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					                           Day 1
                 Common Core State Standards
                         Session 3
                     6-12 Mathematics

•    List words that begin with each letter of the alphabet that
     identify aspects of the Common Core State Standards for
     Mathematics.
                            WELCOME

Agenda

• The Common Core State Standards for Mathematics from A to
    Z

• Exploring the Standards for Mathematical Practice

• Exploring the Common Core State Standards for Mathematics
    Critical Areas

• Exploring the Mathematics Progressions

“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)
                      Expected Outcomes
  1. Enhance knowledge base of the Common Core Standards for
     Mathematics.

  2. Become familiar with the structure of the Common Core State
     Standards for Mathematics.

  3. Enhance knowledge of the Common Core Standards for
     Mathematical Practice.

  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.

                      Day 1 Digital Checkpoint
                     Session 3 6-12 Mathematics


                        Digital Checkpoint
• Poll Everywhere
• http://www.polleverywhere.com/
• To participate in the polls you will need a phone with text
  capability or a computer with Internet access.
• If using your phones, please remember to leave them on
  silent.
• This is a standard rate text message, so it may be free for
  you, or up to twenty cents on some carriers if you do not
  have a text messaging plan.
• The service we are using is serious about privacy. We cannot
  see your phone numbers, and you’ll never receive follow-up
  text messages outside this presentation.

                 How to Vote via Web Response
The texting process uses the example shown on the slide.
  • The URL for Internet Polling is
     http://www.polleverywhere.com/vote
  • In the Box, type in 37607 – this is our session “phone
     number.”
  • Then enter the “Code” for the poll.
  • Remember to leave a space between each entry.

    • Remember if using this Presentation, you will need to create
      your own Polling Account. Poll Everywhere is free for
      educators to use with up to 40 participants. Schools may
      subscribe and create their own account for unlimited usage.



                        Digital Resources
                      for the Common Core
•    Apple
      • http://itunes.apple.com/us/app/common-core-standards/id
         439424555

•    Android
     • https://play.google.com/store/apps/details?id=com.edutat
         er.corestandards

•    Common Core State Standards for Mathematics
     • http://www.corestandards.org/
                          Domains for K-12
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.
Notice the abbreviation for each of the Domains.
Counting and Cardinality (CC) – unique to Kindergarten
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)
Number and Quantity (N) – unique to high school
Algebra (A) – unique to high school
The Common Core Standards for Mathematics structure includes
Domains, Clusters and Standards and in High School there are
Conceptual Categories. The domains progress over several grades and
standards from different domains may sometimes be closely related.

                High School Conceptual Categories
•   Number and Quantity (N)

•   Algebra (A)

•   Functions (F)

•   Modeling (*)
•    Geometry (G)

•    Statistics and Probability (S)

                        Grade 7 Overview
This page provides an overview of the standards, first organized
by domains. Domains describe large groups of related standards.
For grade 7, the domains are:
    • Ratios and Proportional Relationships
    • The Number System
    • Expressions and Equations
    • Geometry
    • Statistics and Probability
Within each domain, you’ll find cluster headings, which describe
smaller groups of related standards. For example, within the
Ratios and Proportional Relationships domain, there is only one
cluster heading:
    • Analyze proportional relationships and use them to solve
      real-world and mathematical problems.
Also note the Statistics and Probability domain has 3 cluster
headings:
         • Use random sampling to draw inferences about a
           population.
         • Draw informal comparative inferences about two
           populations.
      • Investigate chance processes and develop, use, and
        evaluate probability models.

