PA ’s length to that of segment AD. Do that same for segment PB and segment BE . Now do the same for segment PC and segment
CF . What do you notice about the relationship between the lengths?
3. On Grid 1, draw a new point 4 grid spaces below the original point P and label it
P2. Without using the stretcher, predict what a dilated version of ΔABC will look
like and where it will land on the grid when you use P2 instead of P.
a. Use a different-color pencil or marker and draw your predicted triangle on
the grid (but don’t use the stretcher!).
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b. How did you decide where to put the new dilated triangle and what it should
look like? Describe what you thought about as you made your prediction.
c. Check your prediction. Use P2 and your stretcher to create a new dilated triangle based on ΔABC. Use a different-color pencil or marker to draw this
triangle. Was your prediction correct? Explain.
4. Take a look at the Geometer’s Sketchpad applet “Dilations” available in your
facilitator’s Fostering Geometric Thinking Toolkit DVD.1 In this file, ΔDEF is the
dilated version of ΔABC using point P as the center of dilation. Move the point P
around on the grid and explore what happens to the dilated triangle (ΔDEF).
a. What do you notice?
b. What questions come to mind?
1.
If you do not have access to Geometer’s Sketchpad software, you can find a Web-based applet appropriate for this problem
at www.geometric-thinking.org/dilations.htm.
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5. Return to Grid 1 and draw a third point P 3 grid spaces to the right of P. Label
this new point P3. Without using the applet or your stretcher, construct a new
dilated triangle based on P3 and ΔABC. Use a different-color pencil or marker to
draw this triangle. How did you decide where the new triangle should be on the
grid?
6. Maria made the dilated triangle shown on Grid 2 (page 6) using a stretcher like
yours. Someone erased her original triangle. Without using a stretcher, reconstruct her original triangle on Grid 2. Describe how you figured out what her
original triangle would look like and where it would be located.
7. Here is a drawing of the kind of stretcher you’ve been using.
Left dot
Knot
Right dot
You probably figured out that this stretcher doubles the lengths of the sides of
triangles like ΔABC. So, we could call it a 2-stretcher. Suppose you wanted to
make a 3-stretcher—that is, a stretcher that triples the lengths of the sides of
ΔABC. Where would you put the right dot for a 3-stretcher? Explain.
Left dot
Knot
Right dot
for 2-stretcher
Where would you put the right dot for a –21 -stretcher? Explain.
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P
A
C
B
GRID 1
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P
GRID 2
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Mathematical Thinking Record: Dilating Figures
The purpose of the Mathematical Thinking Record is to help you remember important mathematics aspects of this problem at some later time, when you might want
to use this problem with your class, or just to refresh your memory. Some teachers
also find that records of their mathematical thinking inform their thinking on similar problems they encounter.
1. What would you like to recall about the different strategies and/or solutions
used by you and your colleagues? Record the mathematical approaches and
strategies that you would like to remember.
2. What would you like to recall about the geometric thinking used in solving this
problem? Record the specific features of habits of mind that you have seen in
the different solutions.
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INSTRUCTIONS FOR CREATING STRETCHERS
Each student or pair of students will need a stretcher. To create a stretcher, use rubber bands 2.5 inches in diameter or greater and complete the following steps:
1. Cut each rubber band.
2. Tie a knot close to the middle of the rubber band. (The knot will be used to trace
the original figure.)
3. Use a permanent marker to place a dot 2 inches to the left of the knot, and another dot 2 inches to the right of the knot. (One of these dots will be lined up
with the anchor point, point P, and the other dot will indicate where the vertices
of the dilated figure are located.)
2 inches
2 inches
Figure 1
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PREPARING TO CAPTURE STUDENT THINKING
Problem Title: Dilating Figures
In bringing this problem to your students, the expectation is not that all students
will solve it completely and without any struggle. Rather, the goal is to bring a problem to the classroom that all students will be able to engage in enough to expose
their thinking, making it available to you to analyze.
As you prepare to use the problem with students, consider:
1. What tools or resources will best allow your students to demonstrate their geometric thinking?
2. What unclear language (e.g., unclear terms/phrases) do you anticipate your students using? How will you encourage them to clarify their language?
3. What questions will you ask students to probe their thinking?
4. What do you hope to learn from your students as they work on this problem?
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SESSION FEEDBACK
Please reflect on the following questions about today’s session.
1. Which Geometric Habit of Mind (GHOM) was most prominent today in your
roundtable discussion? How did the GHOM support teachers’ work on this
problem?
2. How did the applet affect your understanding of dilations?
3. How do you feel about taking this problem to your students?
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HOMEWORK IN PREPARATION FOR SESSION 9: PART 1
Everyone should use the Dilating Figures problem with some or all of their
students. You will need to make stretchers for your students—instructions are in a
separate handout.
