1. Common Core State Standards
Day 1
Session 2
K-5 Mathematics
What are the Common Core State Standards for Mathematics
2:15-2:45 Common Core State Standards (CCSS) for Mathematics Bell Work, Introductions and Digital Checkup
2:45-3:30 Common Core State Standards for Mathematics from A to Z and Exploring the Standards for Mathematical Practice
3:30-4:15 Exploring the Common Core State Standards for Mathematics Critical Areas and Mathematics Progressions
Common Core State Standards
Session 2
K-5 Mathematics

2. Bell Work Activity Handout #1
The Common Core State Standards for Mathematics from A to Z
List words that begin with each letter of the alphabet that identify aspects of the Common Core State Standards for Mathematics.
Handout #1
Common Core From A to Z
Facilitator Directions:
Before the Session begins, place A to Z Charts on the Wall.
Welcome participants as they enter the room.
Draw the participants attention to the display screen.
Ask participants to find a seat and then to begin the Bell Work Activity displayed on the screen.
The Bell Work Activity is Handout #1 – Common Core State Standards for Mathematics from A to Z.
As participants begin identifying aspects of the Common Core State Standards for Mathematics ask various participants to record a few of these words on the Charts displayed on the wall.
Time for Bell Work Activity: minutes

3. Five Minds for the Future (2007)
“We live in a time of vast changes that include accelerating globalization, mounting quantities of information, the dominating influence of science and technology, and the clash of civilizations. Those changes call for new ways of learning and thinking in school, business, and the professions.”
-Howard Gardner
Five Minds for the Future (2007)
Facilitator Directions:
Read the slide
“We live in a time of vast changes that include accelerating globalization, mounting quantities of information, the dominating influence of science and technology, and the clash of civilizations. Those changes call for new ways of learning and thinking in school, business, and the professions.”
-Howard Gardner ‘s book Five Minds for the Future (2007)
Click to display the Common Core State Standards graphic.
Say: The Common Core State Standards for Mathematics address the call for new ways of teaching, learning, and thinking!
Say: During the next few days, you will be involved in planning your schools steps for helping teachers embrace the call for new ways of teaching and the implementation of the Common Core State Standards for Mathematics.

4. Expected Outcomes
Enhance knowledge base of the Common Core Standards for Mathematics. 1
Become familiar with the structure of the Common Core State Standards for Mathematics. 2
Enhance knowledge of the Common Core Standards for Mathematical Practice. 3
Facilitator Directions:
Read the Expected Outcomes for this session.
Enhance knowledge base of the Common Core Standards for Mathematics.
Become familiar with the structure of the Common Core State Standards for Mathematics.
Enhance knowledge of the Common Core Standards for Mathematical Practice.
Understand how the critical areas bring focus to key mathematical concepts for students to learn at each grade level.
Consider how the learning progressions can be used to inform curriculum and guide instruction. 4
Understand how the critical areas bring focus to key mathematical concepts for students to learn at each grade level. 5
Consider how the learning progressions can be used to inform curriculum and guide instruction.

5. Transition Slide to the Common Core State Standards for Mathematics from A to Z presentation.
Facilitator Directions:
Say: I am now going to turn the microphone over to your presenter - Karol Yeatts, Director of the Office of Mathematics and Science, who will lead you through the Common Core State Standards for Mathematics from A to Z.
Presenter:
Say: Please locate your Bell Work Activity Handout #1: Common Core State Standards for Mathematics from A to Z.
Say: During the next few slides, I will be using a Guided Note-taking strategy highlighting several aspects of the Common Core State Standards for Mathematics. You may want to use Handout #1 to continue adding information about the Common Core State Standards for Mathematics.
Say: Please note, that I will not be progressing sequentially through the alphabet, just like you would not be progressing sequentially through the Common Core State Standards Mathematics. The information that I will be sharing is “clustered” around various overarching topics “domains” with specific components “standards” that will be expanded upon.
Say: So, let’s begin.

