The Common Core State Standards for Mathematics Grades 6-8

The Common CoreState Standardsfor Mathematics

Grades 6-8

The Development of the CCSS

• Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)

• Written by a small group of authors from Sept 2009 – June 2010

• NCTM among the groups that provided feedback and review as part of public review process

Potential Benefits for States and Districts

• Creates potential for collaborative groups to get more mileage from:– Curriculum development, assessment, and

professional development– Encourages collaborative professional

development to be based on best practices– Allows for the development of cross-state

formative and summative assessment tools

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 can be downloaded at www.corestandards.org.

• Adoption is required for Race to the Top funds.

Common Core Implementation

• Each state adopting the Common Core either directly or by fully aligning its state standards may do so in accordance with current state timelines for standards adoption, not to exceed three (3) years.

• States that choose to align their standards with the Common Core Standards accept 100% of the core in English language arts and mathematics. States may add additional standards.

Characteristics Claimed by CCSS (compared to most State Standards)

• Fewer and more rigorous. The goal was increased clarity.• Aligned with college and career expectations – prepare all

students for success on graduating from high school.• Internationally benchmarked, to prepare all students for

succeeding in our global economy and society.• Includes rigorous content and application of higher-order

skills.• Builds on strengths and lessons of current state standards

and on NCTM’s documents: Curriculum Focal Points, PSSM, and Focus on Reasoning and Sense Making

• Research based

Stated Intentions of the Common Core

• Coherence

• Focus

• Clarity and specificity

• The same goals for all students

Coherence According to CCSS

• Articulated progressions of topics and performances that are developmental and connected to other progressions

• Conceptual understanding and procedural skills stressed equally

NCTM states that coherence means that instruction, assessment, and curriculum are aligned.

Focus According to CCSS

• Key ideas, understandings, and skills are identified.

• Deep learning of concepts is emphasized.– That is, adequate time is devoted to a topic to

counter the “mile wide, inch deep” criticism leveled at most current U.S. standards.

CCSS Standards for Mathematical Practice

The Common Core proposes a set of Standards for Mathematical Practice that all teachers should develop in their students.

The Mathematical Practice Standards in the CCSS resonate well with the NCTM Process Standards in Principles and Standards for School Mathematics.

CCSS 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 use of structure.

8. Look for and express regularity in repeated reasoning.

NCTM Process Standards

• Problem Solving is an integral part of mathematics learning.

• Reasoning and Proof are ways of expressing justification.• Communication is an essential part of mathematics

education.• Connections are critical in mathematics, both across

mathematical topics and to contexts outside mathematics.• Representations of mathematical ideas are fundamental

to enhancing mathematical understanding.

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

Clarity and Specificity for CCSS

• Skills and concepts are clearly defined.

• An ability to apply concepts and skills to

new situations is expected.

Common Core Format

High School

Conceptual Category

Domain

Cluster

Standards

K-8

Grade

Domain

Cluster

Standards

(No pre-K Common Core Standards)

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

CCSSM and NCTM Focal Points:Differences in Format

CCSSM

• Five Domains• Clusters of standards

NCTM Focal Points

• Three focal points per grade level

• Connections to other strands

CCSSM Domains

CCSSM-Grade 61. Ratio and

proportional Relationships

2. The Number System

3. Expressions

& Equations

4. Geometry

5. Statistics and Probability

• Ratio and proportional Relationships

• The Number System

• Expressions & Equations

• Geometry• Statistics and

Probability

CCSSM-Grade 7

• Functions• The Number

System• Expressions and

Equations• Geometry• Statistics and

Probability

CCSSM-Grade 8

Domain 1: Ratios and Proportional Relationships

CCSS- Grade 6

• Understand ratio concepts and use ratio reasoning to solve problems.

NCTM Focal Points – Grade 6

• Develop an understanding of and fluency with multiplication and division of fractions and decimals.

• Connect ratio and rate to multiplication and division.

Additional MA StandardMA.3e. – Solve problems that relate the mass of an object to its volume.

Domain 1: Ratio and Proportional Relationships

CCSS- Grade 7

• Analyze proportional relationships and use them to solve real-world and mathematical problems.

NCTM Focal Points – Grade 7

• Develop an understanding of and apply proportionality, including similarity.

• Develop an understanding of operations on all rational numbers and solve linear equations.

Domain 1: Functions

CCSS- Grade 8

• Define, evaluate, and compare functions.

• Use functions to model relationships between quantities.

NCTM Focal Points – Grade 8

• Analyze and represent linear functions and solve linear equations and systems of linear equations

Domain 2: The Number System

CCSS- Grade 6

• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• Compute fluently with multi-digit numbers and find common factors and multiples

• Apply and extend previous understandings of numbers to the system of rational numbers.

