Welcome Academic Math Coaches!
Let’s Mix It Up!
Find a seat at a table. Use the dot on your
nametag to make sure each color is
represented.
Green 1-2 years
Blue 3-4 years
Yellow 5-6 years
Red
7+
Building a Community of Coaches
Share one experience you’ve had in September
that has impacted your work as a coach.
As a table group finish this statement…
Being a math coach is like Autumn
because…
Making Connections across the
Common Core Domains
Beth Schefelker
Bridget Schock
Connie Laughlin
Hank Kepner
Kevin McLeod
October 5, 2012
Content Goals 2012-2013
• Explore how the number system expands across the grades and
how operations are extended in such a way that the properties of
operations remain unchanged.
• Analyze and solve mathematical modeling situations and/or
problems using single and multi-step word problem structures.
• Extend understanding of whole number operations to fractions,
ratios and proportions.
• Focus on Math Practice Standards
 7 Look for and make use of structure.
 8 Look for and express regularity in repeated reasoning.
Academic Coach Math
Meeting Structure
In order to reach the K-12 spectrum:
 First meeting will develop foundational
understandings of a mathematical idea.
 Second meeting will extend those
mathematical understandings across grades
and domains.
1
2
3
4
5
Operations & Algebraic Thinking
6
7
8
HS
Algebra
Expressions and
Equations
Number & Operations in Base Ten
The Number System
Number &
Operations
Fractions
Ratios &
Proportional
Relationships
Measurement & Data
Number
and
Quantity
Functions
Statistics & Probability
Geometry
Modeling
K
Counting
&
Cardinality
Mental Math
• Put your pencils down, it’s time for some
mental math!
48 + 23
93 – 38
• Turn and talk. Share your strategy with a
partner!
Learning Intention & Success Criteria
Learning Intention
We are learning to …
 Understand properties of addition and subtraction to
simplify and solve problems.
 Understand subtraction as a missing addend.
Success Criteria
We will be successful when we can…
 Apply strategies to reason through addition and
subtraction problems.
 Justify how properties support the strategies for addition
and subtraction.
The Importance of
Properties
Addition and Subtraction
“The mathematical foundations for
understanding computational procedures for
addition and subtraction of whole numbers are
the properties of addition and place value.”
Developing Essential Understanding of Addition & Subtraction
Pre-K – Grade 2, p. 28
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
For every a there exists –a so
that a + (–a) = (–a) + a = 0
(a × b) × c = a × (b × c)
Associative property of multiplication
Commutative property of
multiplication
Multiplicative identity property of 1
a×b=b×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
a×1=1×a=a
And in the domain of Operations and
Algebraic Thinking, it is those meanings,
properties, and uses which are the focus…
and it is those meanings, properties, and uses
that will remain when students begin doing
algebra in middle grades [and beyond].
--Jason Zimba
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
Using properties of operations
• 1OA3. Apply properties of operations as strategies to
add and subtract.
• 3OA5. Apply properties of operations as strategies to
multiply and divide.
• 4NBT5. Multiply two two-digit numbers using strategies
based on place value and the properties of operations.
• 5NBT6. Find whole-number quotients and remainders with
… using strategies based on place value, properties of
operations ….
• 5NBT7. Add, subtract, multiply, and divide decimals to
hundredths, using concrete models or drawings and
strategies based on place value, properties of operations….
• 6EE3. Apply the properties of operations to
generate equivalent expressions.
• 7NS2c: Apply properties of operations as
strategies to multiply and divide rational numbers.
• 7EE1. Apply properties of operations as
strategies to add, subtract, factor, and expand
linear expressions with rational coefficients.
• and into high school……
Develop and
use strategies based
on properties of the
operations
Addition Strategies and Properties
Whole Numbers
CCSSM Connection
Understand and apply properties of operations and the relationship
between addition and subtraction.
1.OA.3. Apply properties of operations as strategies to add and subtract.2
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative
property of addition.) To add 2 + 6 + 4, the second two numbers can be added
to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
1.OA.4. Understand subtraction as an unknown-addend problem. For
example, subtract 10 – 8 by finding the number that makes 10 when added to
8. Add and subtract within 20.
Math Practice Standards
• 7 Look for and make use of structure
• 8 Look for and express regularity and repeated reasoning
What’s the relationship between
these two number sentences?
Case 7: Adding 1 to an Addend
Combine tables to form groups of 10.
A. Role players




Person 1:
Person 2:
Person 3:
Person 4:
Teacher
Ester, Connie, Tia
Joe
Terry, Giovanni, Sema
 Use cubes to model out the actions of the students in
the case study.
B. Observers
 Attend to the conversations of student thinking
Case 7: Adding 1 to an Addend
Re-read the case. As you read, consider the
following…
• What mathematical ideas are the students
developing?
• How are students making sense of the
mathematics?
So what’s happening?
We know: 21 + 23 = 44
What about 21 + 24?
21 + 24 = 21 + (23 + 1)
= (21 + 23) + 1
= 44 + 1
= 45
Articulating a Generalization
for Addition
If 21 + 23 is 44, then 21 + 24 is 45.
• If you add 1 to one of the numbers (addends),
then you add 1 to the sum.
• If you add any number to one of the addends,
then you add that same number to the sum.
Does it work for other numbers?
Moving student thinking…
How would a student solve 19+7?
How would you help a student use properties to
make sense of 19 + 7?
 Where would you start?
 What questions would you ask of your students?
Let’s Talk About Questions
• Opportunity for assessment to focus on
questioning….
Subtraction Strategies
and Properties
Whole Numbers
Number Talks
Case 3.3
As you watch the DVD…
• Look for growth and clarification of students’
understanding
• Teacher moves
70-59
70-59
• What strategies did you see students using?
• Why do you plus “it” on again?
• Students were unsure whether 1 should be
added or subtracted. How could you help
students understand this thinking?
• Where are the properties?
Student Generalizations
• What properties underlie computational
strategies for addition and subtraction?
• How do we draw on models for representing
addition and subtraction, such as visual
images, story contexts, and number lines, to
express and justify generalizations?
1.OA.4
Understand subtraction as an unknownaddend problem.
How does the conversation in the DVD support
the use of thinking about subtraction as a
missing addend and using an open number line
to justify thinking strategies?
Math Practice Standards
As you think about what we did today, where do
you see these Math Practice Standards?
 MP7 Look for and make use of structure
 MP8 Look for and express regularity in repeated
reasoning
Learning Intention & Success Criteria
Learning Intention
We are learning to …
 Understand properties of addition and subtraction to
simplify and solve problems.
 Understand subtraction as a missing addend.
Success Criteria
We will be successful when we can…
 Apply strategies to reason through addition and
subtraction problems.
 Justify how properties support the strategies for addition
and subtraction.
Walk Away
• What has to happen in the classroom for
students to develop an understanding of
properties in order to apply them to
strategies?
• What conversations will you have with
teachers as they develop student reasoning
with number strategies?
Professional Practice
• Find at least two ways properties can help you
find solutions to each of the following:
 347 + 454
 135 – 97
• Practice with students
 During the next 2 weeks, work with a small group of
students on one of the two problems
 Chart/Scribe using these representations to capture
student thinking, similar to the teacher in the video:
• Number line
• Associating decomposed and recomposed numbers
Math Practice Standards
Read Think Math article explaining
 MP7 Look for and make use of structure
 MP8 Look for and express regularity in repeated
reasoning
Highlight important ideas in each explanation of
the Practice.
Create a poster for the two standards