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**Study Group 2 – Algebra 2 Welcome Back!**

Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises. Our Bridge to Practice through TNCore Training is linking our classroom instruction to the CCSS!

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**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. Go over Bridge to Practice Part A form last Study Group over Module 1 Part A: Show the next two slides to review the content AND practice standards for the selected CRA tasks. Discuss the EVIDENCE that participants found for the standards they selected. Remind participants that it would help to have a copy of the math practice standards from Module 1 handy (they are on the next slide for reference). There is also an additional Powerpoint of the Math Practice Standards with visuals and clarifications called “CCSS Math Practices” as a reference, but there is not enough time in this session to go through each slide and discuss that Powerpoint. Teachers may want to keep it to refer to as they grow more familiar with the developing these practices in their students.

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**The CCSS for Mathematical Practice**

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. Use this slide as a quick reference to discuss the math practices relative to the two CRA problems. Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

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2. Writing a Polynomial 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 pn is a whole number exponent greater than or equal to 1. Consider the graph of the polynomial function below. 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. (SAY) In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard? (See content standards, slide 40.) What is it about the prompt for the identified item that will require students to use standards for mathematical practice? Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. The task introduces the reasoning of “Lisa” and asks the student to agree or disagree with Lisa’s reasoning. Model with mathematics. Students will write a function to model the situation. However, again, creating a model is actually also a content standard. Use appropriate tools strategically. Attend to precision. Students will attend to precision of language when they identify the zeros of the function and explain why (x - 6) is or is not a factor of the polynomial graphed. Look for and make use of structure. Uses the information from the graph to write a function in factored form to represent the curve. Look for and express regularity in repeated reasoning.

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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 (SAY) In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard? (See content standards, slide 41.) What is it about the prompt for the identified item that will require students to use standards for mathematical practice? Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. The student has to abstract values from the problem situation, perform calculations with them, and then interpret the results of the calculations in the context of the problem. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. When calculating the sum of the circumferences, students must calculate with precision. They will attend to precision of language when explaining the meaning of terms in their function in part a. Look for and make use of structure. In order to be successful on this task students will identify the relationship between circle number and diameter as an exponential decay function and use the structure of exponential functions to represent the relationship. Look for and express regularity in repeated reasoning.

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**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. Go over Bridge to Practice Part B from the last Study Group over Module 1 Lead a brief discussion over what kind of instructional tasks/lessons need to be used and what teaching and learning needs to look like to prepare students for these assessments.

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**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. Go over Bridge to Practice Part C from the last Study Group over Module 1 The remaining part of this session will be spent going through the following slides and analyzing the individual solutions each group member “brought to the table” for the instructional task.

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**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 Overview of the Module: Participants will consider what instruction that is aligned with the CCSS sounds like and looks like. We will engage in a lesson as adult learners. We will not pretend we are students or think about how students will respond. Instead we will engage in the lesson as adult learners. Our goal is to deepen our understanding of the standards and to make sense of the use of models when working with the concept. No Prior Knowledge Necessary. Materials: Slides with note pages Mathematics Common Core State Standards (CSSS) (the Standards for Mathematical Practice and the grade-level Standards for Mathematical Content) Participant handouts (including Notes and Bridge to Practice #2) Chart paper and markers Graph paper Rulers (optional)

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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. (SAY) The CCSS include standards that focus on understanding of mathematical concepts AND the development of skills. We will engage in the lesson with the goal of deepening our understanding of concepts related to the task.

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**What is the difference between the following types of tasks? **

Question to Consider… What is the difference between the following types of tasks? instructional task assessment task Ask participants if they have ever considered this question. Give them a minute to consider it and ask them to briefly share their thoughts.

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**Taken from TNCore’s FAQ Document:**

Allow participants to read about the two types of tasks. Discuss a summary of the differences. (ASK): What type of tasks are the CRAs we worked in our first study group? (Assessment tasks; however, they could be modified to be more open-ended to allow for more solutions paths and discussion) (ASK): What type of task is The Missing Function Task from the Bridge to Practice? (Instructional task)

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**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) Directions: Read the session goals.

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**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) (SAY) We will engage in a task, and then step out and reflect on our engagement in the task. We will consider how our learning was supported, and which standards we had opportunities to think about and use when figuring out the solution path. We will engage with the task for the sake of our thinking and learning about the mathematics. Facilitator Information: If participants start to describe how their students would do the task or how their students think about the mathematics, remind them that for now we are focusing on our thinking and understanding of the task, and the underlying mathematics. Our goal in this module is to deepen our understanding of the standards and to make sense of the use of the mathematical practices when working with the concept(s).

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**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? (SAY) Take a look at the focus cluster standards (on pages 11–12) in the handout. They are also on slides for reference, but you don’t need to select which standards this task addresses now because the Bridge to Practice will be over aligning the content standards with this particular task.

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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. x f(x) -2 -1 1 2 3 4 (SAY): You already solved this task privately and now we are going to look at HOW to facilitate an instructional task by analyzing our solutions in small group and then whole group discussion. Possible Pathways Assessing Questions Advancing Questions Can’t get started What can you tell about the shape of f(x) from the data in the table? Explain. What can you tell about the values of g(x)? What about at the same x-values as are in the table? Uses tables by adding two rows and determining output values of g(x) using the output values of f(x) and h(x). Why did you divide the values in the last column by the values in the f(x) column? What patterns do you see in the g(x) column? What do these patterns tell you about the function? Uses equations f(x)*g(x) = (x+2)(x-1) f(x) = x + 2 g(x) = x – 1 How did you determine the equation for the parabola? What is the relationship between the x-intercepts of the product function and the x-intercepts of each of the factor functions? Do you think this will always occur? Why or why not? Graphs f(x). Uses the graph, circling key points such as the intercepts and vertex and determines values of g(x) based on these points. Why did you circle these points on your graph? What do you know about g(x) from the points you identified?

