1. Calculus AB APSI 2015 Day 3 Professional Development Workshop Handbook
Curriculum Framework Calculus AB and BC
Professional Development
Integration, Problem Solving, and Multiple Representations
Curriculum Module

2. Wednesday Morning (Part 1) Break Morning (Part 2) Lunch
Introducing the Definite Integral Through the Area Model
Investigating How to Find Area Using Riemann Sums and Trapezoids
Developing an Understanding for a Definite Integrals
Numerical Integration
Break  
Morning (Part 2)
Fun Finding Volume
Solids with Known Cross Sectional Area
Fundamental Theorem of Calculus
Activity Sheet for Understanding the 2nd Fundamental Theorem of Calculus
Lunch
Afternoon (Part 1)
Discussion of Homework Problems
Free Response Problem 2014 AB5
Share an Activity
Dan Meyer
 Break
 Afternoon (Part 2)
Calculus Games
Building Understanding for the Average Value of a Function
Mean Value Theorem
Curriculum Module: Motion (w/Smartboard)
Big Idea 3 – The Integral and the Fundamental Theorem of Calculus

3. 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/BC4, AB5

4. Key Ideas to Cover on Integration
A definite integral is the limit of a Riemann sum
The definite integral is the net accumulation of a rate of change or

5. Concept of a Definite Integral
All the important concepts related to definite integrals can be taught and understood without knowing antiderivatives.

6. Calculus AP should include opportunities for students to understand
Area under a graph
Riemann Sum – Definition of a Definite Integral
Ways to Evaluate a Definite Integral
Fundamental Theorem
How integrals accumulate area
How functions can be by integrals
Techniques for finding indefinite integrals
Applications of integrals

7. Deal with graphical and tabular sets of data to find area
That can be represented by a bounded region.
That can be approximated using several methods.
Relationships between approximations.
How more accurate approximation be found
The units of measure for

8. Introducing Integration through the Area Model

9. Introducing the Definite Integral Through the Area Model
Figure 1 shows the velocity of an object, v(t), over a 3-minute interval. Determine the distance traveled over the interval
. The area bounded by the graph of v(t) and the t-axis for represents the distance traveled by this object. The distance can be represented by the
definite integral
Activity

10. The following chart gives the velocity of a particle, v(t), at 0
The following chart gives the velocity of a particle, v(t), at 0.5 second intervals. Estimate the distance traveled by the particle in the three seconds using three different methods. Each method is an approximation for
Activity

11. Investigating How to Find Area using Riemann Sums and Trapezoids
Using the NUMINT program or
LMRRAM and TRAPEZOID program on a TI84
Activity

12. Things You Should have Observed
As the number of rectangles increases on monotonically increasing functions, the left hand sums increase, but remain less then the area.

13. Things You Should have Observed
Which sums are always greater than the actual area? Which sums are always less than the actual area?

14. Things You Should have Observed
The limit of the left hand sum equals the limit of the right hand sum and equals the area of the region.
area of the region or

15. Students should be able to
Set up and evaluate left, right and midpoint Riemann sums from analytical data, tabular data, or graphical data.
Set up and evaluate a trapezoidal sum approximation from analytical data, tabular data, or graphical data.
− 𝑥 2 𝑑𝑥

16. Determine Units of Measure
The units of the definite integral are the units of the Riemann Sum
The units of the function multiplied by the units of the independent variable.

17. Verbal Explanation
Students need to be able to tell what a definite integral represents in the context of the problem and identify the units of measure.
Very common AP question on Free Response Questions

18. Using Technology to Approximate the Definite Integral

19. Developing an Understanding for the Definite Integral
Smartboard File

20. Fun Finding Volume

21. Solids with Known Cross Sectional Area
Activity

22. Volume of a Solid with Known Cross Sectional Area
Create a table and a sketch for
𝑓(𝑥)= 𝑥
scale for the grid is 0.5 cm
on the x and y axes

23. Sketch the graph of f (x). The scale for this grid is 0.25 cm on
both the x and y axes.

24. Select one of the figures
Select one of the figures. Cut out the 9 shapes, keeping the tabs on the shape. Fold the trapezoidal tab. Glue the tab on the graph so that the edge of the shape is the f(x) segment. Face all the colored faces in the same direction.

25. Group Work
Complete the Finding the Volume of the Solid Activity Sheet with your group members.

26. Volume of Solids of Revolution
Section 9 in Notebook
Smartboard File

27. Rotating about a Line Other than the x- or y-axis
Pages 2 to 5

28. Rotating about a Line Above the Region
Pages 5 to 7

29. Rotating about a Line to the Left of the Region
Pages 10 and 11

30. Rotating about a Line to the Right of the Region
Pages 8 and 9

31. Fundamental Theorem of Calculus
Smartboard File

32. Activity Sheet for Understanding the 2nd Fundamental Theorem of Calculus
Section 5 in Notebook
Activity

33. Section 8 in Notebook

34. Areas, Derivatives, and the Fundamental Theorem of Calculus
Worksheet 1: pages 8-11

35. Applying the Second Fundamental Theorem of Calculus: Finding Derivatives of Functions Defined by Integrals
Worksheet 2: page 13

36. Applying the First Fundamental Theorem of Calculus: Definite Integrals as Total Change
Worksheet 3: page 15-16

37. Examples of Multiple Choice Questions
Worksheet 4: page 15-16

38. Applying the Fundamental Theorem of Calculus Exercises
Worksheet 5: page 15-16

39. Tuesday Assignment - AB
Multiple Choice Questions on the test: 9, 11, 15, 19, 21, 22, 23, 27, 28, 82, 88, 89, 90, 91, 92
Free Response:
2014: AB3, AB6
2015: AB2, AB3

40. Scoring Rubric 2014 AB3

41. 2014 AB3

42. 2014 AB6

43. Scoring Rubric-2014 AB6

44. 2015 AB2

45. Scoring Rubric 2015 AB2

46. 2015 AB3

47. Scoring Rubric 2015 AB3

48. Share An Activity from Your Classroom

49. Dan Meyer
Taco Stand
All Examples

50. Calculus Games

51. Building Understanding for the Average Value of a Function
Activity

52. The Mean Value Theorem

53. The Mean Value Theorem - Smartboard
What is guaranteed?
What must be true for the guarantee?
Can parts be true if the conditions are not met?
How does it apply to real data?

54. Motion Smartboard File
College Board has developed a Curriculum Module to assist you in teaching how to use Calculus to study motion.
Dixie Ross
Pflugerville High School
Pflugerville, TX
Motion Smartboard File

55. What You Need to Know about Motion
Worksheet 1: page 5

56. Sample Practice Problems Numerical, Graphical, Analytical
Worksheet 2: pages 7-9

57. Understanding the Relationship Among Velocity, Speed and Acceleration
Worksheet 3: page 13-15

58. What You Need to Know about Motion Along the x-axis
Worksheet 4: page 21

59. Sample Practice Problems
Worksheet 5: page 23-26

60. Big Idea 3: Integrals and the Fundamental Theorem of Calculus
Page

68. 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:
2015: AB4, AB5
2014: AB4, AB6