Riverside Unified School District

COURSE PROPOSAL NVCC COLLEGE-WIDE COURSE CONTENT SUMMARY MTH 291 – DIFFERENTIAL EQUATIONS (3 CR.)

I. Course Purpose and Description:

Differential Equations and Topics in Differential Equations is a proposed rigorous college level course that will be offered to graduating seniors who completed Calculus BC during their junior year. This dual enrollment course will prepare students who are interested in pursuing post secondary studies in science, technology, engineering, or mathematics.

Differential Equations will give students a basic understanding of the techniques used to solve ordinary differential equations. Students will be introduced to first order differential equations, linear differential equations, and numerical methods and applications.

II. Course Goals and/or Major Student Outcomes:

Upon completion of each course students will graduate from T.C. Williams High School with a total of 3 college credits. Students will also be able to:

MTH 291 (3 credits) 1. Display a working knowledge of the terminology of differential equations. 2. Solve first order differential equations using standard techniques. 3. Solve homogeneous and non-homogenous linear differential equations with constant coefficients 4. Solve system of linear differential equations using eigenvalues. 5. Solve applied problems such as growth and decay, oscillatory motion, and electric circuits.

III. Prerequisite

 Successful completion of Calculus BC  NVCC Placement test

IV. Textbook

Differential Equations with Boundary Value Problems (7th edition) – Zill/Cullen

Textbook list price - $242.99

Total Cost based upon student enrollment - $1943.92

V. Dual Enrollment

1 Revised 8/30/11 This course will consist of rising seniors who have completed AP Calculus BC with a score of 3 or better on the AP exam. This course also provides first exposure to college level courses and allows students to earn 3 college credits in addition to a high school math elective credit. Allowing students to take this dual enrollment course fulfills two of our long range goals for college preparedness:

1. Ensure that all ACPS students develop five key college preparation competencies: (a) reading comprehension; (b) writing to promote post-secondary success; (c) data analysis and interpretation; (d) discourse within the disciplines; and (e) speaking and listening. 2. Use these five college preparation competencies as a framework and catalyst for equipping all ACPS students for success in both post-secondary education and the world of work.

VI. Instructor

Louis Kokonis will be the instructor of the course. Mr. Kokonis has taught the course at NVCC in previous years.

VII. Course At A Glance

This course is for students seeking to acquire a year of college level mathematics credit in Differential Equations. This course will focus on the following concepts: a. 2 Revised 8/30/11 Display a working knowledge of the terminology of differential equations. b. Solve first order differential equations using standard techniques. c. Solve homogeneous and non-homogenous linear differential equations with constant coefficients. d. Solve system of linear differential equations using eigenvalues. e. Solve applied problems such as growth and decay, oscillatory motion, and electric circuits.

Quarter 1 Quarter 2 Quarter 3 Quarter 4 Unit 1: Unit 3: Second Order Unit 4: Higher Order Unit 6: Laplace Introduction and First Differential Equations Linear Equations Transform Definitions: In this unit students will learn about In this unit students will understand In this unit students will learn about homogeneous and non-homogeneous In this unit students will learn the that a differential equation is an higher order linear equations. linear differential equations with basic definitions and results of equation involving an unknown Students will explore topics such as: constant coefficients. Students will laplace transform. Students will function and its derivatives. Students explore topics such as:  Introduction to higher order explore topics such as: will also understand that the order of linear equations and basic  Nonlinear equations the differential equation is the order results  Linear equations  Impulse functions of the highest derivative of the  Method of underdetermined unknown function involved in the  Homogenous linear equations coefficients. Linear independence  Convolution product equation. The material in this unit   Method of variation of will be reinforced throughout the  Reduction of order parameters. school year.  Homogeneous equations with  Table of Laplace Transforms Unit 2: First Order constant coefficients  Non-homogeneous linear Unit 5: Applied Unit 7: Systems of Differential Equations equations Problems Differential Equations Students will explore techniques for  Series solutions Students will use techniques for solving first order differential solving differential equations to equations. Topics may include: Students will use techniques for explore systems. Students will  Linear equations solving differential equations to explore topics such as:  Separable equations solve applied problems, to include  Motivation  Qualitative Technique: Slope selections from  Euler’s Method for Systems Fields 1. Growth and decay  Qualitative Analysis  Equilibria and the Phase Line 2. Orthogonal trajectories  Bifurcations 3. Geometrical uses  Bernoulli Equations 4. Mixing of solutions Unit 8:  Riccati Equations 5. Vibrating systems Students will learn  Homogeneous equations 6. Electric circuits Qualitative analysis of systems of  Exact and non-exact equations ODE's - phase plane and stability.  Integrating factor technique Students will also learn about applications such as:  Radioactive decay  Newton’s Law of Cooling  Orthogonal Trajectories  Population Dynamics  Numerical Technique: Euler’s Method  Existence and Uniqueness of Solutions  Picard Iterative Process

T.C. Williams High School

Math 291: Differential Equations Louis Kokonis

Best time to contact: 8:45 a.m. to 9:30 a.m. – Email anytime Office hours: 8:00 am. to 8:30 a.m. daily 3:20 – 5:30 Monday, Tuesday, Wednesday, Thursday, and Friday Email: [email protected] Phone 703-824-6631 (school)

Syllabus 3 Revised 8/30/11 Course Description: Differential Equations is a rigorous dual enrollment course with Northern Virginia Community College designed to introduce students to first order differential equations, linear differential equations, numerical methods, and applications. This course is primarily for the student who has an interest in mathematics, engineering, the sciences and other areas requiring strong mathematical backgrounds. The purpose of this course is to give the student a basic understanding of the techniques for solving ordinary differential equations. In addition, this course will show students how differential equations can be used to make conclusions about the behavior of a function and to integrate analytical, graphical and numerical techniques in making predictions about the future status of a substance based on it rate of change.

