CaliforniaCommon CoreState StandardsMathematicsElectronic EditionAdopted by the CaliforniaState Board of EducationAugust 2010 and modifiedJanuary 2013

CaliforniaCommon CoreState StandardsMathematicsElectronic EditionAdopted by the California State Board of EducationAugust 2010 and modified January 2013

Publishing InformationSenate Bill 1200, Statutes of 2012, called for modification of the California additions to the Common Core State Standardsfor Mathematics. The California Common Core State Standards: Mathematics (CA CCSSM) were modified January 16, 2013,following the recommendation of State Superintendent of Public Instruction (SSPI) Tom Torlakson. SSPI Torlakson consultedthe Mathematics Curriculum Framework and Evaluation Criteria Committee regarding modifications to the CA CCSSM and theorganization of model courses in higher mathematics. SSPI Torlakson and the State Board of Education (SBE) convened twopublic hearings in order for the field to provide input on the recommended modifications. When the CA CCSSM were modified,the members of the SBE were Michael W. Kirst, President; Trish Boyd Williams, Vice President; Sue Burr; Carl A. Cohn; BruceHoladay; Josephine Kao; Aida Molina; Patricia Ann Rucker; Nicolasa Sandoval; and Ilene Straus.Senate Bill 1 from the fifth Extraordinary Session (SB X5 1) in 2010 established the California Academic Content StandardsCommission (Commission) to evaluate the Common Core State Standards for Mathematics developed by the Common CoreState Standards Initiative for rigor and alignment with the California standards. Based on the evaluation, the Commission inserted words, phrases, and select California standards to maintain California’s high expectations for students. On July 15, 2010,the Commission recommended that the SBE adopt the CA CCSSM as amended. The members of the Commission were GregGeeting, Chair; Heather Calahan; Steven Dunlap; Robert Ellis; Eleanor Evans; Bill Evers; Scott Farrand; Mark Freathy; Lori Freiermuth; Bruce Grip; Kathy Harris; Jeanne Jelnick; Deborah Keys; James Lanich; Matt Perry; Pat Sabo; Brian Shay; Alba Sweeney;Hilda Villarreal Writ; Chuck Weis; and Ze’ev Wurman. Support for the Commission was provided by the Sacramento County Officeof Education under the direction of Sue Stickel, Deputy Superintendent of Schools.When the CA CCSSM were adopted by the SBE on August 2, 2010, the members of the SBE were Theodore Mitchell, President;Ruth Bloom, Vice President; Alan Arkatov; James Aschwanden; Benjamin Austin; Yvonne Chan; Gregory Jones; David Lopez; andJohnathan Williams. Jack O’Connell, former State Superintendent of Public Instruction, is also recognized for his leadership during the adoption of the standards in August 2010.The California Common Core State Standards: Mathematics was edited in part by the staff of CDE Press, with the cover and interior design prepared by Tuyet Truong. It was published by the California Department of Education, 1430 N Street, Sacramento,CA 95814-5901. It was distributed under the provisions of the Library Distribution Act and Government Code Section 11096.The Common Core State Standards appear as they were published by the Common Core State Standards Initiative.The electronic edition of this document is slightly different from the print publication issued by the California Department ofEducation in 2013; it includes minor text and typographical corrections. 2013, 2014 by the California Department of EducationAll rights reservedISBN 978-0-8011-1748-8Reproduction of this document for resale, in whole or in part, is not authorizedii

Special AcknowledgmentsSpecial appreciation is extended to Tom Torlakson, State Superintendent of Public Instruction, for support of the revision andupdate of the CA CCSSM. Special commendation is extended to Lupita Cortez Alcalá, Deputy Director, Instruction and LearningSupport Branch; Thomas Adams, Director, Curriculum Frameworks and Instructional Resources Division; and Kristen Cruz Allen,Administrator, and Deborah Franklin, Consultant, Curriculum Frameworks Unit.Special recognition is awarded to Joy Kessel, Analyst, Common Core Systems Implementation Office, for her contribution to theoriginal organization and format design.Ordering InformationCopies of the California Common Core State Standards: Mathematics are available for purchase from the California Departmentof Education. For prices and ordering information, please visit the Department Web site at orcall the CDE Press Sales Office at 1-800-995-4099.NoticeThe guidance in the California Common Core State Standards: Mathematics is not binding on local educational agencies orother entities. Except for the statutes, regulations, and court decisions that are referenced herein, the document is exemplary,and compliance with it is not mandatory. (See California Education Code Section 33308.5).iii

