CCSSI FOR MATHEMATICS “STANDARDS OF PRACTICE” Collegial Conversations HIGH SCHOOL

CCSSI FOR MATHEMATICS“STANDARDS OF PRACTICE”

Collegial Conversations

HIGH SCHOOL

Today’s GoalTo explore the Standards for Content and

Practice for MathematicsBegin to consider how these new CCSS

Standards are likely to impact your classroom practices

What are the Common Core State Standards?

Aligned with college and work expectationsFocused and coherentIncluded rigorous content and application of

knowledge through high-order skillsBuild upon strengths and lessons of current state

standardsInternationally benchmarked so that all students are

prepared to succeed in our global economy and society

Research and evidence basedState led- coordinated by NGA Center and CCSSO

Focus

• Key ideas, understandings, and skills are identified

• Deep learning of concepts is emphasized– That is, time is spent on a topic and on

learning it well. This counters the “mile wide, inch deep” criticism leveled at most current U.S. standards.

Benefits for States and Districts• Allows collaborative professional development based on

best practices• Allows development of common assessments and other

tools• Enables comparison of policies and achievement

across states and districts• Creates potential for collaborative groups to get more

economical mileage for:– Curriculum development, assessment, and

professional development

Common Core Development• Initially 48 states and three territories

signed on

• As of November 29, 2010, 42 states have officially adopted

• Final Standards released June 2, 2010, at www.corestandards.org

• Adoption required for Race to the Top funds

Michigan’s Implementation Timeline

• Held October and November of 2010 rollouts• District curricula and assessments that provide a

K-12 progression for meeting the MMC requirements will require minimal adjustments to meet CCSS

• Curriculum and assessment alignment in SY10-11• Implementation SY11-12• New assessment 2014-15 (Smarter Balanced

Assessment Consortium or SBAC – replaces MEAP and MME)

Background

Each State that is a member of the Consortium in 2014–2015 also agrees to do the following:

Adopt common achievement standards no later than the 2014–2015 school year,

Fully implement the Consortium summative assessment in grades 3–8 and high school for both mathematics and English language arts no later than the 2014–2015 school year,

Adhere to the governance requirements, Agree to support the decisions of the Consortium, Agree to follow agreed-upon timelines, Be willing to participate in the decision-making process and, if a Governing

State, final decisions, and Identify and implement a plan to address barriers in State law, statute,

regulation, or policy to implementing the proposed assessment system and address any such barriers prior to full implementation of the summative assessment components of the system.

Responsibilities of States in the Consortium

Technology Approach

SBAC Item Bank

• Partitioned into a secure item bank for summative assessments and a non-secure bank for the interim/benchmark assessments:

• Traditional selected-response items• Constructed-response items• Curriculum-embedded performance events• Technology-enhanced items (modeled after

assessments in use by the U.S. military, the architecture licensure exam, and NAEP)

Domains are large groups of related standards. Standards from different domains may sometimes be closely related. Look for the name with the code number on it for a Domain.

HOW TO READ THE STANDARDS

Common Core Format

Clusters are groups of related standards. Standards from different clusters may sometimes be closely related, because mathematics is a connected subject.• Clusters appear inside domains.

Common Core FormatStandards define what students should be able to understand and be able to do – part of a cluster.

They are content statements. An example content statement is: “Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2)”.

•Progressions of increasing complexity from grade to grade

Common Core - Clusters

• May appear in multiple grade levels in the K-8 Common Core. There is increasing development as the grade levels progress

• What students should know and be able to do at each grade level

• Reflect both mathematical understandings and skills, which are equally important

Common Core Format

High School

Conceptual Category

Domain

Cluster

Standards

K-8

Grade

Domain

Cluster

Standards

(There are no preK Common Core Standards)

Domains Grade Levels

Counting and Cardinality K only

Operations and Algebraic Thinking

1-5

Number and Operations in Base Ten

1-5

Number and Operations - Fractions

3-5

Measurement and Data 1-5

Geometry 1-5

K – 5 DOMAINS

Domains Grade Levels

Ratio and Proportional Relationships

6-7

The Number System 6-8

Expressions and Equations 6-8

Functions 8

Geometry 6-8

Statistics and Probability 6-8

MIDDLE GRADES DOMAINS

Fractions, Grades 3–6

3. Develop an understanding of fractions as numbers. 4. Extend understanding of fraction equivalence and ordering. 4. Build fractions from unit fractions by applying and extending

previous understandings of operations on whole numbers. 4. Understand decimal notation for fractions, and compare

decimal fractions. 5. Use equivalent fractions as a strategy to add and subtract

fractions. 5. Apply and extend previous understandings of multiplication

and division to multiply and divide fractions. 6. Apply and extend previous understandings of multiplication

and division to divide fractions by fractions.

