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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60
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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61
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
Grade 7 – 2008
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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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Sale!
This problem gives you the chance to:
• work with sales discount offers and percents
The following price reductions are available.
Two for the price of one
Buy one and get 25% off the second
Buy two and get 50% off the second one
Three for the price of two
1. Which of these four different offers gives the biggest price reduction?
____________________________________________________________________
Explain your reasoning clearly.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
2. Which of these four different offers gives the smallest price reduction?
____________________________________________________________________
Explain your reasoning clearly.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
___________________________________________________________________
9
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Sale!
Rubric
The core elements of performance required by this task are:
• work with sales discount offers and percents
Based on these, credit for specific aspects of performance should be assigned as follows
1.
points
Gives correct answer Two for the price of one.
2
Gives an explanation distinguishing which is the best buy.
1
section
points
Ranks all items by sample cost per item, % reduction per item, or fractional
cost per item, such as:
If the original price of one item is $100, then
Two for the price of one means that each item costs $50
or 50% of the original price.
Buy one and get 25% off the second means that each item costs $87.50
or 87.5% of the original price
Buy two and get 50% off the second means that each item costs $75
or 75% of the original price
2.
Three for the price of two means that each item costs $66.67
or 66.7% of the original price
3
Gives correct answer: Buy one and get 25% off the second
2
Gives an explanation distinguishing between the two lowest reductions or
explains why this is the worst choice.
1
6
Total Points
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3
9
82
Sale!
Work the task and look at the rubric. How did you rank the 4 options before making the
comparison? What unit did you choose to make the comparison? What are some other units
that could be used?
______________________________________________________________________
Look at student work for part 1. How many of your students chose :
2 for the
Buy 1 get 25% Buy 1 get 50%
3 for the
Thought that 2 for 1
price of 1
off the second
off the second
price of 2
and 3 for 2 were
equally good
. How many students were able to:
• Rank all the choices using some common unit to make the comparison
• Ranked at least 3 of the 4 choices with a common unit
• Gave a fair justification distinguishing between the top two choices
• Gave an explanation only involving their choice
• Gave an incorrect comparison because of lack of understanding of the situation or
incorrect conclusion
Make a list of the types of reasons or justifications students used for their choice:
• Did students think about changing the options to a similar unit before comparing?
• Did students understand the difference between getting one free in the first answer and
one free in the second? How did they justify or quantify the advantage?
• Did any of your students think about a unit price?
• What were some of the options that didn’t make sense?
• What are some of the errors students made in working with percents?
• What evidence did you find that students couldn’t make sense of the different options?
How often do students get an opportunity to work with problems like this, or rate problems,
where it is important to make units the same to make a comparison? Why is this important?
How has your class worked with the idea of making mathematical comparisons?
How often do students work with problems where they need to make derived calculations
before making a reasoned choice? What are the mathematical norms about using mathematics
to quantify or justify a decision?
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When thinking about providing a balanced curriculum, students need to be exposed to a
variety of task types; including: open investigation, nonroutine problems, design, plan,
evaluation and recommendation, review and critique, re-presentation of information, technical
exercise, and definition of concepts. This task focuses on the evaluation and recommendation.
Now look at student work on part 2. How many of your students chose:
Buy one and get
Two for the price of
Buy two and get
Three for the price
25% off the second
one
50% off the second
of two
What were some of the reasons for their errors? What misconceptions did you see?
Now look at student work for part 2 of the task. How many of your students chose:
Buy 1 get 25% off
3 for the price of 2
2 for the price of 1
Buy 1 get 50% off
the second
the second
Make a list of some of the units used by successful students to rank the choices:
What were some of the most indefensible explanations:
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Looking at Student Work on Sale!
Student A is able to think about the comparison by using a powerful mathematical strategy of
pretending, what if the original price was $10. The student is then able to find the unit price
for each shirt for all of the four options. The relationship between options remains the same no
matter what the pretend price is.
Student A
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85
Student B is able to compare three of the choices by finding the cost for 2 items. The student
was able to eliminate the final option by comparing the price for 6 items between the best buy
from the first three and comparing it to the remaining option. In both sets of comparisons the
student fixes the number of items to compare costs.
Student B
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86
Student C also fixes the number of items to compare the costs of the options. The student
chooses the best buy from the first three options and then compares it to the final option.
Notice that even though the student chooses a different “pretend” price and buys a different
amount of shirts, the relationship between options stays the same.
Student C
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Instead of comparing money spent, Student D compares money saved. The student runs into
trouble on making the final comparison because he doesn’t compare the savings on buying the
same number of shirts.
Student D
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The student correctly orders the options. The student shows that the first choice is 2 items for
$10 and the 4th choice is 3 for $20, so neither the price nor the quantity is fixed. The student
then makes a comparison between option 2 and 3, but does not quantify why they are a worse
buy than option 1 or 4.