Course Essential Questions: 1. What is a differential equation? 2. How do we use the separation of variables method? 3. How do we solve real world problems using differential equations? 4. How can differential equations be used to predict the behavior of a function? 5. How can differential equations be analyzed analytically, graphically, and numerically to make predictions?

Course At A Glance: This course is for students seeking to acquire a year of college level mathematics credit in Differential Equations. This course will focus on the following concepts: a. Display a working knowledge of the terminology of differential equations. b. Solve first order differential equations using standard techniques. c. Solve homogeneous and non- homogenous linear differential equations with constant coefficients. d. Solve system of linear differential equations using eigenvalues. e. Solve applied problems such as growth and decay, oscillatory motion, and electric circuits.

4 Revised 8/30/11 Quarter 1 Quarter 2 Quarter 3 Quarter 4 Unit 1: Unit 3: Second Unit 4: Higher Unit 6: Laplace Introduction and Order Differential Order Linear Transform First Definitions: Equations Equations In this unit students will learn In this unit students will learn In this unit students will In this unit students will learn the basic definitions and results about homogeneous and non- understand that a differential about higher order linear of laplace transform. Students homogeneous linear differential equation is an equation equations. Students will will explore topics such as: equations with constant involving an unknown function explore topics such as: coefficients. Students will and its derivatives. Students  Introduction to higher  Impulse functions explore topics such as: will also understand that the order linear equations and  Nonlinear equations order of the differential basic results  Convolution product  Linear equations equation is the order of the  Method of  Homogenous linear highest derivative of the underdetermined  Table of Laplace unknown function involved in equations coefficients. Transforms Linear independence the equation. The material in   Method of variation of this unit will be reinforced  Reduction of order parameters. Unit 7: Systems of  Homogeneous equations throughout the school year. Differential Unit 2: First Order with constant coefficients  Non-homogeneous linear Unit 5: Applied Equations Differential equations Problems Students will use techniques for Equations  Series solutions solving differential equations to explore systems. Students will Students will use techniques explore topics such as: Students will explore for solving differential  Motivation techniques for solving first equations to solve applied  Euler’s Method for order differential equations. problems, to include selections Systems Topics may include: from  Qualitative Analysis  Linear equations 1. Growth and decay  Separable equations 2. Orthogonal trajectories  Qualitative Technique: 3. Geometrical uses Unit 8: Slope Fields 4. Mixing of solutions Students will learn  Equilibria and the Phase 5. Vibrating systems Qualitative analysis of systems Line 6. Electric circuits of ODE's - phase plane and  Bifurcations stability.  Bernoulli Equations  Riccati Equations  Homogeneous equations  Exact and non-exact equations  Integrating factor technique Students will also learn about applications such as:  Radioactive decay  Newton’s Law of Cooling  Orthogonal Trajectories  Population Dynamics  Numerical Technique: Euler’s Method  Existence and Uniqueness of Solutions  Picard Iterative Process

Grading Criteria:  Exams / Tests / Transfer Tasks/Projects /Quizzes 90 %  Homework / Classwork / Notebook 10%

5 Revised 8/30/11 Materials:

 Textbook, notebook, pen or pencil, and calculator

Supports Available:

Students are expected to use these supports to enhance academic achievement.  Math Center, Writing Center, Saturday Learning Academy, Titan Up, Before and after School Tutoring (See office hours), Online Tutorials (See Blackboard)

PBIS – Titan Pride:

Titan PRIDE: Positive Attitude, Respect/Responsibility, Integrity, Determination, Excellence

Through PBIS we will teach Titan Pride. Students and staff will work together to encourage positive behaviors in the classroom setting and create a positive school environment. Students with Titan Pride demonstrate a positive attitude, respect/responsibility, integrity, determination, and excellence. Students are encouraged to show their Titan Pride in the hallways, cafeteria, bathrooms, classrooms, school bus, and in the community.

Expectations for the Student as Worker:

Students will be expected to engage in bell-to-bell instruction.  Students will be required to access Blackboard as a part of the learning environment. Both assignments and worksheets will be posted.  Students will show effective effort through participation during class activities, including whole group, small group, and independent work.  Students will demonstrate critical thinking in evaluating/solving problems,  Students will be expected to revisit work in a timely manner when standards and expectations are not achieved.  Students will play an active role in monitoring their own progress, including recording formative and summative assessment data in their academic notebooks.  Students will demonstrate efficacy and responsibility, continually affirming their active and direct role in their learning process.  Students are expected to understand and follow the attendance procedures of TCW.