6. Whole Child Approach Sense-making
Building Foundation
Ensuring Education
CCSSO
Focus
Aligned
Daro
Coherence
Clarity
Developmental Level
Evidenced-based
Application
Balanced
Critical Areas
Fluency
Domains
Clusters
Guided
by Principles
Habits of
Mind
Joint
effort
Knowledge
International Benchmarked
Learner-focused Life-long skills
Illustrative Mathematics
McCallum
Progressions
Organized
Robust,
Relevant, Real-world
National Focus NGA
Quality
Procedural fluency
Opportunities
Mathematical Practice
Proficiency
Research-based
Presenter Directions
Refer to the Common Core State Standards for Mathematics notes highlighting various aspects of the Common Core State Standards for Mathematics.
PARCC
Rigor
Teachers
Whole Child Approach
Standards
Vision
Sense-making
Understanding
Zimba X
YOU
Timeline

7. High School Conceptual Categories
Number and Quantity (N)
Algebra (A)
Functions (F)
Modeling (*)
Geometry (G)
Statistics and Probability (S)
Say: The high school standards are set up slightly differently and have Conceptual categories which portray a coherent view of high school mathematics; a student’s work with functions, for example, crosses a number of traditional course boundaries, potentially up through and including calculus.
The Modeling conceptual category does not list any specific standards. Rather they are found throughout the other five conceptual categories, as indicated by a star symbol (*).
A-Z

8. Domains for K-12 Number and Operations in Base Ten (NBT)3 4 5 6 7 8 HS
Counting and Cardinality (CC)
Number and Quantity
Number and Operations in Base Ten (NBT)
The Number System
Number and Operations-Fractions (NF)
Ratios and Proportional Relationships (RP)
Operations and Algebraic Thinking (OA)
Functions (F)
Expressions and Equations (EE)
Algebra
Geometry (G)
Measurement and Data (MD)
Statistics and Probability (SP)
Presenter Directions
Say: This table lists the domains for Kindergarten through grade 12. For each domain, the shaded areas indicate the grade levels where it is addressed. Notice that most of the domains span multiple grades level.
Say: Notice the abbreviation for each of the Domains.
Counting and Cardinality (CC)
Operations and Algebraic Thinking (OA)
Number and Operations in Base Ten (NBT)
Number and Operations – Fractions (NF)
Ratios and Proportional Relationships (RP)
The Number System (NS)
Expressions and Equations (EE)
Functions (F)
Geometry (G)
Measurement and Data (MD)
Statistics and Probability (SP)
Say: The Common Core Standards for Mathematics structure includes Domains, Clusters and Standards.
Say: The domains progress over several grades and standards from different domains may sometimes be closely related.
A-Z

9. Cluster Headings Cluster Headings Cluster Headings Cluster Headings
Domain
Cluster Headings
Domain
Cluster Headings
Domain
Cluster Headings
Presenter Directions
Say: This page provides an overview of the standards, first organized by domains. Domains describe large groups of related standards.
For grade 2, the domains are:
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry”
Say: Within each domain, you’ll find cluster headings, which describe smaller groups of related standards. For example, within the Operations and Algebraic Thinking domain, there are three cluster headings:
Represent and solve problems involving addition and subtraction.
Add and Subtract within 20.
Work with equal groups of objects to gain foundations for multiplication.
Say: Note, the overview includes the Common Core Standards for Mathematical Practice.
Domain
Cluster Headings
A-Z 9

10. Cluster Headings Standards Domain Presenter Directions
Say: This slide shows a Grade 2 example of the domain, cluster headings, and standards. Point out the domain, cluster headings, and then the standards.
Say: Underneath each cluster heading, you will find standards that define what students should understand and be able to do. Again, clusters are groups of related standards.
Say: Standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

11. Florida’s Numbering of the Common Core State Standards
MACC.K.CC.2.5
Subject Grade Domain Cluster Standard
Subject
Mathematics Common Core
Grade K
Domain
Counting and Cardinality
Cluster
Count to tell the number of objects
Standard
Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
Presenter Directions
Say: Florida employs a Statewide Course Numbering System. The Florida Common Core State Standards basically follows the Common Core State Standards for Mathematics numbering system.
In Florida numbering system a number is assigned to the Cluster Headings.
The numbering of the individual standards continues throughout the Clusters. That is the numbering of the standards to not begin again with each new cluster, they continue from the previous cluster’s standards.
A-Z

12. Standards for Mathematical Practice
“The Standards for Mathematical Practice are unique in that they describe how teachers need to teach to ensure their students become mathematically proficient. We were purposeful in calling them standards because then they won’t be ignored.” - Bill McCallum
Presenter Directions
Read the quotation by Professor Bill McCallum, the Coordinator of the Math Team for the Common Core State Standards for Mathematics.
“The Standards for Mathematical Practice are unique in that they describe how teachers need to teach to ensure their students become mathematically proficient. We were purposeful in calling them standards because then they won’t be ignored.”