ContentsA Message from the State Board of Education and the State Superintendent of Public Instruction . vIntroduction .1Standards for Mathematical Practice .6K–8 Standards .9Kindergarten . 10Grade 1 . 14Grade 2 . 18Grade 3 . 23Grade 4 . 28Grade 5 . 34Grade 6 . 40Grade 7 . 46Grade 8 . 52Higher Mathematics Standards .57Higher Mathematics Courses . 59Traditional Pathway. 59Algebra I . 60Geometry . 69Algebra II . 77Integrated Pathway . 85Mathematics I . 86Mathematics II. 95Mathematics III. 105Advanced Mathematics . 113Advanced Placement Probability and Statistics Standards. 114Calculus Standards. 116Higher Mathematics Standards by Conceptual Category . 119Number and Quantity . 120Algebra. 123Functions . 127Modeling . 131Geometry . 133Statistics and Probability. 138Glossary.143iv

A Message from the State Board of Educationand the State Superintendent of Public InstructionThe California Common Core State Standards: Mathematics (CA CCSSM) reflect the importance of focus, coherence, and rigoras the guiding principles for mathematics instruction and learning. California’s implementation of the CA CCSSM demonstrates acommitment to providing a world-class education for all students that supports college and career readiness and the knowledgeand skills necessary to fully participate in the twenty-first-century global economy.The CA CCSSM build on California’s standards-based educational system in which curriculum, instruction, professional learning,assessment, and accountability are aligned to support student attainment of the standards. The CA CCSSM incorporate currentresearch and input from education stakeholders—including other state departments of education, scholars, professionalorganizations, teachers and other educators, parents, and students. California additions to the standards (identified in boldfacetext and followed by the abbreviation “CA”) were incorporated in an effort to retain the consistency and precision of our paststandards. The CA CCSSM are internationally benchmarked, research-based, and unequivocally rigorous.The standards call for learning mathematical content in the context of real-world situations, using mathematics to solve problems,and developing “habits of mind” that foster mastery of mathematics content as well as mathematical understanding. Thestandards for kindergarten through grade 8 prepare students for higher mathematics. The standards for higher mathematicsreflect the knowledge and skills that are necessary to prepare students for college and careers and productive citizenship.Implementation of the CA CCSSM will take time and effort, but it also provides a new and exciting opportunity to ensure thatCalifornia’s students are held to the same high expectations in mathematics as their national and global peers. AlthoughCalifornia educators have implemented standards before, the CA CCSSM require not only rigorous curriculum and instructionbut also conceptual understanding, procedural skill and fluency, and the ability to apply mathematics. In short, the standardscall for meeting the challenges of the twenty-first century through innovation.MICHAEL W. KIRST, PresidentCalifornia State Board of EducationTOM TORLAKSONState Superintendent of Public Instructionv

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IntroductionAll students need a high-quality mathematics program designed to prepare them to graduate from high school ready for collegeand careers. In support of this goal, California adopted the California Common Core State Standards: Mathematics (CA CCSSM)in August 2010, replacing the 1997 statewide mathematics academic standards. As part of the modification of the CA CCSSM inJanuary 2013, the California State Board of Education also approved higher mathematics standards organized into model courses.The CA CCSSM are designed to be robust, linked within and across grades, and relevant to the real world, reflecting theknowledge and skills that young people will need for success in college and careers. With California’s students fully preparedfor the future, our students will be positioned to compete successfully in the global economy.The development of the standards began as a voluntary, state-led effort coordinated by the Council of Chief State SchoolOfficers (CCSSO) and the National Governors Asso