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11/5/2012
1
© ASCD 2012 | Common Core State Standards
Common Core Math: Getting From Standards to Instruction
Nanci Smith, Ph.D.
November 5, 2012
© ASCD 2012 | Common Core State Standards
Welcome! In the next hour, I hope to…
Present methods for chunking standards into teachable units
Provide references for building units and “unpacking” standards
Unpack example standards for Know, Understand, Do, and Essential Questions
Model text alignmentGive next steps for lessons and assessments
through Differentiation and Understanding by Design.
Q and A
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© ASCD 2012 | Common Core State Standards
Building a Scope and SequenceBasic Premises
• Units should be based on the content standards only – not resources or favorite activities. Even “Common Core” resources should be examined very closely!
• All eight of the Standards for Mathematical Practice should be in every unit, although all eight will not necessarily be in every lesson within the unit.
• Hard to shift from what we’ve always done, or what might be assessed during the transition years.
© ASCD 2012 | Common Core State Standards
Building a Scope and SequenceActivity – Part 1
• Get to know the critical areas for each grade level or conceptual area in High School.
• This is the “text” developed in the standards prior to the Overview.
• Make a workable list of non-negotiables.
Mathematics | Grade 4 In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) 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.
(1) Students generalize their
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© ASCD 2012 | Common Core State Standards
Don’t Forget: Required Fluencies in Non-Negotiables
Grade Standard Required Fluency
K K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
22.OA.2
2.NBT.5
Add/subtract within 20 (know single‐digit sums from memory)
Add/subtract within 100
33.OA.7
3.NBT.2
Multiply/divide within 100 (know single‐digit products from memory)
Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
5 5.NBT.5 Multi‐digit multiplication
6 6.NS.2,3Multi‐digit division
Multi‐digit decimal operations
© ASCD 2012 | Common Core State Standards
Building a Scope and SequenceActivity – Part 2
Units are NOT comprised of Domains or Clusters.
Units often combine standards from various domains and clusters.
High school needs to determine which standards will be in which courses, and to what extent if a SBAC state. PARCC has (or will) provided this.
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© ASCD 2012 | Common Core State Standards
Building a Scope and SequenceActivity – Part 2
Spread your “Standards Strips” out on the table. 4 copies of the standards helps!
1. Clump the statements that logically go together.
2. Once all statements are clumped, order the clumps in teaching order from the beginning of the year to the end of the year.
© ASCD 2012 | Common Core State Standards
Building a Scope and SequenceActivity – Part 2
3. Within each clump, order the statements from beginning of clump to end.
4. Give each clump a name.
5. Tape down in order on paper. Title each clump.
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© ASCD 2012 | Common Core State Standards
Snapshot
Source: © 2012 Nanci Smith. Used with permission.
© ASCD 2012 | Common Core State Standards
Building a Scope and SequenceResources
• Dana Center: http://www.ccsstoolbox.com/parcc/PARCCPrototype_main.html
• PARCC: http://www.parcconline.org/mcf/mathematics/parcc-model-content-frameworks-browser
• New Jersey Model Curriculum: http://www.state.nj.us/education/modelcurriculum/math
• Bill McCallum’s Blog: http://commoncoretools.me/2012/03/16/arranging-the-high-school-standards-into-courses/
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© ASCD 2012 | Common Core State Standards
So, How Do You Start?Read through the standards for your grade level. Please mark them as follows:
! Already do this as written
~ Kind of do this; close but not exactly
X Nope, not yet
? What the heck?
The areas that you do or are close should be implemented immediately. Plan steps to include the “Not Yet” areas.
© ASCD 2012 | Common Core State Standards
Unpacking the Units
For further information and deeper understanding of how to develop units, consider Understanding by Design and Differentiated Instruction resources.
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© ASCD 2012 | Common Core State Standards
To Begin Unpacking, Consider…
• What will students come to Understand, Know and Be Able To Do as a result of the learning in this unit?
• What do you think? What is the difference between understanding, knowing and being able to do in math? What are the connections?
Planning a Focused Curriculum
Facts (Columbus came to the “New World”)
Vocabulary (voyage, scurvy)
Concepts (exploration, change)
Principles/Generalizations/Big Ideas (Change can be both positive and negative. Exploration results in change. People’s perspectives affect how they respond to change).
