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Calculus AB APSI 2015
Day 1
Professional Development
Workshop Handbook
Curriculum Framework
Calculus AB and BC
Professional Development
Integration, Problem Solving, and Multiple
RepresentationsCurriculum Module
8
• 46 years of teaching
• grade 7-college
• AP Calculus since 1986
• AP Reader – 7 years
• AP Table Leader – 6 years
• AP Workshops
Jim Rahn
9
Jim Rahn
• BS in Ed – Taylor University
• MA – Ball State University
• Presidential Award for
Excellence in Teaching
Mathematics
• Triple Crown Award - AMTNJ
10
John Christopher
David Wilson
Joseph Molitori
Miroslav Jencik
Gamil Naem
Andrew Siris
Marilyn Coyle
Theresa Shields
Ed Hudak
Jennifer Chase
Cory Fauver
Deborah Maher
Andrew Tufts
Keith Bosler
Komila Sehgal
Katherine Rimakis
Jennifer Shafer
Lauren Hansen
Trisha Seiler
Graham Livingston
Julie Vaccarelli
Margaret Meurer
Who are you?
Welcome
12
Monday
Break Afternoon (Part 2)
Free-Response Problem (2013 AB4)Decimal Answers Sample SyllabiExemplary TextsFour Parts of the TestFree Response/Multiple Choice IndicesIncluded MaterialsRole of Sign ChartsJet Tour of Calculus (Day 5)Activity #1: Concept Outline
Morning (Part 1)IntroductionsJet Tour of Calculus (Day 1, Day 2) Break
Morning (Part 2)Free-Response Problem (2013 AB2)Highlights of New AP Calculus Curriculum Framework (MPAC)AP Equity Policy StatementAudit: Curricular/Resource Requirements Graphing Calculator LunchAfternoon (Part 1)Jet Tour of Calculus (Day 3, Day 4)
13
► Multiple Choice Questions on the 2014 test: 2, 3, 5, 6, 7, 10, 13, 16, 17, 20, 24, 25, 78, 86, 87
► Free Response for AB Track
► 2015: AB1/BC1, AB6
► 2014: AB1
Monday Assignment AB
14
Instantaneous Rate of Change of a Function
Behavior of Functions
What Can Area Represent?
Determining a Definite Integral with Formulas
Introduction to Limits
A Jet Tour of Calculus
15
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 25 seconds, the rocket hits the ground.
Day 1
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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
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2013 AB2
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19
► Professional Development Workshop Materials from College Board – Needed Every Day
► Integration, Problem Solving, and Multiple Representations– Bring back with your laptop/ipad on Thursday
What’s from College Board?
20
► The review of Calculus AP resulted in only minor changes.
► The review resulted in a new curriculum framework that ties course content and mathematical practices to clearly stated learning objectives, giving teachers greater transparency into course and exam expectations. This replaces the topic outline, previously used.
Overview of Update
21
► What AP Calculus students should know and be able to do will be more clearly defined, in the
► Enduring Understandings Statements(EU),
► Learning Objectives Statements (LO), and
► Essential Knowledge statements (EK)
outlined in the new curriculum framework.
Overview of Update
Page 353
22
► The updated courses will place an increased emphasis on conceptual understanding through the Mathematical Practices for AP Calculus (MPAC’s):
► reasoning with definitions and theorems,
► connecting concepts, implementing algebraic/computational processes,
► connecting multiple representations,
► building notational fluency, and
► communicating.
► Each concept and learning objective can be linked to one or more of these MPACs.
Overview of Update
23
► Students can:
► use definitions and theorems to build arguments, to justify conclusions or answers, and to prove results;
► confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem;
► apply definitions and theorems in the process of solving a problem;
► interpret quantifiers in definitions and theorems (e.g., “for all,” “there exists”);
MPAC 1: Reasoning with definitions and theorems
Page 354
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► develop conjectures based on exploration with technology; and
► produce examples and counterexamples to clarify understanding of definitions, to investigate whether converses of theorems are true or false, or to test conjectures.
MPAC 1: Reasoning with definitions and theorems
25
► Students can:
► relate the concept of a limit to all aspects of calculus;
► 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;
► connect concepts to their visual representations with and without technology; and
► identify a common underlying structure in problems involving different contextual situations.
MPAC 2: Connecting concepts
Page 354
26
► 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.
MPAC 3: Implementing algebraic/computational processes
Page 355
27
► Students can:
► associate tables, graphs, and symbolic representations of functions;
► develop concepts using graphical, symbolical, or numerical representations with and without technology;
► identify how mathematical characteristics of functions are related in different representations;
► extract and interpret mathematical content from any presentation of a function (e.g., utilize information from a table of values);
MPAC 4: Connecting multiple representations
Page 355
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► construct one representational form from another (e.g., a table from a graph or a graph from given information); and
► consider multiple representations of a function to select or construct a useful representation for solving a problem.