                 Number and Quantity Overview
        Each of the Conceptual Categories has the same
        format as the elementary and middle school, except
        that the categories are not by grade level, but rather by
        content. Overarching “big ideas” that connect topics
        across the grades/courses.
        Descriptions of the mathematical content to be learned,
        elaborated through clusters and standards
        Clusters indicate WHAT students should know and be
        able to do:
      • May appear in multiple grade levels/courses with
        increasing developmental standards as the grade levels
        progress
      • Reflect both mathematical understandings and skills,
        which are equally important




                  Florida’s Numbering of
             the Common Core State Standards

                      MACC.7.EE.1.1

MACC Mathematics Common Core = Subject
7 = Grade
EE Expressions and Equations = Domain
1 = Cluster (Use properties of operations to generate equivalent
expressions)
1 = Standard (Apply properties of operations as strategies to add,
subtract, factor, and expand linear expressions with rational
coefficients.)
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 the 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 do not begin
again with each new cluster, they continue from the previous cluster’s
standards.


                  Florida’s Numbering of
             the Common Core State Standards
                    MACC.912.N-RN.2.3

This slide shows the difference in the high school standards –
    there is the addition of the Conceptual Category. There are
    six Conceptual Categories:
    Number and Quantity (N)
    Algebra (A)
    Functions (F)
    Modeling (*)
    Geometry (G)
    Statistics and Probability (S)
                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

               8 Standards for Mathematical Practice
  1.   Make sense of problems and persevere in solving them
  2.   Reason abstractly and quantitatively
  3.   Construct viable arguments and critique the reasoning of others
  4.   Model with mathematics
  5.   Use appropriate tools strategically
  6.   Attend to precision
  7.   Look for and make sense of structure
  8.   Look for and express regularity in repeated reasoning

   Florida’s Common Core State Standards Implementation
                         Timeline
2011-2012: K = FL, 1 – 12 = L
2012 – 2013: K - 1 = FL, 2 – 12 = L
2013-2014: K -2 = FL, 3 – 12= BL
2014-2015 CCSS fully implemented and assessed: FL Across
the board
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

             Day 1 Standards for Mathematical Practice
                    Session 3 6-12 Mathematics


                                 Poll Question
Please take out your phones or use your computers to respond to
this Poll: How many Common Core Standards for Mathematical
Practices are there?
Text the Code #129153 and your response to 37607 or submit
your response to http://PollEv.com/vote


                 Standards for Mathematical Practice
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
http://www.corestandards.org/the-standards/mathematics/introduction/stand
ards-for-mathematical-practice

               Standards for Mathematical Practice
Video Clip

•   Develops dispositions and habits of mind
     “Characteristic of an educated person”
•   Precision in thought
•   Precision in the use of language and terms
•   Precision of argument
•   Sense making happens through conversations



                       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…


                                 Task
Your Task is to:

    •At your table, count off 1-8.
    •Read your assigned Mathematical Practice.

    •Identify and underline the words (verbs) that illustrate the
    student actions for this practice.


                 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


              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.

                     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 needs more time.
•    Remember that ELLs from different countries may display
     mathematical functions in different ways.

       Critical Areas and Mathematics Progressions
                          Session 3
                      6-12 Mathematics

                 K-12 Domains and Critical Areas
  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.

6 Grade Domains:

     Ratios and Proportional Relationships

     The Number System

     Expressions and Equations

     Geometry

     Statistics and Probability

6 Grade Critical Areas:

     Connecting ratio and rate to whole number multiplication and division
      using concepts of ratio and rate to solve problems.

     Completing understanding of division of fractions and extending the
      notation of numbers to the system of rational numbers which includes
      negative numbers.

     Writing, interpreting, and using expressions and equations.

     Developing understanding of statistical thinking.

7 Grade Domains:
   Ratios and Proportional Relationships

   The Number System

   Expressions and Equations

   Geometry

   Statistics and Probability

7 Grade Critical Areas:

   Developing understanding of and applying proportional relationships.

   Developing understanding of operations with rational numbers and
    working with expressions and linear equations.

   Solving problems involving scale drawings and informal geometric
    constructions, and working with two-and three-dimensional shapes to
    solve problems involving area, surface area, and volume.

   Drawing inferences about populations based on samples.