13. Standards for Mathematical Practice
Handout
Make sense of problems and persevere in solving them 1
Use appropriate tools strategically 5
Reason abstractly and quantitatively 2
Attend to precision 6
Presenter Directions
Say: These are the main headings for the Common Core Standards for Mathematical Practice. A description of each practice is included in the Common Core document beginning on page 6.
Stress that the Common Core Standards for Mathematical Practices do not stand alone and that they are not intended to be taught as stand alone lessons. The Common Core Standards for Mathematical Practices are an integral part of learning and doing mathematics and need to be taught with the same intention and attention as mathematical content.
Construct viable arguments and critique the reasoning of others 3
Look for and make sense of structure 7
Model with mathematics 4
Look for and express regularity in repeated reasoning 8
A-Z

14. Florida’s Common Core State Standards Implementation Timeline
Year/Grade Level K 1 2
3-8
9-12 FL L
F L
CCSS fully implemented
B L
CCSS fully implemented and assessed
Presenter Directions
Say: Full implementation of CCSS began with Kindergarten in
Say: It is essential that complex, rich, informational text become an integral part of instruction for reading and writing beginning with Kindergarten; informational text should include topics related to History, Social Studies, Science, and Technical Subjects at every grade level.
Say: will be our year of transition for grades 3 through 12. Instruction should be blended (a combination of the CCSS and NGSSS) as appropriate to ensure there are no gaps in presentation of the content.
F - full implementation of CCSS for all content areas
L – begin full implementation of content area literacy standards including: (1) use of informational text, text complexity, quality and range in all grades (K-12), and (2) CCSS Literacy Standards in History/Social Studies, Science, and Technical Subjects (6-12)
B - blended instruction of CCSS with Next Generation Sunshine State Standards (NGSSS); last year of NGSSS assessed on FCAT 2.0
A-Z 14

15. Standards for Mathematical Practice
Reasoning and Explaining
Seeing Structure and Generalizing
Overarching Habits of Mind of a
Productive Mathematical Thinker
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
Modeling
and
Using Tools
4. Model with mathematics
5. Use appropriate tools strategically
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
1. Make sense of problems and persevere in solving them
6. Attend to precision
Presenter Directions
Say: William McCallum, the coordinator of the Common Core State Standards for Mathematics writing has organized the practice standards in the following way.
Overarching Habits of Mind of a Productive Mathematical Thinker
1. Make sense of problems and persevere in solving them
6. Attend to precision
Reasoning and Explaining
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
Modeling and Using Tools
4. Model with mathematics
5. Use appropriate tools strategically
Seeing Structure and Generalizing
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
Adapted from (McCallum, 2011) 15

16. The Standards for Mathematical Practice
Please locate the Common Core State Standards for Mathematics.
Take a moment to examine the first three words of the narrative description for each of the 8 mathematical practices.
What do you notice?
Mathematically Proficient Students…
Page 6
Presenter Directions
Say: Please locate the Common Core State Standards for Mathematics. If you are using the Common Core State Standards document, you can find the Standards for Mathematical Practice on pages If you are using the Common Core App the Standards for Mathematical Practice can be found in the More Resources Section labeled Math.
Say: Please take a moment to examine the first three words of the narrative description for each of the 8 Mathematical Practices.
Ask: What do you notice?
Mathematically Proficient Students…
Time for this activity: 3-4 minutes 16

17. Digital Task
Handout
Your Digital Task is to:
Read your assigned Mathematical Practice.
Identify the words (verbs) that illustrate the student actions for this practice.
Text the words on one continuous line with spaces between each word.
Presenter Directions
Say: Your Digital Task is ….
Read the directions on the slide.
Say: As you read your assigned Practice consider how this practice compares to your current instructional practices?
Assign a mathematical practice to different groups (tables or rows).
One suggestion would be for different sections in the room being labeled with the Mathematical Practice that the groups will be assigned to read.
Another suggestion would be to have cards on the tables that identify the Practice that the participants sitting at that table would read.
Time for this activity: 5 minutes
Example: #..... create analysis model describe demonstrate….

