The Common Core State Standards for Mathematics

1. The Common CoreState Standardsfor Mathematics High School

2. Common Core Development • Initially 48 states and three territories signed on • Final Standards released June 2, 2010, and can be downloaded at www.corestandards.org • As of November 29, 2010, 42 states had officially adopted • Adoption required for Race to the Top funds

3. Common Core Development • 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 corein English language arts and mathematics. States may add additional standards.

5. Benefits for States and Districts • Allows collaborative professional development to be based on best practices • Allows the 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 mileage from: • Curriculum development, assessment, and professional development

6. Characteristics • 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, so that all students are prepared 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. • Research based.

7. Intent of the Common Core • The same goals for all students • Coherence • Focus • Clarity and specificity

8. Coherence • Articulated progressions of topics and performances that are developmental and connected to other progressions • Conceptual understanding and procedural skills stressed equally NCTM states coherence also means that instruction, assessment, and curriculum are aligned.

9. Focus • Key ideas, understandings, and skills are identified • Deep learning of concepts is emphasized • That is, adequate time is devoted to a topic and learning it well. This counters the “mile wide, inch deep” criticism leveled at most current U.S. standards.

10. Clarity and Specificity • Skills and concepts are clearly defined. • An ability to apply concepts and skills to new situations is expected.

11. CCSS Mathematical Practices The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. These practices are similar to the mathematical processes that NCTM addresses in the Process Standards in Principles and Standards for School Mathematics.

12. CCSS Mathematical Practices • 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.

13. Common Core Format K-8 Grade Domain Cluster Standards (No pre-K Common Core Standards) High School Conceptual Category Domain Cluster Standards

14. Domain Format of High School Cluster Standard

15. Regular Standard Format of High School Standards Modeling STEM

16. High School Conceptual Categories • The big ideas that connect mathematics across high school • A progression of increasing complexity • Description of the mathematical content to be learned, elaborated through domains, clusters, and standards

17. 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

18. 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

19. 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

20. 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

21. High School Pathways • 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, consisting of a sequence of 3 courses, each of which treats aspects of algebra; geometry; and data, probability, and statistics.

22. Conceptual Categories • Number and Quantity • Algebra • Functions • Modeling • Geometry • Statistics and Probability

23. 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.

24. 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

25. 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

26. 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

27. Algebra Required for higher math and/or STEM • 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

28. 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

29. Functions Required for higher math and/or STEM • 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

30. Functions Required for higher math and/or STEM • 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

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

32. 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.

33. 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.

34. Geometry • Understanding congruence • Using similarity, right triangles, and trigonometry to solve problems Congruence, similarity, and symmetry are approached through geometric transformations

35. Geometry • Circles • Expressing geometric properties with equations • Includes proving theorems and describing conic sections algebraically • Geometric measurement and dimension • Modeling with geometry

36. Geometry Required for higher math and/or STEM • 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

37. 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

38. 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

39. Additional Information • For the secondary level, please see NCTM’s Focus in High School Mathematics: Reasoning and Sense Making • For grades preK-8, a model of implementation can be found in NCTM’s Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics www.nctm.org/FHSM www.nctm.org/cfp

40. Acknowledgments • Thanks to the Ohio Department of Education and Eric Milou of Rowan University for providing content and assistance for this presentation