Connecting the Threads of Area and Perimeter
Author(s): Jordan T. Hede and Jonathan D. Bostic
Source: Teaching Children Mathematics, Vol. 20, No. 7 (March 2014), pp. 418-425
Published by: National Council of Teachers of Mathematics
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Connecting the
of Area and Perimeter
See how sixth-grade students
design and create quilt squares
for this geometry project.
By Jordan T. Hede and Jonathan D. Bostic
hat is the point of learning this?”
“When will I ever use this again?”
I (Hede) frequently heard these two questions from my
students during a math unit focused on area and perimeter. They consistently asked why it was important to know
the formula for the area of polygons, connections between
formulas, and when to use formulas. Students asked these questions
because they did not value learning to apply their geometry knowledge
in a variety of situations. I realized they needed (1) more meaningful
experiences learning about area and perimeter and (2) understanding
the importance of these concepts. They deserved the opportunity to
experience a worthwhile task that might help them later use their content
knowledge while problem solving.
Their questions also concerned me because my students perceived
problem solving as separate from doing mathematics, a perception that
had been influenced by a greater focus on exercises than on problem
solving in their previous classes (Schoenfeld 2011). Instruction should
allow students to regularly engage with problems that require synthesis
and the creation of novel products (Kilpatrick, Swafford, and Findell
2001). These opportunities to learn through rich tasks ought to encourage
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419
children to (1) adapt to novel situations,
(2) apply efficient and effective strategies, and
(3) collaborate to solve a problem (Bostic and
Jacobbe 2010). To meet these goals, I designed a
project for my students that supported them in
synthesizing their knowledge about perimeter
and area formulas. The Quilting task described
here encourages students to use their imagination to create a quilt square while engaging with
the mathematics found in the Common Core
State Standards for Mathematics (CCSSI 2010).
Classroom expectations
Approximately twenty sixth-grade students
regularly meet in each section of my mathematics classes. Since the beginning of the school
year, they have practiced doing mathematics
independently, in pairs, and in groups of three
or four students. They are accustomed to assisting peers who need help, without expressing
judgment toward one another. Two expectations
Tips for implementing
the Quilting task
Teachers who plan to use this task in their classrooms may find the
following suggestions helpful.
• Before teaching this lesson, provide meaningful opportunities and
contexts for students to explore formulas for the area and perimeter of
different polygons and connections between the formulas. This lesson
supported learning mathematics for problem solving and was designed
to give students experiences with applying knowledge of area and
perimeter of polygons that they had more formally examined during
prior instruction.
•Formatively assess students’ work throughout the project. While formatively assessing students’ progress, have specific goals, such as the ways
in which they engage with the Standards for Mathematical Practice.
To observe students’ progress, walk around the classroom as they are
working. If a student is struggling, immediately help the individual.
Students’ work on the project also showed me the strategies they knew
to calculate area and perimeter. Thus, the project acted as a way to
formatively evaluate their understanding at the end of a unit without
giving another test.
• Encourage creativity and promote self-esteem. I urged students to use
any strategy to calculate area and perimeter and during the design
phase. This seemed to enhance their feelings of self-esteem. One student initially asked me not to display his project because he thought it
looked bad. His project was unique, and his use of strategies to calculate area and perimeter were correct. After I admired his creativity and
hard work, he wanted me to place his project among his peers’ work,
displayed outside our classroom.
in my classroom are that students should ask
questions of their peers if they need assistance
and that students should critique the reasoning
of their peers to promote growth. I also expect
students to share an idea that is still under investigation, even if it is incorrect. Finally, they are
expected to use mathematical reasoning when
giving explanations. These expectations align
with two of the Standards for Mathematical
Practice (CCSSI 2010). A student’s typical time
in my classroom involves independent work as
well as student-to-student small-group collaboration on a worthwhile task to learn mathematics concepts and procedures. I continuously
voice that our classroom is a safe environment
for students to express their ideas during smallgroup work and whole-class discussions. This
encouragement to be creative and use their
imagination while working leads to interesting
comments about connections between mathematics concepts as well as to remarkable student work products.
Purpose of the task
The aim of this lesson was to give students an
experience using area and perimeter to create
a unique quilt square. Students did not necessarily become master quilters, but the task’s
context drew on a situation that was familiar to
them. By the end of the activity, they could do
the following:
• Calculate the area and perimeter of different
regular and irregular polygons
• Justify that summing every shape’s area is
equal to the area of the quilt square
• Design a quilt square using only quadrilaterals and triangles
• Create a quilt square that matched their
design
This project aligns with sixth-grade Common Core geometry topics and was completed
during one academic week. Specifically, it
addresses aspects of Standards 6.G.A.1 and
6.G.A.3 (CCSSI 2010). The task requires only a
few materials and two handouts (see the online
appendix), including grid paper and one piece
of 8 × 8 inch cardstock for each student, a classroom set of glue and scissors, and books of wallpaper samples. A local hardware store provided
discontinued wallpaper sample books for us.
