The Common Core State Standards for Mathematics Grades 6-12
California Teachers Association
Good Teaching Conference
January 21, 2011
California Department of Education
Tom Torlakson, State Superintendent of Public Instruction
CLAB: Developed by SCFIRD with support from ELCSD, SCALD, and AAD
Common Core State Standards for Mathematics
Overview
Structure of the Standards
- Mathematical Practice
- Mathematical Content
Grade 6-8 Content Standards
High School Conceptual Categories
Questions
Common Core State Standards for Mathematics
- What I already know about the Common Core State Standards.
- What I would like to learn about the Common Core State Standards.
- What I learned today about the Common Core State Standards.
California and the Common Core State Standards
Senate Bill 1 from the Fifth Extraordinary Session (SB X5 1):
- established an Academic Content Standards Commission (ACSC) to develop standards in mathematics and English– language arts
- stated that 85 percent of the standards were to consist of the CCSS with up to 15 percent additional material
- directed the State Board of Education to adopt or reject recommendations of the ACSC
Mathematical Proficiency as defined by the California Framework (2006)
A Venn diagram that illustrates mathematical proficiency. The middle of the Venn diagram demonstrates that doing math is the combination of:
Conceptual Understanding
Problem Solving
Procedural Skills
Common Core State Standards for Mathematics
The standards for mathematics:
- are focused, coherent, and rigorous
- aim for clarity and specificity
- stress conceptual understanding of key ideas
- balance mathematical understanding and procedural skill
- are internationally benchmarked
Common Core State Standards for Mathematics
Two Types of Standards:
- Mathematical Practice (recurring throughout the grades)
- Mathematical Content (different at each grade level)
Standards for Mathematical Practice
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
CCSS Standards for Mathematical Practice
A chart showing how similar standards or concepts on reasoning from the CCSS Standards for Mathematical Practices are addressed in three other documents:
Adding It Up Strand of Proficiency
National Council on Teaching Mathematics (NCTM) Process Standards
California Mathematics Framework
Standards for Mathematical Practice
Connecting Practice to Content
- Content Standards include a balanced combination of procedure and understanding
- “Understanding” expectations connect practice to content
Standards for Mathematical Practice
Connecting Practice to Content: Examples
Grade 6: Apply and extend previous understandings of arithmetic to algebraic expressions.
Grade 7: Solve real-life and mathematical problems
Grade 8: Analyze and solve linear equations
Grades 6–8 Mathematics
A screen shot of how the grade level CCSS standards are organized:
Domains
Standards – are organized into clusters
CCSS Domains K-5
Kindergarten
-Counting and Cardinality (CC)
-Operations and Algebraic Thinking (OA)
-Number and Operations in Base Ten (NBT)
-Measurement and Data (MD)
-Geometry (G)
First Grade
-Operations and Algebraic Thinking (OA)
-Number and Operations in Base Ten (NBT)
-Measurement and Data (MD)
-Geometry
Second Grade
-Operations and Algebraic Thinking (OA)
-Number and Operations in Base Ten (NBT)
-Measurement and Data (MD)
-Geometry
Third Grade
-Operations and Algebraic Thinking (OA)
-Number and Operations in Base Ten (NBT)
-Measurement and Data (MD)
-Geometry
- Numbers and Operations - Fractions (NF)
Fourth Grade
-Operations and Algebraic Thinking (OA)
-Number and Operations in Base Ten (NBT)
-Measurement and Data (MD)
-Geometry
- Numbers and Operations - Fractions (NF)
Fifth Grade
-Operations and Algebraic Thinking (OA)
-Number and Operations in Base Ten (NBT)
-Measurement and Data (MD)
-Geometry
- Numbers and Operations - Fractions (NF)
CCSS Domains 6-8
Sixth Grade
-Ratios and Proportional Relationships (RP)
-The Number System (NS)
-Expressions and Equations (EE)
-Geometry (G)
-Statistics and Probability (SP)
Seventh Grade
-Ratios and Proportional Relationships (RP)
-The Number System (NS)
-Expressions and Equations (EE)
-Geometry (G)
-Statistics and Probability (SP)
Eight Grade
-The Number System (NS)
-Expressions and Equations (EE)
-Geometry (G)
-Statistics and Probability (SP)
-Functions (F)
Domain |
K |
1 |
2 |
3 |
4 |
5 |
|---|---|---|---|---|---|---|
Counting and Cardinality (CC) |
Yes |
No |
No |
No |
No |
No |
Operations and Algebraic Thinking (OA) |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
Number and Operations in Base Ten (NBT) |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
Measurement and Data (MD) |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
Geometry (G) |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
Number and Operations – Fractions (NF) |
No |
No |
No |
Yes |
Yes |
Yes |
CCSS Domains 6-8
Domain |
6 |
7 |
8 |
|---|---|---|---|
Ratios and Proportional Relationships (RP) |
Yes |
Yes |
No |
The Number System (NS) |
Yes |
Yes |
Yes |
Expressions and Equations (EE) |
Yes |
Yes |
Yes |
Geometry (G) |
Yes |
Yes |
Yes |
Statistics and Probability (SP) |
Yes |
Yes |
Yes |
Functions (F) |
No |
No |
Yes |
Grades 6-8
CCSS Domains |
CA Strands |
|---|---|
|
|
|
|
|
|
|
|
Grades 6-8 Overview
A screen shot of the Grade 6 overview of page 24 from the Common Core State Standards Mathematical Standards located on the Sacramento County Office of Education
Web site.
