PLEASANT VALLEY SCHOOL DISTRICT
Brodheadsville, Pennsylvania 18322
PLANNED COURSE ADAPTATIONS/MODIFICATIONS
Introduction

PREFACE

PLEASANT VALLEY SCHOOL DISTRICT

Brodheadsville, Pennsylvania 18322

Brodheadsville, Pennsylvania 18322

PLANNED COURSE ADAPTATIONS/MODIFICATIONS

PLANNED COURSE ADAPTATIONS/MODIFICATIONS

Introduction

Introduction

The instructional adaptations that follow are provided as suggestions to be implemented with all students, particularly with those in need of special education services including the gifted. This listing is in no way intended to be exhaustive. Rather, it is reflective of some major considerations in the area of curriculum adaptations/modifications.

These instructional adaptations will work with any student, but are especially beneficial to those in need of learning support. Some may argue that these modifications are simply good teaching. Indeed, modifications of this type do represent good teaching. These principles of good teaching become instructional modifications whenever: (1) certain students in a particular class require such modifications above and beyond what is typically required by most students in that class and (2) without these modifications, these same students would not succeed.

Assessment is an integral part of instruction reflecting student progress as well as achievement. Therefore, also included are suggestions for assessment modifications.

� Peer Support
� Cooperative learning among peers
� Modeling
� Development of Information Organizer
� Development of Graphic Organizer
� Development of structured study guides
� Student selection of instructional material (i.e., reading, writing, math)
� Taped lessons
� Copy notes (peer or teacher)
� Student conferencing
� Combine and vary modes of lesson presentation
� Adjust language level to match the developmental and intellectual levels of students
� Let student practice given examples first. Then assign tasks to be completed.
� Provide opportunity for guided and independent practice in a variety of situations
� Limit number and length of directions
� Have students repeat/review directions (i.e., peer to peer, student to teacher)
� Give feedback that is as immediate, specific, and objective as possible
� Clarify error responses so that students do not make the same errors over and over again
� Reinforce progress towards desired outcomes
� Breakdown complex tasks into smaller, more manageable units
� Use verbal prompts to elicit desired results
� Use manual guidance (i.e., hand over hand) to facilitate correct responses
� Computer assisted instruction
� Assessment based upon teacher observation of student performance (i.e., daily work, portfolio, artifacts, projects)

� Extended test time
� Test read to student by teacher or peer
� Oral testing (i.e., student retelling of information)
� Open book/note test
� Alternate testing (any demonstration of a student's understanding of concepts)
� Retesting
� Reduce the number of responses required on tests
� Use of curriculum based assessment
� Vary test format
� Objectively define mastery as related to each task. Tasks should be learned to mastery
� Reduce or remove distracting stimuli
� Use of concrete objects and manipulatives in all stages of instruction and assessment
� Emphasize important information
� Allow extra time to complete assignments/projects
� Limit the number of assigned tasks in the initial stages of learning. As the student's competency increases, expect the student to complete the same number of tasks as the rest of the class
� Use supplemental materials
� Alternate assignments accepted (i.e., modification to homework assignments)
� Flexible grouping/individual assistance
� Seating to accommodate needs
� Teacher proximity
� Use behavioral management techniques (i.e., contracts, time-out, token system, charts)

PREFACE

Users and information seekers should familiarize themselves with the purpose and terminology of this Planned Course
Curriculum Guide (PCCG). We suggest that you first read the following:

• PCCG PURPOSE AND INTENT
• PCCG DEFINITIONS

The PCCG specifies the unit lesson outcome, essential content, standards, activities, resources, and evaluation of student performance. This sector provides the means to initiate the learning activities to attain the program goal as identified in the course description and intent.

The standards and outcomes are minimal expectations; further embellishment of the course is discretionary with the instructor depending upon the capability of the students.

This PCCG is designed as an ACTIVE document capable of technological modification as required.

The instructional delivery of this curriculum is quality controlled through the lesson plan development of the teacher.

PLANNED COURSE CURRICULUM GUIDE (PCCG)

PURPOSE AND INTENT

The Planned Course Curriculum Guide (PCCG) is a multi-purpose document:

• All staff, particularly new teachers, can understand instructional expectations through the WRITTEN curriculum

• A continuing district-wide instructional process and scope and sequence of subject matter are enhanced. The WRITTEN curriculum is delivered through the TAUGHT curriculum [instructional content and learning activities] and is evaluated through the TESTED curriculum [expected levels of student achievement - learning outcomes]

• Priority student-centered outcomes are identified and attained through suggested learning activities and content designed to help insure a balanced and comprehensive basic curriculum

•• Essential content and course standards provide an efficient basis for selecting appropriate instructional materials and resources