420 March 2014 • teaching children mathematics | Vol. 20, No. 7
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Figu r e 1
Students discussed the visually pleasing contrasting colors
of the quilt squares and the pattern characteristics of lateral,
rotational,
translational, or glide reflective symmetry.
Figure 1. Sample quilts and quilt squares shown to students.
Implementation of the task
Students had learned how to find the area and
perimeter of different polygons during a previous week; now they were asked to complete all
four instructional phases of this project.
Phase 1
(about 45 minutes)
I prepared my students by showing them
examples of real quilts as well as pictures of various quilt squares like those in figure 1. I asked
them to share what they noticed in the quilt
squares. They talked about common patterns
and mathematical features within the quilts and
quilt squares. These comments helped initiate
a whole-class discussion that touched on various topics students had learned previously in
math class and art class. For example, students
mentioned that the quilt squares tended to be
symmetrical. That is, students were able to talk
about the characteristics of a square and that the
pattern within the quilt square showed lateral,
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421
rotational, translational, or glide-reflective symmetry. They also noticed that the more visually
appealing examples of quilt squares used contrasting colors. As I transitioned to the project,
I praised students for their focus in attending to
the qualities of quilt squares. I distributed the
handouts (see the online appendix), described
the requirements for this project, and reviewed
the task with them.
Phase 2
Figure 2
(about 60 minutes)
The next stage of the project focused students’
attention on designing their quilt square on
grid paper that I provided for them. During this
After drawing a quilt square in phase 3 of the project,
students found the area and perimeter of each shape,
showing their calculations on the second page of the
handout.
phase, I walked around the classroom to formatively assess students’ progress and understanding of the task. Specifically, I was looking
at their ability to talk about their designs, trying
to determine whether they were following the
directions of the project and applying formulas
correctly. Students often asked me during this
phase if they must cover the entire piece of grid
paper with shapes. This led me to think that they
did not understand what to do if their designs
and shapes did not fit on the grid paper. We
corrected this misconception with the explanation that not having a shape on the grid paper is
similar to not having fabric in a section of a quilt.
If a student had an unusual shape in his or her
design (e.g., an oval), then I asked for an explanation of how the area and perimeter might be
calculated. When students realized they did not
know how to calculate the area of rounded and
unusual shapes, they reworked their designs to
include shapes with area formulas they knew.
Phase 3
(about 25 minutes)
Next was the synthesis portion of the project.
After completing a drawing of their quilt square,
students examined the second handout. The
chart directed them to find the area and perimeter of each shape and to show their work (see
fig. 2). I expected them to show that the sum
of their polygons was the same as the area of
a square measuring eight inches on each side.
This was an important step because it presented
an opportunity to formatively assess whether
students accurately calculated the sum of
64 units2. After students completed their design
and chart, they submitted them for approval.
During the approval process, I checked
students’ strategies to determine whether they
used appropriate methods for calculating area
and perimeter. This process allowed more time
to probe students’ thinking about the polygons’
area and perimeter. For instance, I asked one
student during the one-on-one review process,
“Why might your total area not be 64 inches2?”
The student replied, “Because some of my
shapes did not cover the entire square.”
Students’ areas usually did not sum to
64 inches2 because they had approximated their
measurements to the nearest fraction of an inch.
When I asked them why their sums did not equal
64 inches2, one student responded as follows:
422
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Other students shared that their designs
required rounding and that they expected their
total area would not equal 64 inches2. I expected
students to be precise with their measurements and calculations and to spend ample
time persevering in making the sum as close to
64 inches2 as possible. Because of this activity,
they showed evidence of attending to precision,
which is the sixth Standard for Mathematical
Practice (CCSSI 2010). During these conversations, I accepted students’ work if the total sum
was not exactly 64 inches2 if and only if they
provided ample evidence indicating why the
sum was slightly more or less than 64 inches2.
The conversations were evidence that students
were able to make sense of the problem, which
is an aspect of the first Standard for Mathematical Practice (CCSSI 2010).
While phase 3 was happening in the classroom, students saw that I was the only person
who checked their work before they proceeded
to the next phase. Because I could verify only
one student’s work at a time, students asked
one another for help to ensure an accurate table
before meeting with me. They reviewed their
peers’ work and showed respect while critiquing others’ designs, reasoning, and arithmetic.
They posed questions to one another and displayed kindness and thoughtfulness with their
remarks. Drawing on the language of the third
Standard for Mathematical Practice, they “justified their conclusions, communicated them to
others, and responded to others’ questions and
concerns” (CCSSI 2010, pp. 6–7 ). Two frequent
occurrences were incorrect arithmetic and
measurement errors. When a student noticed
someone’s error, he or she addressed it with the
peer and asked for clarification. The positive
learning environment and established norms
in my classroom facilitated peer-to-peer mathematical discourse to complete the project successfully and on time.
wallpaper, they created a quilt square. Students
cut out each individual shape from their grid
paper, traced the shapes onto the wallpaper, and
cut the shapes from the wallpaper. This ensured
that students’ polygons fit together.
