Stitching Quilts into Coordinate Geometry Author(s): Susanne K. Westegaard Source:

Stitching Quilts into Coordinate Geometry
Author(s): Susanne K. Westegaard
Source: The Mathematics Teacher, Vol. 91, No. 7 (October 1998), pp. 587-592, 598-600
Published by: National Council of Teachers of Mathematics
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>
Activities
into Coordinate
Quilts
Stitching
Quilts?useful items, visually appealing,
steeped in history, and an integral part of
our culture?can also be the jumping-off
point formany mathematical investigations"
(Morrow and Bassarear 1996). The history behind
many quilt patterns offersan opportunity for cross
disciplinary projects with art and social studies
classes. Such children's books as Eight Hands Round
(Paul 1991)andSelina and theBearPaw Quilt
(Smucker 1995) connectmathematics with literature.
TEACHER'S GUIDE
When my mother asked me to enlarge a quilt-block
pattern forher, I explored transferringthe pattern
onto a coordinate grid as well as using compass
and-straightedge construction.The coordinate-grid
method seemed far easier.
I could not locate formy class activities that
reinforced "writingequations for lines/' so I used
quilt patterns to develop activity sheets for students
and a project to accompany the activity sheets.
These activities reinforce coordinate-geometry
concepts?coordinates, positive and negative
slopes, intercepts, and equations forhorizontal and
vertical lines. For group discussion, this article
includes questions and prompts thatwill lead stu
dents to notions of probability, area, and geometry.
These activities can be used with small groups,
individuals, or large groups.
Prerequisites: These coordinate-geometryactivi
ties can be used with students who are familiar
with coordinates, slope, and equations for lines.
This article includes suggestions formodifications
and extensions.
Grade levels: 7-12
Susanne
.
Westegaard
Geometry
Sheet1:This sheetdealswitha quiltblockthatis
located only in the firstquadrant. Less mathemati
callymature students, or students who need an
introductoryactivity, could be asked to label all the
vertices and find the slope of each segment.More
advanced students could be also asked towrite re
< < 3, foreach segment.
strictions, forexample, 0
is
Of vital importance connecting this activity to
such other ideas as geometric shapes, probability,
and area. Ask students to find the following
geometric shapes: triangle, square, rectangle, par
allelogram, trapezoid, hexagon, right triangle,
isosceles triangle, and regular octagon, as well as a
pair of congruent triangles or congruent squares.
This activity can become open-ended by asking the
students to find as many geometric shapes as they
can. Students could be asked the followingques
tions:How many triangles can you find?How many
squares? How many trapezoids? How many rectan
gles? Are any shapes similar? Students can find the
area of each polygon.
The many possible probability questions include
the following: If a coordinate pair is chosen at ran
of
what is theprobability
domfrom
theblock,
selecting a pair in the shaded square? What is the
probability of selecting a pair in a shaded triangle?
a pair ina
What is theprobability
ofselecting
a
ofselecting
What is theprobability
shadedfigure?
pair in a shaded triangle ifyou select a shaded fig
ure? The probability questions could be expanded to
questions about odds.
Quilts can be
thejumping
off
point
formany
mathematical
investigations
Sheet 2: This quilt block is centered at the origin.
The same modifications could be made for this
activity as for sheet 1.
SusanneWestegaardteachesatHopkins SeniorHigh School,
Materials: Activity sheets; rulers; coloredpencils for
activities 3,4, and 5; and various sizes ofgrid paper
Hopkins,
MN
55305,
[email protected]
She is interestedingraphing calculators, fractals and
chaos, and
real-world
connections.
MS 39701
Mathematicsand Science,Columbus,
Edited byClaudia Carter, [email protected],
Mississippi Schoolfor
mathematicsactivitiesappropriateforstudentsingrades 7-12.
This sectionisdesignedtoprovide inreproducible
formats
This material
may
be reproduced
by classroom
teachers for use in their own classes. Readers
who have developed
success
alreadypublished, to
ful classroomactivitiesare encouragedtosubmitmanuscripts,ina formatsimilarto the"Activities"
are activitiesfocusingon theCouncil'scurriculum
theseniorjournal editorforreview.
standards,its
Ofparticular interest
expandedconceptofbasic skills,problemsolvingand applications,and theuses ofcalculatorsand computers.
Write toNCTM, DepartmentP, or sendee-mailto [email protected],
for thecatalog of educationalmaterials,which
inboundform.?Ed.
listscompilations
of "Activities"
Vol. 91,No. 7 ? October 1998
587
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If students have completed sheet 1 and explored
the connectionsmentioned, divide the class into
small groups to generate their own lists of questions
as well as tomake a solution key. Groups should
exchange lists and find the answers. Discuss any
discrepancies between answers and solution keys.