Teachers volunteering to collect five- to ten-minute video clips for the Dilating
Figures problem will collect video of three or four students working together. In
contrast to the short video clips with accompanying transcripts that focused on a
particular exchange between students, the student artifacts for Session 9 will be
longer clips (five to ten minutes), capturing the evolution of students’ thinking over
time with regard to their: (1) use of one or more of the GHOMs and/or (2) understanding of a particular mathematical concept.
Teachers volunteering to collect video should follow the instructions below.
1. Make sure you have a blank copy of the problem to make copies for your students. Also, make sure you have several copies of the Fostering Geometric
Thinking Video Release Form.
2. Send the Fostering Geometric Thinking Video Release Form home with several
students a few days before you plan to videotape.
3. Videotape the three or four students working together on Dilating Figures. If
possible, the students should be isolated in a quiet area (such as a library, separate room, etc., away from other students) in order to maximize the quality of
the video.
4. Review the video and locate a five- to ten-minute segment you find to be most
representative of productive geometric thinking.
5. Be sure to mark the location of this segment within the video (e.g., if you are
using a VHS tape, set the video to the beginning of the clip before submitting it,
or write down the location of the clip by referring to the counter on a VCR or
time bar on digital video).
6. Write a few notes about how/why you selected this clip:
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
7. Submit this sheet and your video to your facilitator by: _____________________.
Teacher’s Name: ________________________________________________________
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HOMEWORK IN PREPARATION FOR SESSION 9: PART 2
Geometric Transformations Research Summary
What Are Geometric Transformations?
As mathematicians define them, geometric transformations in two dimensions usually map every point in a plane. Many different types of geometric transformations—
or mappings—exist, each altering some geometric properties of figures while
preserving others. Euclidean transformations—translations, reflections, and rotations—map points such that the original figure’s position and/or orientation are
altered while its shape and size are preserved. The mapping process is different for
other, non-Euclidean, transformations. Dilations, for example, map points such that
a figure’s size and position change while its orientation and shape do not. Figure 1
depicts the impact on a triangle of the three Euclidean transformations, as well as
dilation.
Transformations like these four refer to operating on entire planes. Although
some points in a plane appear dark in color and others may be void of color and not
visible, both visible and invisible points are mapped in a geometric transformation.
It is the entire plane of points that is transformed, not only points that combine to
create visible shapes or images. Figure 2 represents this idea of performing these
geometric transformations on entire planes.
Translation
Reflection
Rotation
Dilation
Figure 1
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Translation
Reflection
Rotation
Dilation
Figure 2
How Do Middle-Grades Students Define Geometric Transformations?
Edwards (1991) examined middle-grades students’ interactions with a computer program that afforded them an opportunity to experiment with geometric transformations. Her analysis of their behavior revealed that one-third of the students operated
as if only a pictured figure, rather than the entire plane, was transformed. Interviews
with these students indicated that they thought about transformations as physical
motions of geometric figures rather than the mapping of an entire plane. For the most
part, translations, reflections, rotations, and dilations are the transformations that
present this confusion for middle-grades students. Table 1 more fully delineates the
differences between these students’ understanding of these transformations and the
mathematical definition.
Students who interpret these transformations as the physical motion of geometric figures are often served well by their conceptualization. For example, when centers of rotation and lines of reflection lie on the shape or figure, the position,
orientation, size, and shape of the transformed figures are identical whether
students map the entire plane or just the figure. It is when centers of rotation and
lines of reflection are elsewhere in the plane, however, that students with this understanding have difficulty determining how to perform the transformation. Edwards
found that when she asked students to use a center of rotation outside the boundaries of a figure, many translated the figure to the point of rotation before performing the rotation. As Figure 3 shows, a transformation performed this way and one
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Table 1
How Some Middle-Grades Students Think
About Transformations
Mathematical Definition
of Transformations
• The plane is an empty, invisible background.
• The plane is a set of points.
• Geometric figures lie on top of the plane.
• Geometric figures are subsets of points
within the plane.
• Transformations are physical motions of
geometric figures on the plane.
• Transformations are mappings of every
point in a plane.
performed by rotating the entire plane about the point yield figures situated in very
different locations in the plane.
Translation then rotation
Rotation
Figure 3
Conveying the idea that figures reside in planes and that it is the plane that is
transformed, not just the figure, is especially important when centers of rotation
and lines of reflection lie outside of the boundaries of a figure. You may want to
look for ways to present the notion of a plane to students when working with these
types of transformation problems. Transparencies work really well for exploring rotations around distant points.
