PROBLEM SOLVERS
D o n n a To l l a n d S h e r y l S t u m p
Terrific Trapezoids
Problem
The Terrific Trapezoid family is hosting a summer cookout for family and friends. The family members have
rented six trapezoid-shaped tables, each one the same shape as a red pattern block. They want to join all six
tables together so that a complete side is adjacent to—that is, touches—another complete side; short sides
must connect to short sides and long sides must connect to long sides.
Each short side of a trapezoid will seat one person, while each long side will seat two people. The family
wants to set up the tables for the exact number of people who will be coming to the cookout. Help the Terrific
Trapezoid family figure out the various numbers of guests they can invite to the cookout and the various ways
the tables might be arranged.
• How many different ways can you arrange the tables? How many people can be seated using each arrangement?
• What is the smallest number of people who can be seated around the six tables? What is the largest number?
• Counting consecutively, are there any numbers of people between this smallest number and this largest
number who cannot be seated?
• Is more than one arrangement possible for some of the same numbers? For example, is there more than one
way to arrange the tables to seat four people?
• If the family likes symmetry, can you arrange the tables in a symmetric design for each solution that you
found?
• Are there ways you can organize your solutions to share with the Terrific Trapezoid family? For example, you
could organize your solutions according to these two questions: What kinds of table arrangements seat the
most people? What kinds of arrangements seat the fewest people? Find other ways to organize your data.
Below are some acceptable and unacceptable ways to connect the six trapezoid-shaped tables.
Acceptable ways
Unacceptable ways
Donna Toll, [email protected], and Sheryl Stump, [email protected], teach elementary mathematics content and methods classes at Ball State University in Muncie,
Indiana.
Edited by Joyce Bishop, [email protected], Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL 61920, and Sheryl
Stump, [email protected], Department of Mathematical Sciences, Ball State University, Muncie, IN 47306. Readers are encouraged to submit problems to the
editors to be considered for future “Problem Solvers” columns. Receipt of problems will not be acknowledged; however, problems selected for publication will
be credited to the author.
40
Teaching Children Mathematics / August 2006
Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
T
he goal of the “Problem Solvers” department
is to foster improved communication among
teachers by posing one problem each month
for teachers of grades K–6 to try with their students. Every teacher can become an author: Pose
the problem to your students, reflect on your students’ work, analyze the classroom dialogue, and
submit the resulting insights to this department.
Through contributions to the journal every teacher
can help us all better understand children’s capabilities and thinking about mathematics. Remember that even students’ misconceptions provide
valuable information.
Classroom Setup
Discuss this problem with your students but avoid
giving too much guidance. Allow your students to
work with a partner or in small groups. Encourage
them to experiment with pattern blocks. Ask them
to use pictures, words, lists, or other methods to
record their work and explain their solutions. Collect students’ work, make notes about interactions
and discussions that took place, and document the
variety of student approaches that you observed. As
you reflect on your experience with the problem,
keep in mind the following questions:
• What difficulties did the students have in understanding the problem?
• What strategies did you see students using to
solve the problem?
• Were you surprised by any students’ responses
or interpretations?
• What methods did students use to record their
work?
• Did the students relate this problem to any others that they have investigated?
• What extensions to this problem did you or your
students pose?
• What did your students learn from investigating
this problem?
Share Your Student Work
We are interested in how your students responded
to the problem and how they explained or justified their reasoning. Please send us your thoughts
Teaching Children Mathematics / August 2006
and reflections. Include information about how you posed
the problem and samples of
student work or even photographs showing your problem
solvers in action. Send your
results with your name, grade
level, and school by October 1, 2006, to Sheryl Stump,
Department of Mathematical
Sciences, Ball State University,
Muncie, IN 47306. Selected
submissions will be published
in a subsequent issue of Teaching Children Mathematics
and will be acknowledged by
name, grade level, and school
unless otherwise indicated. s
Where’s the Math?
This problem requires students to collect
and organize data gathered in the context of
building and analyzing geometric figures. It
is related to ideas involving area and perimeter that students will encounter in their future
mathematical experiences—for example, the
idea that objects with the same area may have
different perimeters. A common student error
is confusing the concepts of perimeter and
area. Through their work on this problem,
students will use the concept of perimeter to
analyze table arrangements, establishing a
correspondence between the length of a side
and the number of people who can be seated
on that side. Thus, experience with problems
such as this one can help prevent or remediate students’ confusion about perimeter and
area. The next-to-last question in the list
extends the problem to include the notion of
symmetry. Indirectly, the last question elicits
generalizations about the effects of stringing
out the tables compared with the effects of
clustering them in a block.
(Solutions to a previous problem
begin on the next page.)
41
PROBLEM SOLVERS
C a r l a Ta y e h
Solutions to the
What’s the Overlap?
Problem
T
he problem appearing in the August 2005
“Problem Solvers” section was stated as
follows:
Begin with two squares. They do not have to be
the same size. Lay the two squares down so they
overlap. What shape do you get from the overlap? (See the shaded area in the illustration.) Are
there other ways to overlap the squares to get
a rectangle? A square? A triangle? What other
shapes can you find?
Mary Kay Varley of Fort Worth, Texas, reports
that when she first looked over this problem, she
was certain it would be relatively simple for her
fourth graders. She began the school year with
geometry and thought the problem would be a
good way to review the names of polygons. She
was surprised by the discussion that this problem
generated.
