Problem Solving for the 21st Century
Grade 3 Sample
Instructional Math Task
Pom-poms
Amy is in a craft store looking at packages of pom-poms.
Amy sees packages of four red pom-poms. Amy sees
packages of seven red pom-poms. Amy says she can buy
seven packages of four pom-poms or four packages of
seven pom-poms because she will get the same total of
pom-poms. Is Amy correct? Show all your mathematical
thinking.
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Pom-poms
Problem Solving for the 21st Century
Unit of Study:
Multiplication Unit
Task
Amy is in a craft store looking at packages of pom-poms. Amy sees packages of four red pompoms. Amy sees packages of seven red pom-poms. Amy says she can buy seven packages of
four pom-poms or four packages of seven pom-poms because she will get the same total of
pom-poms. Is Amy correct? Show all your mathematical thinking.
Alternative Versions of the Task
More Accessible Version:
Amy is in a craft store looking at packages of pom-poms. Amy sees packages of four red pompoms. Amy sees packages of three red pom-poms. Amy says she can buy three packages of
four pom-poms or four packages of three pom-poms because she will get the same amount
of pom-poms. Is Amy correct? Show all your mathematical thinking.
More Challenging Version:
Amy is in a craft store looking at packages of pom-poms. Amy sees packages of four red
pom-poms for fifty cents. Amy sees packages of seven red pom-poms for seventy-five cents.
Amy says she can buy seven packages of four pom-poms or four packages of seven pompoms because she will get the same amount of pom-poms. Is Amy correct? Which packages
of pom-poms would be the better buy? 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?
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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.
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 be
familiar with 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
Commutative Property
Number sense to 28
Possible Problem-Solving Strategies
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Model (manipulatives)
Diagram/Key
Table
Tally chart
Arrays
Number line
Possible Mathematical Vocabulary/Symbolic Representation
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Model
Diagram/Key
Table
Tally chart
Number line
Array
Product
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Factor
Set
Total/Sum
Dozen
Greater than (>)/Less than (<)
Equivalent/Equal to
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Odd/Even
Equation
Expression
Row/Column
Rule
Variable
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Possible Solutions
Original Version:
Amy is correct because 4 x 7 = 7 x 4.
Key
1
2
3
4
5
6
Package
Pom-poms
1
4
2
8
3
12
4
16
5
20
6
24
7
28
7
is 1 pom-pom
7 x 4 = 28
is 1 package
1
2
3
4
28 = 28
4 x 7 = 28
Package Pom-poms
1
2
3
4
5
6
7
Package
Pom-poms
Package Pom-poms
1
2
3
4
2
14
3
21
4
28
Packages
1
0
2
2
4
6
3
4
5
6
7
8 10 12 14 16 18 20 22 24 26 28
Pom-poms
Rule
p is package
Packages
t is total pom-poms
4•p=t
7•p=t
1
7
1
0
2
4
2
6
3
4
8 10 12 14 16 18 20 22 24 26 28
Pom-poms
More Accessible Version:
Amy is correct because 4 x 3 = 3 x 4.
More Challenging Version:
Amy is correct. 4 packages of 7 pom-poms is a better buy because they cost $3.00. 7
packages of 4 pom-poms will cost $3.50.
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Possible Connections
Below are some examples of mathematical connections. Your students may discover some
that are not on this list.
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Patterns in table: Packages +1, Pom-poms +7 or +4.
The +4 pom-pom pattern is always even.
The +7 pom-pom pattern is odd, even, odd, even ...
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 0.
The number of equal sets of 4 is extended beyond 7.
The number of equal sets of 7 is extended beyond 4.
Solve more than one way to verify the answer.
Relate to a similar task and state a math link.
4 is an even number. 7 is an odd number. An even number times an odd number is
an even number.
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