Winter Hat
This problem gives you the chance to:
• calculate the dimensions of material needed for a hat
• use circle, circumference and area, trapezoid and rectangle
Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces.
y inches
3.5 inches
3 inches
x inches
Circumference of circle =πd
Area of circle = πr2
2.5 inches
24 inches
1. The rectangular strip is 24 inches long. Eight trapezoids fit together around the rectangular strip.
Find the width (x) of the base of each trapezoid
______________ inches
2. The circle at the top of the hat has a diameter of 3 inches.
a. Find the circumference of the circle. Show your calculation.
_______________ inches
b. Eight trapezoids fit around the circle. Find the width (y) of the top of each trapezoid?
_______________ inches
3. Find the surface area of the outside of the hat. Show all your calculations.
____________square inches
9
Grade 7 – 2008
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Winter Hat
Rubric
•
• The core elements of performance required by this task are:
• • calculate the dimensions of material needed for a hat
• • use circle, circumference and area, trapezoid and rectangle
•
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answer: 3 inches
1
1
2.a. Gives correct answer: 9.4 or 3π inches
1
Shows correct work such as: π x 3
1
b. Gives correct answer: 1.2 or 3/8π inches
1ft
3
3.
Gives correct answer: 126 square inches Allow 125 to 129
1
Shows correct work such as: 24 x 2.5 = 60 (rectangle)
1
π x 1.52 = 2.25 π = 7.1 (circle)
1
(3 + 1.2) / 2 x 3.5 = 7.35 (trapezoid)
1ft
1ft
7.35 x 8 = 58.8 (8 trapezoids)
5
9
Total Points
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Winter Hat
Work the task. Look at the rubric. What are the mathematical concepts
being assessed in this task? ____________________________________
Look at student work for part 2b, finding the width of the top of each
trapezoid using the circumference of the small circle. How many of your
students put:
1.2
1.1 or 1
3
3.5
2.5
1.5
#≥ 15
Other
What is some of the thinking behind these misconceptions? What might the
students with answers of 1.1 or 1 been thinking? How is this misconception
different from that of students with answers of 3 or 3.5?
Now look at work for part 3. How many of your students:
• Labeled calculations so they knew which was the area of the
rectangle, the area of the trapezoid, etc.?
• Correctly found the area of the rectangle?
• Correctly found the area of the circle?
• Correctly found the area of a trapezoid?
• Tried to find the area of a trapezoid but used an incorrect formula?
• Tried to find the area of 8 trapezoids?
• Multiplied areas of different figures together?
• Used dimensions from different figures in attempting to find area?
• Found perimeter of shapes?
• Struggled to interpret the diagram of the hat?
How often are students in your class asked to do a task with a long reasoning
chain? How often do students solve problems where they need to compute
something to use as dimension for something else?
Look in your textbooks. What opportunities do students have to interpret
complex diagrams? How much more practice is devoted to computation
devoid of diagrams, where the measurements are just given? How is the
thinking and understanding significantly different in these two situations?
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What opportunities have students had making nets and unfolding them to
look at the individual pieces? What do you think students understand about
the process of finding surface area?
Looking at Student Work on Winter Hat
Student A uses labels and units to organize work. Notice how the student makes new
diagrams for the shapes and labels the dimensions in order to think through the
calculations in part 3. How do we help students develop this habit of mind?
Student A
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Student B is also able to organize the work in part 3, using diagrams to label the
calculations. Notice the student does not round off numbers. How do we help students to
make sense of numbers from calculators? In making a pattern would it make sense to try
for this level of accuracy?
Student B
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Student C does not label or organize the work. The student understands that surface area
is the total area of all the pattern parts. In part 3 the student can calculate the area of the
rectangle and the small circle. The student does not know the formula for trapezoid (4th
and 5th grade standard) and finds half of one base rather than half the total of the 2 bases.
The student forgets that there are 8 trapezoids.
Student C
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Student D has trouble interpreting the diagram. The student is able to find the
circumference in the small circle in 2a. However when thinking about fitting the 8
trapezoids around the circle, the student divides the diameter by 8 instead of using the
circumference. The student is able to find the area of the rectangle. The student uses the
formula for area of rectangle instead of area of a trapezoid, but does know that there are 8
trapezoids. The student doesn’t square the radius when finding the area of the circle.
Again the student does not think about significant digits in the final answer to 3.