To prepare for the activity, Varley made transparency squares for each of her students. She
Carla Tayeh, [email protected], teaches elementary mathematics methods and content
classes at Eastern Michigan University in Ypsilanti, Michigan.
Edited by Barbara Britton, [email protected], and Carla Tayeh, Carla.tayeh@emich.
edu, at Eastern Michigan University in Ypsilanti, MI 48197.
42
started with two squares of the same size but was
prepared with squares of varying sizes.
I felt the transparencies would help the students
to see the overlap better. In hindsight, it would
have been even better if I had used different color
transparencies so the children could more easily
see the overlap. We used the overhead projector
to look at the problem together. I modeled for
the class what was meant by overlapping. The
students saw that all kinds of rectangles could be
made. They also saw squares and triangles. They
were very interested in getting their own set of
transparency squares to work with. I passed out
the transparency squares and asked them to list
the polygons that they created. I encouraged
them to both draw the picture and label the overlap with the correct name. They were very busy
rotating and sliding the transparency squares. I
asked them to take the squares home and continue to explore the problem.
The next day Varley passed out large sheets of
manila drawing paper and had the students fold
the sheets in half three times so that, when the
sheets were unfolded, the creases would create
eight cells. Each cell was given a label: rectangle,
square, triangle, quadrilateral, pentagon, hexagon,
heptagon, or octagon. Students used the cells to
record shapes under the term that best described
the shape. For example, a square by definition
is also a rectangle; however, the term square is
a more precise label for a figure that is a quadrilateral with equal sides and equal angles. “I
again posed the problem and asked the students
to organize their thinking on the manila sheets,”
wrote Varley (see figs. 1 and 2). They explored
the various shapes that could be created using the
transparent pieces and recorded their findings on
the manila sheet. Now the students were ready to
discuss what they had discovered.
Teaching Children Mathematics / August 2006
Figure 1
Ashley’s analysis of the rectangles that
she found
Figure 3
Possible solutions to the What’s the Overlap? problem
Septagon
Octagon
Right Triangle
Square
Rectangle
Trapezoid
Right Isosceles
Triangle
Figure 2
Emily’s analysis of the quadrilaterals that
she found
Hexagon
“We started with rectangles. We all agreed that it
was easy to create rectangles, and it didn’t matter if
the pieces were congruent or different sizes. I asked
them how many rectangles we could make,” Varley recounted. Shane yelled out, “A lot.” Andrew
chimed in with, “I bet we can make an infinite
number of rectangles.” “There was some discussion
about this—most of the students felt there had to be
a finite number of rectangles because of the limits
of the two figures. We decided to think overnight
on this idea and try to find the other polygons,”
wrote Varley.
The students had no problem finding squares
in the overlap, and they anticipated that Varley
would ask how many squares were possible. “The
students knew there would be a lot but didn’t feel
there would be as many squares as rectangles
because squares had to have equal sides,” observed
Varley. The students really enjoyed making triangles and concluded that they could not make an
obtuse triangle because of the right angles in the
squares. “Quadrangles had them stymied. Since
Teaching Children Mathematics / August 2006
Pentagon
Kite
they had readily found rectangles and squares,
they assumed that it would be as easy to find
trapezoids, kites, rhombuses, and parallelograms,”
Varley noted. “They could not make a rhombus or
parallelogram, and it was somewhat frustrating to
some.” In her judgment it seemed that the students
thought harder about the properties of rhombuses
and parallelograms and concluded that the right
angles in the original overlapping squares were
making it impossible to find a rhombus and parallelogram in the overlap.
inding pentagons, hexagons, heptagons, and
F
octagons in the overlapping squares was tricky
but not impossible. (See fig. 3.) We agreed that
the octagon had to be the polygon with the most
sides that we could make with two overlapping
squares. It was also noted by Emily that the
squares had to be the same size or fairly close
in size or else we could not make the octagon.
Emily explained that to make the octagon, one
square actually had to be placed directly on
top of the other and rotated. If the two squares
differed greatly in size, then one of the squares
would be swallowed up by the other, and we
would have a square within a square.
43
Varley extended the problem by asking her
students to consider what shapes could be found
by overlapping two triangles. Using triangles that
varied in size—some isosceles triangles and some
right triangles—the students discovered that they
could make a rhombus and a parallelogram.
than six sides. They didn’t think that the kind
of triangles used to overlap would change their
results that much.
Later the class continued the discussion of
infinity:
he students predicted that they could make a
T
hexagon by overlapping two triangles but they
would not be able to make a polygon with more
The class was fascinated to think that with
minute slides a new rectangle could be formed.
It baffled the students that there could be an
infinite number of rectangles and an infinite
number of squares. Their “gut” feeling was that
there certainly would be more rectangles than
squares.
This led to an interesting class discussion about
different kinds of infinity in which the class compared the infinite set of whole numbers with the
infinite set of positive even whole numbers.
Searching for shapes in the overlapping squares
reinforces the definitions and characteristics of
shapes. It was especially interesting to hear the
students reason so thoughtfully about the shapes
as they made conjectures and predictions. That
this problem also led these students to an interesting discussion about infinity was something of a
surprise.
Special thanks to Mary Kay Varley and her
fourth-grade students at Fort Worth Country Day
School in Fort Worth, Texas. s
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Teaching Children Mathematics / August 2006