Exemplars
Box Dilemma
As a result of the decline in enrollment, there
is an extra classroom, so the kindergarten will
be moving back to our school building. They
will need to move all of their classroom
materials. We have received large pieces of
cardboard that can be fashioned into boxes
without tops. The cardboard is 40’ by 40’.
Please find the dimensions of a box that would
hold the most (not going over the top of the
box in case we want to stack them). Once you
have found the dimensions of the box that
holds the most, you might decide whether you
think that box is the most practical size to
build. Is there another size you would build?
Why?
Don’t forget to convince me that the box you
came up with holds the most!
Exemplars
TM
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma
- Page 1-
Exemplars
Grade Level 6–8
Box Dilemma
As a result of the decline in enrollment, there is an extra classroom, so the kindergarten will
be moving back to our school building. They will need to move all of their classroom
materials. We have received large pieces of cardboard that can be fashioned into boxes
without tops. The cardboard is 40’ by 40’. Please find the dimensions of a box that would
hold the most (not going over the top of the box in case we want to stack them). Once you
have found the dimensions of the box that holds the most, you might decide whether you
think that box is the most practical size to build. Is there another size you would build? Why?
Don’t forget to convince me that the box you came up with holds the most!
Context
The context of this problem (the declining school enrollment) was a hot topic in our school.
The children could remember when the kindergarten used to be in our school, and they liked
the fact that the class would be returning to the building.
My sixth graders had just finished a unit on perimeter, area, and volume. They had also
completed a unit on density in science earlier in the school year that involved finding
volume. At this point, they had experience working with whole numbers, fractions, and
decimals.
What This Task Accomplishes
This task assesses students’ ability to recognize a volume problem. It also assesses those
students that are capable of organizing sequentially a number of different possible size boxes
and looking for patterns. It also allowed students to make, then test, certain hypotheses like,
“I think the volume will stay the same — after all, you use the same amount of cardboard.” Or,
“I think the volume will keep increasing as the sides of the box increase.”
Time Required for Task
Depending on the approach, it will take either one 45 minute class period with students
finishing up for homework, or plan 2 periods for students to complete the entire task at
school.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma (cont.)
- Page 2-
Exemplars
Interdisciplinary Links
This problem led to a great discussion on population changes in our town and speculation as
to the cause of the decline in beginning school age children. An art project on box design or
containers could also accompany this task.
Teaching Tips
The first thing you need to discuss is how to make a cover-less box given a square sheet of
cardboard. Once everyone understands about cutting out a square from each corner and
lifting the sides, they will all be doing the same problem. I demonstrated on a piece of square
paper.
There will be kids that need to cut out their boxes, but give them guidance because they can
lose sight of the problem in their constructions. Be sure they record important information on
one box before going on to build the next one. A follow-up discussion is essential once
students have completed the task since there is no “biggest box.” The students go crazy
wondering if they got the “right” answer. It didn’t take much encouragement from me to get
some students to investigate fractional heights of the box.
Suggested Materials
You will need calculators, graph paper, scissors, tape, rulers
Possible Solutions
This is really a calculus problem involving limits. There is no “biggest” box, but you can get
close depending on the restrictions the student puts on the solution. For example, if the
student only investigates whole number heights of boxes, then a 26” x 26” x 7” box with a
volume of 4732 cubic inches is the biggest. But you can find a bigger box that has dimensions
of 27” x 27” x 6.5” and a volume of 4738.5 cubic inches. I’ll let you investigate more fractional
heights!
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma (cont.)
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Exemplars
Benchmark Descriptors
Novice
A novice will not recognize this as a volume problem. The novice may be able to build
or sketch a box but cannot put the mathematics with the models.
Apprentice
The apprentice will recognize the task as a volume problem but will not be able to come
up with a correct largest volume for at least the whole number length sides. Apprentices
may find the volume of a few boxes correctly, and pick the box with the largest volume of
the boxes they find, but will not organize their work sufficiently to find the biggest box.
The apprentice may also arrive at incorrect generalizations.
Practitioner
A practitioner will find the box with the largest whole number dimensions (26" x 26" x
7"). The practitioner will have a fairly organized way of showing that it is the largest
whole number box. The practitioner may also have a diagram, an organized list, or a
graph to support their reasoning.
Expert
The expert will find the largest whole number box height and will investigate further to
find one using decimals that has an even larger volume. The expert will support the
reasoning used with an organized list, chart, diagram, or graph, and will make
mathematically relevant comments or observations about the solution.
Author
Clare Forseth has taught sixth grade math at the Marion Cross School in Norwich, Vermont
for 26 years. She is a member of her school’s Mathematics Curriculum Committee,
Technology Committee, Assessment Committee, and Local Standards Board for teacher
re–licensure.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma (cont.)
- Page 4-
Exemplars
Novice
The student does not
recognize this problem
as a volume problem.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student is only finding the
area of one side of the box.
This will never lead to the
solution of the problem.
Box Dilemma (cont.)
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Exemplars
Apprentice
The student uses the
correct formula for the
volume of a prism.
The student only finds the
volume of three boxes, then
picks the biggest of the three.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Although this generalization is correct for
the area of the base, it doesn’t have
anything to do with the resulting volume.
Box Dilemma (cont.)
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Exemplars
Practitioner
The student creates nice
3 dimensional diagrams.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
This student has a
somewhat organized
approach.
Box Dilemma (cont.)
The measurements
are labeled correctly.
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Exemplars
Practitioner (cont.)
This generalization is
correct for heights of
whole number boxes.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma (cont.)
- Page 8-
Exemplars
Expert
The student uses a well–
organized and labeled table.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student has used
fractional heights of boxes.
Box Dilemma (cont.)
- Page 9-
Exemplars
Expert (cont.)
A graph helps to
communicate the
student’s reasoning.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma (cont.)
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Exemplars
Expert (cont.)
The student understands that
you may go to other fractions
and find different solutions, but
s/he chooses not to.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Box Dilemma (cont.)
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