6-12 Mathematics 1 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 General Overview ◦ Focus and Coherence ◦ Mathematical Proficiency Objectives 2  Structure ◦ Organization ◦ Progression of Standards ◦ Grade 8 Options  Similarities  Grade Shifts  Next Steps © CCSESA, California, All rights reserved Mathematics Teacher Overview

3 Intended to avoid the problem of “mile wide and an inch deep” Recognize that “fewer standards” are no substitute for focused standards Aim for clarity and specificity © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Topics and performances are logical over time  Based on research about how students learn math over time  Reflect hierarchical nature of the content  Evolve from particulars to deeper structures 4 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Define what students should understand and be able to do in their study of mathematics ◦ Understanding and procedural skill are equally important and can be assessed using tasks of sufficient richness ◦ Students will be expected to justify, appropriate to their math maturity  Are internationally benchmarked ◦ Reflect rigor, focus and coherence of standards in top-performing countries 5 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Do: ◦ Set grade-level standards K-8 ◦ Identify standards for 8 th Grade Algebra 1 ◦ Provide conceptual cluster standards in high school ◦ Provide clear signposts along the way toward the goal of college and career readiness for all students 6 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Do provide common assessments  Grades 3-8 and 11  Online assessment using “smart” technology  Rapid reporting system to inform instruction  Teachers involved in development and scoring  Variety of item types  Selected response  Short and extended constructed response  Technology enhanced  Performance tasks 7

 Do not: ◦ Define intervention methods or materials ◦ Define the full range of supports for English learners, students with special needs and students who are well above or below grade level expectations ◦ Dictate curriculum or teaching methods 8 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Think about what you have learned so far.  What questions or comments come to mind?  Discuss with your table group. 9 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Mathematical Practices - Recurring throughout the grades  Mathematical Content - Different at each grade level 10 © CCSESA, California, All rights reserved Mathematics Teacher Overview

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 11 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Read your assigned Standard for Mathematical Practice  Highlight key words or phrases in your standard to share with the whole group.  With a partner, discuss: ◦ Your “Aha’s” ◦ A standard that you feel might be a challenge to implement, and why 12 © CCSESA, California, All rights reserved Mathematics Teacher Overview

Reading the Grade Level Overview Domain Cluster 13

Reading the Grade Level Standards 14 © CCSESA, California, All rights reserved Mathematics Teacher Overview Domain Cluster Standards

California’s Additional 15%  Intended to: ◦ Include current California standards not in the Common Core Standards ◦ Enhance the Common Core Standards ◦ Maintain alignment with California’s current standards  Created three courses specific to CA: ◦ 8 th Grade Algebra 1 ◦ Calculus ◦ AP Probability and Statistics 15

 Two sets of standards for grade 8 that will prepare students for college and career ◦ Standards for Algebra 1, taken from 8th grade Common Core, high school content clusters and CA Algebra 1standards ◦ 8th grade Common Core  Goal of grade 8 Common Core is to finalize preparation for students in high school  K-7 standards as augmented prepare students for either set of standards 16 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Arranged by conceptual cluster (NOT by course): Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability  Same K-8 structure of domain, cluster and standard 17 © 2011 California County Superintendents Educational Services Association Mathematics Teacher Overview

 Specify the math that all students should study to be college and career ready  Identify additional math standards that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics. These are indicated by (+).  Development of suggested course pathways will be done by CDE as part of their long-range implementation plan ◦ Traditional vs. Integrated 18 © 2011 California County Superintendents Educational Services Association Mathematics Teacher Overview

Modeling Standard High School Example-Geometry Content Cluster 19 © 2011 California County Superintendents Educational Services Association Mathematics Teacher Overview (+) Standard

 Locate your grade level or course in your standards handout.  Choose one domain and one cluster of standards to review.  Pay attention to the language, content, and structure of the standards.  Share your reactions with a partner. 20 © CCSESA, California, All rights reserved Mathematics Teacher Overview

21 © CCSESA, California, All rights reserved Mathematics Teacher Overview K HS Counting & Cardinality Number and Operations in Base Ten Ratios and Proportional Relationships Number & Quantity Number and Operations – Fractions The Number System Operations and Algebraic Thinking Expressions and EquationsAlgebra Functions Geometry Measurement and DataStatistics and Probability Statistics & Probability Domains and Conceptual Categories K-12

 Choose a domain or conceptual category that covers at least two grade levels, including your own.  Read the standards in each domain or conceptual category.  What connections do you notice? Share with a partner. 22 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 6.EE.9 “…Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity…in terms of the other… Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.…” Develop Conceptual Understanding Across Grade Levels 23

 7.RP.2b “Identify the constant of proportionality (unit rate) in tables, graphs, …”  7.RP.2c “Represent proportional relationships by equations.”  7.RP.2d “Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.” Develop Conceptual Understanding Across Grade Levels 24

How do the skills and concepts involved in completing this task relate to the two previous tasks?  8.F.4 “Construct a function to model a linear relationship between two quantities…Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values…Interpret the rate of change and initial value … in terms of the situation it models...” Develop Conceptual Understanding Across Grade Levels 25

Connecting Standards for Math Practice with Standards for Content  Balanced combination of procedure and understanding  Build in complexity and provide more clarity for expected performance  “Understand” expectations connect practice to content ◦ Lack of understanding prevents students from engaging in the mathematical practices ◦ Weighted toward important concepts that most merit the time, resources, and focus 26 © 2011 California County Superintendents Educational Services Association Mathematics Teacher Overview

GradeCalifornia StandardCommon Core SixthInterpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ). Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. SeventhUse variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three is less than a number, half as large as area A). Use variables to represent quantities in real- world and mathematical problems and construct simple equations and inequalities to solve problems about the quantities. 27 © 2011 California County Superintendents Educational Services Association Mathematics Teacher Overview

Grade/C ourse California StandardCommon Core SeventhConstruct and read drawings and models made to scale. Solve problems involving scale drawings of geometric figures, including actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. AlgebraAlgebra 1: Solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Algebra Content Cluster: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. GeometryGeometry: Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Geometry Content Cluster: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 28 © 2011 California County Superintendents Educational Services Association Mathematics Teacher Overview

Grade Shifts: Examples Developed by SCFIRD Concept1997 StandardsCCCS Compose simple shapes to form larger shapes (e.g., 2 triangles to form a rectangle) Grade 2 K Introduction to Probability Grade 3 Grade 7 Introduction of fractions as numbers Grade 2 Grade 3 Add and subtract simple fractions Grade 3 Grade 4 Introduction of integersGrade 4 Grade 6 29 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Continue to implement current CA standards, as there are more similarities than differences in the standards.  Emphasize developing student conceptual understanding.  Implement our current Math Reasoning Standards as this will support the Standards for Mathematical Practice.  Collaborate with colleagues for Common Core implementation. 30 © CCSESA, California, All rights reserved Mathematics Teacher Overview

 Websites ◦ California Common Core Standards: Visit the California Department of Education’s Common Core State Standards Web page at: The standards Frequently asked questions Informational flyers Additional resources 31 © CCSESA, California, All rights reserved Mathematics Teacher Overview