MPAC 4: Connecting multiple representations
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► Students can:
► know and use a variety of notations;
► connect notation to definitions (e.g., relating the notation for the definite integral to that of the limit of a Riemann sum);
► connect notation to different representations (graphical, numerical, analytical, and verbal); and
► assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts.
MPAC 5: Building notational fluency
Page 355
30
► Students can:
► clearly present methods, reasoning, justifications, and conclusions;
► use accurate and precise language and notation;
► explain the meaning of expressions, notation, and results in terms of a context (including units);
► explain the connections among concepts;
► critically interpret and accurately report information provided by technology; and
► analyze, evaluate, and compare the reasoning of others.
MPAC 6: Communicating
Page 356
31
► No topics will be removed from the AP Calculus program, and the following topics will be added:
► L’Hospital’s Rule will be included in AP Calculus AB.
► The limit comparison test, absolute and conditional convergence, and the alternating series error bound will be added to AP Calculus BC.
What is being added to AB and BC?
Page 351
32
► The framework also contains a concept outline, which presents the subject matter of the updated courses in a table format and is organized around the following four components:
► big ideas
► enduring understandings
► learning objectives
► essential knowledge statements
The New Curriculum Framework
Page 353
33
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.
Equity & Access Policy Statement
Page 0
34
• Eliminate barriers that restrict access to AP for students from ethnic, racial, and socioeconomic groups that have been traditionally underserved.
• Make every effort to ensure their AP classes reflect the diversity of their student population.
• Provide all students with access to academically challenging coursework before they enroll in AP classes.
Equity & Access Policy Statement
Only through a commitment to equitable preparation and access can true equity and excellence be achieved.
Page 0
35
► The teacher has read the most recent AP Calculus AB Course Description.
► The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the Course Description .
► The course provides students with the opportunity to work with functions represented in a variety of ways -- graphically, numerically, analytically, and verbally – and emphasizes the connections among these representations.
► The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.
► The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.
Curricular Requirementsfor the Audit
36
2015-2016 Course Audit Calendar
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► The school ensures that each student has a college-level calculus textbook (supplemented when necessary to meet the curricular requirements) for individual use inside and outside of the classroom.
► The school ensures that each student has a graphing calculator for individual use inside and outside of the classroom, with all the required capabilities listed in the AP Calculus Course Description.
Resource Requirements for the Audit
38
► AP Calculus teachers with previously authorized courses will not be required to revise and resubmit their syllabi for the 2016-17 school year. More information about what current teachers will need to do to retain course authorization will be provided before March 2016, when the AP Course Audit begins for 2016-17.
► During the coming transition period, AP will support teachers and provide many online resources to help plan for 2016-17, including practice exam questions, syllabus development guides, sample syllabi, course planning and pacing guides, and more.
The New Curriculum Framework
39
1. Graph a function in an arbitrary viewing window (BC includes parametric and polar)
2. Find the zeros of a function numerically – solve equations
a) Equation solver
b) Intersection of two functions
c) Set an equation equal to zero and find zeros.
d) Tracing SHOULD NOT BE USED.
3. Find the numerical value of the derivative at a point.
4. Find the numerical value of a definite integral
Graphing Calculators
40
► Test questions will be asked that require the students to complete all four skills.
► P.20 Sample Question 15 (Skill 1, 4)
► P.21 Sample Question 19 (Skill 2, 3)
► On Free-Response questions students must show the set up and then use the calculator to perform the needed calculation (2011 AB 1)
► Programs, such as a Riemann Sum or Trapezoidal Rule Program, can be useful to help build understanding of a new concept, but will not help the student on the exam.
► Students should be encouraged to verify results of paper and pencil computations or other representations.
Graphing Calculators
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2011 AB1
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Scoring Rubric 2011 AB1
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Which Calculator Can You Use?
A teacher must contact the AP Program (609-771-7300 before 4/1/ to inquire whether a student can use a calculator not on the list.
44
► Graphing calculators are valuable tools for achieving multiple components of the Mathematical Practices for AP Calculus, including using technology to develop conjectures, connecting concepts to their visual representations, solving problems, and critically interpreting and accurately reporting information.
Calculator Uses
45
► Appropriate examples of graphing calculator use in AP Calculus include, but certainly are not limited to, zooming to reveal local linearity, constructing a table of values to conjecture a limit, developing a visual represent of Riemann sums approaching a definite integral, graphing Taylor polynomials to understand intervals of convergence for Taylor series, and drawing a slope field and investigating how the choice of initial condition affects the solution to a differential equation
Calculator Uses
46
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 slowdown your speed to a constant rate of 465 feet per minute.
