The Common CoreState Standardsfor 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 www.corestandards.org

• 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 School

DomainDomain

ClusterCluster

StandardStandard

Format of High School Standards

STEMSTEMSTEMSTEMModelingModelingModelingModeling

Regular Regular StandardStandardRegular Regular

StandardStandard

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 www.act.org

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:– http://iu19curriculum.wikispaces.com/home

• 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

www.nctm.org/cfp

www.nctm.org/FHSM

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

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