18. Expectations
Students planning solution pathways, monitoring and evaluating their progress and asking “Does this make sense?” 1
Students knowing and using different properties of operations and objects and creating a coherent representation of the problem at hand. 2
Students understanding and using definitions. Students justifying and explaining their thinking and listening to arguments of others and deciding if they make sense.. 3
Presenter Directions
Say: These statements are aspects of the Standards for Mathematical Practice and describe expectations for the development of mathematical proficiencies.
Say: These Practices and Expectations cut across all content areas.
Say: Let’s identify the UDL components that are aligned with the Common Core State Standards for Mathematical Practice.
Students applying and using mathematics to solve problems connected to real-life situations. Students using models to represent, analyze and interpret results. 4

19. Expectations
Students being familiar with tools appropriate for their grade or course and using technology tools to explore and deepen their understanding of concepts. Students being able to make sound decisions about when each of these tools might be helpful. 5
Students communicating precisely to others. Students calculating accurately and efficiently, expressing numerical answers with a degree of precision appropriate for the problem context. 6
Students being able to look closely to discern a pattern or structure. Students being able to shift perspectives. 7
Presenter Directions
Students evaluating the reasonableness of their results. Student Maintaining oversight of the process, while attending to the details. 8

20. Presenter Directions
Say: You have now had a chance to hear about each of these 8 Common Core Standards for Mathematical Practice. As you listened to the various words that were being identified what did you notice about the different words?
Say: These words emphasize the actions, the changes in instruction, teaching and learning. These actions are the major shifts for the Common Core State Standards for Mathematics.

21. Consider the Learners Over 240,000 ELLs in Florida
Almost every district has ELLs
300 languages are spoken among ELLs
79% of ELLs are in Mainstream/Inclusion model classrooms
ELLs are learning in the same classrooms as non-ELLs
Presenter Directions
Say, How many of you have English Language Learners in your classroom?
Say, With Florida’s population of ELLs, it is important for teachers to design instructional activities that consider the learner’s need.
Read the stats shown on the slide.
Nearly 240,000 ELLs in Florida
Almost every district has ELLs
300 languages are spoken among ELLs
79% of ELLs are in Mainstream/Inclusion model classrooms.
ELLs are learning in the same classrooms as non-ELLs.

22. Making the Content Comprehensible
Use the standards vocabulary as a teaching tool. “Generalize, develop, describe, analyze, apply, measure,” etc. are all words ELLs will hear in the classroom and need to understand.
ELLs may know how to perform the skill using their language, they just may not yet have the English vocabulary.
Use pictures, graphs, and charts whenever possible.
Make use of root words and cognates.
Presenter Directions
Say: The Common Core Standards for Mathematical Practice requires that students start by explaining to themselves the meaning of a problem, that they justify their conclusions, communicate them to others and respond to the arguments of others.
Say: It is important that teachers help students understand the language of mathematics.
Read the information on the slide.
Use the standards vocabulary as a teaching tool. “Generalize, develop, describe, analyze, apply, measure,” etc. are all words ELLs will hear in the classroom and need to understand.
ELLs may know how to perform the skill using their language, they just may not yet have the English vocabulary.
Use pictures, graphs, and charts whenever possible.
Make use of root words and cognates.

23. Classroom Strategies Group ELLs with non-ELLs to work together.
Allow more wait time for ELLs to respond.
Silence does not necessarily mean the student does not know the answer, the ELL may be translating the answer and need more time.
Remember that ELLs from different countries may display mathematical functions in different ways.
Presenter Directions
Say: Employing appropriate instructional/learning strategies that meet the needs of the students is of upmost importance.
Read the instructional strategies on the PPT.
Group ELLs with non-ELLs to work together.
Allow more wait time for ELLs to respond.
Silence does not necessarily mean the student does not know the answer, the ELL may be translating the answer and need more time.
Remember that ELLs from different countries may display mathematical functions in different ways.

24. K-5 Domains and Critical Areas
Handout #2
Kindergarten Domains
Kindergarten Critical Areas
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Representing and comparing whole numbers, initially with sets of objects.
Describing shapes and space.
More learning time in Kindergarten should be devoted to number than to other topics
1st Grade Domains
1st Grade Critical Areas
Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20.
Developing understanding of whole number relationships and place value, including grouping in tens and ones.
Developing understanding of linear measurement and measuring lengths as iterating length units.
Reasoning about attributes of, and composing and decomposing geometric shapes.
2nd Grade Domains
2nd Grade Critical Areas
Extending understand of base-ten notation.
Building fluency with addition and subtraction.
Using standard units of measure.
Describing and analyzing shapes.
3rd Grade Domains
3rd Grade Critical Areas
Number and Operation in Base Ten
Number and Operation: Fractions
Developing understanding of multiplication and division strategies for multiplication within 100.
Developing understanding of fractions, especially unit fractions (fractions with numerator 1).
Developing understanding of the structure of rectangular arrays and of area.
Describing and analyzing two-dimensional shapes.
4th Grade Domains
4th Grade Critical Areas
Number and Operations: Fractions
Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends.
Developing understanding of fractions equivalence, addition and subtraction of fractions with like denominators, multiplication of fractions by whole numbers.
Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
5th Grade Domains
5th Grade Critical Areas
Developing fluency with addition and subtraction of fractions, developing understanding of the multiplication of fractions and of division of fractions in limited case (unit fractions divided by whole numbers and whole numbers divided by unit fractions).
Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations.
Developing understanding volume.
Presenter Directions
Say: The critical areas are designed to bring focus to the standards at each grade level by describing the big ideas that educators can use to build their curriculum and to guide instruction.
Say: For each grade, kindergarten through grade 5, there are two, three, or four critical areas.
Say: It is important that teachers are fully aware of the critical areas for their grade level as well as the critical areas from the prior grade level and the next grade level.
Say: This awareness is important for understanding the learning progression that contribute to the coherency of the Common Core State Standards for Mathematics.
A-Z

25. Two critical areas in Kindergarten
In Kindergarten, instructional time should focus on two critical areas:
Representing, relating, and operating on whole numbers, initially with sets of objects 1.
Describing shapes and space 2.
Presenter Directions
Say: Let’s begin by examining the Critical Areas in Kindergarten.
Say: Please locate the Kindergarten section of the Common Core State Standards for Mathematics. The Kindergarten section begins on page 9.
Say: Please take a few minutes to read the two critical areas.
Say: When you finish reading the Critical areas, glance over the Kindergarten Clusters.
Time for Reading the Critical Areas and Clusters: 3-4 minutes
More learning time in Kindergarten should be devoted to number than to other topics.
Page 9 25

26. Identify the Kindergarten Critical Area
Cluster Heading
Critical Area
Know number names and the count sequence.
Count to tell the number of objects.
Describe and compare measurable attributes.
Compare numbers.
Identify and describe shapes.
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
Work with numbers to gain foundations for place value.
Classify objects and count the number of objects in each category
Analyze, compare, create, and compose shapes.
#1 Numbers
#1 Numbers
#2 Shapes
#1 Numbers
#2 Shapes
#1 Numbers
Presenter Directions
Say: Let’s check for understanding.
Ask: How many critical areas are in Kindergarten? Answer 2 – Numbers and Shapes
Ask: How many Clusters are there in Kindergarten?” Answer 9 - refer to Common Core page 10
Say: Let’s identify the Critical Area associated with each of the Cluster.
Read each Cluster displayed on the slide and have participants raise 1 or 2 fingers and to say aloud the critical area associated with the cluster.
Know number names and the count sequence. #1 Numbers
Count to tell the number of objects. #1 Numbers
Describe and compare measurable attributes. #2 Shapes
Compare numbers. #1 Numbers
Identify and describe shapes. #2 Shapes
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. #1 Numbers
Work with numbers to gain foundations for place value. #1 Numbers
Classify objects and count the number of objects in each category #2 Shapes
Analyze, compare, create, and compose shapes. #2 Shapes
#1 Numbers
#2 Shapes
#2 Shapes

27. Four critical areas in 1st Grade
In Grade 1, instructional time should focus on four critical areas:
developing understanding of addition, subtraction, and strategies for addition and subtraction within 20 1.
developing understanding of whole number relationships and place value, including grouping in tens and ones 2.
developing understanding of linear measurement and measuring lengths as iterating length units 3.
Presenter Directions
Say: Let’s now look at the 1st grade critical areas.
Say: Please locate the First Grade section of the Common Core State Standards for Mathematics. This section begins on page 14 in your Common Core document.
Say: Take a few minutes to read the four critical areas and the clusters.
Time for Reading the Critical Areas and Clusters: 3-4 minutes
reasoning about attributes of, and composing and decomposing geometric shapes 4.
Page 14