• Staff development areas for curriculum improvement are provided

• The PCCG conforms with current Pennsylvania Department of Education curriculum regulations and serves the dual feature of providing both an administrative document and an instructional guide

• Content and subject format remain flexible and adaptable to modification - an “active” document

• Special Pennsylvania Department of Education [PDE] legislation is identified

• Parents and students are provided with an overview of the instructional program and each course in particular

PLANNED COURSE CURRICULUM GUIDE (PCCG)

DEFINITIONS

• Course Description and Intent: a brief overview of the course and program goals

• Instructional Time: frequency of class meetings and time/appropriate credit at the secondary level

• Special Notes: emphatic features or highlights and identification of Department of Education mandates found in the course

• Unit Lesson Outcome: describes the knowledge, skills, attitudes, student performance behaviors and areas of study that have been identified as appropriate to help the student attain the rigorous standards of a quality education

• Teaching-Learning Activities: suggested activities designed to help all students achieve the learning outcomes and standards

• Outcomes/Standards: statements establishing the minimal knowledge, skills, performance behaviors, and essential learning (content) a student must attain

• Expected Levels of Achievement (Learning Outcomes): what students will be expected to do as a result of the application of teaching-learning activities and content

• Evaluation Criteria (Actual Level of Attainment): student performance level achieved and measured through specified evaluation criteria

Subject Title:	MATHEMATICS
Discipline/Grade Level:	AP CALCULUS -- GRADE 12

UNIT LESSON OUTCOME: 1

The learner will identify the concept of a limit and apply limit theorems to solve problems.

RELATIONSHIP TO PA OUTCOMES/STANDARDS (Check Appropriate Graduation Outcomes)

Communications

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Mathematics

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

Science & Technology

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Environment & Ecology

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

Civics & Government

5.1

5.2

5.3

5.4

Economics

6.1

6.2

6.3

6.4

6.5

Geography

7.1

7.2

7.3

7.4

History

8.1

8.2

8.3

8.4

Arts & Humanities

9.1

9.2

9.3

9.4

Health, Safety & PE

10.1

10.2

10.3

10.4

10.5

Family & Consumer Science

11.1

11.2

11.3

11.4

World Language

12.1

12.2

12.3

12.4

12.5

12.6

Career Education & Work

13.1

13.2

13.3

13.4

ESSENTIAL CONTENT OUTCOMES/STANDARD	CONTENT & INSTRUCTIONAL ACTIVITIES/STRATEGIES WITH CORRECTIVES AND EXTENSIONS *(Individually created teaching activities may be used to achieve the standards; however, listed below are activities which may be helpful:*	ACTUAL LEVEL OF ATTAINMENT (EVALUATION CRITERIA) ASSESSMENT	RESOURCES AND MATERIALS
STANDARD 1

� Explore the existence of limits by examining the behavior of algebraic fractions in those areas where they are undefined. � Define limit and use the definition to calculate the limit of a given function. � Prove the limit theorems. � Apply the limit theorems to solving problems. � Discuss one-sided limits. � Define continuity. � Apply the definition of continuity to determine the continuity of a function on an interval.	� Use graphing calculators to enhance the visualization of a limit. � Use graphing calculator to calculate limits that cannot be solved using limit theorems. � Math journal entry describing the definition of limit. � Warm-up problems from previous AP exams. Correctives: � Math tutoring lab. � After school teacher help. � Computer generated worksheets. Extensions: � Bonus problems. � Student generated proofs. � Computer research of calculus web sites.	� Teacher designed tests and quizzes � Worksheets designed to demonstrate knowledge of the concepts taught � Portfolio assessment � Written or oral presentation of projects � Homework assessment � Cooperative group assessment	� Calculus of a Single Variable - Swakowski � Previous AP exam problems � Calculus - Anton � College textbooks � Graphing calculators � Access computer software package � CBL (Calculator Based Lab)

Subject Title:	MATHEMATICS
Discipline/Grade Level:	AP CALCULUS -- GRADE 12

UNIT LESSON OUTCOME: 2

The learner will define, find, and apply the derivative to solve practical problems.

