1. Calculus AB APSI 2017 Day 1 Curriculum Framework Calculus AB and BC
Professional Development Workshop Handbook

2. 48 years of teaching grade 7-college AP Calculus since 1986
Jim Rahn
48 years of teaching
grade 7-college
AP Calculus since 1986
AP Reader – 7 years
AP Table Leader – 6 years
AP Workshops

3. BS in Ed – Taylor University MA – Ball State University
Jim Rahn
BS in Ed – Taylor University
MA – Ball State University
Presidential Award for Excellence in Teaching Mathematics
Triple Crown Award - AMTNJ

4. Who are you? Dawn Heil Jason Eicholtz Mary Brennan Christine Vollrath
Amy Ellsworth                         
Gary Yeager
Laura Mansfield         
Katherine Murphy

5. Welcome

6. Monday
Introductions Lesson 1: The Goals of AP Calculus 1 hr Lesson 2: Developing Student Understanding 1 hr Break Lesson 3: Understanding the Curriculum Framework 1 hr Lunch Afternoon (Part 1) Lesson 4: Understanding the Big Ideas 1 hr Lesson 5: Planning your Course 1 hr Five Day Jet Tour of Calculus

7. Sample FRQ (p. 42-45-Multi-day Workshop Handbook): 1,2
Monday Assignment AB
Multiple Choice Questions on the Calculus AB Practice Exam (p and in Multi-day Workshop Handbook) - Questions1-10 and 76-80
Free Response for AB
2017: 1, 2
Sample FRQ (p Multi-day Workshop Handbook): 1,2

8. Lesson 1 Goals of AP Calculus
1 hr
Lesson 1 Goals of AP Calculus
Key Takeaway 1
Incorporating mathematical practices provides opportunities for students to think and act like mathematicians.

9. Read p. 6 Read about the study Respond to the note at the bortrom
Keep in mind: How can I engage students by making the content relevant? How can I help students build deeper understanding conceptually?

10. Think and Act like a Mathematician
MPAC 1: Reasoning with definitions and theorems
MPAC 2: Connecting concepts
MPAC 3: Implementing algebraic/computational processes
MPAC 4: Connecting multiple representations
MPAC 5: Building notational fluency
MPAC 6: Communicating
Ubd’s goal is to transfer learning to new and novel situations then unpacks the knowledge and skills that will help achieve that goal.
Compare these practices with your responses to the last slide

11. 9 Enduring Understanding for Calculus AB
EU 1.1: The concept of a limit can be used to understand the behavior of functions
EU 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determine using a variety of strategies
EU 2.2: A function’s derivative, which is itself a function, can be used to understand the behavior of the function
EU 2.3: The derivative has multiple interpretations and applications inducing those that involve instantaneous rates of change
EU 2.4: The Mean Value Theorem connects the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval
Do you think your students, or most calculus students know how to do these things intuitively?
Students struggle with these ideas- mathematicians grasp these and take them for granted.
Unless students unpack these themselves they are incapable of transferring the skills to new situations.

12. 9 Enduring Understanding for Calculus AB
EU 3.1: Anti-differentiation is the inverse process of differentiation
EU 3.2: The definite integral of a function over an interval is the limit of a Riemann sum over that interval and can be calculated using a variety of strategies
EU 3.3: The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration
EU 3.4: The definite integral of a function over an interval is a mathematical tool with many interpretations and applications involving accumulation
EU 3.5: Antidifferentiations is an underlying concept involved in solving separable differential equations. Solving separable differentiation equations involves determine a function or relation given its rate of change.
These are the long-term takeaways that a student should have after exploring the content and skills
Complete your observations on page 8.

13. #1 Multiday Workshop Handbook (p. 9)
How does this item require students to apply one or more of the Mathematical Practices?
How does this item require students to demonstrate one or more of the Enduring Understandings?
These problems represent a range of skills students have to develop.
In this problem students are connecting limits, continuity, and differentiability (MPAC2).

