Calculus AB APSI 2017 Day 2 Professional Development Workshop Handbook

Calculus AB APSI 2017 Day 2 Professional Development Workshop Handbook
Curriculum Framework Calculus AB and BC

Tuesday Lesson 6: Understanding the Mathematical Practices – 2 hrs
Break Lesson 7: Aligning Mathematical Practices to Learning Objectives – 30 min Lesson 8: Unpacking the Mathematical Practices – 30 min Lesson 9: Scaffolding the Mathematical Practices – 30 min Lunch Lesson 10: Sequencing the Mathematical Practices – 30 min Lesson 11: Communicating in Mathematics – 1 hr Discussion of Homework Understanding the Derivative Numerically Using Difference Quotients Understanding the Derivative Graphically using Difference Quotients

Tuesday Assignment - AB
Multiple Choice Questions on the Calculus AB Practice Exam (p and in Multi-day Workshop Handbook) Questions and 81-85 Free Response: 2017: AB3, AB4 Sample FRQ (p Multi-day Workshop Handbook): 3,4

Lesson 6 Understanding the Mathematical Practices
Key Takeaway 6 Building student proficiency in Mathematical Practices requires a deep understanding of what each practice means, how it will be assessed, and which teaching strategies can be used to target the development of those skills.

Do you have any specific questions about the MPACs?
The Mathematical Practices capture important aspects of the work that mathematicians engage in, at the level of competence expected of AP Calculus students. Embedding these practices in an AP course enables students to establish mathematical lines of reasoning and use them to apply mathematical concepts and tools to solve problems. Review the MPAC categories and their corresponding subskills on page 8-10 of the Course and Exam Description. Do you have any specific questions about the MPACs? This lesson will serve as an introduction to each of the six MPAC by illustrating how they are assessed on the AP exam, and modeling a teaching strategy that can be used to target a challenging subskill with in each practice.

MPAC 1: Reasoning with Definitions and Theorems (p. 65)
Read MPAC 1 Examine each of sample AP Exam items that target this particular practice and respond to the questions that follow. (pages 65-66) Spend some time discussing participant’s observations.

#19 Sample Exam Questions (p. 65)
Students must be able to recognize and apply the Mean Value Theorem to a function, as well as apply the definition of increasing to the given information. p 65 How does this question require the students to demonstrate that they are proficient in one or more of the subskills in MPAC 1?

Sample Questions #2 How does this question require the students to demonstrate that they are proficient in one or more of the subskills in MPAC 1? Students must be able to recognize the need for and correctly apply the Intermediate Value Theorem. p 66

A challenging subskill MPAC 1b
b. confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem; One instructional strategy that can be used to address the trouble students have with this subskill is to critique reasoning: Through collaborative discussion, students respond to the arguments of others and question the use of mathematical terminology, assumptions, and conjectures to improve understanding and justify and communicate conclusions. How can this be accomplished in your classroom?

Practice Work with your group to determine where the strategy of critiquing reasoning could be applied elsewhere in the curriculum to help develop students’ proficiency with MPAC 1b. Cite the relevant LO Briefly describe how students would practice MPAC 1b in the context of that LO p 67

MPAC 2: Connecting concepts (p. 68)

#5 Sample Exam Questions (p. 51)
Students must connect the concept of local linearity with the derivative of a function How does this question require the students to demonstrate that they are proficient in one or more of the subskills in MPAC 2?

#5 AP Course and Exam Description (p. 51)
Students must be able to connect a concept (increasing/decreasing behavior) to its visual representation. How does this question require the students to demonstrate that they are proficient in one or more of the subskills in MPAC 2?

A challenging subskill : MPAC 2d
Students often have trouble finding an underlying problem structure. This impacts their ability to demonstrate proficiency in MPAC 2d: Identifying a common underlying structure in problems involving different conceptual situations. One instructional strategy that can be used to address this is marking the text. Students highlight, underline, and/or annotate text to focus on key information to help understand the text or solve the problem. This subskill is discussed more on page 35 of the Course and Exam Description and can be applied in other parts of the course.

