Common Core State Standards for Mathematics: Focus at Grade 5

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Common Core State Standards for Mathematics
Focus at Grade 5

• Professional Development Module

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The CCSS Requires Three Shifts in Mathematics

1. Focus Focus strongly where the standards focus.
2. Coherence Think across grades, and link to major
   topics
3. Rigor In major topics, pursue conceptual
   understanding, procedural skill and fluency, and
   application

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Focus

• The two major evidence-based principles on which
  the standards are based are focus and coherence.
• Focus is necessary so that students have
  sufficient time to think, practice, and integrate
  new ideas into their growing knowledge structure.
  
• Focus is also a way to allow time for the kinds
  of rich classroom discussion and interaction that
  support the Standards for Mathematical Practice.

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Shift 1 Focus Strongly where the Standards Focus

• Significantly narrow the scope of content and
  deepen how time and energy is spent in the math
  classroom.
• Focus deeply on what is emphasized in the
  standards, so that students gain strong
  foundations.

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Focus

• Focus is critical to ensure that students learn
  the most important content completely, rather
  than succumb to an overly broad survey of
  content. Focus shifts over time, as seen in the
  following
• In grades K-5, the focus is on the addition,
  subtraction, multiplication, and division of
  whole numbers fractions and decimals with a
  balance of concepts, skills, and problem solving.
  Arithmetic is viewed as an important set of
  skills and also as a thinking subject that, done
  thoughtfully, prepares students for algebra.
  Measurement and geometry develop alongside number
  and operations and are tied specifically to
  arithmetic along the way.

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Key Areas of Focus in Mathematics
Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
K2 Addition and subtraction - concepts, skills, and problem solving and place value
35 Multiplication and division of whole numbers and fractions concepts, skills, and problem solving
6 Ratios and proportional reasoning early expressions and equations
7 Ratios and proportional reasoning arithmetic of rational numbers
8 Linear algebra
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Focus

• Move away from "mile wide, inch deep" curricula
  identified in TIMSS.
• Learn from international comparisons.
• Teach less, learn more.
• Less topic coverage can be associated with
  higher scores on those topics covered because
  students have more time to master the content
  that is taught.

Ginsburg et al., 2005
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The shape of math in A countries
Mathematics topics intended at each grade by at
least two-thirds of A countries
Mathematics topics intended at each grade by at
least two-thirds of 21 U.S. states
1 Schmidt, Houang, Cogan, A Coherent
Curriculum The Case of Mathematics. (2002).
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Traditional U.S. Approach
K 12
Number and Operations

Measurement and Geometry

Algebra and Functions

Statistics and Probability
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Focusing Attention Within Number and Operations
Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Expressions and Equations Expressions and Equations Expressions and Equations Algebra
Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking ? Expressions and Equations Expressions and Equations Expressions and Equations ? Algebra
Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Operations and Algebraic Thinking Expressions and Equations Expressions and Equations Expressions and Equations Algebra
Algebra
Number and OperationsBase Ten Number and OperationsBase Ten Number and OperationsBase Ten Number and OperationsBase Ten Number and OperationsBase Ten Number and OperationsBase Ten ? The Number System The Number System The Number System Algebra
The Number System The Number System The Number System ? Algebra
Number and OperationsFractions Number and OperationsFractions Number and OperationsFractions ? The Number System The Number System The Number System Algebra


K 1 2 3 4 5 6 7 8 High School
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Examples of Opportunities for In-Depth Focus

• 5.NBT.1 The extension of the place value system
  from whole numbers to decimals is a major
  intellectual accomplishment involving
  understanding and skill with base-ten units and
  fractions.
• 5.NBT.6 The extension from one-digit divisors to
  two-digit divisors requires care. This is a major
  milestone along the way to reaching fluency with
  the standard algorithm in grade 6 (6.NS.2).
• 5.NF.2 When students meet this standard, they
  bring together the threads of fraction
  equivalence (grades 35) and addition and
  subtraction (grades K4) to fully extend addition
  and subtraction to fractions.
• 5.NF.4 When students meet this standard, they
  fully extend multiplication to fractions, making
  division of fractions in grade 6 (6.NS.1) a near
  target.
• 5.MD.5 Students work with volume as an attribute
  of a solid figure and as a measurement quantity.
  Students also relate volume to multiplication and
  addition. This work begins a progression leading
  to valuable skills in geometric measurement in
  middle school.

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Example of Opportunities for Connecting
Mathematical Content and Mathematical Practices

• Mathematical practices should be evident
  throughout mathematics instruction and connected
  to all of the content areas addressed at this
  grade level. Mathematical tasks (short, long,
  scaffolded, and unscaffolded) are an important
  opportunity to connect content and practices. The
  example below shows how the content of this
  grade might be connected to the practices.
• When students break divisors and dividends into
  sums of multiples of base-ten units (5.NBT.6),
  they are seeing and making use of structure
  (MP.7) and attending to precision (MP.6).
  Initially for most students, multi-digit division
  problems take time and effort, so they also
  require perseverance (MP.1) and looking for and
  expressing regularity in repeated reasoning
  (MP.8).