After students have completed both sheet 1 and
sheet 2, ask them to respond inwriting to these
prompts:
whichis
I preferred
thesheet1 quiltblock,
Students
could
construct
these
patterns with
geometry
software
...
placed in the first quadrant, because
I preferred the sheet 2 quilt block,which is cen
tered at the origin, because ...
If I were asked to put a quilt block on a coordi
nate grid, I would select the firstquadrant or
select the origin because ...
Sheet 3: In this activity, students are asked to
transfera quilt block to a grid. Note that the origin
isplacedinthemiddleoftheblockand thegrid
marks are lefton the pattern. To make the activity
more challenging, the teachermay decide to remove
the origin hint or give the students differentsizes
ofgrid paper. Such changes will encourage intrigu
ing discussion questions when the probability con
nections are explored. Itmay be interesting to ask
some students to put theirwhole design in one of
the quadrants. Do the geometric shapes change?
The probabilities? The equations? Will students
think that the equations should be identical, no
matter where the segments are placed? What is dif
ferent?What is the same? Dilations and transla
tions could be mentioned.
Sheet 4: Note that all grid lines have been omit
ted from this figure.The teachermay decide to
sketch in the grid lines to assist the students. The
studentsmay place the origin anywhere when they
place this block on a coordinate grid. To further
challenge students, do not indicate the lengths of
the sides. After the students have completed this
block, have them compare the location of their ori
gin and their equations with those used by other
students. They can then sort the quilt blocks and
tape them to thewall.
Be sure to ask students questions about the
areas of the shapes on this block. How does the
area of one triangle compare with the area of one
parallelogram? Compare the area of the shaded
regionswith that of the nonshaded regions.
Sheet 5: Note that all grid lines have been omit
ted on this figure.After completing sheet 3 and
sheet 4, students should be able to overlay a grid
and place the origin. To furtherchallenge students,
do not indicate the lengths of the sides. Be sure
that students explore the areas of the shaded and
nonshaded regions of the figure.What is the ratio
of the area of the shaded regions to that of the non
shaded regions?
EXTENSIONS
Students could construct these patterns with geom
etry software or on the TI-92 calculator. Students
familiarwith the equation fora circle could trans
ferquilt blocks that include circles and arcs to a
coordinate grid and write the equations. Eight Hands
Around is a quilt block that contains quarter circles;
Ernie (1995) mentions several other blocks, as well
as work with patterns, in "Mathematics and Quilt
ing" in the 1995 Yearbook of theNational Council
At the1997
ofTeachersofMathematics(NCTM).
Annual Meeting of theNCTM inMinneapolis
Saint Paul, I attended a session on quilting by
Diana Venters and Elaine Krajenke Ellison; they
had used patterns based on such topics as
Pythagorean triples, the Clifford torus, and spiral
ing squares for their quilts.
Most quilt patterns are based on a nine-square
block or a sixteen-square block; therefore,the
angles are usually 45 degrees. Students can deter
mine the lengths of the various line segments using
the 45-45-90 formulas. Students wanting more
challenging patterns can find books on optical illu
sions or explore such ideas as the Pythagorean
triples, the Clifford torus, and spiraling squares.
ASSESSMENT
Students could develop and name their own quilt
blocks. Less mathematically mature students
should begin with a nine-square block, as shown
on sheet 1, or a sixteen-square block, as shown
on sheets 2,3,4, and 5. They would need towrite
the equations. They could create posters showing
the origin pattern, the coordinatized block, and a
completed block. Students could use colored
paper or fabric.
Have students select a pattern. These patterns
can be found in quilt books, some ofwhich may
be available at your local library,and quilting
magazines. Optical illusionsmay also intrigue
students. Students can write the equations and
develop a poster featuring the original pattern,
their coordinatized pattern, and a colored example.
Ask students to explore their ethnic backgrounds
forpatterns. Students couldmake a poster show
ing their pattern, the coordinatized pattern, and
a colored example. They might include any back
ground history that theyhave discovered.
Students could actually use several differentpat
terns tomake a quilt. "Plane Geometry and
Patchwork," an article inQuilt Almanac (Sisk
1992), featured a special-education teacher
whose studentsmade a quilt.
TEACHER
THEMATHEMATICS
588
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SOLUTIONS
Sheet 4
Sheet 1
I chose to put the bottom-leftcorner of the quilt
pattern at the origin.
y= 9
y= 9
y= 9
y = 16_y
y = 16
= 16
=
.y P
II
y=0
y=0
y=0
Sheet 2
y= 4
y=4
y=4
Sheet 5
I chose to put the bottom-leftcorner of the quilt
pattern at the origin.
y = 16_y
^
= -4
j
= -4
j
= 16
= -4
Sheet 3
y= 8
y= 8
y= 8
y= 8
y=0
rm
BIBLIOGRAPHY
3
y = -8
y = -8
y=0
y = -8
y = -8
Cobb,Mary. The Quilt-BlockHistory ofPioneer Days.
Brookfield,Conn.: Millbrook Press, 1995.
