1. © 2013 UNIVERSITY OF PITTSBURGH Welcome to TNCore Training! Tennessee Department of Education High School Mathematics Geometry Introduction of 2013 CCSS Training

2. © 2013 UNIVERSITY OF PITTSBURGH What this is / What it is not What it isWhat it is not Peer led learningInformation updates from TDOE or expert-delivered training Content focused – we will dive deep into understanding the expectations Generic discussion of teaching strategies Focused on building our capacity (knowledge and skill) as educators Mandating implementation of a recipe for instant success Designed to meet participants at a range of experience with Common Core Redundant of other TNCore trainings - or – dependent on you having done anything thus far Focused on student achievementFocused on compliance Focused on your learningFocused on preparing you to train others

3. © 2013 UNIVERSITY OF PITTSBURGH Core Beliefs Earning a living wage has never demanded more skills. This generation must learn more than their parents to do as well. All children are capable of learning and thinking at a high level. Children in Tennessee are as talented as any in the country and often capable of more than we expect. Our current education results pose a real threat to state and national competitiveness and security. Improving the skills of our children is vital for the future of Tennessee and America. Tennessee is on a mission to become the fastest improving state in the nation. Doing so will require hard work and significant learning for all. We must learn to teach in ways we were not taught ourselves. There is no recipe that will deliver a successful transition. Preparing for Common Core will demand effective leadership focused on student growth. PARCC is coming in two years. We need to use the transition wisely to make sure our students and our state are ready.

4. © 2013 UNIVERSITY OF PITTSBURGH Norms Keep students at the center of focus and decision-making Be present and engaged – limit distractions, if urgent matters come up, step outside Monitor air time and share your voice - you’ll know which applies to you! Challenge with respect – disagreement can be healthy, respect all intentions Be solutions oriented – for the good of the group, look for the possible Risk productive struggle - this is safe space to get out of your comfort zone Balance urgency and patience - we need to see dramatic change and change will happen over time Any other norms desired to facilitate your learning?

5. © 2013 UNIVERSITY OF PITTSBURGH Emily Barton Video

6. © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education High School Mathematics Algebra II Deepening Our Understanding of CCSS Via A Constructed Response Assessment

7. © 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: deepen understanding of the Common Core State Standards (CCSS) for Mathematical Practice and Mathematical Content; understand how Constructed Response Assessments (CRAs) assess the CCSS for both Mathematical Content and Practice; and understand the ways in which CRAs assess students’ conceptual understanding.

8. © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: analyze Constructed Response Assessments (CRAs) in order to determine the way the assessments are assessing the CCSSM; analyze and discuss the CCSS for Mathematical Content and Mathematical Practice; discuss the CCSS related to the tasks and the implications for instruction and learning.

9. © 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards The standards consist of:  The CCSS for Mathematical Content  The CCSS for Mathematical Practice

10. © 2013 UNIVERSITY OF PITTSBURGH Tennessee Focus Clusters Algebra 2  Extend the properties of exponents to rational exponents.  Write expressions in equivalent forms to solve problems.  Understand the relationship between zeros and factors of polynomials.  Build a function that models a relationship between two quantities. 10

11. © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Number and Quantity Common Core State Standards, 2010 The Real Number System(N-RN) Extend the properties of exponents to rational exponents. N-RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. N-RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.

12. © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Common Core State Standards, 2010 Seeing Structure in Expressions (A-SSE) Write expressions in equivalent forms to solve problems A-SSE.B.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression. ★ A-SSE.B.3cUse the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t ≈ 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

13. © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Common Core State Standards, 2010 Arithmetic with Polynomials and Rational Expressions (A-APR) Understand the relationship between zeros and factors of polynomials A-APR.B.2Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

14. © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Functions Common Core State Standards, 2010 Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.1Write a function that describes a relationship between two quantities. ★ F-BF.A.1aDetermine an explicit expression, a recursive process, or steps for calculation from a context. F-BF.A.1bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. F-BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

15. © 2013 UNIVERSITY OF PITTSBURGH Analyzing a Constructed Response Assessment

16. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Assessment Items (Private Think Time) Four assessment items have been provided:  Car Depreciation  Writing a Polynomial  Patterns in Patterns  One Rocket, Three Equations For each assessment item: solve the assessment item; and make connections between the standard(s) and the assessment item.