Day 3
47
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
48
2013 AB 4
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Scoring Rubric – 2013 AB4
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► Decimal answers must be “correct to three places after the decimal point.”
► The answer may be rounded to three decimal places
► The answer may be truncated to three decimal places
► The answer may be left with more than three decimal places
► Ex. If the answer is π you may give the answer as
► π
► 3.1415926535898
► 3.142
► 3.141
► 3.1415999
► Give decimal answers only if you cannot avoid it.
► Don’t round too soon.
Decimals and Arithmetic
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► Radian mode.
► If a function is given, enter and save it in the calculator once.
► If an intermediate value will be used again, store it in memory and recall it when needed.
► Do not find intersections by tracing or integrals by finding the area under the graph on the graph screen.
Other Calculator Techniques
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► Available at http://www.collegeboard.com/apcourseaudit/courses/calculus.htm
Sample Syllabi
Syllabus 1
FDWK AB Syllabus
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► On AP Central
► On your CD
► Part of your Professional Development Workbook (p. 1-50)
► Will change significantly in 2016-2017 with MPAC,Enduring Understandings, Learning Objectives, and Essential Knowledge
Course Description Book
54
► College Board has placed certain topics in AB (1 semester) and others in BC (2 semesters).
► Integration by parts is a BC topic
► L’Hopital’s Rule is a BC topic, but will move to the AB topics in 2016-2017.
► Other topics have been removed from the study of calculus
► Newton’s Method
► Volume using Shells
► New Emphasis on Finding Volume of a Solid with a Known Cross-Section
Former Topics
55
►Textbooks appropriate for the course of study► That are modeled after the Rule of
Four► That develop new understanding
graphically, numerically, algebraically, and verbally
► That introduce transcendentals early in the course
Exemplary Textbooks
56
Finney, Demana, Waits, and Kennedy
Calculus:Graphical, Numerical,Algebraic
4th Edition
Pearson Publishing, 2012
ISBN: 0133178579
Textbooks w/ Rule of Four
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► Step 1: Go to: mymathlabforschool.com
► Step 2: Sign In
► Username: APSIdemo2014
► Password: demo12345
► Step 3: Select your title from the left Course pane
► Step 4: From the Dashboard, on the left navigation, select Chapter Contents and select the chapter you wish to view.
Etext for Demana Calculus 4/eAccess until December, 2015
58
Paul Foerster
Calculus: Concepts and Applications
Kendall-Hunt Publishing, May 2004
Barbara Schoop
ISBN-10: 1559536543
Textbooks w/ Rule of Four
59
James Stewart
Calculus 7th Edition Early Transcendentals
2012
ISBN-10: 0538497815
Textbooks w/ Rule of Four
60
Calculus: Early Transcendentals
Jon Rogawski, Ray Cannon2012
BFW Publishers
ISBN-10: 1429250747
Kellie Rendina
Textbooks w/ Rule of Four
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►PRODUCT: Rogawski’s Calculus for AP
►URL:http://ebooks.bfwpub.com/rogawskiapet2e.php
►•Access code: 9b6-k24-fqp92u77
62
Calculus: Single VariableSixth Edition
Hughes-Hallett, Deborah; Gleason, Andrew M.; McCallum, William G.; Lomen, David O.; Lovelock, David
ISBN: 9780470888537
John Wiley and Sons
Textbooks w/Rule of Four
63
Author: Anton, Bevins, Davis
ISBN10: 9780470647684
Edition/Copyright: 10TH 2012
Publisher: John Wiley & Sons, Inc.
Textbooks w/ Rule of Four
64
Once downloaded, your Wiley E-Textwill appear in the VitalSource Bookshelf on your computer.
Download Bookshelf. http://www.vitalsource.com/downloads )
Ebook Access
65
► Once downloaded, your Wiley E-Text will appear in the VitalSource Bookshelf on your computer.
► Download Bookshelf. (http://www.vitalsource.com/downloads)
► Install Bookshelf. You can find step-by-step instructions online (http://support.vitalsource.com/faqs/gettingstarted/gs‐1001)
► •Once Bookshelf is installed, launch Bookshelf.
► •Click on the "Register for an Account" link.
► •Fill out the form completely, and paste your code into the redemption code field
► Press the register button to create your account and redeem your code.
► Your book will begin to download.
Ebook Access
66
► After the book has downloaded, click on "All Titles" in the collection pane and double click on your book to open it.
► •You have a Bookshelf account:
► •Launch Bookshelf.
► •Go to the Account Menu and Select Redeem Code.
► •Enter in your redemption code #, and press the Redeem button
► •Your updated booklist will be downloaded to your computer, once download your book will begin to download as well.
