Calculus AB APSI 2015
Day 4
Professional
Development
Workshop
Handbook
Curriculum
Framework
Calculus AB and
BC
Professional
Development
Integration, Problem
Solving, and Multiple
Representations
Curriculum Module
Thursday
Morning (Part 1)
• Discussion of Homework
Problems
Break
Morning (Part 2)
• Activity #2 Exploring the
Mathematical Practices
• Free Response 2007 AB3
• Discovering the Relationship
between Slopes at Corresponding
Points on Inverse Functions
• Integration, Problem Solving and
Multiple Representations
• Selected Curriculum Module
Break
• Revising Assessment
Questions
Afternoon (Part 2)
• Balancing Concept and Skill
Lunch
2
Afternoon (Part 1)
• Activity #3 - Scaffolding the
Mathematical Practices
• Discussion about classroom
procedures, grading, homework,
tests, and any other concerns
Closure
Wednesday Assignment - AB
 Multiple Choice Questions on the 2014 test: 1, 4, 8, 12,
14, 18, 26, 76, 77, 70, 80, 81, 83, 84, 85
 Free Response:
2014: AB4, AB6
2015: AB4, AB5
3
2014 AB4
4
Scoring Rubric 2014 AB4
5
2014 AB6
6
Scoring Rubric 2014 AB6
7
Scoring Rubric 2014 AB6
8
2015 AB4
9
Scoring Rubric 2015 AB4
10
Scoring Rubric 2015 AB4
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2015 AB5
12
Scoring Rubric AB5
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Activity #2 Exploring the Mathematical Practices
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Think about this question with your
group. Be prepared to share your
responses with the rest of the class.
15
16
17
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Revising Assessment Questions
One suggestion for preparing students for the types
of questions they see on AP Exams is to use
questions with similar structure, formatting, and
scoring throughout the year. Using the information
you have gained in this workshop, you will revise
existing questions to be similar to the types of
questions and scoring guidelines that are used on
the AP Exams.
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1. Find the derivative of the function 𝒚 = 𝟑𝒙𝟕 − 𝟕𝒙𝟑 + 𝟑
AP Conversion:
Scoring Guideline:
20
2. Given 𝒚′ = 𝟔(𝒙 + 𝟏)(𝒙 − 𝟐)𝟐 , find the points at
which 𝒚 = 𝒇(𝒙) has a local maximum, minimum or
point of inflection.
AP Conversion:
Scoring Guideline:
21
3. Evaluate the integral
AP Conversion
Scoring Guideline:
22
𝟓
𝟒𝒙
𝟐
𝒅𝒙
Balancing
Skill
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Concept
Many students finish Calculus thinking
about a derivative as a process that leads
to a number.
 
2
If f(x)=cos x , then f '  
4 2
They also think about a integration leading
to a number.
If f(x)=cos x , then
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

0
2
cos xdx  1
But we already saw that a derivative
is a function:
f (x )  sin(x )
f '(x )  cos(x )
So likewise, a function can be
represented by an integral.
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Functions Defined
by Integrals
Smartboard File
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Exploring Functions Defined by Integrals
Worksheet 1: Page 3-7
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Exploring Derivatives of Functions Defined by Integrals
Worksheet 2: Page 10-12
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Graphical Analysis of F(x) using F’(x)
Worksheet 3: Page 15-16
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Activity #3 – Scaffolding the Mathematical Practices
Different Learning Objectives could reference the same MPAC, providing multiple
opportunities for students to practice that skill in different contexts. Think about the
following prompts.
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Understanding Scaffolding
1. Label each of the student tasks below to establish a sequence that would help
to support students’ development of MPAC 5a: Know and use a variety of
notations. Use each label only once.
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2. MPAC 6c states that students will be able to explain the meaning of
expressions, notations, and results in terms of a context (including units).
What are some strategies you can use that will help students build that skill
over the course of the year?
3. List three Learning Objectives where you could incorporate those strategies
and provide opportunities for students to practice MPAC 6c in multiple contexts?
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Activity #4 – Exam Items
Course planning can incorporate
instructional strategies that will target
specific misunderstanding and
support students’ ability to
demonstrate mastery of the Learning
Objective.
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• The overall design of the exam has not changed and the Learning Objectives
in the Concept Outline are the target of assessment.
• Review the same item and respond to the given prompts.
• Reference the Curriculum Framework
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1. How does this item require students to demonstrate an understanding
that a function’s derivative, which is itself a function, can be used
to understand the behavior of the function (EU2.2)?
2. How does this tem require students to demonstrate an understanding
that the definite integral of a function over an interval is the limit
of a Riemann Sum over an interval and can be calculated using a
variety of strategies (EU3.2)?
3. How does this item require students to apply one of the subskills in
MPAC 6?
4. What other MPACs would students need to apply in order to be
successful on this item?
37
Discussion about classroom procedures, grading,
homework, tests, and any other concerns
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Integration, Problem Solving and Multiple
Representations Curriculum Module
Lesson 1
Lesson 2
Lesson 3
Lesson 4
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Meeting Code/Consultant Code
Lasalle University
Jim Rahn
► Event
Code:
► Consultant
► Session
40
Code 0602
Number:
Meeting Code/Consultant Code
Middlesex County College
Jim Rahn
Event Code:
Consultant Code 0602
Session Number:
41
Meeting Code/Consultant Code
Ocean County College
Jim Rahn
Event Code:
Consultant Code 0602
Session Number:
42