Download Circles and Stars

Problem Solving for the 21st Century
Grade 3 Sample
Assessment Math Task
Circles and Stars
Sophie is playing a game called Circles and Stars. Sophie
rolls one number cube and gets the number six. Sophie
draws six circles on her paper. Sophie rolls the number
cube again and gets the number three. Sophie draws
three stars in each of the six circles. Sophie writes 6 x 3 on
her paper. Sophie plays the game again. Sophie rolls one
number cube and gets the number three. Sophie draws
three circles on her paper. Sophie rolls the number cube
again and gets the number six. Sophie draws six stars in
each of the three circles. Sophie writes 3 x 6 on her paper.
Sophie says she got the same total number of stars both
times! Is Sophie correct? Show all your mathematical
thinking.
© 2016
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Circles and Stars
Problem Solving for the 21st Century
Unit of Study:
Multiplication Unit
Task
Sophie is playing a game called Circles and Stars. Sophie rolls one number cube and gets the
number six. Sophie draws six circles on her paper. Sophie rolls the number cube again and
gets the number three. Sophie draws three stars in each of the six circles. Sophie writes 6 X
3 on her paper. Sophie plays the game again. Sophie rolls one number cube and gets the
number three. Sophie draws three circles on her paper. Sophie rolls the number cube again
and gets the number six. Sophie draws six stars in each of the three circles. Sophie writes 3
x 6 on her paper. Sophie says she got the same total number of stars both times! Is Sophie
correct? Show all your mathematical thinking.
Multiplication Unit
The Multiplication Unit involves identifying a variety of models to represent the process of
multiplication in order to learn how to use it to solve problems. Questions to answer may
include:
• How do multiplication situations differ from addition situations?
• How do equal-sized groups model multiplication situations in the world
outside the classroom? What real-world examples of equal-sized groups can
you think of?
• How do arrays and area models represent multiplication situations in the
world outside the classroom? What real-world examples of arrays can you
think of?
• Given a multiplication equation, how can you create a situation to match it?
Math Concepts and Skills:
The student develops and uses strategies for multiplying whole numbers in order to solve
problems.
The student:
• finds the total number of objects when equal-sized groups of objects are
joined or arranged in arrays up to 10 by 10.
• represents multiplication facts using a variety of methods.
• uses a variety of strategies to multiply a two-digit number by a one-digit
number. Strategies may include mental math, partial products, and the
commutative, associative, and distributive properties.
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Exemplars Task-Specific Evidence
This task requires the students to know that multiplication involves finding the whole when
they know the number of equal parts and the number in each part. Students must also use
a variety of models to represent multiplication situations such as equal groups, rectangular
arrays and/or equal jumps on a number line.
Underlying Mathematical Concepts
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•
•
•
Creating multiplication situations to match an expression
Finding the product when both factors are known
Number sense to 18
Commutative Property
Possible Problem-Solving Strategies
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•
•
•
•
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Model (manipulatives)
Diagram/Key
Table
Tally chart
Number line
Array
Possible Mathematical Vocabulary/Symbolic Representation
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•
•
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•
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Model
Diagram/Key
Table
Tally chart
Product
Factor
Set
Array
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•
•
•
•
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Row
Column
Number line
Total/Sum
Dozen
Greater than (>)/Less than (<)
Equivalent/Equal to
Odd/Even
•
•
•
•
•
•
•
•
1/2
Rule
Variable
3·c=s
6·c=s
Equation
Commutative Property
Expression
Possible Solutions
Original Version:
Sophie is correct, she did get the same total number of stars both times.
Key
is 1 circle
6 x 3 = 18
is 1 star
3 x 6 = 18
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Possible Solutions (cont.)
Circles
Stars
Circles
Stars
1
3
1
6
2
6
2
12
3
9
3
18
4
12
5
15
6
18
Circles
1
0
1
2
2
3
4
3
5
6
7
4
8
5
6
9 10 11 12 13 14 15 16 17 18
Stars
Circles
2
1
0
Possible Connections
1
2
3
4
5
6
7
8
3
9 10 11 12 13 14 15 16 17 18
Stars
Below are some examples of mathematical connections. Your students may discover some
that are not on this list.
• Repeat the activity with other rolls of the number cubes.
• 6 is a half dozen.
• 6 threes is 1 1/2 dozen.
• Patterns: Stars +3 or +6, Circles +1.
• When you add equal groups on a number line, you jump over the same number of
spaces each time moving to the right, away from zero.
• Extend the number of equal sets of 3 beyond 6.
• Solve more than one way to verify answer.
• Relate to a similar task and state a math link.
• Rewrite the story with a new expression.
• Explain how 6 x 3 and 3 x 6 are both 18 but are used differently to represent the
situation in the game.
• 6 x 3 is an even number times an odd number which gives you an even product.
• Generalize and prove the rules 3 · c = s and 6 · c = s (key: c is circles, s is stars).
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Novice Scoring Rationales
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Criteria and
Performance Level
Assessment Rationales
Problem Solving
Novice
The student’s strategy of diagramming 18 circles with nine
S’s (stars?) in each circles would not work to solve the task.