Outcomes from the Quilting task
Students created amazing quilt squares (see
fig. 3). They applied their knowledge of area
Figure
3. One class’
quilt square
creations. their
students
proudly
displayed
FI gure 3
I did not get 64 inches2. I got very close. The
area of one of my shapes was not a whole
number, so I got a decimal for my total area,
which was close to 64 inches2.
The positive learning
environment and
established norms
in my classroom
facilitated peer-to-peer
math discourse to
complete the project
successfully and on time.
bulletin board for all to see.
quilt squares on a classroom
Phase 4
(about 60 minutes)
After designs and charts had been approved,
students began phase 4 of the project. Using
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423
Students’
conceptual
and procedural
understandings
are enhanced
by mathematics
projects and
collaboration.
and perimeter while problem solving, creatively
applied mathematical procedures, and produced
artwork that we displayed in the school. I
encouraged them to use valid approaches for
calculating area and perimeter rather than
algorithms. Although some used algorithms,
others employed alternate strategies, such as
counting the squares on grid paper, evidence
of learning to adapt their reasoning and use
efficient, effective strategies when appropriate—
two features of mathematical proficiency. When
a student could not remember a formula for a
specific polygon, instead of giving it to them, I
provided scaffolding by asking questions to push
his or her thinking forward. For example, when I
asked, “How else could you find the area without
using the formula?” a student responded, “By
counting up the number of whole spaces a
shape takes up and then adding any half spaces
the shape takes up.” Their comments and
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424
actions suggested that they were aware that
multiple ways exist for finding area, albeit some
are less precise approaches than others.
One student struggled with accurately calculating the area because he did not count the half
of each square in his drawing. When questioned
about how he could do so, he told me, “I could
count the number of half-squares I have and
add them up and then add that to the number
of whole squares I have.” I praised him for his
approach and asked if he might be able to use
his answer to remember the area formula for a
parallelogram. He paused to think for a moment
and replied, “Oh, yeah! I need to multiply the
length by the height and not the side. I was trying to multiply the length by the side and kept
getting a decimal.”
Scaffolding and encouragement supported
students in using various methods for calculating area and making connections to formulas.
Creating a new product
Many students complete exercises flawlessly
and yet often struggle when asked to apply this
knowledge (Schoenfeld 2011). They may be able
to complete exercises with accuracy but cannot
use this knowledge to create something. Students
should be able to apply their math knowledge.
Math tasks that involve creativity while problem
solving and that draw on multiple mathematics
concepts and procedures can support the idea
that doing math involves more than completing
exercises (Bostic and Jacobbe 2010). The benefits
of such tasks are greatest when students know
how to work together to solve a problem (Bostic
and Jacobbe 2010). Thus, students’ conceptual
and procedural understandings are enhanced by
mathematics projects and collaboration.
The Quilting task gives students an
opportunity to synthesize their knowledge of
area and perimeter with an application problem.
Creating a new product is the highest level of
Bloom’s taxonomy (Kratwohl 2002). Hence,
students need to have novel opportunities to
Common Core
Connections
6.G.A.1
SMP 1
SMP 4
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SMP 5
SMP 6
SMP 8
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demonstrate their mathematical knowledge.
Just imagine what you might learn about your
students when they make their quilt squares.
R EFER E NCE S
Bostic, Jonathan, and Tim Jacobbe. 2010. “Promote Problem-Solving Discourse.” Teaching
Children Mathematics 17 (August): 32–37.
Common Core State Standards Initiative (CCSSI).
2010. Common Core State Standards for
Mathematics. Washington, DC: National
Governors Association Center for Best
Practices and the Council of Chief State School
Officers. http://www.corestandards.org/assets
/CCSSI_Math%20Standards.pdf
Kratwohl, David R. 2002. “A Revision of Bloom’s
Taxonomy: An Overview.” Theory into Practice
41 (4): 212−18. http://dx.doi.org/10.1207
/s15430421tip4104_2
National Research Council (NRC). 2001. Adding
It Up: Helping Children Learn Mathematics,
edited by Jeremy Kilpatrick, Jane ­Swafford,
and Bradford Findell. Washington, DC:
National Academies Press.
Schoenfeld, Alan H. 2011. How We Think: A
Theory of Goal-Oriented Decision Making
and Its Educational Applications. Studies in
Mathematical Thinking and Learning Series.
New York: Routledge.
Jordan T. Hede, hede
[email protected], a
seventh-grade math
educator at Carter
Middle School in
Knoxville, Tennessee, is interested in teaching students
how to integrate math content with problem-solving
skills in real-life situations. Jonathan D. Bostic, bosticj
@bgsu.edu, an assistant professor of math education
at Bowling Green State University in Ohio, investigates
mathematics instruction that promotes problem
solving and multiple-strategy use. He enjoys giving
professional development that focuses on the
Standards for Mathematical Practice.
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425