14. #2 Multiday Workshop Handbook (p. 10)
How does this item require students to apply one or more of the Mathematical Practices?
How does this item require students to demonstrate one or more of the Enduring Understandings?
MPAC1 AND MPAC2
eu 3.3

15. #3 AP Course and Exam Description (p. 51)
How does this item require students to apply one or more of the Mathematical Practices?
How does this item require students to demonstrate one or more of the Enduring Understandings?
MPAC 2, MPAC6, AND MPAC4
EU2.2

16. Discussion p. 12 Workshop Handbook
Based on the sample questions you just reviewed, what is the relationship between content and skills in the AP Calculus course? Can a student answer these questions without having conceptual understanding, knowledge, and skills? Explain.
Understanding content knowledge and skills are strongly tied within the curriculum framework. Students who only apply rate memorization or a “plug and chug” method will not be as successful as students who can make connections between concepts and apply mathematical practices in a variety of context.

17. Discussion p. 12 Workshop Handbook
What impact will this have on how you plan and design your instruction? Explain.
Lessons will have to focus on building conceptual understanding grounded in both content and skill.

18. Read the Equity and Access Policy on page 13
Read the Equity and Access Policy on page 13. Reflect on how the skills and practices allow all student to engage with the content in meaningful ways, and what implications this could have on your planning.
Mark the text for key principles

19. College Board's Equity and Access Policy Statement
The College Board strongly encourages educators to make equitable access a guiding principle for their AP programs by giving all willing and academically prepared students the opportunity to participate in AP.
We encourage the elimination of barriers that restrict access to AP for students from ethnic, racial, and socioeconomic groups that have been traditionally underserved.
Schools should make every effort to ensure their AP classes reflect the diversity of their student population.
The College Board also believes that all students should have access to academically challenging coursework before they enroll in AP classes, which can prepare them for AP success.
It is only through a commitment to equitable preparation and access can true equity and excellence be achieved.

20. College Board's Equity and Access Policy Statement
The College Board strongly encourages educators to make equitable access a guiding principle for their AP programs by giving all willing and academically prepared students the opportunity to participate in AP.
We encourage the elimination of barriers that restrict access to AP for students from ethnic, racial, and socioeconomic groups that have been traditionally underserved.
Schools should make every effort to ensure their AP classes reflect the diversity of their student population.
The College Board also believes that all students should have access to academically challenging coursework before they enroll in AP classes, which can prepare them for AP success.
It is only through a commitment to equitable preparation and access can true equity and excellence be achieved.

21. Reflect p. 14 Workshop Handbook
How do the mathematical practices and enduring understandings help students in AP Calculus learn to think and act like a mathematician? How should the MPAC’s and how might the EU’s influence your teaching?

22. Self Assessment
Complete the self assessment on pages 15 and 16 to rate your own comfort level/proficiency with teaching each MPAC.

23. Lesson 2 Developing Student Understanding
1 hr
Lesson 2 Developing Student Understanding
Key Takeaway 2
Students must demonstrate a deep understanding of the concepts, in addition to a solid knowledge base and proficiency in the mathematical practices, to receive qualifying scores on the AP exam.

24. 2008 FRQ 4 (p. 17) Solve this problem.
Identify what understanding is needed to answer this problem.

25. List the understandings you would expect a student at each proficiency level to demonstrate
Little Conceptual Understanding
(responses on exams earning a 1 or a 2)
Some Conceptual Understanding (responses on exams earning a 3)
Deep Conceptual Understanding (responses on exams earning a 4 or a 5)
Work in groups to complete the chart.

26. Analyzing Student Responses (p. 19)
Solve this problem in your group. Come to an agreement about which content understandings they expect to be demonstrated in a student response based on the stem of the problem.

27. Read Sample Response A and B
Discuss with your partners the student’s understanding and the evidence they provide.
Place the three responses on a continuum from least understanding to greatest understanding.

