Module 1: A Closer Look at the Common Core State Standards for Mathematics High School Session 2: Matching Clusters of Standards to Critical Areas in one HS course

In this session you will Become familiar with model pathways which were developed to address standards progressively across high school courses Deepen understanding of the mathematical concepts in the critical areas of the Common Core State Standards (CCSS) for Mathematics Align “clusters of standards” with the critical areas 2

Principle #1: Increases in student learning occur only as a consequence of improvements in the level of content, teachers’ knowledge and skill, and student engagement. Richard Elmore, Ph.D., Harvard Graduate School of Education Principle #2: If you change one element of the instructional core, you have to change the other two. The Instructional Core

Adapted from the Public Education Leadership Project at Harvard University STRUCTURES POLICIES, PROCESSES & PROCEDURES RESOURCES HUMAN, MATERIAL, MONEY STAKEHOLDERS CULTURE Organizational Elements

Focused and Coherent For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards must address the problem of a curriculum that is “a mile wide and an inch deep.” - Common Core State Standards for Mathematics, page 3 6

Coherence Content standards and curricula are coherent if they are … articulated over time as a sequence of topics and performances that are logical and reflect … the sequential or hierarchical nature of the disciplinary content … What and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. - Common Core State Standards for Mathematics, page 3 7

Reminder … – Conceptual categories: themes that connect mathematics across high school and contain a set of domains Domains: overarching “big ideas” that connect topics across high school courses Clusters: groups of standards that describe coherent aspects of the content category within a domain Standards: define what students should know and be able to do at each grade level – Critical Areas: units that organize the standards within courses as recommended in Appendix A

Remember… The course structures in Appendix A illustrate possible approaches—models, not mandates. The standards are required; however, the organization and course structure are not. In Appendix A, there are pages that show how standards are organized within each course. There are also pages that show how the standards are organized over three years of courses (Session 3).

High School Courses Critical areas or units Clusters of standards Standards Instructional notes

Critical area/or Unit and Overview Standards associated with clusters Clusters Instructional Notes Algebra I

Critical area/Unit and Overview Standards associated with clusters Clusters Instructional Notes Mathematics I

Task: Matching Clusters and Critical Areas Read through the clusters for your selected course. Examine each cluster. Discuss with your partner how you interpret the cluster. Decide which critical area the cluster would address. On large chart paper, using scissors and tape, organize the clusters by critical area. NOTE: Some clusters may fall in more than one critical area.

Match Clusters with Critical Areas: Using the worksheet (Option 1) Match clusters to critical areas (cut out available handout B)

Match Clusters with Critical Areas (Option 2) Critical Area 1: Relationships Between Quantities and Reasoning with Equations By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now, students analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. Critical Area 2: Linear and Exponential Relationships In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Write clusters here. Traditional Algebra I Write clusters here.

Small Group Discussion : Matching Clusters with Critical Areas: How do the clusters of standards illuminate the concepts in the critical areas? – What did you learn about each of the mathematical big ideas in the critical areas? How does this content compare to the course you currently teach? – In general, how much alike or different is this from the course you teach now?

Whole Group Discussion Were you able to match each cluster of standards with one of the critical areas? – What challenges did you have in matching the clusters? – What questions arose for you and your team about the organization of the standards? How do the clusters illuminate the concepts in the critical areas? In general, how much alike or different is this from the course you teach now?

Reflection Questions How will your knowledge of the standards in the model courses inform your curriculum and guide your instruction? What will be some major changes? What questions do you still have about course structure? 18