Problem Solving for the 21st Century
Grade 5 Sample
Assessment Math Task
Prize Ribbons With Sequins
Mrs. Mason volunteers to make one prize ribbon for each
of the forty-seven students who will attend math camp.
Mrs. Mason will decorate each prize ribbon with sequins.
Mrs. Mason buys packages of sequins. Each package
holds one-fourth of a pound of sequins. Each package
has enough sequins to decorate six prize ribbons. How
many packages of sequins does Mrs. Mason need to buy
to make the prize ribbons? How many pounds of sequins
does Mrs. Mason have to buy to make the prize ribbons?
What part of a pound of sequins will be used to make
each prize ribbon? Show all your mathematical thinking.
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Prize Ribbons With Sequins
Problem Solving for the 21st Century
Unit of Study:
Dividing With Fractions Unit
Task
Mrs. Mason volunteers to make one prize ribbon for each of the forty-seven students who will
attend math camp. Mrs. Mason will decorate each prize ribbon with sequins. Mrs. Mason buys
packages of sequins. Each package holds one-fourth of a pound of sequins. Each package
has enough sequins to decorate six prize ribbons. How many packages of sequins does Mrs.
Mason need to buy to make the prize ribbons? How many pounds of sequins does Mrs.
Mason have to buy to make the prize ribbons? What part of a pound of sequins will be used
to make each prize ribbon? Show all your mathematical thinking.
Dividing With Fractions Unit
The Dividing with Fractions Unit develops meaning for situations involving dividing a whole
number by a unit fraction and dividing a unit fraction by a whole number. Questions asked
when dividing whole numbers provide meaning for situations involving division with fractions:
• When a quantity is shared equally, what is the size of each share? When 1/4 is
shared equally among 5, what is the size of each share: 1/4 ÷ 5 = £?
• How many groups of a given size are in this whole number? How many 1/4s
are in 5: 5 ÷ 1/4 = £ ?
Math Concepts and Skills:
The student develops and uses strategies for positive rational number computation in order
to solve problems.
The student:
• describes situations in which a unit fraction is divided by a whole number and
a whole number is divided by a unit fraction.
• represents the division of a unit fraction by a whole number and the division
of a whole number by a unit fraction such as 1/4 ÷ 5 and 5 ÷ 1/4 using
objects and pictorial models such as area models.
Exemplars Task-Specific Evidence
This task requires students to divide a unit fraction by a whole number. Students also need to
find the sum of like fractions.
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Underlying Mathematical Concepts
•
•
•
•
•
Fractional part of a whole
Multiples of a whole
The meaning of division
Division of a unit fraction by a whole number
Addition of like fractions
•
•
•
•
Model (manipulatives)
Diagram/Key
Table
Number line
Possible Problem-Solving Strategies
Possible Mathematical Vocabulary/Symbolic Representation
•
•
•
•
•
•
•
•
3
Model
Diagram/Key
Table
Number line
Pound (lb)
Ounce (oz)
Multiples
Equivalent/Equal to
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•
•
•
•
•
•
•
Unit
Whole
Fraction
1/4, 1/24 ...
Numerator/Denominator
Weight
Decimals
•
•
•
•
•
•
•
0.25 ...
Pattern
Per
Rules: 6 • p = r, 1/4 • p = t
Variable
Graph
Axis
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Possible Solutions
Original Version:
Mrs. Mason needs to buy 8 packages of sequins, which is 2 pounds of sequins. 1/24 of a
pound of sequins will make one prize ribbon.
Prize Ribbons With Ribbons
1
4 ÷ 6
Ribbons per Pounds per
Package
Package
Total Pounds
Packages
1
6
1
1
2
6
1
2
4 x 6 = 24 ribbons
1 pound of sequins
3
6
1
3
4
6
1
24 + 24 = 48 ribbons
2 pounds of sequins
5
6
1
5
6
6
1
6
7
6
1
7
8
6
1
1
1
1
•
=
4
6
24
of a pound
Key
1
is / 4 lb package of sequins
6
is 6 ribbons
/4
/4
/4
/4
/4
/4
4
/4
/4 = 1
/4
/4
/4
/4
/4
/4
8
/4
/4 = 2
6
12
18
24
30
36
42
48
6
6
6
6
6
6
6
6
Ribbons
1
2
3
0
4
5
6
7
1
/4
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1
/2
3
/4
1
Pounds
Possible Connections
Below are some examples of mathematical connections. Your students may discover some
that are not on this list.
