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Study Group 2 – Algebra 2

Study Group 2 – Algebra 2. Welcome Back! Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises. Part A From Bridge to Practice # 1:. Practice Standards

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Study Group 2 – Algebra 2

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

  2. 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.

  3. The CCSS for Mathematical Practice Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

  4. 2. Writing a Polynomial • Lisa claims that, since the point (0, 6) is on the graph, (x – 6) is a factor of this polynomial. Explain why you agree or disagree with Lisa’s claim. Identify all the zeroes of the function and use that information in your explanation. • Suppose a = . Write a function in factored form to represent this graph. Justify your equation mathematically. Recall that polynomial functions with only real number zeros can be written in factored form as follows: where each zn represents some real root of the function, and each pnis a whole number exponent greater than or equal to 1. Consider the graph of the polynomial function below.

  5. 3. Patterns in Patterns Laura creates a design of circles embedded in each other for a poster. The largest circle has a diameter of 28 inches, and each successive circle has a diameter of the previous circle. • 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. • 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

  6. 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.

  7. 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.

  8. Supporting Rigorous Mathematics Teaching and Learning Engaging In and Analyzing Teaching and Learning through an Instructional Task Tennessee Department of Education High School Mathematics Algebra 2

  9. 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.

  10. Question to Consider… What is the difference between the following types of tasks? instructional task assessment task

  11. Taken from TNCore’s FAQ Document:

  12. 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)

  13. 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)

  14. 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?

  15. 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.

  16. The Structures and Routines of a Lesson MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning 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. 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

  17. Solve the Task(Private Think Time and Small Group Time) • Work privately on theMissing 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).

  18. 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.

  19. 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.

  20. 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.

  21. Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction?

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

  23. Language Context Table Graph Equation Five Different Representations of a Function Van De Walle, 2004, p. 440

  24. The CCSS for Mathematical ContentCCSS Conceptual Category – Number and Quantity Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

  25. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

  26. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

  27. The CCSS for Mathematical ContentCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

  28. Bridge to Practice #2: Time to Reflect on Our Learning 1. Using the Missing Function Task: a. Choose the Content Standards from pages 11-12 of the handout that this task addresses and find evidence to support them. • Choose the Practice Standards students will have the opportunity to use while solving this task and find evidence to support them. • Using the quotes on the next page, Write a few sentences to summarize what Tharp and Gallimore are saying about the learning process. • 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.

  29. Research Connection: Findings by Tharp and Gallimore Tharp & Gallimore, 1991 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.

  30. 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, x1, the point (x1, f(x1)+g(x1)) 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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