Statistics and Probability, Grade 6

Develop understanding of statistical variability• Recognize a statistical question as one that anticipates variability in the

data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

• Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

• Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Algebra, Grade 8

Graded ramp up to Algebra in Grade 8• Properties of operations, similarity, ratio and proportional relationships, rational number system.

Focus on linear equations and functions in Grade 8• Expressions and Equations

– Work with radicals and integer exponents.– Understand the connections between proportional relationships, lines, and linear equations.– Analyze and solve linear equations and pairs of simultaneous linear equations.

• Functions– Define, evaluate, and compare functions.– Use functions to model relationships between quantities.

High SchoolConceptual themes in high school• Number and Quantity• Algebra• Functions• Modeling• Geometry• Statistics and Probability

College and career readiness threshold• (+) standards indicate material beyond the threshold; can

be in courses required for all students.

Format of High School

DomainDomain

ClusterCluster

StandardStandard

Format of High School Standards

STEMSTEMSTEMSTEMModelingModelingModelingModeling

Regular Regular StandardStandardRegular Regular

StandardStandard

Common Core - Domain

• Overarching “big ideas” that connect topics across the grades

• Descriptions of the mathematical content to be learned, elaborated through clusters and standards

Common Core - Clusters

• May appear in multiple grade levels with increasing developmental standards as the grade levels progress

• Indicate WHAT students should know and be able to do at each grade level

• Reflect both mathematical understandings and skills, which are equally important

Common Core - Standards

• Content statements

• Progressions of increasing complexity from grade to grade – In high school, this may occur over the course

of one year or through several years

HS Pathways

1.) Traditional (US) – 2 Algebra, Geometry and Data, probability and statistics included in each course

2.) International (integrated) three courses including number , algebra, geometry, probability and statistics each year

3.) Compacted version of traditional – grade 7/8 and algebra completed by end of 8th grade

4.) Compacted integrated model, allowing students to reach Calculus or other college level courses

High School Pathways

• The CCSS Model Pathways are NOT required. The two sequences are examples, not mandates

• Two models that organize the CCSS into coherent, rigorous courses

• Four years of mathematics: – One course in each of the first two years

– Followed by two options for year 3 and a variety of relevant courses for year 4

• Course descriptions – Define what is covered in a course

– Are not prescriptions for the curriculum or pedagogy


• Four years of mathematics: – One course in each of the first two years– Followed by two options for year three and a

variety of relevant courses for year four

• Course descriptions – Define what is covered in a course – Are not prescriptions for the curriculum or

pedagogy


• Pathway A: Consists of two algebra courses and a geometry course, with some data, probability and statistics infused throughout each (traditional)

• Pathway B: Typically seen internationally that consists of a sequence of 3 courses each of which treats aspects of algebra, geometry and data, probability, and statistics.

Interrelationships

• Algebra and Functions– Expressions can define functions– Determining the output of a function can

involve evaluating an expression

• Algebra and Geometry– Algebraically describing geometric shapes– Proving geometric theorems algebraically

Numbers and Quantity

• Extend the Real Numbers to include work with rational exponents and study of the properties of rational and irrational numbers

• Use quantities and quantitative reasoning to solve problems.

Numbers and Quantity

Required for higher math and/or STEM

• Compute with and use the Complex Numbers, use the Complex Number plane to represent numbers and operations

• Represent and use vectors

• Compute with matrices

• Use vector and matrices in modeling

Algebra and Functions

• Two separate conceptual categories

• Algebra category contains most of the typical “symbol manipulation” standards

• Functions category is more conceptual

• The two categories are interrelated

Algebra

• Creating, reading, and manipulating expressions– Understanding the structure of expressions– Includes operating with polynomials and

simplifying rational expressions

• Solving equations and inequalities– Symbolically and graphically

Algebra


• Expand a binomial using the Binomial Theorem

• Represent a system of linear equations as a matrix equation

• Find the inverse if it exists and use it to solve a system of equations

Functions

• Understanding, interpreting, and building functions– Includes multiple representations

• Emphasis is on linear and exponential models

• Extends trigonometric functions to functions defined in the unit circle and modeling periodic phenomena

Functions


• Graph rational functions and identify zeros and asymptotes

• Compose functions

• Prove the addition and subtraction formulas for trigonometric functions and use them to solve problems

Functions


• Inverse functions– Verify functions are inverses by composition– Find inverse values from a graph or table– Create an invertible function by restricting the

domain– Use the inverse relationship between

exponents and logarithms and in trigonometric functions

High School - Modeling

• Linking mathematics and statistics to everyday life, work, etc.

