Domains of Study/Conceptual Categories Learning Progressions/Trajectories

 Aligned with college and work expectations  Written in a clear, understandable, and consistent format  Designed to include rigorous content and application of knowledge through high-order skills  Formulated upon strengths and lessons of current state standards  Informed by high-performing mathematics curricula in other countries to ensure all students are prepared to succeed in our global economy and society  Grounded on sound evidence-based research

 Coherent  Rigorous  Well-Articulated  Enables Students to Make Connections

 Articulated progressions of topics and performances that are developmental and connected to other progressions.  Conceptual understanding and procedural skills stressed equally.  Real-world/Situational application expected.

 Key ideas, understandings, and skills are identified.  Deep learning stressed.

K Grade Domain Cluster Standard Course Conceptual Category Domain Cluster Standard

 Domain  Cluster  Standards

Domain: Overarching “big ideas” that connect content across the grade levels. Cluster: Group of related standards below a domain. Standards: Define what a student should know (understand) and do at the conclusion of a course or grade.

 Overarching big ideas that connect mathematics across high school  Illustrate progression of increasing complexity  May appear in all courses  Organize high school standards

Number & Quantity AlgebraFunctionsModelingGeometry Statistics & Probability The Real Number System Seeing Structure in Expressions Interpreting Functions Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ( ★ ). Congruence Interpreting Categorical and Quantitative Data Quantities Arithmetic with Polynomials & Rational Expressions Building Functions Similarity, Right Triangles, and Trigonometry Making Inferences and Justifying Conclusions The Complex Number System Creating Equations Linear, Quadratic and Exponential Models Circles Conditional Probability and the Rules of Probability Vector and Matrix Quantities Reasoning with Equations and Inequalities Trigonometric Functions Expressing Geometric Properties with Equations Using Probability to Make Decisions Geometric Measurement and Dimension Modeling with Geometry Domains

 Multiple Courses  Illustrate Progression of Increasing Complexity from Grade to Grade

Algebra IAlgebra II with Trigonometry Interpret the structure of expressions. (Linear, exponential, quadratic.) 7. Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] a.Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a] b.Interpret complicated expressions by viewing one or more of their parts as a single entity. [A-SSE1b] 8. Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Interpret the structure of expressions. (Polynomial and rational.) 6. Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] a.Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a] a.Interpret complicated expressions by viewing one or more of their parts as a single entity. [A-SSE1b] 7. Use the structure of an expression to identify ways to rewrite it. [A-SSE2] 9-12 Cluster

 Content standards in this document contain minimum required content.  Each content standard completes the phrase “Students will.”  Reflect both mathematical understandings and skills, which are equally important.

 Conceptual Categories Cross course boundaries Span high school years  Standards – “Core” for common mathematics curriculum for all students to be college and career ready – “College and Career Ready” for entry-level, credit- bearing academic college courses and work-force training programs. – “STEM” (+) Additional mathematics that students should learn in order to take courses such as calculus, discrete mathematics, or advanced statistics.

◦ Algebra I ◦ Geometry ◦ Algebraic Connections ◦ Algebra II ◦ Algebra II with Trigonometry ◦ Precalculus ◦ Analytical Mathematics

Geometry Modeling Algebra Functions Number & Quantity Statistics & Probability

3.Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. [N-RN3] AI.3.1.Explain why the sum of two rational numbers is rational. AI.3.2.Explain why the product of two rational numbers is rational. AI.3.3.Explain that the sum of a rational number and an irrational number is irrational. AI.3.4.Explain that the product of a nonzero rational number and an irrational number is irrational.

K-2 Number and number sense. 3-5 Operations and Properties (Number and Geometry) Fractions 6-8 Algebraic and Geometric Thinking Data Analysis and using Properties High School Functions, Statistics, Modeling and Proo f

Confrey (2007) “Developing sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” CCSS, p. 4 “… the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.”

