Welcome Academic Math Coaches! Let’s Mix It Up! Find a seat at a table. Use the dot on your...

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 yearsBlue 3-4 yearsYellow 5-6 yearsRed 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 SchefelkerBridget SchockConnie LaughlinHank KepnerKevin 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.

• Put your pencils down, it’s time for some mental math!

48 + 2393 – 38

• Turn and talk. Share your strategy with a partner!

Mental Math

Learning Intention & Success Criteria

Learning IntentionWe are learning to …

Understand properties of addition and subtraction to simplify and solve problems.

Understand subtraction as a missing addend.

Success CriteriaWe 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

Associative property of multiplication (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

Distributive property of multiplication over addition

a × (b + c) = a × b + a × c

Not just learning them,

but learning to use them.

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”?

28 standards

Grade 1: OA, NBT

Grade 2: NBT

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 ConnectionUnderstand 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?

4 + 4 = 8

4 + 5 = 9

Case 7: Adding 1 to an Addend

Combine tables to form groups of 10.A. Role players

Person 1: Teacher Person 2: Ester, Connie, Tia Person 3: Joe Person 4: 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 = 44What 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 tomake 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 PropertiesWhole Numbers

Number TalksCase 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.4Understand subtraction as an unknown-addend 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 IntentionWe are learning to …

Understand properties of addition and subtraction to simplify and solve problems.

Understand subtraction as a missing addend.

Success CriteriaWe 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