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Group Discussion

• Shift 1 Focus strongly where the Standards
  focus.
• In your groups, discuss ways to respond to the
  following question, Why focus? Theres so much
  math that students could be learning, why limit
  them to just a few things?

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Engaging with the shift What do you think
belongs in the major work of each grade?
Grade Which two of the following represent areas of major focus for the indicated grade? Which two of the following represent areas of major focus for the indicated grade? Which two of the following represent areas of major focus for the indicated grade?
K Compare numbers Use tally marks Understand meaning of addition and subtraction
1 Add and subtract within 20 Measure lengths indirectly and by iterating length units Create and extend patterns and sequences
2 Work with equal groups of objects to gain foundations for multiplication Understand place value Identify line of symmetry in two dimensional figures
3 Multiply and divide within 100 Identify the measures of central tendency and distribution Develop understanding of fractions as numbers
4 Examine transformations on the coordinate plane Generalize place value understanding for multi-digit whole numbers Extend understanding of fraction equivalence and ordering
5 Understand and calculate probability of single events Understand the place value system Apply and extend previous understandings of multiplication and division to multiply and divide fractions
6 Understand ratio concepts and use ratio reasoning to solve problems Identify and utilize rules of divisibility Apply and extend previous understandings of arithmetic to algebraic expressions
7 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers Use properties of operations to generate equivalent expressions Generate the prime factorization of numbers to solve problems
8 Standard form of a linear equation Define, evaluate, and compare functions Understand and apply the Pythagorean Theorem
Alg.1 Quadratic inequalities Linear and quadratic functions Creating equations to model situations
Alg.2 Exponential and logarithmic functions Polar coordinates Using functions to model situations
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Examples of Key Advances from Grade 4 to Grade 5

• In grade 5, students will integrate decimal
  fractions more fully into the place value system
  (5.NBT.14). By thinking about decimals as sums
  of multiples of base-ten units, students begin to
  extend algorithms for multi-digit operations to
  decimals (5.NBT.7).
• Students use their understanding of fraction
  equivalence and their skill in generating
  equivalent fractions as a strategy to add and
  subtract fractions, including fractions with
  unlike denominators.

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Example from SBA p.16-17

• Students apply and extend their previous
  understanding of multiplication to multiply a
  fraction or whole number by a fraction (5.NF.4).
  They also learn the relationship between
  fractions and division, allowing them to divide
  any whole number by any nonzero whole number and
  express the answer in the form of a fraction or
  mixed number (5.NF.3). And they apply and extend
  their previous understanding of multiplication
  and division to divide a unit fraction by a whole
  number or a whole number by a unit fraction.13
• ?

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Content Emphases by Cluster Grade Five

Key Major Clusters Supporting Clusters
Additional Clusters
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Examples of Opportunities for Connections among
Standards, Clusters or Domains

• The work that students do in multiplying
  fractions extends their understanding of the
  operation of multiplication. For example, to
  multiply a/b x q (where q is a whole number or a
  fraction), students can interpret a/b x q as
  meaning a parts of a partition of q into b
  equal parts (5.NF.4a). This interpretation of the
  product leads to a product that is less than,
  equal to or greater than q depending on whether
  a/b lt 1, a/b 1 or a/b gt 1, respectively
  (5.NF.5).
• Conversions within the metric system represent an
  important practical application of the place
  value system. Students work with these units
  (5.MD.1) can be connected to their work with
  place value (5.NBT.1).

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Task Matching Clusters and Critical Areas

• Read through the cluster headings for your
  grade.
• Discuss each cluster heading and decide which
  critical area it falls within.
• Cut and paste the cluster heading on the page
  with the appropriate critical area.

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Matching Clusters with Critical Areas Small
Group Discussion

• Were you able to match each cluster heading with
  one of the critical areas? How did you decide
  which area to place it under? What challenges did
  you have?
• How do the cluster headings help clarify the
  concepts in the critical areas?

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The Standards

• Find the critical areas for your grade.
• Find the cluster headings for your grade.
• Find and read the standards that fall under each
  cluster heading.
• Write down two first impressions you have about
  the standards.
• Write down two questions you have about the
  standards.

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Reflection Journal

• How have the cluster headings helped clarify
  the important mathematical concepts in the
  critical areas?
• How will you use this information to guide your
  curriculum and instruction? What changes will you
  make?
• What questions do you still have about the
  standards?

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Your Assignment

• Choose one of the critical areas to investigate
  back in the classroom
• Find a lesson in your curriculum addressing the
  critical area
• What evidence will convince you that students
  understand this concept?
• What common misconceptions do students have when
  studying this critical area?
• What challenges have you had in teaching these
  critical area concepts?

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Thank you