Student E
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89
Student F is able to show a comparison between option 1,2, and 3 by finding the total cost of 2
items. The student chooses the wrong option because he does not convert the final option into
the cost of two items. The student forgets that $3 is less than $6. If the student had fixed the
cost: For $6 option 1 will get 4 items and option 4 will only get 3 items. If the student fixes the
number of items, option 1 will get 2 items for $3 and option for will get 2 items for $4.
Student F
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In part one Student G only talks about the good features of the best buy. The student makes
no comparison to the other options. Students need to learn the structure of comparison and
making a convincing argument. In part 2 the student is able to think about cost between the
choice in one and the choice in two. The options are still not eliminated. How do we provide
students opportunities to listen to convincing arguments or develop their own convincing
arguments? How do they learn the logic of making a sound argument?
Student G
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Student H has “pretended’ the cost is $2. The student then finds the cost of buying items with
each option. The problem is that the student is comparing buying 2 items in the first three
amounts and the cost of 3 items in the final option. When making a comparison either the
amount of items or the money spent needs to be equivalent before making the comparison.
Student H
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Student I might be thinking that the cost for each item is $1, although that is not clearly stated.
While the logic seems reasonable on first read for part, if the student quantified the options by
comparing the same number of items she would see the error in thinking.
Student I
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Student K has a strategy that could have led to the correct answer. Because the student is
trying to do the calculations in his head, he may be losing track of the amounts. For example
if the item is $40, then for option 2 he should have subtracted $10 from the cost rather than
adding $10 to the cost. It doesn’t get more expensive. What should the costs be for the other
options?
Student K
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In part 1 and part 2, Student L only justifies the qualities of the option picked. The student
does not understand the language and logic of a comparison. What opportunities do you
provide for students in your classroom to help develop these thinking skills?
Student L
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Student M
Student N
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Student O
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7th Grade
Student Task
Core Idea 1
Number and
Operation
Task 5
Sale!
Work with sales discount offers and percents. Make mathematical
comparisons of different options by fixing one quantity.
• Develop, analyze and explain methods for solving problems
involving proportional reasoning, such as scaling and finding
equivalent ratios.
• Work flexibly with fractions, decimals, and percents to solve
problems.
• Understand the meaning and effects of operations with rational
numbers.
Mathematics of the task:
• Making equivalent units for comparison
• Understanding the logic of comparison
• Using percents, fractions, etc. to find cost
Based on teacher observation, this is what seventh graders knew and were able to do:
• Getting something for free
• 25% is less than 50%, and could find 50% of a number
• Almost everyone could pick the best buy
• Understood that two for the price of one means buy one and get one free
Areas of difficulty for seventh graders:
• Making all four options in comparable units, many could change the first 3 into equivalent
units, but didn’t understand how to deal with the extra shirt in the final option
• Did not do calculations or confused percent off with percent paid
• Thought buy two and get 50% off second one means you’re buying three items
• Understanding the logic of a comparison: one quantity needs to be fixed to make the
comparison. In this case the students could either fix the number of items or fix the amount
paid
Strategies used by successful students:
• Using numerical values to determine the discounts or price paid
• Make up a starting cost for the items to quantify sales price or discount
• Using common multiples to set the number of items the same
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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, 81%, could identify either the best or the worst option for buying the items. Less
than half the students, 45%, could identify both the best and the worst option. A few students, 19%,
could also give a reason for their choices. 7% of the students could meet all the demands of the task
including ranking all the options using equivalent units for comparison. Almost 19% scored no
points on this task. 73% of those students attempted the task.
Grade 7- 2008
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99
Sale!
Points
Understandings
0
73% of the students with this
score attempted the task.
2
Students could identify the best
and worst buy.
3
Students with this score could
identify the worst buy and give a
reason why it was worse than the
next lowest option.
4
Students could identify both the
best and the worst option.
Students could identify best and
worst options and give some
supporting reasons for their
choices.
6
9
Misunderstandings
Students were confused by the number of
shirts. 19% thought getting 3 shirts was a
better deal than buy one get one free. 13%
thought 50% off was the best deal.
Students could not make convincing
arguments to support their ideas. Many
students only made comments about the
option they chose, not even attempting a
comparison with the other options.
15% of the students though 3 for the price
of 2 was the worst option. 10% thought
50% off was the worst option. Almost 6%
thought 2 for the price of one was the worst
option.
Students could not give convincing reasons
for their choices.
Students did not rank all the options by
making some kind of equivalent unit. 69%
of the students did not attempt to rank the
solutions. 10% attempted to rank the items
but didn’t know how to deal with the three
for the price of 2.
Students could find a way to rank
the options using equivalent units
and use this ranking to chose the
best and worst buys. Students
understood the idea of fixing
either the number of items or the
price to make the comparison.