Regardless of whether students understand transformations to be actions carried out on figures or mapping entire planes, performing transformations can be a
cognitively challenging task. This is true for the mapping kinds of transformations
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like translations, reflections, rotations, and dilations, as well as more localized
transformations like dissections. A number of variables factor into the difficulty of
performing a transformation. Those variables are outlined in the following section,
which describes the cognitive processes involved in performing geometric
transformations.
How Do Students Perform Geometric Transformations?
How students perform geometric transformations depends to a great extent on:
1. the amount of visual support provided
2. the complexity of the geometric figures
3. students’ strategy preferences
The amount of visual support students are given when asked to perform transformations can vary. Some problems allow solvers to physically manipulate geometric
figures and, therefore, observe the figures throughout the transformations; others
require that the transformation is performed mentally. Consider the two problems
below:
■
Create a rectangle with the square and 2 small triangles from a set of tangrams
(see Figure 4). Decide the transformation(s) you would perform to change the
rectangle into a right triangle.
Figure 4
■
Reflect this triangle over the x-axis and draw the resultant triangle (see Figure 5).
8
6
4
2
–5
5
–2
–4
–6
–8
Figure 5
In the first problem, students can physically work with the tangram pieces as
they consider possible transformations. However, in the second problem, students
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have a visual display of the original figure but must perform the transformation and
keep track of the shape’s orientation and side lengths in their minds. Research by
Kidder (1976) has demonstrated that middle-grades students’ performance on
transformation tasks is markedly affected by the amount of visual support provided,
such that tasks become more challenging as the level of visual support decreases.
This suggests that most students would find the second problem presented above
more challenging than the first because of its stronger reliance on spatial skills.
Students’ abilities to work with mental representations are assessed to varying
degrees, depending on the level at which problems are posed. When posing transformations problems to students, you may want to consider how you want students
to be working with transformations at that particular point—physically or mentally.
The complexity of figures and students’ strategy preferences are two factors, in addition to
the amount of visual support, that affect students’ performance on geometric transformations. Figure complexity and strategy choice, together, determine the type of
approach students will employ during the transformation process. A holistic approach
is defined as one in which mental operations are performed on complete figures,
and an analytic approach is an approach in which the components of a figure are
transformed separately. Most middle-grades students employ the holistic strategy to
perform mental transformations when figures are simple. As the figures become
more complex, those students most successful in accomplishing transformations
tend to switch to an analytic approach (Rosser 1994). With the analytic approach to
problems, students convert figures into sets of features to be manipulated both spatially and logically. For example, a right triangle would be thought of in terms of its
three angles and sides. Students might focus in on a particular feature and ask themselves something like, “What should happen when I reflect a right angle pointing left
across a vertical line?” With this strategy, the task becomes less spatial and more
verbal and logical than it was with the holistic approach.
Although a figure’s complexity is certainly important, the approach students will
take to a transformation problem is not solely determined by the complexity of the
figure at hand. Some students express a preference for the holistic or analytic approach and, therefore, gravitate toward that strategy more often than figure complexity would dictate (Boulter and Kirby 1994).
The different ways of approaching mental transformations are rarely presented
to students, with most students who use the analytic approach discovering it on
their own. Students who struggle with transformations may benefit from an explicit
instruction related to this strategy.
In Conclusion . . .
■
Geometric transformations like translations, reflections, rotations, and dilations
involve mapping every point in a plane, not just those points that make up figures. Many students find this a difficult concept to understand.
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■
■
Not all problems that ask students to perform transformations assess the same
skills. Students’ abilities to work with mental representations are assessed to
varying degrees, depending on the level at which problems are posed.
The complexity of a figure impacts the approach students will take in performing
mental transformations. Most students use the holistic approach when translating simple figures. As figures increase in complexity, however, those students
who are most successful adopt an analytic approach. They attend individually to
different components of the figure and impart logic in addition to using spatial
reasoning.
References
Boulter, D. R., and Kirby, J. R. 1994. “Identification of Strategies Used in Solving Transformational
Geometry Problems.” Journal of Educational Research 87: 298–303.
Edwards, L. D. 1991. “Children’s Learning in a Computer Microworld for Transformational Geometry.”
Journal for Research in Mathematics Education 22: 122–37.
Kidder, R. F. 1976. “Elementary and Middle School Children’s Comprehension of Euclidean Transformations.” Journal of Research in Mathematics Education 7: 40–52.
Rosser, R. A. 1994. “Children’s Solution Strategies and Mental Rotation Problems: The Differential
Salience of Stimulus Components.” Child Study Journal 24: 153–68.