RELATIONSHIP TO PA OUTCOMES/STANDARDS (Check Appropriate Graduation Outcomes)

Communications

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Mathematics

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

Science & Technology

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Environment & Ecology

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

Civics & Government

5.1

5.2

5.3

5.4

Economics

6.1

6.2

6.3

6.4

6.5

Geography

7.1

7.2

7.3

7.4

History

8.1

8.2

8.3

8.4

Arts & Humanities

9.1

9.2

9.3

9.4

Health, Safety & PE

10.1

10.2

10.3

10.4

10.5

Family & Consumer Science

11.1

11.2

11.3

11.4

World Language

12.1

12.2

12.3

12.4

12.5

12.6

Career Education & Work

13.1

13.2

13.3

13.4

ESSENTIAL CONTENT OUTCOMES/STANDARD	CONTENT & INSTRUCTIONAL ACTIVITIES/STRATEGIES WITH CORRECTIVES AND EXTENSIONS *(Individually created teaching activities may be used to achieve the standards; however, listed below are activities which may be helpful:*	ACTUAL LEVEL OF ATTAINMENT (EVALUATION CRITERIA) ASSESSMENT	RESOURCES AND MATERIALS
STANDARD 2

� Define the derivative in terms of the slope of the tangent to a function on a given interval. � Make an application of the derivative on an open interval to the physical properties of velocity. � Connect the idea of one-sided limits to the definition of right handed and left handed derivatives on a closed interval. � Discover the connection between the existence of a derivative and the continuity of the function on the interval. � Prove the theorems defining the rules for finding derivatives. � Define increment as the change in x, and differential by the dependent variable. � Apply increment to volume problems. � Define average error, percentage error, and relative error. � Define the Chain Rule for finding the derivative of composite functions. � Extend the Chain Rule definition to include power functions. � Apply the rules of differentiation to algebraic functions. � Define higher order derivatives. � Calculate first, second, and third derivatives of functions where they exist. � Establish the intervals on which a function is increasing or decreasing. � Utilize the concepts of increasing and decreasing to establish relative maximum and minimum points of a function. � Prove Rolle's Theorem and the Mean Value Theorem. � Using higher order derivatives establish the concavity of the graph of a function. � Find the points of inflection, if they exist, of the graph of a function. � Establish the existence of horizontal and vertical asymptotes wherever they occur in the graph of a function. � Connect the concept of horizontal or vertical asymptotes with those areas where the function increases or decreases without bound. � Develop a strategy for applying the idea of extreme to practical problems. � Find the velocity and acceleration of a particle moving along a line. � Calculate the average and instantaneous rates of change. � Establish the derivative as a rate of change. � Apply the rules of differentials to physical problems involving related rates. � Use the concepts of inverses to determine the antiderivatives. � Make application of the derivative and antiderivatives to problems in economics. � Find derivative of logarithmic and exponential functions.	� Use graphing calculator to have students discover the derivatives of trig functions and exponential functions. � Use graphing calculator to enhance understanding of relative max and min values. � Use calculus "match game" to have students identify a graph based on its derivative and vice-versa. � Guided discovery. � Cooperative learning activities. � Warm-up problems from previous AP exams. Correctives: � Math tutoring lab. � Teachers after school help. � Computer generated worksheets. Extensions: � Bonus problems. � Student generated max/min problems. � Student generated proofs.	� Teacher designed tests and quizzes � Worksheets designed to demonstrate knowledge of the concepts taught � Portfolio assessment � Written or oral presentation of projects � Homework assessment � Cooperative group assessment	� Calculus of a Single Variable - Swakowski � Previous AP exam problems � Calculus - Anton � College textbooks � Graphing calculators � Access computer software package � CBL (Calculator Based Lab)

Subject Title:	MATHEMATICS
Discipline/Grade Level:	AP CALCULUS -- GRADE 12

UNIT LESSON OUTCOME: 3

The learner will define, find, and apply the definite integral to solve practical problems.

RELATIONSHIP TO PA OUTCOMES/STANDARDS (Check Appropriate Graduation Outcomes)

Communications

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Mathematics

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

Science & Technology

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Environment & Ecology

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

Civics & Government

5.1

5.2

5.3

5.4

Economics

6.1

6.2

6.3

6.4

6.5

Geography

7.1

7.2

7.3

7.4

History

8.1

8.2

8.3

8.4

Arts & Humanities

9.1

9.2

9.3

9.4

Health, Safety & PE

10.1

10.2

10.3

10.4

10.5

Family & Consumer Science

11.1

11.2

11.3

11.4

World Language

12.1

12.2

12.3

12.4

12.5

12.6

Career Education & Work

13.1

13.2

13.3

13.4

ESSENTIAL CONTENT OUTCOMES/STANDARD	CONTENT & INSTRUCTIONAL ACTIVITIES/STRATEGIES WITH CORRECTIVES AND EXTENSIONS *(Individually created teaching activities may be used to achieve the standards; however, listed below are activities which may be helpful:*	ACTUAL LEVEL OF ATTAINMENT (EVALUATION CRITERIA) ASSESSMENT	RESOURCES AND MATERIALS
STANDARD 3

� Establish summation notation. � Use the idea of the limit of the sum of a series to calculate the area under a curve. � Define the definite integral. � Use Riemann's definition of an integral to calculate the area under a curve. � Relate the area under a curve to the definition of a definite integral. � Make the connection between the properties of the definite integral and the limit theorems of the previous unit. � Prove the Mean Value Theorem for definite integrals. � Prove the Fundamental Theorem of Calculus. � Define indefinite integrals. � Use the power rule for indefinite integration. � Prove the change in variable theorem. � Define the Trapezoidal Rule and the Error Estimate for same. � Define Simpson's rule and Error Estimate for same. � Apply properties of antiderivatives to growth and decay problems. � Solve problems of the form f(y)dy=g(x)dx. � Use the Trapezoidal Rule and Simpson's rule to approximate the definite integrals for stated values. � Apply the idea of integrals to find the area between two curves. � Find the volume of a solid of revolution by washers and discs, cylindrical shells, and by slicing. � Define Hooke's Law. � Use Hooke's Law to solve work problems. � Apply the rules of the integrand to force problems. � Use the methods of integration to find the length of irregular arcs. � Use the integral as an accumulation function.	� Use graphing calculator to enhance student visualization of calculating area under a curve by use of inscribed and circumscribed polygons. � Use graphing calculator programs to do various techniques of numeric integration. � Guided discovery. � Cooperative learning activities. � Warm-up problems from previous AP exams. Correctives: � Math tutoring lab. � Teachers after school help. � Computer generated worksheets. Extensions: � Bonus problems. � Student generated proofs. � Student generated problems involving volume, area, or work.	� Teacher designed tests and quizzes � Worksheets designed to demonstrate knowledge of the concepts taught � Portfolio assessment � Written or oral presentation of projects � Homework assessment � Cooperative group assessment	� Calculus of a Single Variable - Swakowski � Previous AP exam problems � Calculus - Anton � College textbooks � Graphing calculators � Access computer software package � CBL (Calculator Based Lab)

Subject Title:	MATHEMATICS
Discipline/Grade Level:	AP CALCULUS -- GRADE 12

UNIT LESSON OUTCOME: 4

The learner will define logarithmic and exponential functions and their properties and apply transcendental functions to models of growth and decay.

RELATIONSHIP TO PA OUTCOMES/STANDARDS (Check Appropriate Graduation Outcomes)

Communications

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Mathematics

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

Science & Technology

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Environment & Ecology

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

Civics & Government

5.1

5.2

5.3

5.4

Economics

6.1

6.2

6.3

6.4

6.5

Geography

7.1

7.2

7.3

7.4

History

8.1

8.2

8.3

8.4

Arts & Humanities

9.1

9.2

9.3

9.4

Health, Safety & PE

10.1

10.2

10.3

10.4

10.5

Family & Consumer Science

11.1

11.2

11.3

11.4

World Language

12.1

12.2

12.3

12.4

12.5

12.6

Career Education & Work

13.1

13.2

13.3

13.4

ESSENTIAL CONTENT OUTCOMES/STANDARD	CONTENT & INSTRUCTIONAL ACTIVITIES/STRATEGIES WITH CORRECTIVES AND EXTENSIONS *(Individually created teaching activities may be used to achieve the standards; however, listed below are activities which may be helpful:*	ACTUAL LEVEL OF ATTAINMENT (EVALUATION CRITERIA) ASSESSMENT	RESOURCES AND MATERIALS
STANDARD 4

� State the laws of exponents. � State the laws of logarithms. � Define the natural logarithmic function. � Using the idea of inverse to find the natural exponential function. � Apply the above laws and definitions to a variety of problems. � Prove the theorems for differentiation and integration of natural exponential functions. � Extend the laws of logarithms and exponents to others bases. � Apply the laws of exponents to problems of growth and decay. � Use L'Hapitals rule to find limits of functions that have indeterminate forms. � Use integration by parts to find integrals.	� Use graphing calculators to enhance student comprehension of growth and decay problems. � Guided discovery. � Cooperative learning activities to enhance curriculum. � Warm-up problems from previous AP exams. Correctives: � Math tutoring lab. � Teachers after school help. � Computer generated worksheets. Extensions: � Bonus problems. � Student generated growth and decay problems. � CBL activity on Newton's Law of Cooling.	� Teacher designed tests and quizzes � Worksheets designed to demonstrate knowledge of the concepts taught � Portfolio assessment � Written or oral presentation of projects � Homework assessment � Cooperative group assessment	� Calculus of a Single Variable - Swakowski � Previous AP exam problems � Calculus - Anton � College textbooks � Graphing calculators � Access computer software package � CBL (Calculator Based Lab)

1
5