28. Identify the 1st Grade Critical Areas
Cluster Heading
Critical Area
Represent and solve problems involving addition and subtraction.
Use place value understanding and properties of operations to add and subtract.
Represent and interpret data.
Add and subtract within 20.
Tell and write time.
Work with addition and subtraction equations.
Reason with shapes and their attributes.
Understand place value.
Measure lengths indirectly and by iterating length units.
Extend the counting sequence.
Understand and apply properties of operations and the relationship between addition and subtraction.
#1 Operations
#2 Base Ten
#3 Measurement
#1 Operations
#3 Measurement
#1 Operations
#4 Geometry
#2 Base Ten
Presenter Directions
Say: Let’s check for understanding.
Ask: How many critical areas are in 1st grade? Answer 4 – Operations, Base Ten, Measurement, Geometry.
Ask: How many Cluster are there in 1st grade? Answer refer to Common Core page 14
Say: Let’s identify the Critical Area associated with each of the Clusters.
Read each Cluster displayed on the slide and have participants raise 1, 2, 3, or 4 fingers and to say aloud the critical area associated with the cluster.
Represent and solve problems involving addition and subtraction. - #1 Operations
Use place value understanding and properties of operations to add and subtract #2 Base Ten
Represent and interpret data. #3 Measurement
Add and subtract within #1 Operations
Tell and write time. #3 Measurement
Work with addition and subtraction equations. #1 Operations
Reason with shapes and their attributes. #4 Geometry
Understand place value. #2 Base Ten
Measure lengths indirectly and by iterating length units. #3 Measurement
Extend the counting sequence. #2 Base Ten
Understand and apply properties of operations and the relationship between addition and subtraction. #1 Operations
#3 Measurement
#2 Base Ten
#1 Operations

29. Four critical areas in 2nd Grade
In Grade 2, instructional time should focus on four critical areas:
extending understanding of base-ten notation 1.
building fluency with addition and subtraction 2.
using standard units of measure 3.
Presenter Directions
Say: Please locate the 2nd Grade section of the Common Core State Standards for Mathematics. The critical areas for 2nd Grade are found on page 17.
Read the 4 Critical Areas shown on the Slide.
Say: Take a few minutes to read the four critical areas and the clusters.
Say: As you read the critical areas what are some of the learning progressions that you noticed from 1st grade to 2nd grade?
Possible Answers:
Extension of the Base-Ten system
Develop fluency with addition and subtraction within 100.
Mentally calculate sums and differences with only tens or hundreds.
Recognize the need for standard units of measure.
Use rules or other measurement tools with an understanding of iteration of units.
Describes shapes examining their sides and angles.
Decompose and combine shapes to make other shapes.
Begin to develop an understanding of area, volume, congruence, similarity and symmetry.
Time for Reading the Critical Areas and Clusters: 3-4 minutes
describing and analyzing shapes 4.
http; ://
Page 17 29

30. Four critical areas in 3rd Grade
In Grade 3, instructional time should focus on four critical areas:
developing understanding of multiplication and division and strategies for multiplication and division within 100 1.
developing understanding of fractions, especially unit fractions (fractions with numerator 1) 2.
developing understanding of the structure of rectangular arrays and of area 3.
Presenter Directions
Say: Please locate the 3rd Grade section of the Common Core State Standards for Mathematics. The critical areas for 3rd Grade are found on page 21.
Read the 4 Critical Areas shown on the Slide.
Ask: What is one of the new critical areas for 3rd Grade? Fractions!
Possible Answers include:
Fluently add and subtract within 1,000
Fluently multiply multi-digit whole numbers
Develop an understanding of the meanings of multiplication
and division of whole numbers.
Use properties of operations to calculate products of whole numbers.
Develop an understanding of fractions, beginning with unit fractions.
Begin to develop an understanding of area
Compare and classify shapes by their sides and angles and connect these with definitions of shapes.
Time for Reading the Critical Areas and Clusters: 3-4 minutes
describing and analyzing two-dimensional shapes 4.
http; ://
Page 21 30

