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© 2013 UNIVERSITY OF PITTSBURGH Study Group 2 – Algebra 2 Welcome Back! Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.

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© 2013 UNIVERSITY OF PITTSBURGH Part A From Bridge to Practice #1: 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.

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© 2013 UNIVERSITY OF PITTSBURGH 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. 3 Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

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© 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.

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© 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

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© 2013 UNIVERSITY OF PITTSBURGH Part B from Bridge to Practice #1: 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.

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© 2013 UNIVERSITY OF PITTSBURGH Part C From Bridge to Practice #1: 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.

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© 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education High School Mathematics Algebra 2 Engaging In and Analyzing Teaching and Learning through an Instructional Task

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Rationale By engaging in an instructional task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.

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© 2013 UNIVERSITY OF PITTSBURGH Question to Consider… What is the difference between the following types of tasks? instructional task assessment task

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© 2013 UNIVERSITY OF PITTSBURGH Taken from TNCore’s FAQ Document:

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© 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: develop a shared understanding of teaching and learning through an instructional task; and deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics. (This will be completed as the Bridge to Practice)

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© 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: engage in a lesson; and reflect on learning in relationship to the CCSS. (This will be completed as the Bridge to Practice #2)

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© 2013 UNIVERSITY OF PITTSBURGH Looking Over the Standards Briefly look over the focus cluster standards. We will return to the standards at the end of the lesson and consider: What focus cluster standards were addressed in the lesson? What gets “counted” as learning?

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© 2013 UNIVERSITY OF PITTSBURGH Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. xf(x)

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The Structures and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving 1.Generate and Compare Solutions 2.Assess and Advance Student Learning MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: Different solution paths to the same task Different representations Errors Misconceptions SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: By engaging students in a quick write or a discussion of the process. Set Up of the Task Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

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© 2013 UNIVERSITY OF PITTSBURGH Solve the Task (Private Think Time and Small Group Time) Work privately on the Missing Function Task. (This should have been completed as the Bridge to Practice prior to this session) Work with others at your table. Compare your solution paths. If everyone used the same method to solve the task, see if you can come up with a different way. Consider what each person determined about g(x).

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© 2013 UNIVERSITY OF PITTSBURGH Expectations for Group Discussion Solution paths will be shared. Listen with the goals of: –putting the ideas into your own words; –adding on to the ideas of others; –making connections between solution paths; and –asking questions about the ideas shared. The goal is to understand the mathematics and to make connections among the various solution paths.

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© 2013 UNIVERSITY OF PITTSBURGH Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. xf(x)

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© 2013 UNIVERSITY OF PITTSBURGH Discuss the Task (Whole Group Discussion) What do we know about g(x)? How did you use the information in the table and graph and the knowledge that h(x) = f(x) · g(x) to determine the equation of g(x)? How can you use what you know about the graphs of f(x) and g(x) to help you think about the graph of h(x)? Predict the shape of the graph of a function that is the product of two linear functions. Explain from the graphs of the two functions why you have made your prediction.

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© 2013 UNIVERSITY OF PITTSBURGH Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction?

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Linking to Research/Literature Connections between Representations Pictures Written Symbols Manipulative Models Real-world Situations Oral Language Adapted from Lesh, Post, & Behr, 1987

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Five Different Representations of a Function Language TableContext GraphEquation Van De Walle, 2004, p. 440

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The CCSS for Mathematical Content CCSS Conceptual Category – Number and Quantity The Real Number System (N-RN) Extend the properties of exponents to rational exponents. N-RN.A.1 Explain 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.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

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The CCSS for Mathematical Content CCSS Conceptual Category – Algebra 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 the expression. ★ A-SSE.B.3c Use 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 t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.B.4 Derive 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. Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

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The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Arithmetic with Polynomials and Rational Expressions (A–APR) Understand the relationship between zeros and factors of polynomials. A-APR.B.2 Know 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.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

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The CCSS for Mathematical Content CCSS Conceptual Category – Functions Building Functions (F–BF) Build a function that models a relationship between two quantities. F-BF.A.1 Write a function that describes a relationship between two quantities. ★ F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. F-BF.A.1b Combine 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.2 Write 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. Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

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© 2013 UNIVERSITY OF PITTSBURGH Bridge to Practice #2: Time to Reflect on Our Learning 1. Using the Missing Function Task: a. Choose the Content Standards from pages of the handout that this task addresses and find evidence to support them. b.Choose the Practice Standards students will have the opportunity to use while solving this task and find evidence to support them. 2.Using the quotes on the next page, Write a few sentences to summarize what Tharp and Gallimore are saying about the learning process. 3.Read the given Essential Understandings. Explain why I need to know this level of detail about quadratics to determine if a student understands the structure behind quadratics.

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© 2013 UNIVERSITY OF PITTSBURGH Research Connection: Findings by Tharp and Gallimore For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.” They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support. For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation. Tharp & Gallimore, 1991

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© 2013 UNIVERSITY OF PITTSBURGH Underlying Mathematical Ideas Related to the Lesson (Essential Understandings) The product of two or more linear functions is a polynomial function. The resulting function will have the same x- intercepts as the original functions because the original functions are factors of the polynomial. Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, f(x 1 )+g(x 1 )) will be on the graph of the sum f(x)+g(x). (This is true for subtraction and multiplication as well.)

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