• Mrs. Mason has 1/24 lb of sequins left to make 1 more prize ribbon.
• Patterns: Packages +1, Total ribbons +6, Total pounds per package +1/4
• Rules: 6 · p = r, 1/4 · p = t (p is package, t is total pounds, r is ribbons).
• Generalize and apply the rules, and graph functions.
• 2 pounds of sequins is 32 ounces.
• Relate to a similar task and state a math link.
• Solve more than one way to verify the answer is correct.
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Novice Scoring Rationales
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Novice
The student does not appear to address the first part of the
task. The student’s strategy of multiplying 47 students by
16 does not work to solve the second part of the task. The
student’s answer, “The answer would be 752 pounds,” is not
correct. The student does not address the third part of the
task.
Reasoning & Proof
Novice
The student demonstrates no correct reasoning of the underlying concepts of the task. The student does not apply
adequate mathematical arguments. The student is not able
to identify the whole and the number of fractional parts in
multiples of a whole. The student does not understand that
each 1/4 pound package of sequins decorates six prize ribbons. The student states “a quarter of a pound would be
16,” which leads one to think the student may incorrectly
think sixteen ounces are in a quarter of a pound. The student does not determine how many packages of sequins are
needed and is not able to determine what part of a pound of
sequins will be used to make each prize ribbon.
Communication
Practitioner
The student correctly uses the mathematical term pound
from the task. The student also correctly uses the term
quarter to refer to 1/4 of a pound.
Note: Although the student does not know that 1/4 of a
pound is 4 ounces, the student does use the terms pound
and quarter to refer to fractional part of a weight.
5
Connections
Novice
The student does not make a mathematically relevant
observation.
Representation
Novice
The student does not use a mathematical representation in
her/his solution.
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Novice
Achievement Level: Novice 1
P/S R/P Com Con Rep A/Level
N
6
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P
N
N
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Apprentice Scoring Rationales
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Criteria and
Performance Level
Assessment Rationales
Problem Solving
Apprentice
The student’s strategy of using a table would work to solve
the first two questions of the task, but the student follows an
incorrect pattern for the total ribbons. The student’s answers,
“4 packages,” and “4/4 = 1 pound,” are not correct. The
student confuses ribbons with part of a pound of sequins
in addressing the third question. The student’s answer, “48
parts,” is not correct.
Reasoning & Proof
Apprentice
The student demonstrates some correct reasoning of the
underlying concepts of the task. The student demonstrates
understanding of 1/4 pound of sequins per package for six
ribbons. The student ends her/his reasoning at four packages because the student is following a doubling pattern for
the number of ribbons.
Communication
Practitioner
The student correctly uses the mathematical term pounds,
from the task. The student correctly uses the mathematical
notation 1/4, 2/4, 3/4, 4/4.
Connections
Apprentice
The student makes a mathematically relevant observation.
The student states, “I know 1 pound is 16 ounces.”
Representation
Apprentice
The student’s table is appropriate to the task but is not
accurate. The second column should be labeled total
sequins and the third column should be labeled total
ribbons. The data in the third column is not correct for the
third and fourth packages.
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Apprentice
Achievement Level: Apprentice 2
P/S R/P Com Con Rep A/Level
A
8
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A
A
A
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Practitioner
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Criteria and
Performance Level
Assessment Rationales
Problem Solving
Practitioner
The student’s strategy of using a diagram to find how many
packages of sequins and how many pounds of sequins are
used to make 47 prize ribbons works to solve the first two
questions of the task. The student’s answers, “8 packages,”
and “2 pounds sequins,” are correct. The student’s strategy
of using number lines to find that 24 ribbons can be made
with one pound of sequins works to solve the third question.
The student’s answer, “1/24 pound,” is correct.