31. Mathematics Progressions
Whole Numbers to Fractions in Grades 3-6
3-5 Number and
Operations -
Fractions
Common Core Progressions Project
Presenter Directions
Say: Fractions is an important concept that is linking to an understanding of whole numbers. The Common Core Progressions Project has researched the mathematical progression for developing an understanding of fractions.
Say: Let’s take a moment to listen to what Professor William McCallum says about the learning progression for fractions.
Say: Fractions are introduced in a way that shows the unity of the number system. Fractions are an extension of the number system. Linked to the understanding of whole numbers.
1st Grade - Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. (MACC.1.G.1.3)
2nd Grade - Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. (MACC.2.G.1.3)
3rd Grade - Number and Operations—Fractions (MACC.3.NF)
Develop understanding of fractions as numbers.
4th Grade – Number and Operations – Fractions (MACC.4.NF)
Extend understanding of fraction equivalence and ordering
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Understand decimal notation for fractions, and compare decimal fractions.
5th Grade - Number and Operations—Fractions (MACC.5.NF)
• Use equivalent fractions as a strategy to add and subtract fractions.
• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

32. Three critical areas in 4th Grade
In Grade 4, instructional time should focus on three critical areas:
developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends 1.
developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers 2.
Presenter Directions
Say: Please locate the 4th Grade section of the Common Core State Standards for Mathematics. The critical areas for 4th Grade are found on page 27.
Read the 4 Critical Areas shown on the Slide.
Ask: What are some of areas of focus?
Possible answers include:
Generalize understanding of place value to 1,000,000.
Use properties of operations, distributive property.
Develop fluency with efficient procedures for multiplying whole numbers.
Estimate and mentally calculate answers.
Develop understanding of fraction equivalence and operations with fractions.
Deepen their understanding of properties of 2-dimensional shapes.
Say: Let’s take a few minutes to discuss what is meant by Fluency in Mathematics.
Time for Reading the Critical Areas and Clusters: 3-4 minutes
understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. 3.
Page 27

33. What is meant by fluency?
Building Fluency
What is meant by fluency?
Presenter Directions
Ask: What is Meant by Fluency?
Have participants turn to someone sitting next to them to discuss what fluency in mathematics implies.
Time for this activity: 2-3 minutes
Say: The Common Core State Standards for Mathematics explicitly sets a demand for students to attain a variety of fluencies at specific grade levels. The Common Core State Standards for Mathematics document uses the phrase “fast and accurate” to describe the notion of fluency.
To further clarify the idea they use an analogy of being fluent in a foreign language – the ability to communicate and understand with automaticity.
By adhering to the Common Core State Standards for Mathematics, students will receive instruction through a progression of stages surrounding a target concept that ultimately provides them with the knowledge and practice needed to acquire the endpoint goal of fluency.
Say: Let’s listen now to Professor William McCallum and Jason Zimba, the Coordinators of the Math Team for the Common Core State Standards for Mathematics as each discusses the meaning of fluency.
Watch Video Clip: Mathematics Fluency: A Balanced Approach
Mathematics Fluency: A Balanced Approach (1:56)

34. Key Fluencies Handout #3 Grade Standard Key Fluency K MACC.K.OA.1.5
Add/subtract within 5 1
MACC.1.OA.3.6
Add/subtract within 10 2
MACC.2.OA.2.2
MACC.2.NBT.2.5
Add/subtract within 20
Add/subtract within 100 (pencil and paper) 3
MACC.3.NBT.1.2
MACC.3.OA.3.7
Add/subtract within 1,000
Multiply/divide within 100 4
MACC.4.2.4
Critical Area #1
Add/subtract within 1,000,000
Develop fluency with efficient procedures for multiplying whole numbers 5
MACC.5.NBT.2.5
Multi-digit multiplication
Developing fluency with addition and subtraction of fractions 6
MACC.6.NS.2.2
MACC.6.NS.2.3
Multi-digit division
Multi-digit decimal operations 7
MACC.7.EE.2.4a
Solve px + q = r, p(x + q) = r 8
MACC.8.EE.3.8b
Solve simple 22 systems by inspection
Presenter Directions
Say: Let’s take a few minutes to examine the Key Fluencies for grades K-5 found on Handout #3
Say: In Kindergarten the progression of learning is from representing addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations (K.OA.1.1). To solving addition and subtraction word problems (K.OA.1.2) to Fluently adding and subtracting within 5 (K.OA.1.5)
Say: In 1st grade the progression of learning is from Adding and subtracting within 20, to demonstrating fluency for addition and subtraction within 10 (1.OA.3.6).
Say: In 2nd grade students fluently add and subtract within 20 using mental strategies (2.OA.2.2). They also Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (2.NBT.2.5).
Say: In 3rd grade students have progressed to being fluently with adding and subtracting within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction (3.NBT.1.2). The progression of learning for developing fluency in multiplication and division is from Interpreting products and quotients of whole numbers (3.OA OA.1.4) to Applying properties of operations as strategies to multiply and divide (3.OA.2.5) to fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division (3.OA.3.7).
Say: In 4th grade students fluently add and subtract multi-digit whole numbers using the standard algorithm. (4.NBT.2.4). They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems (Critical Area #1)
Say: In 5th grade students fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.2.5). They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them (Critical Area #1)
A-Z