Skills
Basic (literacy, numeracy)
Thinking (analysis, evidence of reasoning,
questioning)
Of the Discipline (graphing/math/social
studies)
Planning (goal setting; use of time)
Social
KNOW
UNDERSTAND
BE ABLE TO DO
In general, these are held steady as a core for nearly all learners
in a differentiated classroom.*
*Exception--linear skills and information that can be assessed for mastery in the sequence (e.g. spelling) Source: Tomlinson, C. A. (1999). The differentiated classroom:
Responding to the needs of all learners. Alexandria, VA: ASCD. Used with permission.
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© ASCD 2012 | Common Core State Standards
Unpacking Units
A form of the word “understand” appears 120 times in the cluster headings or Standards
statements of the Common Core State Standards for Mathematics
A necessary and frequently asked question is what is meant by the word
understand…..
© ASCD 2012 | Common Core State Standards
Unpacking UnitsUnderstandings
• These are the written statements of truth, the core to the meaning(s) of the lesson(s) or unit.
• Understandings are purposeful. They focus on the key ideas that require students to understand information and make connections.
• Understandings are conceptual, not factual or procedural.
• Understandings show what the mathematician values or reasons about within a context.
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© ASCD 2012 | Common Core State Standards
Unpacking UnitsWriting Understandings
• Begin with the stem “Students will understand that…”
• Explain the “why or so what” about the understanding.
• Are not just truisms are statements of facts by definition (e.g., triangles have three sides)
• Do not use the phrase, “Students will understand how to…” this would be a skill.
© ASCD 2012 | Common Core State Standards
Unpacking UnitsKnow
These are the facts, formulas, vocabulary, steps to a procedure (“how-to”) and examplesyou want students to learn.
The KNOW can be looked up and is usuallyin a text book.
Know is binary: youeither know or youdon’t.
Facts 2X3=6, D = rtVocabularynumerator, slope
Facts Steps in factoring a quadraticy=ax2+bx+cVocabularyparabola, vertex
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© ASCD 2012 | Common Core State Standards
Unpacking UnitsDo
• Basic skills including thinking skills
The “DO” portion encourages the students to “think” like the professionals who use the knowledge and skill daily as a matter of how they do business. It answers the question, if students have truly come to Know and Understand, what should they now be able to do?
This is what it means to “be like” a mathematician, an analyst, or an economist.
© ASCD 2012 | Common Core State Standards
Standard8th Grade
8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
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© ASCD 2012 | Common Core State Standards
Evidence that a student has attained the Standard
Students:
• Model a function relationship from a given context with multiple representations and explain the relationships among the representations.
• Interpret the meaning of the rate of change and intercepts in terms of the context
• Create a context for a given representation.
© ASCD 2012 | Common Core State Standards
Knowledge that students should have while attaining this Standard
Students Know:
• Function representations
• Definition of function
• Function families
• Rate of change
• Slope Formula
• Intercepts
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© ASCD 2012 | Common Core State Standards
Skills and reasoning that students should have while attaining this standard
Students are able to DO:
• Move among function representations
• Interpret function representations
• Calculate and interpret slope as a rate of change
• Calculate with Slope Formula
• Calculate and Interpret Intercepts
© ASCD 2012 | Common Core State Standards
Understandings that students have upon attaining this Standard
Students understand that:
• There are multiple representations for the same function, each with their own strengths and weaknesses, but all represent the same function.
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© ASCD 2012 | Common Core State Standards
Understandings that students have upon attaining this Standard
Students understand that:
• The rate at which change occurs can be quantified and analyzed.
• Functions model real world situations and can be used to understand and make predictions about the situation.
© ASCD 2012 | Common Core State Standards
Another Example: 2nd Grade Place Value
Standards
• 2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: – 100 can be thought of as a bundle of ten tens — called a “hundred.”
– The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
• 2.NBT.5 Add and subtract within 20 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction
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© ASCD 2012 | Common Core State Standards
Another Example: 2nd Grade Place Value
Standards
• 2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
• 2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram. (Use as a strategy for addition)
© ASCD 2012 | Common Core State Standards
Another Example: 2nd Grade Place Value
Know
K1: Vocabulary: digits, Symbols, Expanded form, Hundreds, tens, ones, Compare, Base ten
K2: exchanges among place values (10 tens to one hundred)
K3: names and value of place values through hundreds
K4: expanded form (87 is the same as 80 + 7)
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© ASCD 2012 | Common Core State Standards
Another Example:2nd Grade Place Value
Understand
• U1: The value of a digit will change based on its placement in a number.