The student’s answer, “NO,” is not correct.
Reasoning & Proof
Novice
The student does not show correct reasoning of the underlying concepts of the task. The student is not able to place
three stars in each of six circles or six stars in each of three
circles. It appears that the student may have added six and
three for a total of nine stars and multiplied six and three for
a product of 18 circles to form the structure of her/his diagram.
Communication
Novice
The student does not earn credit for the term key because it
is used incorrectly in the student’s diagram.
Connections
Novice
The student does not make a mathematically relevant
observation about her/his solution.
Representation
Apprentice
The student attempts to make a diagram but it is not
accurate. The student diagrams 18 circles with nine S’s
(stars?) in each circle. The student’s diagram does not record
or communicate correct reasoning.
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Novice
Achievement Level: Novice 1
P/S R/P Com Con Rep A/Level
N
6
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N
N
A
N
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Apprentice Scoring Rationales
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Criteria and
Performance Level
Assessment Rationales
Problem Solving
Practitioner
The student’s strategy of making two arrays to determine
that 6 x 3 and 3 x 6 have the same product works to solve
the task. The student’s answer, “Sophie is correct,” is correct.
Reasoning & Proof
Practitioner
The student shows correct reasoning of the underlying concepts of the task. The student uses arrays to confirm that 18
stars is the product for the first and second game number
rolls.
Communication
Apprentice
The student correctly uses the mathematical term array.
Connections
Practitioner
The student makes mathematically relevant observations
about her/his solution by stating, “She makes 36 stars,” and,
“You can do 6 + 6 + 6 = 18 or 3 + 3 + 3 + 3 + 3 + 3 = 18 if
you want.”
Representation
Practitioner
The student’s arrays are appropriate to the task and accurate.
The circles and stars are labeled and the arrays are correctly
constructed.
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Apprentice
Achievement Level: Apprentice 1
P/S R/P Com Con Rep A/Level
P
P
A
P
P
A
,
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Apprentice, cont.
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Practitioner
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Criteria and
Performance Level
Assessment Rationales
Problem Solving
Practitioner
The student’s strategy of making diagrams to determine that
6 x 3 and 3 x 6 have the same product works to solve the
task. The student’s answer, “18 = 18 so she is correct,” is correct.
Reasoning & Proof
Practitioner
The student shows correct reasoning of the underlying concepts of the task. The student uses diagrams to confirm that
18 stars is the product for the first and second game number
rolls.
Communication
Practitioner
The student correctly uses the mathematical terms diagrams,
key, circle, number line.
Connections
Practitioner
The student uses a different strategy to compare 6 x 3 and
3 x 6 by making three number lines. The student does not
earn Expert credit for verification because the student does
not link the two strategies/representations to confirm that
Sophie was correct.
Representation
Practitioner
The student’s diagrams are appropriate to the task and
accurate. The circles and stars are labeled in a key and each
circle contains the correct number of stars. The student’s
number lines are also appropriate to the task and accurate.
The stars and circles are labeled and the jumps are correctly
indicated.
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Practitioner
Achievement Level: Practitioner 1
P/S R/P Com Con Rep A/Level
P
P
P
P
P
P
,
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Practitioner, cont.
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Expert Scoring Rationales
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Criteria and
Performance Level
Assessment Rationales
Problem Solving
Expert
The student’s strategy of making tables to determine that 6
x 3 and 3 x 6 have the same product works to solve the task.
The student’s answer, “Sophie is correct,” is correct. The
student verifies her/his answer by generalizing and proving
two rules.
Reasoning & Proof
Expert
The student shows correct reasoning of the underlying concepts of the task. The student uses tables to confirm that 18
stars is the product for the first and second game number
rolls. The student justifies that her/his answer is correct by
applying two rules, 3 x C = S and 6 x C = S.
Communication
Expert
The student correctly uses the mathematical term total from
the task. The student also correctly uses the terms tables,
dozen, multiples, pattern, rules, key, circles. The student
uses the symbolic notation 3 x C = S, 6 x C = S.
Connections
Expert
The student makes the mathematically relevant Practitioner
observations, “She actually draws a total of 36 stars,” “That
is 3 dozen stars,” “3, 6, 9, 12, 15, 18 are multiples of 3-the
pattern game 1,” “6, 12, 18 are multiples of 6-the pattern
game 2,” and, “circle pattern is +1.” The student makes the
Expert connection by verifying that her/his answer is correct.
The student generalizes the rules, 3 x C = S and 6 x C = S.
The student uses the rules to solve for two, six, 10, and 100
circles with three per circle and one, two, three, 11, and 10
circles with six stars per circle. The student states, “The rules
match the tables. I am correct, Sophie is correct.”
Representation
Expert
The student’s tables are appropriate to the task and
accurate. All labels are provided and the data is correct. The
student uses her/his tables to generalize two rules to support
the data on the tables and to verify that her/his answer is
correct.
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Expert
Achievement Level: Expert 1
14
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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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