NCTM Focal Points –Grade 6

• Develop an understanding of and fluency with multiplication and division of fractions and decimals. (page 18)

• Negative integers in grade 7 (page 19)

Additional MA StandardMA.4a: Apply number theory concepts, including prime factorization and relatively prime numbers to the solution of problems.

Domain 2: The Number System

CCSS- Grade 7

• Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

NCTM Focal Points –Grade 7

• Develop an understanding of and apply proportionality, including similarity.

Domain 2: The Number System

CCSS- Grade 8

• Know that there are numbers that are not rational, and approximate them by rational numbers.

NCTM Focal Points –Grade 8

• Students now use division to express any fraction as a decimal, including fractions that they must represent with infinite decimals.

• Students connect their work with dividing fractions to solving equations of the form ax = b, where a and b are fractions.

• Students continue to develop their understanding of the structure of numbers by determining if a counting number greater than 1 is a prime.

Domain 3:Expressions and Equations

CCSS- Grade 6

• Apply and extend previous understanding of arithmetic to algebraic expressions

• Reason about and solve on-variable equations and inequalities

• Represent and analyze quantitative relationships between dependent and independent variables

NCTM Focal Points – Grade 6

• Write, interpret, and use mathematical expressions and equations

Domain 3: Expressions and Equations

CCSS- Grade 7

• Use properties of operations to generate equivalent expressions

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations

NCTM Focal Points – Grade 7

• Develop an understanding of operations on all rational numbers and solve linear equations

Additional MA StandardMA.4c: Extend analysis of patterns to include analyzing, extending, and determining an expression for simple arithmetic and geometric sequences (e.g., compounding, increasing area) using tables, graphs, words, and expressions.

Domain 3:Expressions and Equations

CCSS- Grade 8

• 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.

NCTM Focal Points – Grade 8

• Analyze and represent linear functions and solve linear equations and systems of linear equations

Domain 4: Geometry

CCSS- Grade 6

• Solve real-world and mathematical problems involving area, surface area, and volume.

NCTM Focal Points – Grade 6

• Solve problems that involve areas and volumes, calling on students to find areas or volumes from lengths or to find lengths from volumes or areas and lengths, are especially appropriate.

Additional MA Standards MA.1.a: Use the relationship between radius, diameter, and center of a circle to find the circumference and area.

MA.1.b: Solve real-world and mathematical problems involving the measurements of circles.

Domain 4: Geometry

CCSS- Grade 7

• Draw, construct, and describe geometrical figures and describe the relationships between them.

• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

NCTM Focal Points – Grade 7

• Understand and use formulas to determine surface areas and volumes of three-dimensional shapes.

Additional MA Standard:Please note: MA.7: Solve real-world and mathematical problems involving the surface area of shapes.

Domain 4: Geometry

CCSS - Grade 8

• Understand congruence and similarity using physical models, transparencies, or geometry software.

• Understand and apply the Pythagorean Theorem.

• Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

NCTM Focal Points – Grade 8

• Analyze two- and three-dimensional space and figures by using distance and angle.

Domain 5:Statistics and Probability

CCSS- Grade 6

• Develop understanding of statistical variability.

• Summarize and describe distributions.

NCTM Focal Points – Grade 6

• None Listed

MA.4.a – Read and interpret circle graphs

Domain 5:Statistics and Probability

CCSS- Grade 7

• 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.

NCTM Focal Points – Grade 7

• Students understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of outcomes in which the event occurs. Students use theoretical probability and proportions to make approximate predictions.

Implementation Support for Teachers from NCTM

Making It Happen:

A Guide to Interpreting and Implementingthe Common Core State Standards

for Mathematics

Making It Happen Will Include:

• How the content of CCSSM fits with NCTM’s Standards documents—including a road map to NCTM’s resources

• A deep analysis of the connections between NCTM’s Process Standards and the CCSS Standards for Mathematical Practice

• Examples of using NCTM resources to interpret and implement CCSS

Implementation Support for Teachers from NCTM

Possible actions under current discussion• Development of resource toolkit linked to a

mobile or e-conference package• A focus on Professional Development with

respect to both content and mathematical processes

• Grade band specific PowerPoints available on the NCTM Web site

NCTM Standards and CCSS

The pre-eminent message from both these sets of standards is, and for NCTM always has been, the importance of nurturing the mathematical thinking and reasoning processes as set forth in the NCTM Process Standards, and in the CCSS Standards for Mathematical Practice.

No collection of specific standards bullets will hold together without interweaving mathematical processes for our students.