Practice MPAC 2d: Identifying a common underlying structure in problems involving different conceptual situations. Look in Section 4 for Optimization Problems (APSI 2). Mark up the text in the problems by circling certain key words such as “minimize” and underlining commonalities between the problems. Work in your groups to determine where the strategy of marking the text could be applied in the curriculum to help develop students’ proficiency with MPAC 2d. Cite the relevant Learning Objective, then briefly describe how students would practice MPAC 2d in the context of that LO.

MPAC 3: Implementing algebraic/computational processes (p. 71)
Read MPAC 3 and make some observations about this MPAC Examine each of the sample AP Exam items and answer the question that follows.

How does this question require students to demonstrate that they are proficient in one or more subskills listed for MPAC 3?

Course and Exam Description FRQ 2

A Challenging subskill MPAC 3b
Turn to page 72 What is meant by Error Analysis (2nd Paragraph)? Think about Error Analysis and its application in the course. (See p. 34 in Course and Exam Description) Turn to the activity on page 73 Work in your group to determine where the strategy of Error Analysis could be applied elsewhere in the curriculum.

MPAC 4: Connecting multiple representations (p. 74)

Students must be able to calculate a limit using information provided in a graph.

Students must be able to calculate a derivative analytically using values and information provided about a function given only in graphical form. How does this question require students to demonstrate that they are proficient in one or more subskills listed for MPAC 4?

A challenging subskill : MPAC 4c
Students often have trouble associating information in one representation with a different representation. This impacts their ability to demonstrate proficiency in MPAC 4c: Identifying how mathematical characteristics of functions are related in different representations. One instructional strategy that can be used to address this is a graphic organizer: Students arrange information into charts and diagrams. This helps to build comprehension and facilitates discussion by representing information in a visual form. For example, providing a diagram that has arrows connecting values in a table to their locations on a graph can help students better understand how a particular concept (i.e. the “zeros” of a function) are represented in two different ways. This subskill is discussed more on page 35 of the Course and Exam Description and can be applied in other parts of the course.

Riemann Sums – Graphic Organizer Section 2 (APSI – 174)
Using a Graphic Organizer for Riemann Sums

Slope along Inverse Functions

Practice MPAC 4c: identify how mathematical characteristics of functions are related in different representations Work in your groups to determine where the strategy of a graphic organizer could be applied in the curriculum to help develop students’ proficiency with MPAC 4c. Cite the relevant Learning Objective, then briefly describe how students would practice MPAC 4c in the context of that LO.

Related Rates – Graphic Organizer – Section 2 (APSI -171)

MPAC 5: Building notational fluency (p. 77)
Read MPAC 5 Examine each of sample AP Exam items that target this particular practice and respond to the questions that follow. (pages 79-80)

#8 Course and Exam Description (p. 54)
Students must be able to associate the Riemann sum notation for an integral with the standard notation for a definite integral (e.g. the integrand and the limits of the integral. How does this question require the students to demonstrate that they are proficient in one or more of the subskills in MPAC 5?

FRQ #2 Curriculum Framework (p. 78)
Students must be able to take a limit that includes a named function, given only in graphical form. How does part c of this question require students to demonstrate that they are proficient in one or more subskills listed for MPAC 5?

Practice MPAC 5d: assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts. Work in your groups to determine where the strategy of a notation read aloud could be applied in the curriculum to help develop students’ proficiency with MPAC 5d. Cite the relevant Learning Objective, then briefly describe how students would practice MPAC 5d in the context of that LO.

A challenging subskill : MPAC 5d
Students often have trouble assigning meaning to the symbols in calculus problems. This impacts their ability to demonstrate proficiency in MPAC 5d: assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts. One instructional strategy that can be used to address this is a notation read aloud: Students read symbols and notational representations out loud. This helps to ensure that students are accurately interpreting symbolic representations, and actively engaging with each part of the notational expression (rather than just skipping over it.) This subskill is discussed more on page 35 of the Course and Exam Description and can be applied in other parts of the course.