Grades 6-8 Overview
A screen shot of page 25 from the Common Core State Standards Mathematical Standards located on the Sacramento County Office of Education
Web site that highlights how the standards are abbreviated. This example shows how the Grade 6 standards in the Number System is abbreviated 6.NS.
Grades 6-8
CCSS with California additions
A screen shot of page 26 from the Common Core State Standards Mathematical Standards located on the Sacramento County Office of Education
Web site showing an example of California additions to the Common Core State Standards in grades 6-8.
Grades 6-8
Example: CCSS Grade Level Topics
Topic |
Grade |
|---|---|
|
6 |
|
6 |
|
7 |
|
7 |
|
8 |
|
8 |
|
8 |
Grade Shifts
Examples 6-8
Concept |
1997 Standards |
CCSS |
|---|---|---|
Dividing fractions by fractions |
Grade 5 |
Grade 6 |
Concepts of mean and median to summarize data sets |
Grade 5 |
Grade 6 |
Operations with numbers in scientific notation |
Grade 7 |
Grade 8 |
Pythagorean Theorem |
Grade 7 |
Grade 8 |
Grades 6-8
Instructional Considerations Activity 6-8
Discuss implications for our work as we move toward full implementation of the Common Core State Standards
Grades 6-8
A screen shot of the activity page on comparing the 1997 California Content Standards and the Common Core State Standards audience used during the presentation.
Mathematics Grade 8
“The California State Board of Education acknowledges that the goal for 8th grade students is Algebra I. However, they also recognize that not all 8th grade students have the necessary prerequisite skills for Algebra I. Consequently, the State Board of Education adopted two sets of standards for 8th grade. The first set describes standards for Algebra I.”
Source: CCSS Mathematical Standards, page 33, located on the Sacramento County Office of Education Web site.
Mathematics Grade 8 (continued)
“The second set of standards is from the 8th grade Common Core document published June 2, 2010. These standards are for 8th grade students who do not have the necessary prerequisite skills for Algebra I. The goal of the 8th grade Common Core is to finalize the mathematics preparation for students in high school. There is some duplication of standards between grades and courses that will be resolved in the frameworks/instructional materials development process.”
Source: CCSS Mathematical Standards, page 33, located on the Sacramento County Office of Education Web site.
Grade 8 Algebra 1 Activity
A screen shot of a blank activity chart that audience used to fill in the standards comprising the Grade 8 Algebra 1 Option.
Grade 8 Algebra 1 Activity
A screen shot of a completed activity chart of standards comprising the Grade 8 Algebra 1 Option .
Grade 8 Algebra 1 Activity
- How would your teaching of Algebra 1 be different, compared to the current California Algebra 1 standards, if you taught the subject using those SBE-adopted grade eight Algebra 1 standards? Discuss in table groups and record your findings to report out to the group. (10 minutes)
Common Core State Standards with California Additions
Grade 8 – Algebra 1 Overview
A screen shot of page 36 of the Common Core State Standards showing the following standards:
Number and Quantity
The Real Number System
- Extend the properties of exponents to rational exponents (CCS N-RN-1 and N-RN-2)
- Use Properties of rational and irrational numbers (CCSS N-RN-3)
Quantities
- Define appropriate quantities for the purpose of descriptive modeling (CCS N-Q2*)
Algebra
Expressions and Equations
- Work with radicals and integer exponents. (CCS 8EE-1 and 4)
- Understand the connections between proportional relationships, lines and linear equations. (CCSS 8EE-5 and 6)
- Analyze and solve linear equations and pairs of simultaneous linear equations. (CCS 8EE-7, 7a, 7b, and 8, 8a, 8b, 8c)
Seeing Structure in Expressions
- Interpret the structure of expressions. (CCS A-SSE-1*, 1a, 1b, and 2, 2a, 2b)
- Write expressions in equivalent forms to solve problems. (CCS A-SSE-3*, 3a, 3b)
High School Mathematics
The CCSS high school standards are organized in six conceptual categories:
- Number and Quantity
- Algebra
- Functions
- Modeling (*)
- Geometry
- Statistics and Probability
California additions:
- Advanced Placement Probability and Statistics
- Calculus
Modeling standards are indicated by a (*) symbol. Standards necessary to prepare for advanced courses in mathematics are indicated by a (+) symbol.