8 Grade Domains:

   The Number System

   Expressions and Equations

   Functions

   Geometry

   Statistics and Probability

8 Grade Critical Areas:

   Formulating and reasoning about expressions and equations,
    including modeling an association in bivariate data with a linear
    equation, and solving linear equations and systems of linear
    equations.
   Grasping the concept of a function and using functions to describe
    quantitative relationships.

   Analyzing two-and-three dimensional space and figures using
    distance, angle, similarity, and congruence, and understanding and
    applying Pythagorean Theorem.

Algebra 1 Domains:

   The Real Number System

   Quantities

   Seeing Structure in Expressions

   Arithmetic with Polynomials and Rational Expressions

   Creating Equations

   Reasoning with Equations and Inequalities

   Interpreting Functions

   Building Functions

   Linear, Quadratic, and Exponential Models

   Interpreting Categorical and Quantitative Data

Algebra 1 Critical Areas:

   Relationships between Quantities and Reasoning with Equations;
    Analyze and explain the process of solving an equation.

   Linear and Exponential Relationships; Learn function notation and
    develop the concept of domain and range.

   Descriptive Statistics; Learn formal means of assessing how a model
    fits (data regression techniques, graphical representations and
    goodness of fit.)
   Expressions and Equations; Create and solve equations, inequalities,
    and systems of equations involving quadratic expressions.

   Quadratic Functions and Modeling; Comparing key characteristics.



                 Four critical areas in 6th Grade

1. Connecting ratio and rate to whole number multiplication and division
and using concepts of ratio and rate to solve problems

2. Completing understanding of division of fractions and extending the
notion of number to the system of rational numbers, which includes
negative numbers

3. Writing, interpreting, and using expressions and equations

4. Developing understanding of statistical thinking



              Identify the 6th Grade Critical Area
  • 1 - Ratio: Understand ratio concepts and use ratio reasoning to solve
    problems.

  • 4 - Statistics: Summarize and describe distributions.

  • 2 - Rational #: Compute fluently with multi-digit numbers and find common
    factors and multiples.

  • 3 - Equations: Reason about and solve one-variable equations and
    inequalities.

  • 3 - Expressions: Apply and extend previous understandings of arithmetic to
    algebraic expressions.

  • 2 - Rational #: Apply and extend previous understandings of numbers to the
    system of rational numbers.
   • 3 - Equations: Represent and analyze quantitative relationships between
     dependent and independent variables.

   • 4 - Statistics: Develop understanding of statistical variability.

   • 2 - Rational #: Apply and extend previous understandings of multiplication
     and division to divide fractions by fractions.

Solve real-world and mathematical problems involving area, surface area, and
volume was not included in the above list. It is included as one of the areas to
build on their work in elementary school.



                   Four critical areas in 7th Grade
1. Developing understanding of and applying proportional relationships

2. Developing understanding of operations with rational numbers and working
with expressions and linear equations

3. Solving problems involving scale drawings and informal geometric
constructions, and working with two- and three-dimensional shapes to solve
problems involving area, surface area, and volume

4. Drawing inferences about populations based on samples



               Identify the 7th Grade Critical Areas
1-Proportions, 2-Rational: Analyze proportional relationships and use them to
solve real-world and mathematical problems.

3-Geometry: Draw, construct and describe geometrical figures and describe the
relationships between them.

2-Rational #: Apply and extend previous understandings of operations with
fractions to add, subtract, multiply, and divide rational numbers.

4-Statistics: Use random sampling to draw inferences about a population.
3-Geometry: Solve real-life and mathematical problems involving angle measure,
area, surface area, and volume.

4-Statistics: Draw informal comparative inferences about two populations.

2-Rational #: Use properties of operations to generate equivalent expressions.

2-Rational #: Solve real-life and mathematical problems using numerical and
algebraic expressions and equations.

Additional: Investigate chance process and develop, use, and evaluate probability
models.