Student D
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Student E is able to calculate the width of the trapezoid and the diameter of the small
circle. She divides the circumference by 8 to find the width of the top of the trapezoid
(1.17) but rounds incorrectly. In part 3 the student only calculates the area of the circle.
The student does not think about surface area as the sum of all the sides.
Student E
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Student F is good at calculations, when told explicitly what to find. Notice that in part
three the student breaks the trapezoid into a rectangle (moving one triangle to the other
side) to calculate the area for the trapezoid. It is unclear how the student decided on the
size of the base (4) or if that is a rounded number (3.5 ≈ 4). The student adds in the
circumference to the area of the other shapes. What types of experiences would help this
student? What questions might you ask?
Student F
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Student G calculates the circumference of the small circle and uses that as one of the
dimensions of the large rectangle, replacing the 2.5. Do students get enough
opportunities to think about and deconstruct diagrams as part of their regular class
program? How do we help students develop their visual thinking? Again, the student
struggles with interpreting the diagram when thinking about 2b. The student thinks now
about using the circumference of the circle as the top dimension of the trapezoid. Would
a habit of mind, like labeling diagrams with dimensions, have helped this student? Why
or why not? Finally in part 3 the student multiplies the “width” of the trapezoid by 8
instead of the area of the trapezoid. The student adds this calculation to the other top side
of the trapezoid. There is no use of area in any of the calculations in part 3.
Student G
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Student H tries to find the area of the rectangle in part one, but struggles with multiplying
decimals. How has the student dealt with multiplying by 0.5? So the student then divides
the area by 8 instead of the circumference by 8. In part 2a the student finds the radius
instead of the circumference and uses that as a dimension of the trapezoid. Is this student
struggling with understanding the diagram? What other issues are at play? In part 3 the
student uses the derived width of the trapezoid times 8 rather than multiplying an area
times 8. Would labels help this student? What experiences might help the student make
sense of the context and what is being asked?
Student H
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Student I appears to multiply pi by the length of the rectangle rather than the diameter of
the small circle in part 2a. The student has a correct answer for 2b, but there is no
supporting work. In part 3 the student calculates the area of the rectangle in the work
above the question. The student then also calculates the area of the circle (again in the
work above the prompt for 3.) The student also appears to have an area for the trapezoid,
but doesn’t use it below. In the final work the student seems to multiply the area of the
circle times the eight trapezoids and the circle), but then doesn’t use that calculation. The
final total could be either the area of the rectangle and the area of the circle or the area of
the rectangle and the area of the trapezoid. Students need to have practice organizing
large tasks for themselves to develop the logic of tracking calculations. Students also
need to see and compare examples of how to organize work in order to improve their
skills.
Student I
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Student J does not understand the demands of the task. Most answers have no attached
calculations or the student uses the dimensions from part of the diagram. In part 3 the
student appears to have measured the picture of the hat and used those dimensions to find
the perimeter instead of thinking about surface area. What resources are currently
available at your school site to help students who are missing this much background
knowledge? What are reasonable steps you can take within the classroom? How can you
help the student get other services?
Student J
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7th Grade
Task 4
Winter Hat
Student Task
Calculate the dimensions of material needed for a hat. Use circle,
measures of circumference and area. Calculate area for rectangles and
trapezoids.
Core Idea 4
Analyze characteristics and properties of two-dimensional
Geometry
geometric shapes. Apply appropriate techniques, tools, and
and
formulas to determine measurements.
Measurement
• Develop, understand, and use formulas to determine area of
quadrilaterals and the circumference and area of circles.
• Investigate, describe, and reason about the results of subdividing,
combining, and transforming shapes.
• Select and apply techniques and tools to accurately find length,
area, and angle measures to appropriate levels of precision.
The mathematics of this task:
• Calculating geometric units, such as area, circumference, radius, surface area
• Composing and decomposing 3-dimensional shapes
• Diagram literacy, being able to read and interpret a diagram and match parts of
the diagram to given dimensions, such as, seeing how circumference relates to the
rectangular shape, understanding what parts of the trapezoids connect to other
parts of the figure
Based on teacher observations, this is what seventh graders know and are able to do:
• Calculate the circumference of a circle
• Divide 24 by 8 to get the width of the trapezoid
• Give a value for pi
Areas of difficulty for seventh graders:
• Knowing the formula for the area of a trapezoid
• Understanding what dimensions or measurements are needed to find the area of a
trapezoid
• Visualizing how the sides of the trapezoid connect to the rest of the diagram
• Confusing area and circumference of a circle
• Understanding a diagram and breaking it down into separate parts
• Understanding of how to find surface area
• Organizing work to keep track of what is known, what is being calculated, what
else needs to be calculated
Strategies used by successful students:
• Labeling answers and defining what is being calculated each time
• Writing dimensions on the diagram as they are calculated for quick reference for
future parts of the task
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The maximum score available for this task is 9 points.