28. Sample Student Response A

29. Sample Student Response B

30. Read Sample C
Think about the understandings demonstrated in the response and identify where on the understanding continuum they believe the sample should be.
Contrast with samples A and B

31. Sample Student Response C
Think about the understandings demonstrated in the response and identify where on the understanding continuum they believe the sample should be. Contrast with samples A and B.
Respond to the two questions on page 22.

32. Interventions to Guide Understanding
Select one problem on pages 24-25
Fill in the chart on page 23 with a sample student response you would think demonstrates each level.
Describe the misunderstanding and the intervention or strategy that could be used to address the gap in understanding. Refer to the list of Representative Instructional Strategies on pages (Course and Exam Description)
Understanding demonstrated
Misunderstandings
Intervention or strategy
Low
Medium
High

33. 2013 FRQ #3

34. 2013 FRQ #4

35. 2013 FRQ #5

36. Summary Questions
Page 26
Reflect
Page 27

37. Lesson 3 Understanding the Curriculum Framework
Key Takeaway 3
The UBD model of the updated AP calculus curriculum framework supports planning that helps to build students conceptual understanding.

38. EU 2.4
The Mean Value Theorem connects the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval.
What would students need to learn about and practice in order to walk away with that understanding?
Connect EK statements with their corresponding LO and the LO with the EU. p. 16 in framework.

39. Sorting Activity
You will receive an assortment of Enduring Understandings and a mixture of Learning Objectives and Essential Knowledge statements to identify what students must “do” versus what they must “get” in order to achieve conceptual understanding.
Work in your teams to group the cards based on common characteristics you decide upon as a group. Then create a label for each group of cards.
In the next activity participants will be exploring the relationship between what students must “do” and what students must “get”.

40. Each card is a statement taken from the concept outline.
Turn to pages 7-23 of the AP Calculus Course and Exam Description
Review the concept outline and record your observations on page 30.

41. Share your observations about the concept outline. What are EK’s?
What are LO’s?
What are EU’s?
How are the Learning Objectives related to the Enduring Understandings?
Ek’s: something a student must know in order to master the learning objectives and to recall in order to demonstrate mastery of each learning objective
Lo’s: what a student needs to do to apply the enduring understanding to a new context
EU’s: a statement that summarizes important ideas and core processes central to calculus

42. Enduring Understanding
Regroup the cards into EU’s, LO’s, and EK’s.
Essential Knowledge
Learning Objectives
Enduring Understanding

45. Advantages of Including Disadvantages of Including
Page 33
Exclusion Statements
Advantages of Including
Disadvantages of Including
EK 1.1A1

46. What is the difference between a curriculum and a framework?
How do the Enduring Understandings ensure focus on long-term learning and transferring that learning to new situations beyond the score of the course?
What do the verbs in the learning objectives ask students to do?
How can you see the Mathematical Practices being applied in different contexts throughout the course?

47. Complete the Reflection Question on page 34

48. Lesson 4 Understanding the Big Ideas
1 hr
Lesson 4 Understanding the Big Ideas
Key Takeaway 4
Revisiting concepts in multiple contexts helps students make stronger mathematical connections and deepen their understanding of the concepts.

49. Open your Course and Exam Description to the Curriculum Framework beginning on page 7
Use sticky notes to tag all the places where limits are referenced besides Big Idea 1.
Generate a list on page 35 of all the places that you tagged.

50. Reccuring Concepts: Limits
While the concept of limits is introduced in Big Idea 1, it is a concept that I revisited at multiple points in each of the Big Ideas.
It is explicitly supported by two Mathematical Practices

51. Recurring Concept Graphic Organizer p 38-41
You will work in your groups to create a similar graphic organizer to illustrate the recurrences of other concepts in this course.
Possible concepts or themes are Theorems, continuity, differentiation, rate of change

52. Reflect page 43
How can a focus on Big Ideas help to develop students’ conceptual understanding?
Knowing that Big Ideas must be revisited at multiple points throughout the course, how will this impact the way you plan? What challenges or opportunities might this present?
page 42

53. Lesson 5 Planning Your Course
1 hr
Lesson 5 Planning Your Course
Key Takeaway 5
Setting up a full year course plan helps to highlight focus areas and to mitigate instructional challenges.