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MATH NOTES: DILATING FIGURES
Dilation is a transformation that magnifies or shrinks figures in the plane to make
geometrically similar figures. (In vector space terminology, dilation involves “scalar
multiplication.” Positive scalars greater than 1 magnify; positive scalars less than 1
shrink.) In this problem, we focus mainly on dilations that magnify, mostly by a dilation factor of 2. The last part of the problem focuses on factors of 3 and –12 .
A prominent concept underlying the problem is geometric similarity, a very important mathematical concept. Similarity is, for example, a fundamental relationship behind the algebraic ideas of linear function and slope. However, the problem
does not stress the formal mathematical definition of similarity. Rather, it asks the
solver to Investigate invariants, comparing figures to determine what has changed and
what has stayed the same. We believe that pushing on this habit of mind, in the
context of solving the problem, allows the solver to engage more deeply with the
concept of similarity. Having said that, we encourage you to use the problem as an
opportunity (perhaps after the problem solving is completed, or in a future class) to
sharpen in your students’ minds the definitions of similarity and congruence.
The Importance of Precise Definitions
Sometimes, congruence is defined for middle graders as “same size, same shape,” and
similarity is defined as “same shape, not necessarily same size.” These are imprecise
and, for some students, may produce faulty conceptions. For example, two students
may have different conceptions of what “same shape” means. Consider the following two figures (see Figure 1). The student whose conception of “same shape” allows reflections will think of these as congruent; the student whose conception
does not allow reflections will not.
Figure 1
A comparable dilemma dogs the informal definition of similarity. One student’s
conception of “same shape” may not match another student’s conception. (See Figure 2.) Besides different orientations of figures, other features may lead to different
meanings for “same shape.” For example, a 3-4-5 triangle and a 5-12-13 triangle are
both right triangles, so does that mean they have the same shape? In some
students’ minds, it may mean that.
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Figure 2
We believe that students should be given more precise definitions, to enhance
their Reasoning with relationships, as well as other GHOMs. For example, two figures are
congruent if one can be obtained from the other by a sequence of rotations, translations, and reflections. And two figures are similar if one is congruent to a dilated version of the other. These definitions presume that students have experience with
rotations, translations, reflections, and dilations.
Understanding Similarity Through Dilations
An important line of inquiry in parts 2a and 2b of this problem asks the solver to
describe differences between the original figures and the dilated figures. Answers—
particularly students’ answers—can reveal the solvers’ focus when thinking about
geometric scaling. Generally, answers will concentrate on linear effects, as in “all
the sides get doubled.” Even though the problem itself doesn’t try to push that further, you may want to ask: “What is the effect on the area inside the triangle?” By
focusing problem solvers’ attention on comparisons of various attributes in figures,
the problem and your questions provide rich opportunities to develop Reasoning with
relationships.
We have found that parts 3 and 5 of the problem push on solvers’ skills in visualizing, in two dimensions. This is valuable experience. In responding to part 6,
solvers will be asked to do some undoing as they use what they have learned about
the effects of transformations on figures to determine what the original figure might
have looked like. Particularly for middle-grades students, this can elicit Balancing
exploration and reflection to locate the original figure (e.g., estimating a location), then
taking stock to see that the size of the figure is correct, that it is oriented correctly,
that it lines up appropriately between P and the given triangle on Grid 2, then trying
another location, if necessary. Students also demonstrate the Investigating invariants
GHOM in this problem, as evidenced by the backup video for Session 9. One of the
students in it considers what changed and what stayed the same during a dilation
transformation. His statement, “You know how similar shapes are . . . they have
everything the same but like their size is different, right?” suggests he understood
area and/or perimeter (size) changed, but other attributes, like angle size, stayed
the same.
We have found that solvers—both adult and adolescent—sometimes seem to
apply the wrong metaphor to dilation. They appear to think of dilating a figure in
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terms of what happens when light projects a figure onto a screen. There, too, the
size and location of the original figure can vary, as can the location of the projecting
light source and its distance from the original figure. The piece that makes the
metaphor inappropriate is the dilation factor. It is tempting for the solver to imagine that when the anchor point moves closer to the original figure, the resulting
figure will be larger. Of course, keeping the dilation factor constant ensures that the
resulting figure will be twice as large, though now in a different location. This becomes apparent when using the applet we make use of in part 4 of the problem.
As presented, the problem mainly concerns dilations that double, or factor-2
dilations. In part 7 we introduce other factor dilations, a 3-dilation and a –12 dilation.
If you like, you can go one step further and combine dilations (e.g., if I apply a 2dilation to a triangle, then apply a 3-dilation to the resulting triangle, is the end
result the same as I’d get from a 6-dilation? What if I apply a 2-dilation to a triangle,
then apply a –12 -dilation to the result? What dilation is that equivalent to?).
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