The Common Core State Standards for Mathematics Kindergarten through Grade 5
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 K-5
- Overview Structure of Standards
- Mathematical Practice
- Mathematical Content
- K-5 Content Standards
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.
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 (SBE) 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 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 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 Content K-8
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)
Grades K-5
CCSS Domains |
1997 CA Strands |
|---|---|
|
|
|
|
|
|
Problem Solving
- Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding (3.OA.8)
Develop Conceptual Understandings
- Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. (K.OA.2)
- Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. (2NBT.7)
Emphasis on Fluency
- Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers. (3.OA.7)
- Fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.5)
Grades K-5
Example: CCSS Grade Level Topics
Topic |
Grade |
|---|---|
Count to 100 |
K |
Add within 100 |
1 |
Fluently add and subtract within 100 |
2 |
Fluently multiply and divide within 100 |
3 |
Introduce fractions |
3 |
| Add and subtract fractions Introduce decimals |
4 |
Add, subtract, multiply and divide fractions |
5 |
Grades K-2: Overview
A screen shot of page 3 from the CCSS Mathematical Standards located on the Sacramento County Office of Education (Outside Source) Web site.
Grades K-2
A screen shot of page 5, from the CCSS Mathematical Standards located on the Sacramento County Office of Education (Outside Source) Web site, that highlights how the standards are abbreviated. This example shows that Kindergarten standards in Number and Operations in Base Ten are abbreviated K.NBT.
Grades K-2: Activity
- Read the kindergarten standard in the Number and Operations in Base Ten domain (K.NBT.1).
- How might the content in this standard be familiar or unfamiliar to a kindergarten teacher?
Grades K - 2
Example: CCSS Topics Across Grades
K |
1 |
2 |
|---|---|---|
Count to 100, write numbers (to 20) |
Count, read and write numbers (to 120) |
Count, read and write numbers (to 1,000) |
Place value to 19 |
Place value to 100 |
Place value to 1,000 |
|
Fluently add and subtract (within 5) |
Fluently add and subtract (within 10) |
Fluently add and subtract (within 100) |
Correlating Standards
How do specific Common Core State Standards compare to the 1997 California Standards?
Kindergarten Example
1997 CA Standard |
CCSS |
|---|---|
Use concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10). (K.NS.2.1) |
Solve addition and subtraction word problems, and add and subtract within 10, e.g. by using objects or drawings to represent the problem. (K.OA.2) |
Grade 1 Example
1997 CA Standard |
CCSS |
|---|---|
Count, read, and write whole numbers to 100. (1. NS. 1.1) |
Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. (1.NBT.1) |
Grade Shifts Examples: K-2
Concept |
1997 Standards |
CCSS |
|---|---|---|
Count from 30 to 100 |
Grade 1 |
K |
Skip count by 2s, 5s, and 10s to 100 |
Grade |
Grade 2 |
Know from memory the multiplication tables for 2s and 5s *CCSS 3.OA.7-Know from memory all products of two one-digit numbers |
Grade 2 |
Grade 3* |
Introduction to fractions as numbers |
Grade 2 |
Grade 3 |
Grades K-2
Instructional Considerations Activity
Discuss implications for our work as we move toward full implementation of the Common Core standards
Grades K-2
A screen shot of group activity that the audiences participated in during the presentation. This activity compares standards from Grade 1 and CCSS might affect how the teacher may teach the standard.
Grades 3 – 5
A Focus on Fractions
- Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. (3.NF.2.a)
- Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g. by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5+ 1/2 = 3/7, by observing that 3/7 < 1/2. (5.NF.2)
Grades 3 – 5 Overview
A screen shot showing the introduction of fractions beginning in Grades 3-5, page 12, from the CCSS Mathematical Standards located on the Sacramento County Office of Education (Outside Source) Web site.
Grades 3 – 5 Overview
Number and Operations - Fractions |
Grade |
|
3 |
|
4 |
|
5 |
Grades 3 – 5
A screen shot from the third grade CCSS page 14, from the CCSS Mathematical Standards located on the Sacramento County Office of Education (Outside Source) Web site, that highlights two of the California Additions to the Number and Operations – Fractions in the CCSS.
Grades 3 - 5: Activity
- Read the standards on fractions in 3.NF
- Discuss with a neighbor how the "California additions” might help maintain California’s rigorous expectations for mathematics.
Grades 3 – 5 Example: Fractions Across Grades
3 |
4 |
5 |
|---|---|---|
Represent unit fractions on a number line |
Add and subtract mixed numbers with like denominators |
Add and subtract fractions with unlike denominators |
Generate simple equivalent fractions |
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) |
Multiply a fraction or whole number by a fraction |
Compare two fractions with the same numerator or denominator |
Compare two fractions with different numerators or denominators |
Divide unit fractions by whole numbers and whole numbers by unit fractions |
Grade 3 Example
1997 CA Standard |
CCSS |
|---|---|
Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4). (3.NS.3.1) |
Compare two fractions with the same numerator or the same denominator, by reasoning about their size, Recognize that valid comparisons rely on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. (Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.) (3.NF.3d) |
Grade 4 Example
1997 CA Standard |
CCSS |
|---|---|
Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form. (5.NS.2.3) |
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. (4.NF.3d) |
Grade 5 Example
1997 CA Standard |
CCSS |
|---|---|
Understand the concept of multiplication and division of fractions. (5.NS.2.4) |
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. (5.NF.4) Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (5.NF.7) (Note: Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.) |
Grade Shifts
Examples Grades 3 - 5
Concept |
1997 CA Standard |
CCSS |
|---|---|---|
Add and subtract simple fractions |
Grade 3 |
Grade 4 |
Multiply a fraction by a whole number |
Grade 5 |
Grade 4 |
Introduce Probability |
Grade 3 |
Grade 7 |
Graphing points on a coordinate plane |
Grade 4 |
Grade 5 |
Grades 3-5
Instructional Considerations Activity
Discuss implications for our work as we move toward full implementation of the Common Core standards
Grades 3-5 Activity
A screen shot of the activity page on comparing fractions that the audience used during the presentation.
Fractions: Sample Solution
A illustration showing how fractions can be represented on a number line.
Fractions: Sample Solution
A second illustration showing how fractions can be represented on a number line. The illustration is from Math Is Fun (Outside Source) Web site.
Fractions: Sample Solution
A third illustration showing how fractions can be represented on a number line. The illustration is from Math Is Fun (Outside Source) Web site
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.
- 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?
Resources
Visit the California Department of Education’s Common Core State Standards Resources Web page.
Contacts:
Standards, Curriculum Frameworks and Instructional Resources Division,
916-319-0881 or
Mathematics and Science Leadership Office,
English Learner and Curriculum Support Division,
916-323-6189
Curriculum, Learning and Accountability Branch
California Department of Education
1430 N Street, Sacramento, CA 95814
Contact Us
Tony Quirarte
Standards, Curriculum Frameworks &
Instructional Resources Division
tquirarte@cde.ca.gov
916-319-0449 [Note, the preceding phone number is no longer valid. For more information, please contact the Curriculum Frameworks and Instructional Resources Division at 916-319-0881.]
Mary Sprague
Standards, Curriculum Frameworks &
Instructional Resources Division
msprague@cde.ca.gov
916-319-0510
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