MPAC 6: Communicating (p. 80)
Read MPAC 6 Examine the sample AP Exam item that target this particular practice and respond to the questions that follow. (page 80)

#8 Course and Exam Description (p. 54)
Students must be able to explain how the notation for a definite integral relates to information given in the context of the problem. How does part c of this question require students to demonstrate that they are proficient in one or more subskills listed for MPAC 6?

A challenging subskill : MPAC 6c
Students often have trouble explaining the meaning of the results after they’ve solved a problem. This impacts their ability to demonstrate proficiency in MPAC 6c: explain the meaning of expressions, notation, and results in terms of a context (including units) One instructional strategy that can be used to address this is working backwards. This allows student to not only check their solutions for accuracy, but to ensure that their explanation of the results is both reasonable for and consistent with the original context of the problem. This subskill is discussed more on page 35 of the Course and Exam Description and can be applied in other parts of the course.

Practice MPAC 6c: explain the meaning of expressions, notation, and results in terms of a context (including units) Work in your groups to determine where the strategy of working backwards could be applied in the curriculum to help develop students’ proficiency with MPAC 6c. Cite the relevant Learning Objective, then briefly describe how students would practice MPAC 6c in the context of that LO.

Reflect on lesson 6 on page 84

Lesson 7 Aligning Mathematical Practices to Learning Objectives: Focus on MPAC 1
Key Takeaway 7 Teachers have choice and flexibility in determining when to embed skills-based instruction into their courses, and those choices will drive their instructional planning.

The MPAC’s articulate the behaviors in which students need to engage in order to achieve conceptual understanding in the AP Calculus course. Notice how the MPAC’s are not embedded into the Concept Outline (pp ) of the Curriculum Framework. Why do you think this is the case?

Uncovering the MPAC’s Solve the next problem on your own. Show your work. Compare your answers with others in your group. Make a list of all the steps in your process when you worked out our solution – Be specific. Explicitly consider what questions you asked yourself in your head what key words you made note of in the problem what you typed into your calculator, etc. Consider which of the six MPAC’s students need to have practiced in order to answer this question. Reinforce the importance that both they and their students are able to assign meaning to the notation being used and that they understand the connection between definite integrals and the limit of a Riemann Sum- washers, discs, volumes with known cross-sections, area for which those understandings are foundational.

#16 Curriculum Framework (p. 84)
What MPAC’s must students have practiced in order to answer this question?

How Does This Make Sense The same LO 1.1B and a different MPAC 1B?
Students can confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem. MPAC 1B In order to determine the limit of a function, students must demonstrate awareness of the conditions under which a limit exists.

Use the diagram to illustrate how aligning a particular LO with a subskill from MPAC 1 can help build towards one of the EU’s. Note that the subskills for MPAC 1 are listed on page 8 of the Course and Exam Description. Brainstorm a list of instructional strategies or approaches that could be used to incorporate that MPAC skill when teaching that particular LO. How would these strategies help build students’ conceptual understanding of the EU?

Reflect In what ways will the choice and flexibility you have to determining when to embed skills-based instruction impact your instructional planning? Revisit your planning schedule from Day 1- What are some things you would rearrange or plan differently now that you have explored ways to teach MPAC’s in alignment with LO’s?

Lesson 8 Unpacking the Mathematical Practices: Focus on MPAC’s 3 and 4
Key Takeaway 8 Students must be explicitly taught to think and act like a mathematician, not just perform complex mathematical problems.

In your Groups (p. 91) For your assigned problem:
describe the process a student could use to answer the question describe how students must demonstrate proficiency with MPAC 1 and 3 in order to answer the problem given. MPAC 3 Have participants respond in books and then briefly discuss in their groups. Verify that

In your Groups (p. 91) For your assigned problem:
describe the process a student could use to answer the question describe how students must demonstrate proficiency with MPAC 4 in order to answer the problem given. MPAC 4 Have participants respond in books and then briefly discuss in their groups. Suppose f is an odd function. Does ?