Most commonly students
compared the first three options,
then made a different unit to
compare the best of those choices
to the final choice. Many
students understood that they
could “pretend” the original
price. They then could compare
total cost, savings, fractional or
percent amount paid.
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Implications for Instruction
Students need more practice reasoning about percents and quantifying relationships to make
comparisons between different options. Many students could pick out the best or worst option, but
gave little or no justification for their choice. They don’t understand the logic or language of making
a comparison. Students need to be able to find a common unit for comparison, such as a fraction, the
money saved, or the total cost for buying the same quantity to help verify their choice. Students need
to be able to fix one quantity in order to compare it to the other options.
Ideas for Action Research – Writing a Re-engagement Lesson
One of the effective tools developed by MAC through Action Research is the idea of re-engagement,
or using student work to plan further lessons. After the initial task the teacher or a group of teachers
sits down to examine student work. As teachers read several papers, a story emerges. Sometimes
the story is about strategies used by successful students. Sometimes the story is about the different
demands in working with a concept in context versus being given the information or data. Other
times the story is about a common recurring misconception. Sometimes the story is about the
difference in cognitive demand in the task, such as finding derived measurements in Winter Hat or
finding height in last year’s Parallelogram task, and how students are presented with similar tasks in
their textbooks. Sometimes the story is about recognizing significant digits, understanding
remainders, understanding a big mathematical idea, such as what an average does or how to make a
comparison. In this part of the tool kit, a larger sample of student work has been provided to give
you an opportunity to develop your own re-engagement lesson.
Sit down with a group of colleagues to examine the work on “Sale!” or look at your own student
work. What are the important mathematical ideas that students are struggling with in this task?
What do you think is the story of this task?
Now think, what is the minimum that most students in the class can understand and think about a
series of 3 to 5 questions to help move student thinking to the more desired goal. What pieces of
student work could you use to pose the questions? It helps to personalize the questions by attaching
student names (not names from your class, because you don’t want to put someone on the spot) or by
posing a dilemma to get students intrigued about finding the solution.
For example:
Julie says that two for the price of three is a better deal than two for the price of one. Because in both
cases you get one free item, but in her choice you get more items. Do you think Julie is correct?
Why or Why not?
Or
Ken says suppose the item was a video game, which cost $40. Then buy 1 get 50% off would be
$60. Can Ken pick a price? What would be the price for the other options? Would the best buy be
the same if Ken had picked a different price?
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Or
If the items cost $9 each, Martha says that 3 for the price of 2 would cost $18. Juanita thinks it
would cost $27. Diedra says the price per item would be $6. What are they thinking? Who do you
think is correct?
When using student work, it is important not to show too much or use labels. The
purpose is to make all students in the class have a reason or re-think the mathematics
from a slightly different perspective.
For example, a teacher might make a poster (without the correction mark) of Tabitha’s
work.
What do you think this student was thinking? Where do the numbers come from?
What are the figures in the drawing?
Hopefully, class discussion will get students to think about each item originally
costing $1. The common misconception that the option buy 2 and get 50% off the
second is about 3 items will be surfaced. Also it can bring into play how to compare
buying 2 items versus 3 items. Students or the teacher may then pose a further
question about what would happen to the price if we bought the same number of items
for each option. This type of minimal student work, or naked question, provokes
thinking by all students. The cognitive demand of trying to get into someone else’s
head is engaging and much deeper than just solving the problem originally.
The purpose of the re-engagement lesson is not to give students another worksheet,
but provide a forum for the exchange of ideas so that students develop the logic of
making convincing arguments, confront their misconceptions explicitly, learn
strategies used by others through the feedback of listening to their peers.
Now try writing or designing a series of questions that you might pose for your class
to develop the important mathematics of this task.
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Reflecting on the Results for Seventh Grade as a Whole:
Think about student work through the collection of tasks and the implications for instruction.
What are some of the big misconceptions or difficulties that really hit home for you?
_______________________________________________________________________
If you were to describe one or two big ideas to take away and use for planning for next year,
what would they be?
_______________________________________________________________________
What are some of the qualities that you saw in good work or strategies used by good students
that you would like to help other students develop?
_______________________________________________________________________
Four areas that stand out for the Collaborative as a whole are:
1. Increasing the Cognitive Demands and Rigor - As students move through the grades the
expectations for thinking about a topic and the mathematics they use should deepen and
become more complex. In early grades, usually fourth, students learn to use words like
likely, unlikely, certain to describe probabilities. They also learn to write numerical
probabilities for simple events like drawing a marble out of a bag or rolling a die. What
is new and different for middle grades is to be able to think about the sample space for
compound events and be able to quantify the numerical probability. Students had
difficulty with this in Will it Happen? At third and fourth grade students are learning to
draw and extend patterns and work backwards from a total to a stage number. What is
new and different for middle grades is the idea of searching for generalizations.