Expectations for the Instructor as Teacher and Coach:

It is my fervent wish and my absolute commitment that each of you learn all that this course includes and that you find satisfaction in your accomplishments, just as I find satisfaction in knowing that I did all I possibly can to help you achieve to the highest possible levels. As your Titan Teacher, I am committed to the following:  Teachers will facilitate an instructional environment to support the Student As Worker.  Teachers will create an instructional environment that encourages 21st Century Skills such as critical thinking, problem solving, communication, collaboration, creativity, and innovation

6 Revised 8/30/11  Teachers will model, guide, facilitate, monitor, provide feedback, and assess students to support their active thinking and learning.  Teachers will design their learning environment to support and challenge student thinking.  Teachers will develop strategies to help students direct their learning.

Assessment:

Pre-Assessment/Diagnostic:  Diagnostic Test/Quiz – Used to determine students’ readiness or requisite knowledge and skills  Written performance – Used when assessing students’ background knowledge (e.g. – academic prompts or essays).  Informal Diagnostic – Listen/ think /pair /share, voting activities, brainstorming, visual representation, anticipation guides, mini-seminars, pairs share, and initial response to unit essential questions.

Formative Assessment:  Daily criterion based feedback to students to help them self-monitor, self- regulate, and self-adjust.  Teacher observation of student behaviors, criterion based coaching feedback, student engagement in modeling and shaping tasks, voting activities, reciprocal teaching strategies, checklists, graphic organizers, student exit slips, reflective journal entries, think/log entries, peer coaching activities, and peer response groups.

Summative Assessment:  Unit Transfer Tasks, academic prompts, projects, unit tests, constructed- response test and quiz items (i.e. not multiple choice/true false), and selected- response test and quiz items (i.e. multiple choice/true false)

Grading Policy: According to ACPS IFA Regulations:

ACPS Division Grading Policy (IFA) Students shall be expected and permitted to revisit work when, in the teacher’s professional judgment, it is clear that additional study, effort and time will produce higher achievement.”

 All students shall be given opportunities to succeed in meeting or exceeding the standards and related benchmarks.  Students shall be expected and permitted to revisit and review their work (e.g., tests, quizzes, essays) in all instances where it is clear that additional study, effort, and time will produce improved performance and achievement.

7 Revised 8/30/11  The teacher and student shall revisit the learner’s work if, in the teacher’s professional judgment, progress and effort are being made to meet or exceed the standards.  The final grade awarded for reassessment performance shall reflect the student’s mastery of identified course standards and task-specific performance indicators.  Grades will be based on students’ achievement of identified standards and related performance criteria.

Assigning an Incomplete (I) If the assignment / assessment has not been submitted the teacher may assign a grade of Incomplete (I) as a placeholder so that the student can work with the teacher to complete the assignment.  The teacher shall use professional judgment when assigning a grade of Incomplete (I).  When a teacher has determined that a student has worked hard and still cannot demonstrate mastery, a grade of 50-59 can be assigned.  When the teacher has determined that the student has made no effort on the assignment and does not plan on completing the assignment, a numerical grade of 40 should be applied.

Grading Scale: High School Courses Numerical Letter Advanced Standard Receives Include Weighted Honors Grade Grade Placement Classes Credit in GPA Credit 93-100 A 5.0 4.5 4.0 Y Y Y 90-92 A- 4.7 4.2 3.7 Y Y Y 87-89 B+ 4.3 3.8 3.3 Y Y Y 83-86 B 4.0 3.5 3.0 Y Y Y 80-82 B- 3.7 3.2 2.7 Y Y Y 77-79 C+ 3.3 2.8 2.3 Y Y Y 73-76 C 3.0 2.5 2.0 Y Y Y 70-72 C- 1.7 1.7 1.7 Y Y N 67-69 D+ 1.3 1.3 1.3 Y Y N 60-66 D 1.0 1.0 1.0 Y Y N 59-40 F 0.0 0.0 0.0 N Y N I N N N

8 Revised 8/30/11 NG N N N NR N N N P N N N WP N N N WF N N N I Incomplete Given when a student has not completed sufficient assignments and assessments to calculate a valid quarter grade. Students will work with the teacher to resolve the incomplete within 10 days of the end of the quarter. NG Not Graded Given when a student exits a class in the middle of the quarter to enroll in a more appropriate placement. NR Not Given when a particular assignment is not required of an individual Required student for such reasons as medical, IEP accommodation, senior experience, etc. P Pass Given for pass/fail classes such as PACE, SOL Prep, etc. WF Withdraw Given when a student withdraws from a class with a current grade of F. Failing WP Withdraw Given when a student withdraws from a class with a passing grade. Passing

Alexandria Access:

Alexandria Access is a parent portal that allows parents to view their students’ grades and attendance. This portal is used as a tool to assist parents and students in monitoring their academic progress. Teachers are required to update Alexandria Access biweekly.

9 Revised 8/30/11 10 Revised 8/30/11