The Role of MPAC’s – Thinking and acting like a mathematician (p. 92)
Consider Problem B. How could you help guide a student to “think like a mathematician” by “making connections across multiple representations”? What question prompts could you provide that would guide the student’s understanding?

MPAC 3: Implementing algebraic/computational processes
Students can: select appropriate mathematical strategies; sequence algebraic/computational procedures logically; complete algebraic/computational processes correctly; apply technology strategically to solve problems; attend to precision graphically, numerically, analytically, and verbally and specify units of measure; and connect the results of algebraic/computational processes to the question asked. Note each MPAC contains a list of subskills that are independent from one another. How do these subskills work together to help students build proficiency within MPAC 3?

MPAC 3a: select appropriate mathematical strategies;
What are some things students must be able to do in order to demonstrate proficiency in this subskill across different contexts? decide an appropriate method for anti-differentiation; know when to use a calculator to help find a solution

MPAC 3 MPAC 3 Implementing algebraic/computational processes
3e: attend to precision graphically, numerically, analytically, and verbally and specify units of measure. What are some things students must be able to do in order to demonstrate proficiency in this subskill across different contexts? 3f: connect the results of algebraic/computational processes to the question asked. decide an appropriate method for anti-differentiation; know when to use a calculator to help find a solution

MPAC 4 MPAC 4 Connecting Multiple Representations
4e: construct one representational form from another (e.g., a table from a graph or a graph from given information). What are some things students must be able to do in order to demonstrate proficiency in this subskill across different contexts? MPAC 4 Connecting Multiple Representations 4f: consider multiple representations of a function to select or construct a useful representation for solving a problem. decide an appropriate method for anti-differentiation; know when to use a calculator to help find a solution

Unpacking the subskills for students
Students must be taught how to unpack a subskill and how to recognize when it would apply to a particular topic. To help students learn how to “unpack” MPAC 3b- sequence algebraic/computational procedures logically – they could be provided with a problem and a series of steps to solve that problem. Students determine an appropriate sequence for the steps and explain their logic for selecting that sequence.

Solving Differential Equations through Sorting – Section 2 – p. 191

Design an instructional activity or demonstration that would help students “unpack” one of the subskills from MPAC 3 or MPAC 4. Remember there are several instructional strategies listed on pages of the Course and Exam Description which might be helpful to reference.

Obstacles What are some obstacles that keep you from taking this instructional approach more frequently with your students in your classroom? How can you overcome those obstacles? Revisit your planning calendar from Day 1 Select a calculus topic that has not already been discussed in this lesson. Brainstorm ways in which certain MPAC subskills might be used to introduce the topic, develop the topic, and/or practice the topic? What are some things you would rearrange or plan differently now that we’ve explored each of the MPACs in greater depth? How would the emphasis on certain skills change now that you know how those skills need to be demonstrated on the exam?

Lesson 9 Scaffolding the Mathematical Practices: Focus on MPAC’s 2 and 5
Key Takeaway 9 Thoughtful scaffolding of skills allows students to access a skill at an appropriate level of challenge and to build that skill progressively throughout the year.

#6 Curriculum Framework (p. 29)
Answer the following question without using a calculator: I and III are false. Remember f(0)=-4 Slope from the left is not 2.

Student Samples Sample A “Options I and II look like the definition of the derivative. The slope of (i.e. from the left) is clearly 2, and the slope of (i.e. the slope from the right is clearly 2. Option III is the notation for the derivative, but this is clearly a discontinuous function, so III cannot be true. Thus the correct answer is C. Sample B “Options I and II look like the definition of the derivative, but they’re not really. In Option II the slope of (i.e. the slope from the right) is 2 because f(3) is defined here, but in Option I, So only II is true, and the correct answer is B.” Sample C Options I and II look like the definition of the derivative. The slope of (i.e. from the left) is clearly 2, and the slope of (i.e. the slope from the right) is clearly 2. Option III is the notation for the derivative, which is 2, so the correct answer is D. Student A: Intermediate Student B: advanced Student C: beginner proficiency