► Once the eTextbook has been downloaded, click on "All Titles" in the collection pane and double click on your book to open it
Ebook Access
67
►Ron Larson and Bruce Edwards
►Calculus: Early Transcendental Functions
►2013
A Very Popular Text
68
Four Parts of the Calculus AB Test
69
Comparison of Tests
2016-2017
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
2015-2016
Section 1 A - 28 questions – No Calculator - 55 minutes
Section 1B - 17 questions – Calculator required - 50 minutes
Section 2A -2 Free Response questions – Calculator required
Section 2B – 4 Free Response questions – no Calculator – 60 minutes
70
• Section I: a multiple-choice section testing proficiency in a wide variety of topics
• Part A: 28 questions – No Calculator -55 minutes
• Part B: 17 questions – Calculator permitted - 50 minutes
• 50% of the score – Total Number correct
• 45 x 1.2 = 54 points
Comparison of Tests
71
► Section 2 - 6 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
Weight of Questions
72
What’s in the Notebook?
► Section 8: Motion, FTC, Functions Defined by Integrals
► Section 9: Volume of Solids of Revolution, Approximations
► Section 10: Last Five Years- Free Response Questions
► Section 11: 2014 AB/BC Practice Test Multiple Choice Questions
► Section 12: 2997 Calculus Teacher Guide
► Section 13: Articles
► Section 14: WinPlot Manual
► Section 15: Scoring Distributions
► Section 1: Introductory Handout
► Section 2: Limits
► Section 3: The Derivative
► Section 4: Slope Fields
► Section 5: Integration and Definite Integral
► Section 6: Miscellaneous Materials
► Section 7: MVT, Reasoning with Tables, and Extrema
73
1. 2014 Handouts
2. AB-BC Syllabus
3. Audit
4. Calculus Free Response Questions
5. Calculus Multiple Choice Questions
6. Etexts
7. Lin McMullin Limit Powerpoint
8. Slope Field Materials
9. Special Focus
10. TI-Programs
What’s on the Flash Drive
74
Excel Spreadsheet Files
75
Section 1 in Notebook
Role of Sign Charts
76
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
77
► Limits are a way of explaining what happens near a point.
► The formal definition of limits is necessary to give meaning to imprecise terms like near, gets closer to, approaches, etc.
► The formal definition is not on the AP Course Description. – This does not mean you cannot or should not teach it.
► It is difficult to test this idea.
► Except for linear functions, the computations always involve some convoluted “tricks”.
Limits
78
►The Course Description indicates that students should have an “intuitive understanding of the limiting process.”
► Students should be able to calculate limits
► Understand “what not having a limit means”
► Have familiarity with vertical and horizontal asymptotes.
79
►Develop an intuitive understanding for when a function has a limit and when it does not.
►http://www.calculus-help.com/tutorials/
►Lin McMullin’s Limit Powerpoint
80
What Happens as X Approaches Zero?What Happens as X Approaches Infinity?
0
sinlimx
xx
sinlimx
xx
Near 0 Near Infinity
81
►Section 2.1 in FDWK
►Section 2.2 in Rogawski/Cannon
►Section 1.5 in Stewart
Other Limit Activities
82
Key Limits that are helpful to know
1lim 0x x
0
1limx x
0
1limx x
0
sinlim 1x
xx
0
1 coslim 0x
xx
1
0lim 1 x
xx e
1lim 1
x
xe
x
0
1lim ( ) limx x
f x fx
83
Activity # 1
Concept Outline
84
Task 1
Each group will receive a set of pre-cut cards The cards have an assortment of statements from multiple Big Ideas in the concept.
Work with your group organize the cards based on common characteristics that you identify, and create a label for each grouping.Remind participants that sorting the cards into a particular arrangement does not imply a particular sequencing or chunking for those concepts.
85
Task 2
Each card is a statement from the concept outline.• Some might be
• An Enduring Understand (EU) – Long Term Takeaways a student should have after studying the content and skills
• A Learning Outcome (LO) - What a student must be able to do in order to develop an enduring understanding – serve as targets for assessment
• Essential Knowledge (EK) – describe facts and basic concepts that students should know or be able to recall to demonstrate mastery of each learning objective
• Regroup your cards as needed so that the groupings are by EU, LO, and EK.
86
Task 3
Line up each group of cards horizontally so that EKs align with their corresponding LOs, and LOs align with their corresponding EUs. For example
87
88
89
► Multiple Choice Questions on the 2014 test: 2, 3, 5, 6, 7, 10, 13, 16, 17, 20, 24, 25, 78, 86, 87
► Free Response for AB Track
► 2015: AB1/BC1, AB6
► 2014: AB1
Monday Assignment AB
