Exemplars
Quilting Quandary
The other day Mrs. Williamson was bored and decided to make a wall hanging. She decided
she could just take the pattern below and enlarge it so that 1 centimeter on the pattern would
equal 1 decimeter on the wall hanging. So instead of the 12 cm x 12 cm hanging, she would
have a 12 dm x 12 dm wall hanging.
Her next step was to go buy fabric for the hanging. She needed to buy 3 types: 2 prints and 1
solid. Following the design given in the pattern below, determine how many square
decimeters of fabric Mrs. Williamson needed to buy of the print, and of the solid material to
make a 12 dm x 12 dm wall hanging.
Mrs. Williamson needed to buy...
____ square decimeters of
____ square decimeters of
____ square decimeters of
Show all of your work and explain your reasoning.
Exemplars
TM
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary
- Page 1-
Exemplars
Grade Level 6–8
Mrs. Williamson’s Quilting Quandary
The other day Mrs. Williamson was bored and decided to make a wall hanging. She decided
she could just take the pattern below and enlarge it so that 1 centimeter on the pattern would
equal 1 decimeter on the wall hanging. So instead of the 12 cm x 12 cm hanging, she would
have a 12 dm x 12 dm wall hanging.
Her next step was to go buy fabric for the hanging. She needed to buy 3 types: 2 prints and 1
solid. Following the design given in the pattern below, determine how many square
decimeters of fabric Mrs. Williamson needed to buy of the print, and of the solid material to
make a 12 dm x 12 dm wall hanging.
Mrs. Williamson needed to buy...
____ square decimeters of
____ square decimeters of
____ square decimeters of
Show all of your work and explain your reasoning.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 2-
Exemplars
Context
This task was given to my sixth graders during the first few weeks of school after a review of
area and perimeter. I had provided them with instruction and practice with combining the
areas of shapes to form figures for which they could easily determine the area, and more
specifically, that 2 congruent right triangles can form a rectangle.
What This Task Accomplishes
This task allows the teacher to assess students’ measurement skills including preciseness and
familiarity with the metric system. It is an excellent task for getting students to conceptualize
shapes, and how to combine them to create new, more easily to manipulate figures. There are
also several ways to solve the task, inspiring stimulating class discussions.
Time Required for Task
This task takes a total of 45 minutes.
Interdisciplinary Links
This task links nicely to social studies units involving colonial days and quilting. The type of
quilt square made is called a “flower basket.” In Consumer Science class, students could
actually make the quilt square and create their own pillows!
Teaching Tips
Students who are not well versed in finding area and perimeter will attempt to draw a grid
on the diagram provided to find the number of square centimeters. This approach would
work if students were precise in their measurements, and may be a good way for students
who choose a different method to verify their solutions.
Students who are better at finding area and perimeter will find the dimensions of smaller
shapes and add them to find the amount of material needed for each pattern type. The most
challenging part of the task is in determining the amount of material needed for the darkest
pattern. Students will come up with a variety of strategies for dealing with this challenge.
Give students who are not familiar with finding the area of triangles experience with finding
the area and perimeter of unusual shapes. Each morning I put a few warm up math problems
on the board to get students to practice mathematics concepts and skills. As a precursor to
this task, I had students find the area and perimeter of shapes like this:
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 3-
Exemplars
This enabled the students to come up with strategies with which to be successful with this
task. This task does have a variety of ways in which to solve it. I took the time to have
students share their strategies once the problem was complete so that students would be aware
of the different strategies. This was a very productive discussion and will assist
students
in thinking more broadly on the next task they solve. This was also a great time to review
such math terms as congruent, right angle, etc. in a meaningful context.
Suggested Materials
centimeter rulers, scissors, graph paper
Possible Solutions
Some students will block out the diagram in square units and count them. Others will find
the area and perimeter of the white and wavy pattern, but be challenged by finding the area
of the dark pattern. Some find the total area of the figure and then subtract the amounts they
have found so far to find the area of the dark pattern. Some students used this to verify their
solutions. Other students found the solution by combining triangles to form rectangles, and
then taking half to find the area. When it came to the 2 triangles that overlapped, students
simply subtracted 1 square centimeter from one of the triangle areas. The correct solution
should be:
Mrs. Williamson needed to buy ...
64.5 square decimeters of
35.5 square decimeters of
44 square decimeters of
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 4-
Exemplars
Benchmark Descriptors
Novice
The novice will not have an approach that will work for the problem and finds solutions that
are not even close to the actual amounts needed. The novice will use little or no math
language to communicate the solution, and the solution will lack reasoning.
Apprentice
The apprentice will find an approach but that approach will not lead to an accurate solution.
The apprentice’s measurements may be off, or the apprentice may not be able to correctly
determine all 3 amounts of material needed. The apprentice will use some basic math
language to communicate, and may label the diagram presented in the task.
Practitioner
The practitioner will find an approach that will lead to finding the correct amounts of
material needed of all three types. The practitioner will use accurate and appropriate math
language to communicate the solution, and will use correct mathematical reasoning.
Expert
The expert will find an approach that will lead to finding the correct amounts of material
needed of all three types, and may use more than one approach to solving the problem to
verify the solution. The expert will use precise math language such as the formula for finding area. The expert will create an additional mathematical representation in which to display data. The expert will explain her/his correct reasoning, and will make mathematically
relevant observations about the solution.
Author
Carol Amico McNair teaches grade six at the Camel’s Hump Middle School in Richmond,
Vermont. She has worked as the mathematics assessment consultant for the Vermont
Department of Education’s Portfolio assessment program, and acts as a consultant to school
districts and to publishing companies. She is also an associate editor for Exemplars.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 5-
Exemplars
Novice
It is unclear how
the student
achieved these
answers.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 6-
Exemplars
Novice (cont.)
No mathematical reasoning is
used.
Little or no math
language is used.
There is little evidence of the
student’s work.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
These answers are incorrect
and are far from close to the
correct amounts needed.
- Page 7-
Exemplars
Apprentice
The student’s approach does
not allow for an accurate
solution.
There are gaps
in the student’s
mathematical
reasoning.
Little or no math
language is used
to communicate
the solution.
All work is labeled with
appropriate units.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 8-
Exemplars
Practitioner
The student changes
the initial approach
realizing it will not
lead to an accurate
answer.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 9-
Exemplars
Practitioner (cont.)
The student organizes the work
used to obtain a solution, and all
parts are labeled.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 10-
Exemplars
Practitioner (cont.)
Accurate and
appropriate math
language is used.
The student clearly
explains the approach and
the reasoning used.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 11-
Exemplars
Expert
The student expressed the formula
for finding an area.
The student
provides reasons
behind
discussions made.
There is clear
communication
on how the
problem was
solved.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 12-
Exemplars
Expert (cont.)
The student creates an
accurate and
appropriate
representation in
which to record the
data.
The student
verifies the
solution.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Quilting Quandary (cont.)
- Page 13-