Levels of Proficiency within the MPACs
Each MPAC has its own inherent spectrum of proficiency, ranging from low levels of mastery to high levels of mastery in different contexts. What are some examples of student tasks that would demonstrate varying levels of proficiency within MPAC 6c? MPAC 6c: explain the meaning of expressions, notation, and results in terms of a context (including units) Beginner Proficiency Examples Intermediate Proficiency Examples Advanced Proficiency Examples Beginner: matching expressions to given meaning; selecting from a list which units would be appropriate for the given problem Intermediate: explaining the error in a given explanation, filling in blanks in a verbal explanation Advanced: writing complete explanations including units in context, explaining the meaning of expression in their own words The flexible nature of the MPACs allows teachers to revisit, scaffold, and reinforce skills in multiple situations

Practice Label each of the student tasks below to designate what proficiency level a student might be demonstrating for the given MPAC. What types of scaffolds or supports are embedded in the task to help build to the next level of understanding? Knowing what the next understanding is for the student is a critical piece that should inform teachers’ determination of what would make a task more or less challenging.

Scaffolding the MPACs The sequential nature of the LO provides multiple opportunities to apply mathematical practices in various contexts and scaffold the development of students’ critical thinking, reasoning and problem solving skills throughout the course. LO 2.3A Interpret the meaning of a derivative within a problem. LO 2.3 C Solve problems involving related rates, optimization LO3.4 A Apply definite integrals to problems involving movtion

MPAC 5: Building notational fluency Students can:
5a) know and use a variety of notations; 5b) connect notation to definitions (e.g., relating the notation for the definite integral to that of the limit of a Riemann sum); 5c) connect notation to different representations (graphical, numerical, analytical, and verbal); and 5d) assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts. assign an MPAC subskill to each group

In the chart below, provide examples of activities that could be used to scaffold and support students’ development of that subskill across three different LO in the course. High scaffolding means a lot of support Low scaffolding mean very little support Try to select Los across different Big Ideas

Reflect How can you use thoughtful scaffolding to help your students access a skill at an appropriate level of challenge and build that skill progressively throughout the year? Revisit your planning calendar from Day what are some things you would rearrange or plan differently now that you have explored strategies for scaffolding the MPACs?

Lesson 10: Sequencing the Mathematical Practices
Key Takeaway 10 Teachers must thoughtfully sequence the explicit instruction of skills in order to account for the inherent levels of challenge within and interdependencies among the Mathematical Practices, and to help students apply those skills in different contexts.

MPAC 2b: Connecting concepts
Students can use the connection between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, antidifferentiation) to solve problems MPAC 6e: Communicating Students can analyze, evaluate, and compare the reasoning of others. Write MPAC 2b on three stickies and MPAC 6e on three stickies. Then revisit your planning calendar from Day 1. For each of these two MPAC subskills, use the stickies to tag three places in your calendar where it would be relevant to practice that skill.

Interdependence among MPACs
These two subskills have dependence on other skills and thus cannot necessarily be taught without addressing those other skills first. MPAC 2b: Connecting concepts Students can use the connection between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, antidifferentiation) to solve problems This refers to students’ ability to connect concepts or processes, so the students need to have learned about multiple concepts or processes before being able to apply this skill. MPAC 6e: Communicating Students can analyze, evaluate, and compare the reasoning of others. This skill refers to students’ ability to analyze, evaluate, and compare reasoning so the students will need to have learned how to reason with MPAC 1 before learning how to critique the reasoning of others.

MPAC 6e is dependent upon MPAC’s above it.
Do you agree with the dependencies noted for these MPACs? If not, what would you change?

The least dependent MPACs (the ones at the top) are depended upon by other MPACs for student proficiency. Discuss which topic, unit, or lesson you could first introduce MPAC 1e and MPAC 2c? How would you spiral exposure and/or practice with the specific MPAC throughout the course?