Reasoning & Proof
Practitioner
The student demonstrates correct reasoning of the underlying concepts of the task. The student demonstrates understanding of identifying the whole and the number of fractional parts in multiples of a whole. The student correctly finds
eight packages of sequins, two pounds of sequins, and that
1/24 of a pound of sequins is used to decorate each prize
ribbon.
Communication
Practitioner
The student correctly uses the mathematical term pounds,
from the task. The student also correctly uses the terms per,
diagram, key, total, whole, ounces. The student correctly
uses the mathematical notation 1/4, 4/4, 1/24, 24/24.
Connections
Practitioner
The student makes the mathematically relevant observations,
“24/24 = 1 whole pound,” “16 ounces is 1 pound,” and, “2
pounds is 32 ounces or 32 oz.”
Representation
Practitioner
The student’s diagram is appropriate to the task and
accurate. A key provides the necessary labels and all entered
data is correct. The student’s four number lines representing
six ribbons per 1/4 pound are appropriate to the task and
accurate. All necessary labels are provided.
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Practitioner
Achievement Level: Practitioner 1
P/S R/P Com Con Rep A/Level
P
10
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P
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Practitioner, cont.
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Expert Scoring Rationales
12
Criteria and
Performance Level
Assessment Rationales
Problem Solving
Expert
The student’s strategy of using a table to find how many
packages of sequins, total ribbons made, and total weight
of sequins works to solve the first two questions of the task.
The student’s answers, “8 packages,” and “2 lbs,” are correct. The student’s strategy of using division to find what part
of a pound of sequins is used to decorate each prize ribbon
works to solve the third question in the task. The student’s
answer, “1/24 lb per ribbon,” is correct. The student uses the
patterns of the table to generalize and apply two rules.
Reasoning & Proof
Expert
The student demonstrates correct reasoning of the underlying concepts of the task. The student displays understanding
of identifying the whole and the number of fractional parts
in multiples of a whole. The student uses patterns of add 1,
6, and 1/4 to determine Mrs. Mason needs 8 packages of
sequins for a total weight of two pounds to decorate fortyeight ribbons. The student uses division to find that 1/24 of a
pound of sequins is used to decorate each prize ribbon. The
student verifies her/his answers by generalizing and applying a rule to find total ribbons and to find total pounds of
sequins.
Communication
Expert
The student correctly uses the mathematical term pounds,
from the task. The student also correctly uses the terms
table, total, weight, per, ounces, patterns, input, output,
rules, key, summer. The student correctly uses the
mathematical notation 1/4, 2/4, 3/4, 4/4, 1 1/4, 1 2/4, 1 3/4,
1 4/4, 1/6, 1/24, .25, 1 4/4, 8/4, 6 · p = t, 1/4 · p = lb (with all
variables defined).
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Expert Scoring Rationales, cont.
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Criteria and
Performance Level
Assessment Rationales
Connections
Expert
The student makes mathematically relevant Practitioner
connections. The student states, “I noticest that she has
enough for 1 extra prize ribbon,” “1/4 is also .25,” “1/4 lb
is 4 ounces,” “She buys 32 ounces of sequins,” and “1 4/4
= 8/4 lbs = 2 lb.” The student also states, “I think 2 lbs of
sequins is a lot. I think the ribbons must be wide and long
and have sequins on both sides if Mrs. Mason uses 1/24 lb
per ribbon.” The student makes Expert connections. The
student generalizes a rule with the variables defined, 6 · p =
t. The student uses this rule to find the total ribbons for two,
eight, and ten packages of sequins. The student generalizes
a rule with the variables defined, 1/4 · p = lb. The student
uses this rule to find the total pounds for two, five, eight,
and 20 packages of sequins. The student also writes, “I am
correct,” next to 6 · 8 = 48 and 1/4 · 8 = 8/4 = 2 lb to verify
that her/his answers are correct.
Representation
Expert
The student’s table is appropriate to the task and accurate.
All labels are provided and the entered data is correct.
The student uses the “input” and “output” in the table to
generalize and apply two rules.
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Expert
Achievement Level: Expert 1
P/S R/P Com Con Rep A/Level
E
E
E
E
E
E
,
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Expert, cont.
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Expert, cont.
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