• Process of choosing and using appropriate mathematics and statistics

• Examples: pg 72

Modeling

Modeling has no specific domains, clusters or standards. Modeling is included in the other conceptual categories and marked with a asterisk.

Modeling

Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Technology is valuable in modeling.

A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical object.

Modeling

• Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player.

• Analyzing stopping distance for a car.

• Modeling savings account balance, bacterial colony growth, or investment growth.

Geometry, High School

Middle school foundations• Hands-on experience with transformations.• Low tech (transparencies) or high tech (dynamic

geometry software).

High school rigor and applications• Properties of rotations, reflections, translations, and

dilations are assumed, proofs start from there.• Connections with algebra and modeling

Geometry

Geometry

• Circles

• Expressing geometric properties with equations– Includes proving theorems and describing

conic sections algebraically

• Geometric measurement and dimension

• Modeling with geometry

Geometry


• Non-right triangle trigonometry

• Derive equations of hyperbolas and ellipses given foci and directrices

• Give an informal argument using Cavalieri’s Principal for the formulas for the volume of solid figures

Statistics and Probability

• Analyze single a two variable data

• Understand the role of randomization in experiments

• Make decisions, use inference and justify conclusions from statistical studies

• Use the rules of probability

Key Advances

Focus and coherence• Focus on key topics at each grade level.• Coherent progressions across grade levels.Balance of concepts and skills• Content standards require both conceptual

understanding and procedural fluency.Mathematical practices• Foster reasoning and sense-making in mathematics.College and career readiness• Level is ambitious but achievable.

Design and Organization

Mathematical Practice – expertise students should acquire: (Processes & proficiencies)

• NCTM five process standards: • Problem solving• Reasoning and Proof• Communication • Connections• Representations

NCTM Process Standards CCSS Mathematical Practices

Problem Solving Make sense of problems and persevere in solving them.Use appropriate tools strategically

Reasoning and Proof Reason abstractly and quantitatively.Critique the reasoning of others.Look for and express regularity in repeated reasoning

Communication Construct viable arguments

Connections Attend to precision.Look for and make use of structure

Representations Model with mathematics.

NCTM Process Standards and theCCSS Mathematical Practice Standards

Design and Organization

• Mathematical proficiency (National Research

Council’s report Adding It Up) – Adaptive reasoning– Strategic competence– Conceptual understanding (comprehension of

mathematical concepts, operations, relations)– Procedural fluency (skill in carrying out procedures

flexibly, accurately, efficiently, and appropriately)– Productive disposition (ability to see mathematics as

sensible, useful, and worthwhile

Mathematics/Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning

of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Mathematics/Standards for Mathematical Practice

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.

These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.” CCSS, 2010

Standards for Mathematical Practice• Carry across all grade levels• Describe habits of a mathematically expert student

Standards for Mathematical Content• K-8 presented by grade level• Organized into domains that progress over several grades• Grade introductions give 2-4 focal points at each grade level• High school standards presented by conceptual theme (Number &

Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability

Standards of Mathematical Practice

1.Choose a partner at your table and “Pair Share” the Standards of Practice between you and your partner.

2. When you and your partner feel you understand generally each of the standards, discuss the following question:

What implications might the standards of practice have on your classroom?

Transition from Current State Standards & Assessments to New Common Core Standards

• Develop Awareness• Needs Assessment/Gap Analysis• Planning• Capacity Building• Job-embedded Professional Development

Transition PlanningNext Steps:• Alignment of CCSS with curriculum• Gap analysis (content and skills that vary from

the MEAP and MME)• What instructional practices will facilitate the

transition?• What new assessment strategies will be

needed?• Professional development needs?

Transition Planning• Gather in teams from your schools and discuss

– What are your immediate needs as a classroom teacher being asked to implement the CCSS?

– What professional development is needed?– What initial gaps come to mind and how do you address these

gaps?– As a school team, look at the eight Standards for Mathematical

Practice. What do they look like? Sound like? What will students need in order to implement them? What will teachers need? What are the implications for assessment and grading?

Select a recorder, time keeper and someone to report out for your group.

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Have a great day!

Documents

CCSSI FOR MATHEMATICS “STANDARDS OF PRACTICE” Collegial Conversations HIGH SCHOOL