54. After reviewing the curriculum framework, what factors should you consider when planning your course?
learning objectives, mathematical practices, the school year calendar, date of the exam, course audit requirements, students background, state standards, local requirements, challenging content, resources available, what to do with the time after the exam.

55. Course Approaches (p 25)
Turn to AP Calculus Course and Exam Description and review the Organizing the Course Section
page 25

56. Course Planning and Pacing Guides (CPPG’s)
Sample CPPG
Which instructional approach is being used?
Observations
Lindsey Bibler
Section 5 of your notebook
p. 47

57. High-Level Course Pacing Overview
Unit
Hours/Days of Instruction
Notes
Page 48

58. Course Calendar

59. Instantaneous Rate of Change of a Function
A homemade rocket is fired initially from a platform 400 feet above ground at a velocity of 300 ft/sec. After 20 seconds, the rocket hits the ground.
Day 1

60. Behavior of Functions
Make a sketch of each function in the given window. At x = 1 draw a tangent line that approximates the steepness of the function at x = 1. Approximate the slope of the tangent line. Describe how this tangent line describes the behavior of the graph at x = 1.
Day 2

61. What Can Area Represent?
As you pull out on the highway on your road bike you gradually increase your speed according the graph at the right. Then you notice your speedometer approaching 465 ft per minute so you tap hand brake to slow
down your speed to a constant rate of 465 feet per minute.
Day 3

62. Determining a Definite Integral with Formulas
Water is being pumped into a large storage tank at a rate, R(t) = (x - 2)3 +12 thousands of gallons/day.
Draw a sketch of R(t) in Figure 1 for time 0 ≤t ≤ 4 days .
The definite integral of R(t)
from t = 0 to t = 4 represents the thousands of gallons pumped into the tank during the four days.
Day 4

63. Composition of Tests
Section 1 A - 30 questions – No Calculator - 60 minutes Section 1B – 15 questions – Calculator required - 45 minutes Section 2A -2 Free Response questions – Calculator required – 30 minutes Section 2B – 4 Free Response questions – no Calculator – 60 minutes

64. Weight of Questions Section 1: 45 Questions
Section Free Response Questions
Section 1:
45 Questions
1.2 points each or 54 points
Section 2:
6 questions
9 points available each or 54 points
50% of the score –
9 points each – 54 points

65. What’s in the Notebook? Section 1: Introductory Handout
Section 2: Student Activities
Section 3: Slope Fields
Section 4: Miscellaneous Materials
Section 5: Course Planning and Pacing Guides (CPPG)
Section 6: Audit/Self-Evaluation Checklist; Syllabus Development Guide
Section 7: Free Response Questions
Section 8: WinPlot Manual

66. What’s on the Dropbox Folder
2017 Handouts
Calculus Free Response Questions
Calculus Multiple Choice Questions
TI-Programs
Other added files

67. Mark Howell’s FRQ Index

68. Introduction to Limits
Consider the graph of the function .
Make a sketch of this function.
How does the graph differ from what you expected to see?
What does your graph indicate about the value of f(2)?
Why is this?
Algebraically calculate f(2). Why is there no value for f(2)?
Day 5

69. Key Limits that are helpful to know

70. Sample FRQ (p. 42-45-Multi-day Workshop Handbook): 1,2
Monday Assignment AB
Multiple Choice Questions on the Calculus AB Practice Exam (p and in Multi-day Workshop Handbook) - Questions1-10 and 76-80
Free Response for AB
2017: 1, 2
Sample FRQ (p Multi-day Workshop Handbook): 1,2