• U2: A number can be represented in different ways based on its placement. (The 1 in the number 216 could be expressed as 10 or one group of ten ones.)
• U3: combining groups of 10s can make the next place value level (ten ones equals one ten, ten tens equals one hundred, etc)
• U4: Only things that are alike (place value) can be added together.
© ASCD 2012 | Common Core State Standards
Another Example:2nd Grade Place Value
Be Able to Do
• D1: read and write numbers to 100 using base ten numerals, number names and expanded form.
• D2: combine groups of ten ones to form ten and groups of ten tens to form one hundred.
• D3: interpret the value of a digit within a number based on place value.
• D4: explain the significance of “10” in our place value system – why is it called Base 10?
• D5: Expand numbers
• D6: Add using correct place value.
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© ASCD 2012 | Common Core State Standards
Unpacking UnitsThemes in K-12 Content Standards - U
1. Properties of operations: their role in arithmetic and algebra
2. Mental math and [algebra vs. algorithms]3. Units and unitizing4. Operations and the problems they solve5. Quantities-variables-functions-modeling6. Number-Operations-Expressions-Equation7. Modeling8. Practices
© ASCD 2012 | Common Core State Standards
Unpacking UnitsPractices – U and D
1. Make sense of complex problems and persevere in solving them.
2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning
of others.4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning.
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© ASCD 2012 | Common Core State Standards
Unpacking UnitsResources
• North Carolina Department of Education: http://www.ncpublicschools.org/acre/standards/common-core-tools/
• Arizona Department of Education: http://www.azed.gov/azcommoncore/mathstandards/
© ASCD 2012 | Common Core State Standards
Resource Alignment
• After the units have been determined and unpacked, align the resource.
• Common core has the shift of “Focused” and “Coherent.” You can not be focused if you teach every page of a resource. Pick and choose, and even go out of sequence!
• Align your resource to the units. Do not write your units from the resource!
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© ASCD 2012 | Common Core State Standards
From Unit to Assessment
• If the Do list is complete, assessments follow easily.
• What is the evidence that students have attained the K and U?
• What tasks will best demonstrate this? Performance? Authentic? Traditional?
© ASCD 2012 | Common Core State Standards
Another Example:2nd Grade Place Value
Be Able to Do
• D1: read and write numbers to 100 using base ten numerals, number names and expanded form.
• D2: combine groups of ten ones to form ten and groups of ten tens to form one hundred.
• D3: interpret the value of a digit within a number based on place value.
• D4: explain the significance of “10” in our place value system – why is it called Base 10?
• D5: Expand numbers
• D6: Add using correct place value.
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© ASCD 2012 | Common Core State Standards
Assessment Sample
• (D1, D3, Practice 1and 3) Mary says that a 2 is a 2, no matter where it is in a number. It is always worth the same.– How would you answer Mary?– Expand 23 and 52. Explain to Mary what the 2 is
worth in each number.
• (D2, D4, Practice 2 and 3) Explain why groups of 10 are important to place value. Give an example.
© ASCD 2012 | Common Core State Standards
From Unit to Instruction
• Knowing the “Mary” problem is coming, instruction would include:– Many samples of critiquing incorrect answers
– Use the wording of “worth” with value, quantity, etc.
– Practice expanding numbers and determining how you find the value of the digit based on place value.
– Roll dice (or draw number cards) to make the largest and smallest numbers possible. How do you know they are the largest and smallest?
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© ASCD 2012 | Common Core State Standards
From Unit to Assessment and InstructionResources
• PARCC– http://www.parcconline.org/samples/item-task-
prototypes
• Smarter Balanced– http://www.smarterbalanced.org/sample-items-and-
performance-tasks/
• Common Core Tools (McCallum’s blog)– http://Commoncoretools.me
• Illustrative Mathematics Project– www.illustrativemathematics.org
© ASCD 2012 | Common Core State Standards
From Unit to Assessment and InstructionResources
• Milwaukee Math Project– http://www4.uwm.edu/Org/mmp/_resources/ccss_r
esources.html
• Inside Mathematics– www.insidemathematics.org
• Shell Organization – MARS project– http://map.mathshell.org
• Educore– http://educore.ascd.org