CCSSM: Operations and Algebraic
Thinking (OA) Progression
Common Core State Standards for Mathematics
Pathways to Teacher Leadership in Mathematics
Wednesday, July 2, 2014
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Learning Intention & Success Criteria
We are learning to:
Understand three essential aspects of operations
important to arithmetic and algebra.
We will be successful:
When we can identify the three essential aspects in
work with whole numbers, fractions, and variable
expressions.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
OA Domain
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Operations and Algebraic Thinking
Dr. Jason Zimba
Professor of Physics and Mathematics
Bennington College, Vermont
Lead Writer, Common Core Standards for Mathematics
The Hunt Institute Video Series
Common Core State Standards: A New Foundation for Student Success
www.youtube.com/user/TheHuntInstitute#p
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Meanings of
the Operations
Properties of
the Operations
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Contextual
Situations
Meanings of the Operations
Meanings of
the Operations
Properties of
the Operations
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Contextual
Situations
What do students say?
Addition

Addition means plus.

It means to put two things together and then add
them like to see what the amount is at the end.
Subtraction

Subtraction means borrow.

Take away.

Take the number at the top and the number at the
bottom and subtract how many the number is at
the bottom. And then put the answer down.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
What do students say?
Multiplication

Times.

It means to take the number at the top and take it
how many ever times that the bottom number is.
Division

It’s something like the easy way to subtraction.

It’s to see how many numbers are in a number.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
‘“Addition, subtraction, multiplication, and
division have meanings, mathematical
properties, and uses that transcend the
particular sorts of objects that one is operating
on, whether those be multi-digit numbers or
fractions or variables or variables expressions.”
--Jason Zimba
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Meanings for the Operations
Each group selects one operation.
Addition
or
Multiplication
• Discuss and define using language that would
be meaningful to your students.
• Write your definition on chart paper and post.
Pathways to Teacher Leadership in Mathematics Project
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Criteria: Definitions that Work Well
Visualize actions on or relationships among quantities.
Encompass many interpretations, uses, and situations
(not limiting to just one view).
Accurate in the long run (doesn’t set up misconceptions).
Support seeing relationships among the operations.
Pathways to Teacher Leadership in Mathematics Project
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Comment on aspects of the definitions that seem
capable of serving students well across grades.
Comment on aspects that might need further
revision to avoid leading to misconceptions or
limited views of operations and their uses.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Contextual Situations
Meanings of
the Operations
Properties of
the Operations
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Contextual
Situations
Addition and Subtraction Situations
Add to
Take from
Put
together/Take
apart
Compare
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Multiplication and Division Situations
Equal
Groups
Arrays,
Area
Compare
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
In Grades K-8, how many standards reference
“real-world contexts” or “word problems”?
Grade K: OA
Grade 1: OA
54
standards
Grade 2: OA, MD
Grade 3: OA, MD
Grade 4: OA, NF, MD
Grade 5: NF, MD, G
Grade 6: RP, EE, NS, G
Grade 7: RP, EE, NS, G
24% of K-8 standards
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Grade 8: EE, G
Properties of
the Operations
Meanings of
the Operations
Properties of
the Operations
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Contextual
Situations
72 – 29 = ?
24 x 25 = ?
Mental Math
Solve in your head.
No pencil or paper!
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
72 – 29 = ?
24 x 25 = ?
Turn and share your reasoning.
Discuss how you used:
 Composing and decomposing
 Place value in base ten
 Properties of the operations
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
24 x 25 = ?
I would think what 25 x 25 is
then subtract 25 or I would
think what 20 x 20 is then
add it to 5 x 4.
I thought 25 x 25 = 625
and then I subtracted 25.
625 – 25 = 600.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
I figured that there are 4
twenty-fives in 100, and
there are 6 fours in 24,
so 100 x 6 = 600.
I thought 24 x 100 = 2400,
and 2400 ÷ 4 = 600.
24 x 25 = ?
25 x 4 = 100, 6 x 100 =
600, 600 + 100 = 700.
Well, 10 x 25 = 250,
2(10 x 25) = 500,
500 x 4 = 2000.
“I would try to multiply in my
head, but I can't do that.”
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
The properties of operations.
Associative property of addition
(a + b) + c = a + (b + c)
Commutative property of addition
a+b=b+a
Additive identity property of 0
a+0=0+a=a
Existence of additive inverses
Associative property of multiplication
For every a there exists –a so that
a + (–a) = (–a) + a = 0
(a × b) × c = a × (b × c)
Commutative property of multiplication
a×b=b×a
Multiplicative identity property of 1
a×1=1×a=a
Existence of multiplicative inverses
For every a ≠ 0 there exists
1/a so that a × 1/a = 1/a × a = 1
a × (b + c) = a × b + a × c
Distributive property of multiplication
over addition
Pathways to Teacher Leadership in Mathematics Project
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In Grades K-8, how many standards
reference “properties of the operations”?
Grade 1: OA, NBT
Grade 2: NBT
28
standards
Grade 3: OA, NBT
Grade 4: NBT, NF
Grade 5: NBT
Grade 6: NS, EE
Grade 7: NS, EE
Grade 8: NS
12% of K-8 standards
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Standard 3.OA.5
Apply properties of operations as strategies to
multiply and divide.
Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also
known. (Commutative property of multiplication.)
3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30,
or by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7
as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(Distributive property.)
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
In Grades K-8, how many standards
reference using “strategies”?
Grade K: CC
26
standards
Grade 1: OA, NBT
Grade 2: OA, NBT
Grade 3: OA, NBT
Grade 4: NBT, NF
Grade 5: NBT
11% of K-8 standards
Pathways to Teacher Leadership in Mathematics Project
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Grade 7: NS, EE
CCSSM Glossary
Computation strategy
Purposeful manipulations that may be chosen
for specific problems, may not have a fixed
order, and may be aimed at converting one
problem into another.
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Develop & use
strategies
•
•
•
Develop and use strategies
for single-digit computation facts... before
any expectation of knowing facts from memory.
Develop and use strategies to add, subtract,
multiply, and divide multi-digit whole numbers,
fractions, decimals…. before use of standard
algorithms.
As Hank noted, “CCSSM assumes about three
years of development of concepts and strategies
before demonstrating fluency.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Homework
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Readings
Due Monday, July 7, 2014
• Carpenter: Chapters 4 & 6
• Revisit: PtA: Representations p. 24-29.
• Revisit OA Progressions, pp. 3-20, 36-38. Appendix.
• EE Progressions, p. 6-7.
------------------------• Thornton (1978). Thinking strategies for basic facts.
• PtA: Fluency p. 42-48.
• Russell (2000). Computational fluency.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Homework
Email by Saturday night:
• One key idea related to “fluency” and one
question or wondering about developing fluency
with your students.
• One key message from the Thornton article on
basic facts.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Course Assignment: Sequence of Equations
Email by Sunday night:
Sequence of T/F or Open Number Sentences and
Rationale (5% of grade)
Equations
Rationale
z
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee
Disclaimer
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee, 2014-2017
This material was developed for the Pathways to Teacher Leadership in Mathematics project through the
University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This
material may be used by schools to support learning of teachers and staff provided appropriate attribution and
acknowledgement of its source. Any other use of this work—including reproduction, modification, distribution, or
re-publication and use by non-profit organizations and commercial vendors—without prior written permission is
prohibited.
This project was supported through a grant from the Wisconsin ESEA Title II Improving Teacher Quality Program.
Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee