The Common Core State Standards for Mathematics Appendix A: Designing High School Mathematics Courses Based on Common Core State Standards

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

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 core in English language arts and mathematics. States may add additional standards.

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.

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

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.

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.

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

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.

CCSS Mathematical Practices 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.

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

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

High School Conceptual Categories Number and Quantity Algebra Functions Geometry Modeling Probability and Statistics

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 –In high school, this may occur over the course of one year or through several years

Format of High SchoolDomainDomain ClusterCluster StandardStandard

Format of High School StandardsSTEMSTEM ModelingModeling Regular Standard

High School Pathways - Overview The CCSS Model Pathways are NOT required. The pathways are models, not mandates The models that organize the CCSS into coherent, rigorous courses Four years of mathematics: –One course in each of the first three years –Followed relevant courses for year 4 –Intended to significantly increase coherence of HS Math Course descriptions –Delineate mathematics standard to be covered –Are not prescriptions for the curriculum or instruction

High School Pathways - Overview Units within Courses –Suggest possible grouping of standards into coherent blocks –May be considered “big ideas” or “critical areas” (terms used interchangeably) Within each pathway, modules are virtually identical, but arranged differently –Order of standards within each unit follows order of standards, not the order in which they should be taught –Course names for organizational purposes Pathways organize standards for Math Content –Must connect content to standards for mathematical Practice

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 includes number, algebra; geometry; and data, probability, and statistics. Compacted Versions of Integrated Pathway and Traditional Pathway for Grades 7 and 8 also presented.

Evidence to Support Pathways ACT and Educational Trust Study High poverty schools with high percentages of students reaching and exceeding ACT’s college readiness benchmarks Opened doors to these classrooms, observation and course content Commonalities established grounding for pathways On Course for Success at

Understanding the Pathways Each pathway, traditional and integrated, consists of two charts –Overview of the pathway: organized by course and conceptual category –All HS Standards in CCSS are in at least one course within chart

Understanding the Pathways Second component shows clusters and standards as they appear in the course Each course contains the following –Introduction to course and list of units Traditional pathway course 1: HS Algebra I (p.15) –Unit titles and overview (p 16) –Units that show the cluster titles, associated standards and instructional notes* (p. 17)

Comparison of Traditional Pathway to Integrated Pathway Algebra I –U1: Relationships between Quantities and Reasoning with Equations –U2: Linear and Exponential Relationships –U3: Descriptive Statistics –U4: Expressions and Equations –U5: Quadratic Functions and Modeling Mathematics I –U1: Relationships between Quantities –U2: Linear and Exponential Relationships –U3: Reasoning with Equations –U4: Descriptive Statistics –U5: Congruence, Proof, and Constructions –U6: Connecting Algebra and Geometry through Coordinates

Comparison of Traditional Pathway to Integrated Pathway Geometry –U1: Congruence, Proof and Construction –U2: Similarity, Proof, and Trigonometry –U3: Extending to Three Dimensions –U4: Connecting Algebra and Geometry through Coordinates –U5: Circles With and Without Coordinates –U6: Applications of Probability Mathematics II –U1: Extending the Number System –U2: Quadratic Functions and Modeling –U3: Expressions and Equations –U4: Applications of Probability –U5: Similarity, Right Triangle, and Proof –U6: Circles With and Without Coordinates

Comparison of Traditional Pathway to Integrated Pathway Algebra II –U1: Polynomial, Rational and Radical Relationships –U2: Trigonometric Relationships –U3: Modeling and Functions –U4: Inferences and Conclusions from Data –U5: Quadratic Functions and Modeling Mathematics III –U1: Inferences and Conclusions from Data –U2: Polynomial, Rational and Radical Relationships –U3: Trigonometry of General Triangles and Trigonometric Functions –U4: Mathematical Modeling

Considerations Both HS Pathways include all CC math content standards Crosswalks between current course programming and CC math, PA Standards, Keystone Anchors CC math and PA standard/anchor crosswalks: – Scroll down to “Transition to Common Core” documents

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

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.

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

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

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

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

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

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

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

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

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.

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.

Geometry

Circles Expressing geometric properties with equations –Includes proving theorems and describing conic sections algebraically Geometric measurement and dimension Modeling with geometry

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

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

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