Cohen, Luanne Seymour. Quilt Design Masters. Palo
Alto, Calif.: Dale Seymour Publications, 1996.
Ernie, Kathryn T. "Mathematics and Quilting." In
ConnectingMathematics across theCurriculum,
1995Yearbook of theNational Council ofTeachers
ofMathematics (NCTM), edited by PeggyA. House
and Arthur F. Coxford, 170-76. Reston,Va.: NCTM,
1995.
Hasting, Kathy. "APatchworkAlphabet." Matrix
Design Stamps.
Morrow, Charlene, and Tom Bassarear. "Mathemati
cal Ideas Embedded inQuilts." Paper presented at
theAnnual Meeting of theNational Council of
Vol. 91,No. 7 ? October 1998 589
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Teachers ofMathematics, San Diego, Calif, April
1996.
Paul, AnnWhitford.Eight Hands Round: A Patch
workAlphabet. New York: HarperCollins, 1991.
Sisk,Martha J. "Plane Geometry and Patchwork." In
Quilt Almanac. New York: Harri Publications, 1992.
test
ou'll neverhave tocreateanothertestorworksheet
again!MathCheck
algorithm
Barbara. Selina and theBear Paw Quilt.
Smucker,
?
or testvariations
all at thetouchofa
worksheets
software
printscountless
generator
New York: Crown Publishers, 1995.
?
most important
teach.
Nowyouare freetodo your
button.
job
Sykes,Mabel. A Source Book ofProblems forGeome
Based upon Industrial Design and Architecture
try,
JustConsidertheBenefits
Ornament.
Boston,Mass.: Allyn & Bacon, 1912.
Print
oftestsandworksheetsinstantly
countless
variations
Venters, Diana, and Elaine Krajenke Ellison. "Mathe
Print
andanswerkeysforeachworksheet
scoringrubrics
matical Quilts?Geometry Topics fromPythagoras
Writeandstoreyourownproblems
toFractals as Quilt Patterns." Paper presented at
andpromote
Eliminate
learning
cooperative
answer-sharing
calculus
theAnnual Meeting of theNational Council of
Covergrades3 through
Teachers ofMathematics, Minneapolis-Saint Paul,
MathCheckisDesignedforTeachers
Minn., April 1997.
You
-.
control.
WithMathCheck,theteacheris incomplete
Mathematical Quilts: No Sewing Required.
can selectjusttheproblems
youwantusingMathCheck's
Berkeley, Calif: Key Curriculum Press, 1998.
Or,write,save,and
WoodrowWilson National Fellowship Foundation.
easy-to-use
draganddropgraphicinterface.
or giveidentical
You can printa different
worksheet
foreach student,
insert
yourownproblems.
Shapes and Dimension, 1991 CurriculumModule.
andpreserves
Whatever
MathCheckreduces
foreveryone.
worksheets
your
youchoose,
paperwork
Princeton,N.J.;Woodrow Wilson National Fellow
brochure.
callorwritetodayforyourfreeinformative
valuabletime.Formore information,
ship Foundation, 1991,294-98.
Zaslavsky, Claudia. Africa Counts:Number and Pat
terninAfrican Culture. Brooklyn,N.Y.: Lawrence
Hill Books, 1990.
no
an
I
is
me
understand
there
about
MathCheck.
informative
brochure
Please rush
obligation.
-.
Multicultural Mathematics, Interdisciplinary
*
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590 THEMATHEMATICS
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CHURN-DASHQUILT BLOCK
On each
SHEET 1
line segment, write itsequation. You may need to use an equation several times.
x = 0
y
= -x + 3
=
y 0
=
=
x y9 3
=
y x + 6
x-3
= =
y y6 9
y-x-6
x-6
y
= -x+~\5
From theMathematics Teacher, October 1998
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ANVILQUILT BLOCK
On each
SHEET 2
line segment, write itsequation. You may need to use an equation several times.
= -4
=
y x-4
y-x-2
=
y 0
=
x =
y20
=2
=
y 4
=
y -4
= -2
y
-A
= -2
=
y x+ 4
y=x+2
From theMathematics Teacher, October 1998
(Continuedonpage 598)
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(Continuedfrompage 592)
BACHELOR'SPUZZLEQUILTBLOCK
SHEET 3
of each side of the
Transferthepatternto thecoordinategrid.Note thattheoriginand the length
quilt block are
labeled. On each
line segment, write itsequation.
From theMathematics Teacher, October 1998
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SHEET 4
QUILT BLOCK
WINDBLOWN-SQUARE
Transfer the pattern to a coordinate grid. You will need to determine where you want to locate the
origin. On each line segment, write itsequation.
16 units
From theMathematics Teacher, October 1998
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MONKEY-WRENCH
QUILT BLOCK
SHEET 5
Transfer the pattern to a coordinate grid. You will need to determine where you want to locate the
origin. On each line segment, write itsequation.
16 units
From theMathematics Teacher, October 1998
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`