                 Three critical areas in 8th Grade
1. Formulating and reasoning about expressions and equations, including
modeling an association in bivariate data with a linear equation, and solving linear
equations and systems of linear equations

2. Grasping the concept of a function and using functions to describe quantitative
relationships

3. Analyzing two- and three-dimensional space and figures using distance, angle,
similarity, and congruence, and understanding and applying the Pythagorean
Theorem



             Five critical areas (units) in Algebra 1
   1. Relationships Between Quantities and Reasoning with
      Equations
   2. Linear and Exponential Relationships
   3. Descriptive Statistics
   4. Expressions and Equations
5. Quadratic Functions and Modeling


         Six critical areas (units) in Geometry
1. Congruence, Proof, and Constructions
2. Similarity, Proof, and Trigonometry
3. Extending to Three Dimensions
4. Connecting Algebra and Geometry Through Coordinates
5. Circles With and Without Coordinates
6. Applications of Probability

        Four critical areas (units) in Algebra 2
1. Polynomial, Rational, and Radical Relationships

2. Trigonometric Functions

3. Modeling with Functions

4. Inferences and Conclusions from Data


                 Building Fluency
• What is meant by fluency?
  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.

     When it comes to measuring the full range of the Standards, usually the
     first things that come to mind are the mathematical practices, or perhaps
     the content standards that call for conceptual understanding. However, the
     Standards also address another aspect of mathematical attainment that is
     seldom measured at scale either: namely, whether students can perform
     calculations and solve problems quickly and accurately. At each grade level
     in the Standards, one or two fluencies are expected:.

     Fluent in the Standards means “fast and accurate.” It might also help to
     think of fluency as meaning the same thing as when we say that somebody
     is fluent in a foreign language: when you’re fluent, you flow. Fluent isn’t
     halting, stumbling, or reversing oneself. Assessing fluency requires
     attending to issues of time (and even perhaps rhythm, which could be
     achieved with technology).

     The word fluency was used judiciously in the Standards to mark the
     endpoints of progressions of learning that begin with solid underpinnings
     and then pass upward through stages of growing maturity. In fact, the
     rarity of the word itself might easily lead to fluency becoming invisible in the
     Standards—one more among 25 things in a grade, easily overlooked.
     Assessing fluency could remedy this, and at the same time allow data
     collection that could eventually shed light on whether the progressions
     toward fluency in the Standards are realistic and appropriate.



                              Key Fluencies
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 - Add/subtract within 20

  MACC.2.NBT.2.5 - Add/subtract within 100 (pencil and paper)

3 - MACC.3.NBT.1.2 - Add/subtract within 1,000

   MACC.3.OA.3.7 - Multiply/divide within 100

4 - MACC.4.2.4 - Add/subtract within 1,000,000

 Critical Area #1 - Develop fluency with efficient procedures for multiplying
whole numbers

5 - MACC.5.NBT.2.5 - Multi-digit multiplication



   Critical Area #1 - Developing fluency with addition and subtraction of
fractions

6 - MACC.6.NS.2.2 - Multi-digit division

   MACC.6.NS.2.3 - 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



                    Mathematics Progressions
  • The Importance of Mathematics Progressions

                                       VIDEO

          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.



           Mathematics Progressions Project
     • Kindergarten Counting and Cardinality

     • K-5 Number and Operations in Base Ten

     • 3-5 Number and Operations—Fractions

     • K–5 Operations and Algebraic Thinking

     • K-5 Measurement and Data (Data part)

     • Geometry Progression (coming soon!)

     • 6-8 Expressions and Equations

     • 6-7 Ratios and Proportional Relationships

     • 6-8 Statistics and Probability

     • High school Statistics and Probability

• http://commoncoretools.me/category/progressions/


                      Reflective Thoughts
1. How will you use the Common Core Standards for Mathematical
   Practices to inform your curriculum and guide your instruction?

2. How will the Critical Areas and the Cluster headings help to inform
   your curriculum and guide your instruction?
3. How will you use the Learning Progressions to inform your
   curriculum and guide your instruction?

				
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