Read the excerpt from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction  Identify 3 ideas that you are willing to talk about with colleagues.  Highlight the location in the text where these ideas appear.

 Designate a facilitator and timekeeper.  A volunteer begins by reading the sentence(s) from the text that embody one of his/her selected ideas. The speaker does not comment on the text at this point.  The individual to right of first speaker takes up to one minute to comment on the selected text.  The next two individuals also take up to one minute to comment on the initial speaker’s idea.  The individual selecting the idea has up to 1 minutes to react to colleagues’ ideas and to talk about why she or he thought this was important.  Another group member introduces one idea, and the group follows the same protocol. Continue until all members have shared or until time is called.

Learning Trajectories – sometimes called learning progressions – are sequences of learning experiences hypothesized and designed to build a deep and increasingly sophisticated understanding of core concepts and practices within various disciplines. The trajectories are based on empirical evidence of how students’ understanding actually develops in response to instruction and where it might break down. Daro, Mosher, & Corcoran, 2011

Starting Point Ending Point Starting Point Ending Point K HS Counting and Cardinality Number and Operations in Base TenRatios and Proportional Relationships Number and Quantity Number and Operations – Fractions The Number System Operations and Algebraic Thinking Expressions and Equations Algebra Functions Geometry Measurement and DataStatistics and Probability

Investigating the Domains/Conceptual Categories  Domains provide common learning progressions.  Curriculum and teaching methods are not dictated.  Standards are not presented in a specific instructional order.  Standards should be presented in a manner that is consistent with local collaboration.

K HS Counting and Cardinality Number and Operations in Base TenRatios and Proportional Relationships Number and Quantity Number and Operations – Fractions The Number System

 Beginning at the lowest grade examine the domain and conceptual category, cluster and standards at your grade level - identify how the use of numbers and number systems change from K- 12. ◦ Counting & Cardinality (CC)– K only ◦ Number and Operations in Base Ten (NBT) – K-5 ◦ Number and Operations – Fractions (NF) – 3-5 ◦ The Number System (NS) – 6-8 ◦ Number and Quantity (N)– 9-12  Look at the grade level above and grade level below (to see the context).  Make notes that reflect a logical progression, increasing complexity.  As a table group share a vertical progression (bottom–up or top-down) on chart paper.

 Beginning at the lowest grade examine the domain and conceptual category, cluster and standards at your grade level - identify how the use of numbers and number systems change from K- 12. ◦ K-2 - Counting & Cardinality (CC) Number and Operations in Base Ten (NBT) ◦ Number and Operations in Base Ten (NBT) Number and Operations – Fractions (NF) ◦ The Number System (NS) ◦ Number and Quantity (N)  Look at the grade level above and grade level below (to see the context).  Make notes that reflect a logical progression, increasing complexity.  As a table group share a vertical progression (bottom–up or top-down) on chart paper.

 Summary and/or representation of how the concept of the use of numbers grows throughout your grade band.  Easy for others to interpret or understand.  Visual large enough for all to see.  More than just the letters and numbers of the standards – include key words or phrases.

 Display posters side-by-side and in order on the wall.  Begin at the grade band you studied. Read the posters for your grade band.  Discuss similarities and differences between the posters.  Establish a clear vision for your grade band.

 As a table group, consider your journey through the 2010 ACOS as you studied the concept of the use of numbers K-12.  What did you learn?  What surprised you?  What questions do you still have?

 Know what to expect about students’ preparation.  More readily manage the range of preparation of students in your class.  Know what teachers in the next grade expect of your students.  Identify clusters of related concepts at grade level.  Clarity about the student thinking and discourse to focus on conceptual development.  Engage in rich uses of classroom assessment.

2003 ACOS2010 ACOS Contains bulletsDoes not contain bullets Does not contain a glossaryContains a glossary

ALSDE Office of Student Learning Curriculum and Instruction Cindy Freeman, Mathematics Specialist  Phone: