Curriculum Design of Advanced Placement Calculus AB

Curriculum Design of Advanced Placement Calculus AB

Cameron Crowson

Keiser University

Dr. VanDeventer

EDU 740 – Curriculum Design

12/11/2016

Curriculum Design of Advanced Calculus AB 2

Curriculum Design of Advanced Placement Calculus AB

This project will breakdown an advanced high school curriculum for students interested

in taking Calculus. For most teachers inclined to teach Advanced Placement (AP) classes, the

first step is to become AP certified by attending a CollegeBoard workshop or seminar based on

the course the instructor is eager to teach. These workshops, provided by CollegeBoard (2016),

offer extensive one to two-day training sessions perfect for new AP teachers, experienced AP

teachers, K-12 teachers, and administrators and AP coordinators (CollegeBoard, AP Central,

2016). However, with the state and professional development programs, currently available

through the state of Florida, any instructor – certified in the 6-12 grade level - could easily

become comfortable with the material and the lab projects applicable to each AP classroom. In

regards to the type of instruction needed in AP classes, most use a type of applied curriculum- in

which, the students learn about a topic and by the end of the unit complete a laboratory

associated with the information. As far as the appeal of these type of courses, all are geared

toward students who intend to go to college and therefore, provide students with the knowledge

needed to succeed in a similar class at the college level.

Aims

However, most of the experience in AP classes tend to help students to realize that there

are different career paths, associated with these AP courses, open to each student. In such

experiences, students enjoy more of an applicable aspect rather than a theoretical. In other words,

the investigations and laboratory experience allow the students to dive deeper into an

understanding of the topics in each course and help relate the information to the real-world. For

example, in relation to the AP Calculus AB course, instructors teach students to “learn problem


solving methods that … can apply to real world problems involving theorems, definitions, and

functions represented in different ways” (CollegeBoard, AP Calculus AB, 2016, para. 1).

Phase 1

Needs and Purpose of the Curriculum

The AP Calculus AB curriculum has been derived to assist with students majoring in the

fields of applied mathematics, physics, engineering, and other math and science disciplines in

college. Students taking this course at the high school level will gain fundamental knowledge

needed to succeed in advanced courses passed introductory college calculus. Therefore, the

meaning of calculus, as explained by MIT (2016), is to study “… how things change. It provides

a framework for modeling systems in which there is change, and a way to deduce the predictions

of such models” (MIT, 2016). In other words, the calculus course gives a solution to solving

systems of change. In studying systems of change, students will be more equipped to

understanding how the real-world is modeled in various scientific fields like for instance physics

-which is the study of motion and matter.

Audience

The curriculum’s purpose is focused around teaching students who are college ready that

are currently in high school. CollegeBoard (2016) explains, “AP Calculus AB is roughly

equivalent to a first semester college calculus course devoted to topics in differential and integral

calculus” (CollegeBoard, AP Calculus AB, 2016, para. 1). Therefore, the main audience of the

course is focused around advanced students at the high school level.

Financial/Equipment

The curriculum is designed to have a lab with the completion of each unit; in which the

students learn about the applications of calculus in respect to physics and other fields of


engineering. However, as with the public-school contracts, the state will most likely fund the

school to support and give incentives for students enrolled in AP classes. Besides the basic lab

equipment needed for the major applications of the course, students will need access to graphing

calculators, computers in the classroom, and textbooks to reassure the course objectives and to

prepare students for the assessments.

Goals of the Course

The state of Florida, partnered with CollegeBoard, has determined that the AP Calculus

AB course to be an advanced mathematics course that explains the concepts related to the

understanding of the following topics as related standards:

1. Limits and Continuity – MAFS.912. C.1.1 – MAFS.912. C.1.13

2. Differential Calculus – MAFS.912. C.2.1 – MAFS.912. C.2.11

3. Applications of Derivatives – MAFS.912. C.3.1 – MAFS.912. C.3.12

4. Integral Calculus – MAFS.912. C.4.1 – MAFS.912. C.4.8

5. Applications of Integration – MAFS.912. C.5.1 – MAFS.912. C.5.8

Objectives of the Course

Florida’s adoption of Common Core Standards (CCS) have provided “high-quality

academic standards in mathematics” (Initiative, 2016, para. 2) and other subject areas. As

provided by the Florida Common Core curriculum, the objectives of the AP Calculus AB course

are divided into 8 units – Limits and Continuity, Derivatives, Implicit Differentiation,

Application of Derivatives, Integral Accumulation/Approximation and the Fundamental

Theorem of Calculus, Transcendental Functions, Differential Equations, and Area/Volume of

Revolution. Therefore, each of the objectives will be stated based on the individual unit in the


class. The objectives found below are found on the Florida Standards website and can also be

found on CPalms – where educators go to find basic unit standards:

Limits and Continuity. Students will develop learning skills needed to understand the

concepts of limits by way of estimation, graphical interpretation, and numerical evaluation and

extend their understanding to the concepts related the basic objectives found below:

a) Students will be able to understand the concept of a limit by

being able to estimate limits from graphs and tables.

b) Students will be able to find limits by way of substitution.

c) Students will be able to find limits of sums, differences,

products, and quotients.

d) Students will be able to find limits of rational functions that

are undefined at a point.

e) Students will be able to find one-sided limits.

f) Students will be able to find limits at infinity.

g) Students will be able to decide whether a limit is infinite and

use limits involving infinity to describe asymptotic behavior.

h) Students will be able to find special limits.

i) Students will be able to understand continuity in terms of

limits.

j) Students will be able to decide if a function is continuous at a

point.

k) Students will be able to find discontinuities of a function.


l) Students will be able to use the Intermediate Value Theorem

on a function over a closed interval.

m) Students will be able to apply the Extreme Value Theorem.

Derivatives. Students will be able to understand the derivative as an instantaneous rate of

change. Using geometric, numeric, and analytical methods and extend their understanding to the

concepts related the basic objectives found below:

a) Students will understand the concept of derivative

geometrically, numerically, and analytically, and interpret

the derivative as an instantaneous rate of change or as the

slope of the tangent line.

b) Students will be able to state, understand, and apply the

definition of a derivative.

c) Students will be able to find the derivatives of functions.

d) Students will be able to find the derivatives of sums,

products, and quotients.

e) Students will be able to find the derivatives of composite

functions using the Chain Rule.

f) Students will be able to find the derivatives of inverse

functions.

g) Students will be able to find second derivatives and

derivatives of higher order.

h) Students will be able to find derivatives using logarithmic

differentiation.


i) Students will be able to use the relationship between

differentiability and continuity.

j) Students will be able to apply the Mean Value Theorem.

Implicit Differentiation. Students will be able to dive deeper into the understanding of

derivatives through understanding when to use the Chain Rule for special functions and extend

their understanding to the concepts related the basic objectives found below (the student will also

revisit most of the concepts again for differentiation – found above):

a) Students will be able to find the derivatives of implicitly-

defined functions.

b) Students will be able to use implicit differentiation to find

the derivative of an inverse function.

Applications of Differentiation. Students will be able to apply the concepts of

derivatives to find slopes of curves and the related tangential lines to functions, analyze and

graph functions – finding where the functions is increasing, decreasing, or constant-, find

maximum points, minimum points, points of inflection, and their concavity, and more related to

the understanding of the concepts related to the basic objectives found below:

a) Students will be able to find the slope of a curve at a point.

Including points at which there are vertical tangent lines

and no tangent lines.

b) Students will be able to find an equation for the tangent line

to a curve at a point and a local linear approximation.

c) Students will be able to decide where functions are

decreasing and increasing.


d) Students will be able to find the local and absolute

maximum and minimum points.

e) Students will be able to find points of inflection of

functions and understand the relationship between

concavity of a function.

f) Students will be able to use the first and second derivatives

to help sketch graphs.

g) Students will be able to solve optimization problems.

h) Students will be able to find average and instantaneous

rates of change.

i) Students will be able to find the velocity and acceleration

of a particle moving in a straight line.

j) Students will be able to model rates of change, including

related rates problems.

k) Students will be able to solve problems using the Newton-

Raphson method.

Integration Approximation. Students will be able to understand that integration is used

to find areas, and evaluate through several methods of approximation and add to the concept that

integration is an inverse process of differentiation. The understanding of the concepts related to

the basic objectives found below are related to the understand of integrals:

a) Students will be able to use rectangle approximations to

find approximate values of integrals.


b) Students will be able to calculate the values of Riemann

sums over equal subdivisions.

c) Students will be able to interpret a definite integral as a

limit of Riemann sums.

d) Students will be able to interpret a definite integral of the

rate of change of a quantity over an interval as the change

of the quantity over an interval.

e) Students will be able to use the Fundamental Theorem of

Calculus.

f) Students will be able to apply the properties of definite

integrals.

g) Students will be able to use integration by substitution to

find values of integrals.

h) Students will be able to use Riemann sums, the Trapezoidal

Rule, and technology to approximate definite integrals.

Transcendental Functions. Students will be able to understand that a transcendental

function is a function that cannot be expressed in algebraic terms. Therefore, the investigation

will dive deeper into the main objectives of the derivatives unit with more emphasis on special

functions.

a) Students will be able to understand the concept of a

Transcendental Function

b) Students will be able to understand Natural Logarithmic

Functions in terms of a derivative


c) Students will be able to understand Natural Logarithmic

Functions in terms of integration

d) Students will be able to understand Inverse Functions in

terms of a derivative

e) Students will be able to understand Inverse Functions in

terms of integration

f) Students will be able to understand Exponential Functions

– Derivatives and Integration

g) Students will be able to investigate bases other than “e”

h) Students will be able to understand Trigonometric

Functions in terms of a derivative

i) Students will be able to understand Trigonometric

Functions in terms of integration

Differential Equations. Students will be able to apply knowledge about the major

concepts in Calculus to the understanding of basic Differential Equations and Slope Fields.

Therefore, the student will be able to apply knowledge of integrals to finding solutions to

differential equations. The understanding of the concepts related to the basic objectives are found

below:

a) Students will be able to find specific antiderivatives using

conditions.

b) Students will be able to solve separable differential

equations, and use them in modeling.


c) Students will be able to solve differential equations of the

form dy/dt = ky as applied to growth and decay problems.

d) Students will be able to use slope fields to display a graphic

representation of the solution to a differential equation.

Area/Volume of Revolution. Students will be able to apply knowledge of integrals to

finding areas and volumes. The understanding of the concepts related to the basic objectives are

found below:

a) Students will be able to use definite integrals to find areas

between a curve and the x-axis or between two curves.

b) Students will be able to use definite integrals to find the

average value of a function over a closed integral.

c) Students will be able to find the volume of a solid with

known cross-sectional area, including solids of revolution.

d) Students will be able to apply integration to model, and

solve problems in physical, biological, and social sciences.

Scope and Sequence

The class structure is designed for either a traditional bell-schedule in which the students

have 8 periods of classes that meet for 45-minutes per day or a block schedule in which students

meet for 90-minute blocks 2-3 times per week. This course is designed to be over a period of 36

weeks – or two high school semesters that last for 18 weeks each. In which the last few weeks of

the schedule is devoted to review for the AP examination. To further explain, at the completion

of each unit will be a major examination to assess the student’s understanding of the overall

goals and benchmarks of the chapter. Below is a table to explains the course sequence of each


individual unit, percent of time on each unit, and Florida Common Core Benchmarks associated

with each unit:

Table 1: Course Sequence and Objectives of AP Calculus ABUnit Number of

Weeks (every 5 Days = 1

Week)

Percent of Time

Florida Common Core Benchmarks

Limits and Continuity 5 13.9% 1

Derivatives 2.8 7.8% 2

Implicit Differentiation 2 5.6% 2, 3

Applications of Differentiation/ Curve Sketching

4.2 11.7% 3

Integration approximation/ Fundamental Theorem of Calculus

3.8 10.6% 4

Transcendental Functions – Derivatives and Integration

5.8 16.1% 2

Differential Equations/Slope Fields

2 5.6% 5

Area/Volume of Revolution – Applications of Integration

3.4 9.4% 5

AP Review 7 19.3% 1, 2, 3,4, 5

Along with each unit, which will be explained at a later portion of the paper, are subunits that

microscopically explain each objective of the course.

Assessment

This course will offer cumulative assessment throughout each semester to prepare

students for the ultimate exam at the end of the fourth quarter or second semester. The course

will have a multitude of homework assignments and quizzes to compensate for the information

in between each unit to test for student understanding of each objective. An assessment will be

based off the knowledge in the previous mathematics courses taken. An example of a pre-


assessment can be found on the Patrick Henry Preparatory Academy website (Academy, n.d.).

This assessment tests student’s knowledge in the areas of…

1. Basic Algebraic Skills

a. Rule of Exponents

b. Simplifying Expressions

c. Factoring Expressions

d. Evaluating Expressions at a Point

2. Functional Analysis

a. Composition of Functions

b. Asymptotes

c. Inverse Functions

d. Domain and Range of Functions

3. Geometry and Trigonometry

a. Rules of Trigonometric Functions

b. Triangle Inequalities, Angles, Degrees, and Radians

Phase II

Along with the various unit-breakdown listed above pertaining to student goals, time

frames of units, and major course benchmarks, the second part of this paper will explain the units

in further detail by explaining how the lessons will be taught/explained: events, instructional

methods used, and instructional treatment and strategies to achieve the standards assessed in the

unit.


Semester 1 Lesson Plans of Unit 1-4

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Limits and ContinuityTime Frame: 5 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.1.1 – MAFS.912. C.1.13Objectives: Students will be able to…

1. Understand the concept of a limit2. Find limits by way of substitution3. Find limits of sums, differences, products, and quotients4. Find limits of rational functions that are undefined at a

point5. Find one-sided limits6. Find limits at infinity7. Decide whether a limit is infinite and use them to describe

asymptotic behavior8. Find special limits9. Understand continuity in terms of limits10. Decide if a function is continuous at a point11. Find discontinuities of a function12. Use the Intermediate Value Theorem (IVT)13. Apply the Extreme Value Theorem (EVT)

Procedures:Event Method/Media Instructional Treatment

or StrategyGain Attention of Learner

Instructor demonstrates the concepts of previous concepts in Pre-Calculus by way of talking about the how they connect in Calculus; therefore, connecting concepts of graphical analysis, algebra, and geometry to limits.

Instructor will use the Smart-Board in the classroom to demonstrate basic skills the learner needs to know before starting the unit.

Inform Learner of Objective

Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; along with the concepts related to various ways to calculate limits (using algebra vs. using geometry).

Stimulate Recall of Prerequisites

Students will take a summative assessment of the skills learned in Pre-Algebra.

1. Students will take a summative assessment of skills learned in Pre-Algebra

2. Students will then look


over their results to determine if they have the adequate skills needed to move forward in the course.

Present Stimulus Material

Instructor/Course Materials Instructor demonstrates the main concepts of limits and continuity by way of small group explanations and demonstrating the various ways of reading graphs and finding limits.

Provide Learning Guidance

Instructor/Course Materials Instructor explains the various algebraic and geometric ways of finding limits and expresses the correct ways vs. wrong ways of interpreting graphs of continuous vs. inconsistent functions.

Elicit Performance Students/Course Materials 1. Students, by way of completing homework assignments, class work, and taking quizzes/tests, will show a base understanding of the unit and show the various ways of solving problems in the unit.

2. Students are encouraged to attend tutoring sessions after school on specific days to express their concerns in the understanding of the materials of the unit.

Provide Feedback Instructor: Grading/Observation Instructor will provide feedback on all graded assignments and group activities; in which, the student will be able to assess their understanding from the instructor’s comments.

Assess Performance Instructor observes student’s assignments and assessments

Students continue their work with limits and continuity. Instructor assess each student’s work. Students are


evaluated on their ability to understand and apply their knowledge of limits and continuity on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.

Enhance Retention and Transfer

Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of finding limits at a point.

LESSON PLAN FORMAT

Subject: AP Calculus AB Lesson Plan Title: Unit 2-DerivativesTime Frame: 3 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.2.1 – MAFS.912. C.2.11Objectives: Students will be able to…

1. Understand the concept of a derivative2. State, understand, and apply the definition of a derivative3. Find the derivatives of functions4. Find the derivatives of sums, products, and quotients5. Find the derivative of a composite function using the “Chain

Rule”6. Find second derivatives and derivatives of higher order7. Find derivatives using logarithmic differentiation8. Use and understand the relationship between

differentiability and continuity9. Apply the Mean Value Theorem (MVT)

Procedures:Event Method/Media Instructional Treatment or

StrategyGain Attention of Learner

Instructor demonstrates the concepts of previous concepts regarding limits and continuity by way of talking about slopes of functions and connecting it to the definition of a derivative: Lim h-> 0; (f(x+h) -f(x)) / h

Instructor will use the Smart-Board in the classroom to demonstrate basic skills the learner needs to know before starting the unit – like demonstrating the definition of a derivative and showing that it represents the slope of a function at a point.


Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the


curriculum; along with the concepts of calculating slopes (using algebra vs. using geometry).


Students will take a summative assessment of the skills learned in unit 1.

1. Students will take a summative assessment of skills learned in unit 1 – Limits and Continuity

2. Students will then look over their results to determine if they have the adequate skills needed to move forward in the course.


Instructor/Course Materials Instructor demonstrates the main concepts of derivatives by way of small group explanations and demonstrating the various ways of reading graphs and finding derivatives of functions as a function approaches a point.


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of finding derivatives the correct ways vs. wrong ways of using various theorems in this unit.

Elicit Performance

Students/Course Materials 1. Students, by way of completing homework assignments, class work, and taking quizzes/tests, will show a base understanding of the unit and show the various ways of solving problems in the unit.


Provide Feedback

Instructor: Grading/Observation Instructor will provide feedback on all graded assignments and group activities; in which, the student will be able to assess their understanding from the instructor’s comments.

Assess Performance

Instructor observes student’s assignments and assessments

Students continue their work with limits and continuity. Instructor assess each student’s work. Students are evaluated on their


ability to understand and apply their knowledge of derivatives on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.


Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of finding derivatives at a point

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Unit 3-Implicit DifferentiationTime Frame: 2 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.2.6, MAFS.912. C.3.7Objectives: Students will be able to…

1. Find the derivatives of implicitly defined functions 2. Use implicit differentiation to find the derivative of an

inverse functionProcedures:

Event Method/Media Instructional Treatment or Strategy

Gain Attention of Learner

Instructor demonstrates the concepts of previous concepts regarding differentiation and explains the difference between explicitly vs. implicitly-defined functions.



Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; along with the concepts of calculating derivatives of implicit functions in comparison to explicit functions (using algebra vs. using geometry).



1. Students will take a summative assessment of skills learned in unit 2 – Differentiation

2. Students will then look over their results to determine if they have the adequate skills needed to move forward in the


course. Present Stimulus Material

Instructor/Course Materials Instructor demonstrates the main concepts of slopes/derivatives by way of small group explanations and demonstrating the various ways of reading graphs and finding derivatives of functions that are not dependent on a 1-1 relationship.


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of finding derivatives the correct ways vs. wrong ways of using various theorems in this unit.

Elicit Performance



Provide Feedback


Assess Performance


Students continue their work with differentiation. Instructor assesses each student’s work. Students are evaluated on their ability to understand and apply their knowledge of finding derivatives of implicitly-defined functions on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.


Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems


by way of finding derivatives of inverse functions.

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Unit 4-Applications of DifferentiationTime Frame: 2 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.3.1 - MAFS.912. C.3.12Objectives: Students will be able to…

1. Find the slope of a curve at a point 2. Find an equation of a tangent line3. Decide where functions are increasing or decreasing4. Find local and absolute maximum and minimum points5. Find points of inflection of functions6. Use the first and second derivatives to help sketch graphs7. Solve optimization problems8. Find average and instantaneous rates of change9. Find the velocity and acceleration of a particle moving in a

straight line10. Model rates of change, including related-rates problems11. Solve problems using the Newton-Raphson method

Procedures:Event Method/Media Instructional Treatment or

StrategyGain Attention of Learner

Instructor demonstrates the concepts of previous concepts relating to graphical analysis and understanding zeros of a function and derivatives of areas of a graph.



Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; along with the concepts of calculating derivatives of functions in specific areas of the function in relation to the extreme values and zeros (using algebra vs. using geometry).



1. Students will take a summative assessment of skills learned in unit 3 – Implicit Differentiation

2. Students will then look over their results to determine if


they have the adequate skills needed to move forward in the course.


Instructor/Course Materials Instructor demonstrates the main concepts of slopes/derivatives by way of small group explanations and demonstrating the various ways of reading graphs and finding derivatives at important points; in which, the student will be able to explain if the function is increasing or decreasing.


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of analyzing a graph of a function.

Elicit Performance



Provide Feedback


Assess Performance


Students continue their work with differentiation. Instructor assesses each student’s work. Students are evaluated on their ability to understand and apply their knowledge of finding derivatives of implicitly-defined functions on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.



Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of modeling a event in science relating it back to graphing.

Semester 2 Lesson Plans of Units 5-8

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Integration ApproximationTime Frame: 4 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.4.1 – MAFS.912. C.4.8Objectives: Students will be able to…

14. Use rectangle approximations to find approximate values of integrals

15. Calculate the values of “Riemann” sums over equal subdivisions

16. Interpret a definite integral as a limit of “Riemann” sums17. Interpret a definite integral of the rate of change of a

quantity over an interval as the change in the quantity over an interval

18. Use the “Fundamental Theorem of Calculus”19. Apply the properties of definite integrals20. Use integration by way of substitution to find the values of

integrals21. Use “Riemann” sums, the Trapezoidal rule, and technology

to approximate definite integralsProcedures:



Instructor demonstrates the previous concepts relating to area of 2-D shapes by way of explaining “Riemann” sums.



Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; along with the concepts of calculating areas under the curve by way of approximation techniques.



Students will take a summative assessment regarding the material from the last units (1-4)

1. Students will take a summative assessment of skills learned in the units prior.



Instructor/Course Materials Instructor demonstrates the main concepts of approximation techniques regarding area, by way of small group explanations and demonstrating the various concepts of integration.


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of analyzing a graph of a function (by way of finding areas under the curve).




Assess Performance Instructor observes student’s Students will begin their work


assignments and assessments with anti-differentiation. Instructor assesses each student’s work. Students are evaluated on their ability to understand and apply their knowledge of finding anti-derivatives of well-defined functions on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.


Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of modeling an event in geometry and trigonometry

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Transcendental Functions- Derivatives and IntegrationTime Frame: 6 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.3.1 – MAFS.912. C.4.8Objectives: Students will be able to…

1. Understand the concept of a Transcendental Function2. Understand Natural Logarithmic Functions in terms of a

derivative3. Understand Natural Logarithmic Functions in terms of

integration4. Understand Inverse Functions in terms of a derivative5. Understand Inverse Functions in terms of integration6. Understand Exponential Functions – Derivatives and

Integration7. Investigate bases other than “e”8. Understand Trigonometric Functions in terms of a

derivative9. Understand Trigonometric Functions in terms of integration


or StrategyGain Attention of Learner

Instructor demonstrates the previous concepts relating to derivatives and integrals and recaps the various

Instructor will use the Smart-Board in the classroom to demonstrate basic skills the learner needs to know before


forms/formulas starting the unit.Inform Learner of Objective

Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; along with the concepts of calculating derivatives and integrals of special functions


Students will take a summative assessment regarding the material relating to taking derivatives and integrals of functions




Instructor/Course Materials Instructor demonstrates the main concepts of special functions by way of small group explanations and demonstrating the various concepts of derivatives and integration.


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of analyzing a graph of a function.






Students continue their work with differentiation and integration. Instructor assesses each student’s work. Students are evaluated on their ability to understand and apply their knowledge of finding derivatives and integrals of special functions on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.


Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of modeling various special functions

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Differential EquationsTime Frame: 2 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.5.1 – MAFS.912. C.5.8Objectives: Students will be able to…

1. Find specific anti-derivatives using conditions2. Solve separable differential equations and use them in

modeling3. Solve differential equations of the form dy/dy = ky as

applied to growth and decay problems4. Use slope fields to display graphic representation of the

solution to a differential equationProcedures:


Gain Attention of Instructor demonstrates the Instructor will use the Smart-


Learner previous concepts relating to functions and finding derivatives and integrals of functions

Board in the classroom to demonstrate basic skills the learner needs to know before starting the unit.


Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; along with the calculating solutions to differential equations.


Students will take a summative assessment regarding the material in the previous unit – Transcendental Functions




Instructor/Course Materials Instructor demonstrates the main concepts of differential equations by way of small group explanations and demonstrating the various concepts of derivatives and integrals


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of analyzing slope fields


2. Students are encouraged to attend tutoring sessions after school on specific days to express their concerns in the understanding of the


materials of the unit. Provide Feedback Instructor: Grading/Observation Instructor will provide

feedback on all graded assignments and group activities; in which, the student will be able to assess their understanding from the instructor’s comments.


Students continue their work with differentiation and integration. Instructor assesses each student’s work. Students are evaluated on their ability to understand and apply their knowledge of solving basic differential equations on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.


Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of modeling differential equations

LESSON PLAN FORMATSubject: AP Calculus AB Lesson Plan Title: Area/Volume of RevolutionTime Frame: 4 Weeks (2-3 days per week) at 90 minutes/per dayStandards Assessed: MAFS.912. C.5.1 – MAFS.912. C.5.8Objectives: Students will be able to…

1. Use definite integrals to find areas between a curve and the x-axis or between two curves

2. Use definite integrals to find the average value of a function over a closed interval

3. Find the volume of a solid with known cross-sectional area, including solids of revolution

4. Apply integration to model and solve problems in the physical, biological, and social sciences


or Strategy



Instructor demonstrates the previous concepts relating to functional area and volume.



Instructor-Lecture Instructor will teach the unit objectives above so that the students will be able to apply them to basic applications in the curriculum; giving the students the basic foundations of how integration plays a part in 2-D and 3-D geometry - with revolution of shapes.


Students will take a summative assessment regarding the material in the previous unit – Differential Equations




Instructor/Course Materials Instructor demonstrates the main concepts relating to area and volume of revolution by way of small group explanations and demonstrating the various concepts of derivatives and integrals


Instructor/Course Materials Instructor explains the various algebraic and geometric ways of finding areas and volumes.


2. Students are encouraged to attend tutoring sessions


after school on specific days to express their concerns in the understanding of the materials of the unit.



Students continue their work with integration. Instructor assesses each student’s work. Students are evaluated on their ability to understand and apply their knowledge of solving basic geometric- equation problems on classroom assessment, which is measured by the student’s satisfactory understanding of each objective listed above.


Students - Laboratory Student will work with a group of his/her peers to evaluate expressions of real-world problems by way of modeling 3-D areas and volumes.

Conclusion


In conclusion of the course sequence – limits and continuity, derivatives, implicit

differentiation, applications of differentiation, integration approximation, transcendental

functions, differential equations, and area/volume of revolution – students should be able to

adequately perform on the AP Calculus AB exam given by CollegeBoard. By way of testing the

students on the material, when the course is completed, the teacher should give a practice AP

Calculus AB exam found on the CollegeBoard website. By giving the students a practice exam,

they should be able to understand the testing format that is expected the day of the exam.

References


Academy, P. H. (n.d.). AP Calculus AB Pre-Exam. Retrieved from Patrick Henry Preparatory

Academy: http://academy.hslda.org/files/Calculus_AB_%20Pre-Exam.pdf

CollegeBoard. (2016). AP Calculus AB. Retrieved from Collegeboard.org:

https://apstudent.collegeboard.org/apcourse/ap-calculus-ab

CollegeBoard. (2016). AP Central. Retrieved from CollegeBoard.com:

http://apcentral.collegeboard.com/InstitutesAndWorkshops

CPalms. (2016). Standards. Retrieved from CPalms: http://www.cpalms.org/Public/

Diamond, R. M. (2008). Designing and Assessing Courses and Curricula. San Francisco:

Jossey-Bass.

Glatthorn, A. A., Boschee, F., Whitehead, B. M., & Boschee, B. (2016). Curriculum Leadership:

Strategies for Development and Implementation. Los Angeles: Sage.

Initiative, C. C. (2016). About the standards. Retrieved from Common Core State Standards

Inititative: http://www.corestandards.org/about-the-standards/

Marzano, R. (2007). The Art and Science of Teaching. United States of America: Association for

Supervision and Curriculum Development.

MIT. (2016). What is Calculus and why do we study it? Retrieved from MIT: http://www-

math.mit.edu/~djk/calculus_beginners/chapter01/section02.html

Sharon E. Smaldino, D. L. (2015). Instructional Technology and Media for Learning. United

States of America: Pearson.

Documents

Curriculum Design of Advanced Placement Calculus AB