The High School Mathematics Standards:
- Students applying mathematical ways of thinking to real world issues and challenges
- Developing a depth of understanding and ability to apply mathematics to novel situations, as college students and employees are requested to do
- Emphasizing mathematical modeling
- Preparing students to be college and career ready.
Conceptual Category Overview
A screen shot of page 49 of the CCSS Mathematical Standards located on the Sacramento County Office of Education Web site showing the Number and Quantity Overview.
Format of High School Standards
A screen shot of page 50 of the CCSS Mathematical Standards located on the Sacramento County Office of Education Web site showing how high school standards are organized in the Common Core State Standards by:
Conceptual Category
Domain
Standards - organized into clusters
Common Core State Standards for Mathematics
A screen shot of the Common Core State Standards Appendix A:
Designing High School Mathematics Courses Based on the Common Core State Standards from the Common Core State Standards Initiative
Web site.
Model Course Pathways for Mathematics
Graphic which illustrates two possible pathways to higher level mathematics course.
Courses in higher level mathematics: Precalculus, Calculus (upon completion of Precalculus), Advanced Statistics, Discrete Mathematics, Advanced Quantitative Reasoning, or other courses to be designed at a later date, such as additional career technical courses.
Pathway A (Traditional in U.S.): Algebra I to Geometry to Algebra II.
Pathway B (International Integrated approach (typical outside of U.S.): Mathematics I to Mathematics II to Mathematics III.
High School Courses
- CBEDS has approximately 40 codes and descriptions for middle and high school mathematics courses.
- Code 2498 is for a mathematics course not identified in the series of courses outlined on the CBEDS assignment code list.
- Available from the California Department of Education, Data and Statistics Web page.
Group Activity: High School Courses
- At your tables, create an outline for a Intermediate Algebra/Algebra II course using standards from the Common Core High School Conceptual Categories. You can use the current California Algebra II Standards as a reference. Be prepared to discuss with the larger group why you selected certain standards and topics. (10-15 minutes)
Group Activity: High School Courses
- How could the Common Core State Standards for Mathematical Practice be taught as part of your course? Again in your table groups, develop a number of suggestions for how those standards would fit into the teaching of the content standards you selected. (5 minutes)
Common Core Traditional Pathway: Algebra II
- Number and Quantity
- The Complex Number System (5 standards)
- The Complex Number System (5 standards)
- Algebra
- Seeing Structure in Expressions (4)
- Arithmetic with Polynomials and Rational Expressions (7)
- Creating Equations (4)
- Reasoning with Equations and Inequalities (2)
- Functions
- Interpreting Functions (8)
- Building Functions (3)
- Linear, Quadratic, and Exponential Models (1)
- Trigonometric Functions (4)
- Statistics and Probability
- Interpreting Categorical and Quantitative Data (1)
- Making Inferences and Justifying Conclusions (6)
- Using Probability to Make Decisions (2)
Group Activity: High School Courses
- There is no “right” answer!
- Recommended courses for high school should be addressed in the next revision of the Mathematics Framework.
- We do not currently know how the high school mathematics standards will be assessed.
A screen shot of the Common Core State Standards Resources Web page
Source: Common Core State Standards Resources
A screen shot of the CDE on iTunes U Web page
Source: CDE on iTunes U
A screen shot of an iTunes U page showing a selection of videos and other materials about the Common Core State Standards.
Common Core State Standards for Mathematics
This is the follow up to the beginning K – W – L activity highlighting what the participants learned from the presentation.
Resources
Visit the California Department of Education’s Common Core State Standards Resources Web page
Contacts:
Curriculum Frameworks and Instructional Resources Division
916-319-0881 or
Mathematics and Science Leadership Office
English Learner and Curriculum Support Division
Curriculum, Learning and Accountability Branch
California Department of Education
1430 N Street, Sacramento, CA 95814
916-323-6189
Contact Us
Jim Greco
English Learner & Curriculum Support Division
jgreco@cde.ca.gov
916-323-6189
Ken McDonald
Curriculum Frameworks &
Instructional Resources Division
kmcdonal@cde.ca.gov
916-319-0447