35. Three critical areas in 5th Grade
In Grade 5, instructional time should focus on three critical areas:
developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions) 1.
extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations 2.
Presenter Directions
Say: Please locate the 5th Grade section of the Common Core State Standards for Mathematics. The critical areas for 5th Grade are found on page 33.
Read the 3 Critical Areas shown on the Slide.
Ask: What are some of the learning progressions that you noticed from Kindergarten to 5th grade?
Possible answers include:
A more in depth understanding of the place values system
Performing operations from single digits to multi-digit wholes numbers and with decimals to hundredths.
Applying previous understanding of whole operations to multiply and divide fractions.
Converting like measurements units within a given measurement system.
Representing and interpreting data.
Understanding concepts of volume.
Graphing points on the coordinate plane.
Classifying 2-dimensional figures into categories based on their properties.
Time for Reading the Critical Areas and Clusters: 3-4 minutes
developing understanding of volume 3.
Page 33

36. Mathematics Progressions
Handout #4
The Importance of Mathematics Progressions
Presenter Directions
Say: The mathematics progressions are a narrative description of how a particular domain plays out over a grade level. They build from grade to grade and topic to topic, connecting topics logically and sequentially - providing K-12 focus and coherence. The mathematics progressions are research-based learning detailing what is known about students’ mathematical knowledge, skill, and understanding. The mathematics progression offer explanations for the sequence of the standards, potential cognitive difficulties, and pedagogical solutions which may be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.
Watch the Video clip: The Importance of Mathematics Progressions
Say: Please locate Handout #4 – Operations on Whole Number Progressions. Take a few minutes to observe the progressions from Kindergarten-5th grade.
Review a few of the Progressions together with the participants.
Operations on Whole Number Progressions

37. Mathematics Progressions Project
Kindergarten Counting and Cardinality
Number and Operations in Base Ten
Number and Operations—Fractions
K–5 Operations and Algebraic Thinking
Measurement and Data
Geometry Progression (coming soon!)
Presenter Directions
Say: Drafts of additional Progressions can be found on the website:
Expressions and Equations
Ratios and Proportional Relationships
Statistics and Probability, Grades 6–8

38. Give me a lever long enough and a fulcrum to place it on,
and I can move the world.
‐‐Archimedes
Presenter Directions
Say: This session was designed to provide you with an overview of the Common Core State Standards for Mathematics and to enrich your understanding of the Standards for Mathematical Practices, the Critical Areas of instruction and learning and the Mathematics Progressions.
Say: The Common Core State Standards for Mathematics address the call for new ways of teaching, learning, and thinking!
Say: As mentioned at the beginning of the session, during the next few days, you will be involved in planning your schools steps for helping teachers embrace the call for new ways of teaching and the implementation of the Common Core State Standards for Mathematics.

39. Reflective Thoughts
Handout #5
How will you use the Common Core Standards for Mathematical Practices to inform your curriculum and guide your instruction? 1
How will the Critical Areas and the Cluster headings help to inform your curriculum and guide your instruction? 2
Presenter Directions
Say: As we bring this session to a close, please take a few minutes to reflect on what you have heard and how this information can be used when planning your schools steps for helping teachers embrace the call for new ways of teaching and the implementation of the Common Core State Standards for Mathematics.
Read each of the Reflective Thoughts aloud. 3
How will you use the Learning Progressions to inform your curriculum and guide your instruction?