2014 FRQ #3 (p. 109) Solve this problem in your notebook.
Explore sequencing the MPACs in a way that helps scaffold the skills needed to be successful on that problem.

Explore sequencing MPAC in a way that
helps scaffold the skills needed to be successful. Solve this problem in your notebook. Explore sequencing the MPACs in a way that helps scaffold the skills needed to be successful on that problem.

Step FRQ #3 (p. 110) ___2b use the connection between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, antidifferentiation) to solve problems; ___2c connect concepts to their visual representations with and without technology; ___3a select appropriate mathematical strategies ___3c complete algebraic/computational processes correctly ___4a associate tables, graphs, and symbolic representations of functions ___4d extract and interpret mathematical content from any presentation of a function (e.g., utilize information from a table of values); ___5a know and use a variety of notations; ___5b connect notation to definitions (e.g., relating the notation for the definite integral to that of the limit of a Riemann sum); ___5c connect notation to different representations (graphical, numerical, analytical, and verbal); ___5d assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts. ___6a clearly present methods, reasoning, justifications, and conclusions ___6b use accurate and precise language and notation; ___6d explain the connections among concepts These MPAC correspond to the skills and knowledge needed to be successful on 2014 FRQ 3 Which part of the question uses each MPAC? Write that subpart in the line next to the MPAC

Step FRQ 3 (p. 110) Create a tree diagram stressing the dependencies of the MPACs for frq #3 for 2014.

Spiraling the MPACs Some skills are so challenging, students need early and frequent exposure in order to build proficiency in that skill by the end of the year. MPAC 5d: Students can assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts. In the table below are three different contexts (LO) to which this subskill could be applied. Learning Objective Application of MPAC 5d LO 1.1A(b): Interpret limits expressed symbolically Students learn to “read” the symbols in a limit expression as representing something: e.g. reading the symbol → to mean approaches. LO2.3A: Interpret the meaning of a derivative within a problem. Students learn to interpret that the notation f’(x) means the “instantaneous rate of change” of a particular quantity with respect to an independent variable LO3.4B: Apply definite integrals to problems involving motion Students learn to interpret the expression as the average value of a function f over an interval [a,b]. Walk through the example. Assigning meaning to notation is a challenging skill for many students, so introducing that skill early with limits will help build that skill from the very first unit. Students must be able to apply these skills across multiple contexts. Knowing where in the course these skills will be applicable later in the course can help participants spiral back to those skills that were introduced early in the course.

Application of subskill
Identify two other subskills you believe need to be revisited with your students at multiple points throughout the course. Then examine the curriculum framework to determine where and how those subskills could be revisited at different point in the year. MPAC___ Learning Objective Application of subskill MPAC___ Learning Objective Application of subskill

Reflect How has this lesson helped you think differently about sequencing MPACs in order to account for inherent levels of challenge within and interdependencies between them, and the support your students need in order to apply those skills in different contexts? Revisit your planning calendar from Day 1: How will you ensure challenging skills are introduced early and revisited often?

Lesson 11: Communicating in Mathematics: Focus on MPAC 6
Key Takeaway 11 A student who truly understands a mathematical concept should be able to effectively communicate reasoning and justification. Therefore, students need to be explicitly taught how to communicate about their solution as much as they need to be taught how to find the solution.

p. 114 What are some of the key characteristics you might find in a sound mathematical argument? How does someone justify something in normal conversations versus in a mathematical context? Using definitions/theorems; using a proof or sequence of logical steps; drawing conclusions based on theorems, explicitly restating assumptions or given information More precise with descriptions, use definitions/theorem, use descriptors such as “for every’ or “there exists”; consider all possible cases; explicitly restate assumptions or given information.