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**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 This slide is a model of how the Structures and Routines of a lesson should unfold as teachers facilitate an Instructional Task. (SAY) This is how we will engage together. Structures and routines are patterned ways of working that help students know what to expect. When we engage in lessons, we first set up the task. This usually takes a few minutes. Be careful not to provide students with too much scaffolding so they can develop their own ideas about how to solve it. Then, you will have approximately 5 minutes of private time to solve the task independently. It is VERY important to give students this time prior to breaking up into groups so they can process the problem for themselves. (For the purpose of our training, we completed the private think time as our Bridge to Practice) Next, you will work in small groups for about 15 minutes. While you are working, I will circulate asking assessing and advancing questions. During this time, I will be looking for a variety of solution paths to have shared with the whole group. I may be asking you to write your method on chart paper. Finally, we will engage in a group discussion of the different solution paths and make connections between the paths to arrive at the essential understandings of the standards related to this task.

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**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). Directions: Read through the directions on the slide. Focus participants on the characteristics of the function. Remind them that the goal is not to determine the symbolic representation for the function. Instead the goal is to determine the characteristics of the function and to explain the reasoning.

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**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. Directions: Use the information on the slide to describe the Share, Discuss, and Analyze Phase of the lesson.

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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. x f(x) -2 -1 1 2 3 4 Facilitator Information: This slide is here for you to reference during the group discussion.

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**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. Directions: The following sequence of mathematical concepts and questions can be used to guide your facilitation of the Share, Discuss, and Analyze Phase. Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms Using the distributive property We knew that g(x) had to be linear, because the product is quadratic. So, we guessed g(x) = x first and multiplied by f(x) and then we kept guessing until we got g(x) = x – 1 How did you use the equations to find the product of f(x) and g(x)? How did you determine if their product matched the graph provided in the task? Does the equation representing the product give us any information about what the graph of the product will look like? Do we have to plug in points to know whether or not it matches the graph? Finding missing numeric factors We found the y value for g(x) by figuring out what would have to be multiplied with the f(x) y-value to get the product. We saw g(x) was linear. What about the data in the table indicates that g(x) is linear? How did this group come up with the equation? How does the graph of the product function compare to f(x) and g(x)? 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. Relating factors to zeros f(x) = x + 2 g(x) = x – 1 f(x)*g(x) = (x+2)(x-1) We started with a table and found the equation g(x) = x – 1 Then we wanted to check and make sure it was right so, we wrote (x+2)(x-1) and we saw that this equals 0 at x = -2 and 1 and that matches the graph in the problem. How do the zeros of the product relate to f(x) and g(x)? Do you think this will always happen? Explain. How does this group’s method relate to the first group’s method? Are they using equations in the same way? Are the equations y = x2 + x – 2 and y = (x+2)(x-1) equivalent? Do they give us the same information about the graph of the product function? Summary What do we know about the product of two linear functions? Explain. How can we tell from the two linear functions where the product will cross the x-axis? What can we tell about the factor functions by looking at a graph of the product function? Explain.

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**Reflecting on Our Learning**

What supported your learning? Which of the supports listed will EL students benefit from during instruction? Directions: Chart responses. Probing Facilitator Questions and Possible Responses: What supported your learning? Private think time Small group problem solving time The whole group discussion The design of the task—It was open and we were able to come up with our own ways of thinking about the problem. The Turn and Talk time—Because sometimes a question was hard to think about and I needed the time to think. Did the talk help anyone else? What about the talk? Did anyone else find it helpful to hear others repeat ideas and to add on to other’s ideas? Which of these supports would be helpful to English learners? All of the things that supported our learning are good for all learners. ELs will especially benefit from the structures built in that allow them to talk to their peers. Multiple representations are also helpful. Cognates would be helpful if any apply to the problem. Cognates are the use of English words that have a word in Spanish that sounds similar. ELs might also benefit from hearing the problem before other students hear it. Teaching the vocabulary words in advance of the problem or having picture available would also be helpful.

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**Linking to Research/Literature Connections between Representations**

Pictures Written Symbols Manipulative Models Real-world Situations Oral Language (Say) Many of you moved between representations when solving and discussing the solution paths to the task. Research has shown that some of the better problem solvers are those who, when they are struggling to figure out a problem, have the resources or know how to use different representations in order to solve a problem. If this is true, then what are the implications for instruction in our classrooms? Adapted from Lesh, Post, & Behr, 1987

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**Five Different Representations of a Function**

Language Table Context Graph Equation (SAY) This diagram varies slightly from the one shown on the earlier slide. This one includes equations and graphs, which are more appropriate for middle school and high school students. Both diagrams are included so that we can consider how students can benefit from looking at multiple representations of function relationships at all levels of mathematics learning. 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 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2 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 1.15t can be rewritten as (1.151/12)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. Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2 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. Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2 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. Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2 Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

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**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. 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. Bridge to Practice #2: Make sure participants have a copy of the Bridge to Practice #2 handout that slides explain.

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**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. Bridge to Practice #2: 2) Read the Tharp and Gallimore Quotes. Write a few sentences to summarize what Tharp and Gallimore are saying about the learning process. Tharp & Gallimore, 1991

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**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.) Bridge to Practice #2: 3) 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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Supporting Rigorous Mathematics Teaching and Learning

Supporting Rigorous Mathematics Teaching and Learning

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