17. © 2013 UNIVERSITY OF PITTSBURGH 1. Car Depreciation After you purchase a new car, it begins to lose value. As the years pass, more and more value is lost. This process is known as depreciation. a.Carmen buys a new car for $24,500. Carmen’s new car loses 14% of its value each year through depreciation. Write a function that can be used to model the value of the car at the end of each year that Carmen owns the car. Justify your equation mathematically. b.After Carmen buys the car, he adds an audio and speaker system worth $500. The audio system loses 15% of its value the minute it is installed into the car, and 7% of the remaining value each year through depreciation. Write a function that can be used to model the value of the car with the audio system and speakers at the end of each year that Carmen owns the car.

18. © 2013 UNIVERSITY OF PITTSBURGH 2. Writing a Polynomial Recall that polynomial functions with only real number zeros can be written in factored form as follows: where each z n represents some real root of the function, and each p n is a whole number exponent greater than or equal to 1. Consider the graph of the polynomial function below.

19. © 2013 UNIVERSITY OF PITTSBURGH 3. Patterns in Patterns a.Write a function that can be used to determine the diameter of any circle drawn in the poster in this way. Explain the meaning of each term in your expression in the context of the problem. b.Laura eventually draws 10 circles. Write and use a formula for the sum of a series to find the sum of the circumferences of the 10 circles, accurate to two decimal places. Show your work. 28 inches

20. © 2013 UNIVERSITY OF PITTSBURGH 4. One Rocket, Three Equations

21. © 2013 UNIVERSITY OF PITTSBURGH Discussing Content Standards (Small Group Time) For each assessment item: With your small group, find evidence in tasks 2 and 4 for the content standard(s) that will be assessed.

22. © 2013 UNIVERSITY OF PITTSBURGH 2. Writing a Polynomial Common Core State Standards, 2010 Arithmetic with Polynomials and Rational Expressions (A-APR) Understand the relationship between zeros and factors of polynomials A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.1Write a function that describes a relationship between two quantities. ★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

23. © 2013 UNIVERSITY OF PITTSBURGH 4. One Rocket, Three Equations Common Core State Standards, 2010 Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression. ★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

24. © 2013 UNIVERSITY OF PITTSBURGH David Williams Video

25. © 2013 UNIVERSITY OF PITTSBURGH Determining the Standards for Mathematical Practice Associated with the Constructed Response Assessment

26. © 2013 UNIVERSITY OF PITTSBURGH Getting Familiar with the CCSS for Mathematical Practice (Private Think Time) Count off by 8. Each person reads one of the CCSS for Mathematical Practice. Read your assigned Mathematical Practice. Be prepared to share the “gist” of the Mathematical Practice. 26

27. The CCSS for Mathematical Practice 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. 27 Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

28. © 2013 UNIVERSITY OF PITTSBURGH Discussing Practice Standards (Small Group Time) Each person has a moment to share important information about his/her assigned Mathematical Practice. 28

29. © 2013 UNIVERSITY OF PITTSBURGH Bridge to Practice: Practice Standards Choose the Practice Standards students will have the opportunity to use while solving these tasks we have focused on and find evidence to support them. Using the Assessment to Think About Instruction In order for students to perform well on the CRA, what are the implications for instruction? What kinds of instructional tasks will need to be used in the classroom? What will teaching and learning look like and sound like in the classroom? Complete the Instructional Task Work all of the instructional task “Missing Function Task” and be prepared to talk about the task and the CCSSM Content and Practice Standards associated with it.