Categories of justification in AP Calculus AB
There are many AP calculus context in which students may be expected to provide a reasoning or justification for their conclusions. Generate a list of the types of problems or topics for which a reasoning or justification might be required. - p. 115 function characteristics/relationships between f, f’ and f” and slopes of graphs particle motion candidate’s test IVT MVT Other Theorems Meaning of a derivative at a point in context Citing a definition is not the same as justifying their answer. Explanation must be “personalized to the specific problem. Meaning of definite integral and/or integrand This exercise could be repeated with students just prior to the exam

Using Model Questions 2015 FRQ 1 (p. 116)
Parts b and c Because of the phrases “give a reason for your answer” and “justify your answer.” Part b requires justification regarding the relationship between f and f’ or interpretation of a derivative at a point. Part C requires the Candidate’s test. Which of the parts require justification? How do you know? For each part that requires justification, what category or topic would each justification fall under?

Your group will be assigned several problems to study.
Group Activity p. 119 Your group will be assigned several problems to study. Is any justification reasoning, or explanation needed? How do you know? For each part that requires a justification, what category or topic would each justification fall under? Give the correct justification phrase (doing the problem is optional).

Group FRQ 3B (p. 119)

Group FRQ 4b and 4c (p. 120)

Group FRQ 5a, 5b, 5c (p. 120)

Group FRQ 1b (p. 121)

Group FRQ 3b (p. 122)

Group FRQ 4b (P. 122)

Group FRQ 5a and 5b (P. 123)

Group FRQ 1a and 1d (P. 123)

Group FRQ 2c and 2d (P. 123)

Group FRQ 3b and 3c (P. 124)

Group FRQ 4a, 4b and 4c (P. 124)

Group FRQ 4c (P. 125)

Group FRQ 5a and 5b (P. 125)

Comparing Student Responses 2015 FRQ #5 (P. 129)
Study the three sample student responses Determine if the student wrote a correct justification and explain why. Identify the main misconception and compose a brief statement that would help redirect the student’s thinking process.

Sample Responses (P ) Student Response A: “It has a point of inflection at 1 because at that point it equals zero but it never passes the x-axis.” Student Response B: “The graph of f has points of inflection at x = -1, x=1, and x = 3 because this is where f’ has a slope of 0.” Student Response C: “The x-coordinate of the points of inflection for the graph of f are x=-1, x=1, and x=3. This is because at these points f”(x)=0. Is the justification fully correct? Why or why not? incorrect because the student used “it” instead of specifically naming f’ in their justification. The student is not clear about what feature of f’ indicates a point of inflection on f and does not identify all of the points. The student does not understand that points of inflection are determined by setting f”=0 not f’=0 and then determining whether and how the second derivative changes sign. This student does not make the connection that f” needs to change sign. It is not sufficient for f” to just equal 0. While setting f’=0 is one method of finding possible points of inflection, these points are not guaranteed to be points of inflection.

Comparing Student Responses 2015 FRQ #5 (P. 129)
Study the three sample student responses Determine if the student wrote a correct justification and explain why. Identify the main misconception and compose a brief statement that would help redirect the student’s thinking process.

Comparing Student Responses
Select either part A or B and answer the questions that follow based on the part you selected: How might a student who is not explicitly trained in answering this type of question attempt to justify their answer to part a or b? Write three correct justifications. Compare your incorrect justifications with those of a partner. What observations do you have? How do you teach your students the differences between necessary and sufficient criteria for a response? What instructional strategies can you use to help your students build overall proficiency in mathematical communication skills? A criterion might be necessary but not sufficient for example, f” must be 0 or undefined at a point of inflection, but this is its on its own does not guarantee a point of inflection.

Reflect How can the explicit teaching of mathematical communication skills help students build an understanding of (and demonstrate proficiency with) mathematical concepts. Revisit you planning calendar from day 1. Into which units can you specifically incorporate “justify your answer” instructions? Refer to the LOs listed in the Curriculum Framework on pages of the Course Exam and Description.

Tuesday Assignment - AB
Multiple Choice Questions on the Calculus AB Practice Exam (p and in Multi-day Workshop Handbook) Questions and 81-85 Free Response: 2017: AB3, AB4 Sample FRQ (p Multi-day Workshop Handbook): 3,4

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