The minimum score needed for a level 3 response, meeting standard, is 4 points.
Many students, about 77%, could divide the width of the rectangle by 8 to find the
bottom dimension for the trapezoids. More than half the students, 50%, could also
calculate the circumference of the small circle and show their calculations. Some
students, 35%, could also divide the circumference of the small circle by 8 to find the top
dimension for the trapezoid. Less than 3% of the students could meet all the demands of
the task including finding the surface area of a 3-dimensional shape composed of a
rectangle, a circle, and 8 trapezoids. 37% of the students scored no points on this task.
60% of the students with this score attempted the task.
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Winter Hat
Points
Understandings
Only 60% of the students with
0
this score attempted the task.
2
Students could divide the width
of the long rectangle by 8 to find
the width of the base of the
trapezoid.
Students with this score could
usually find the circumference of
the small circle and show their
work. (These students could not
find the answer in part 1.)
4
Students could solve parts 1 and
2a, showing their work.
7
Students could use the given
measurements to find the area of
the rectangle, the dimensions of
the trapezoid, and find the
circumference of the small circle.
They understood that surface
area meant adding together the
parts.
9
Students could reason about a
complex 3-dimensional shape,
using a series of calculations to
derive needed dimensions, and
using the dimensions to calculate
surface area.
1
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Misunderstandings
Students could not reason about how the
trapezoids attached to the rectangle.
Answers for part 1 ranged from 1.5 to
221.7. The most common error was 3.5
Students had difficulty finding the
circumference of the small circle. Common
errors were 6,9, 1.5, and 5.5.
Students struggled with decomposing
shape to understand the relationship
between the circumference and the top of
the trapezoid. Some students did not round
properly (3%). 9% found radius instead of
circumference. 3% gave the height of the
trapezoid instead of the length of the top.
Students had difficulty organizing their
thinking to find surface area of a complex
shape. 15% did not attempt to find the area
of a rectangle. Many students did not think
about the fact that 8 trapezoids were
needed to make the complete hat. (74%)
Students struggled with the longer
reasoning chain and organizing their
thinking. Students with this score could not
find the area of the circle. 54% did not
attempt to find the area of a circle. 10%
used the circumference instead of the area
to add with the other shapes. 3% just
squared the radius and forgot to multiply
by pi. Students did not remember or could
not use the formula for area of a trapezoid.
43% did not attempt to make this
calculation. 11% multiplied the lower base
by the height. 6% just used the height for
the area.
75
Implications for Instruction
Students at this grade level should be challenged frequently to work on larger problems
involving longer chains of reasoning. Students have been working with geometric
concepts, such as area and perimeter of rectangles since 3rd grade, area of trapezoids in 5th
grade, and area of circles in 5th and 6th grade. The challenge for this grade level is to think
about complex shapes and geometric relationships. Students need to learn to organize
their thinking and do their own scaffolding. Students should develop tools and habits of
mind for making sense of what they know and what they need to find out. Using labels
for their calculations can help them think through the reasoning process.
Students had trouble making sense of diagrams. Students need more concrete
experiences with building and decomposing 3-dimensional shapes to help them think
about moving between a figure drawing and a net. Students at all grade levels seem to
fear writing on diagrams. This can be a powerful tool to aid in thinking. Writing
dimensions directly on the diagrams helps students track their thinking and plan what
needs to come next.
To prepare students for algebraic thinking, students at this grade level should start to
make generalizations about geometric formulas and understand how they are derived.
Instead of memorizing lots of different formulas, students should look at the trapezoid
and think about averaging the two bases to make a rectangle.
Ideas for Action Research – Problems of the Month
One interesting task to help students stretch their thinking about 3-dimensional shapes is
the problem of the month: Piece it Together, from the Noyce Website:
www.noycefdn.org/math/members/POM/pom.html
Ask students to work individually or in teams to solve the problem. Have them make
posters of calculations they made, their conclusions, and graphics or visuals to support
their thinking. The poster might also include other ideas they want to explore or
conjectures they haven’t had time to test. The purpose is to give them some complex
mathematical thinking, that requires persistence, willingness to make mistakes, edit and
revise, and is worth understanding the thinking of others.
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