Students should not need to draw every shape or add on in long strings to find an
answer. They should be letting go of this comfortable habit and learning some new
strategies.(Odd Numbers) In geometry students have been working with area and
perimeter since third grade. At this level students need to work with longer chains of
reasoning, where it is necessary to look at geometric relationships to derive some of the
measurements or dimensions needed for achieving the final solution. They need to be
able to relate their calculations to specific parts of the diagram to reason about the
problem solution. Students should also be able to think about the relationships involved
in how formulas are put together (generalization), so they should be able to understand
why the 2 bases of the trapezoid need to be added together and averaged. (Winter Hat)
2. Using Academic Language and Number Theory – Students struggled with academic
language and number theory in both Odd Numbers and Pedro’s Tables. Students did
not recognize categories of numbers, such as multiples of 6, consecutive odd numbers,
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square numbers, and prime numbers. Students confused factors and multiples. Students
need opportunities to use academic language in conversation for a purpose, like making
a convincing argument or explaining their thinking. Language acquisition is developed
with practice. In probability students confused the mathematical idea of likely (having
a more than 50% chance of happening) with the common language of likely(it is
possible that it could happen).
3. Logic of Making a Justification– In Pedro’s Table students had difficulties justifying
why multiples of 5 were not factors of 12 or why there was only one factor of 3 that
was a prime number. Students didn’t know how to tie facts or information back to the
original assertion or how to connect different parts of the argument into a cohesive
whole. In Sale! Students did not understand the format or language of making a
comparison. Students might choose an option and explain why it is great without any
references to the other choices. Students didn’t understand that to make a comparison,
there needs to be one quantity that is fixed so the other quantities can be compared. In
this case either the number of items purchased needed to be the same so that the cost,
the savings, or unit price could be compared or the cost needed to be fixed so that the
number of items could be compared.
4. Diagram Literacy– Students did not know how to make models for sample space to
help them reason about compound probability in Will It Happen?. Students did not
understand how the labels for rows and columns worked together in the table for
Pedro’s Table. Students could not think about how to decompose the diagram of the hat
in Winter Hat, to think about the individual pieces. The students could not visualize
how the trapezoid connected to the large rectangle or the circumference of the small
circle. Students did not feel comfortable writing dimensions on the diagram as they
were derived to help them think about the relationships, what was needed next, and
what needed to be found.
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Examining the Ramp: Looking at Responses of the Early 4’s (29-32)
The ramps for the seventh grade test:
Will it Happen?–
• Part 3 – Finding probability of a compound event
o Understanding the difference between the number of combinations that
create a favorable outcome and the number of ways that combination can
be made
o Finding sample space for a compound event
o Quantifying the probability with a numerical expression
Odd Numbers • Part 6 – Working backwards from a total to a figure number
o Understanding relevant features of the pattern
o Making connections between the number being squared and the number of
elements in the pattern
Pedro’s Table • Part 4 – Making a mathematical justification
o Quantifying the relevant multiples of 5 and the factors of 12
o Showing the connection between multiples of 5 and factors of 12
o Relating the information back to the assertion being proved
Winter Hat–
• Part 3 – Finding the surface area of the hat
o Recognizing all the pieces in the hat and finding the area of each
o Being able to see how the trapezoid connects to other parts in the pattern
to find the appropriate dimensions (especially the connection between the
circumference of the circle and the top of the trapezoid
o Being able to identify and use the appropriate dimensions in the trapezoid
needed to calculate area
o Realizing that there are 8 trapezoids in the diagram
o Being able to organize and keep track of information in a longer chain of
reasoning
Sale!
• Part 1 – Finding equivalent units in order to compare
o Fixing either the cost or the quantity to make comparisons between
options
o Understanding how to compare 3 items with 2 items
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With a group of colleagues look at student work around 30 – 33 points. Use the papers
provided or pick some from your own students.
How are students performing on the ramp?
What things impressed you about their performance?
What are skills or ideas they still need to work on?
Are students relying on previous arithmetic skills rather than moving up to more
grade level strategies?
What was missing that you would hope to see from students working at this level?
When you read their words, do you have a sense of understanding from them
personally or does it sound more like parroting things they’ve heard the teacher
say?
How do you help students at this level step up their performance or see a standard
to aim for in explaining their thinking?
Are our expectations high enough for these students?
For each response, can you think of some way that it could be improved?
How do we provide models to help these students see how their work can be
improved or what they are striving for?
Do you think errors were caused by lack of exposure to ideas or misconceptions?
What would a student need to fix or correct their errors?
What is missing to make it a top-notch response?
What concerns you about their work?
What strategies did you see that might be useful to show to the whole class?
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Student 1
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Student 2
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Student 2, part 2
Student 3
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Student 3, part 2
Student 4
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Student 4, part 2
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Student 5
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Student 5, part 2
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