PRE-CALCULUS / TRIGONOMETRY

Course Name: Pre-Calculus / Trigonometry

Course Number: 250

Level: Honors

Credit: 1

Prerequisites: Minimum of "B" in Algebra II Honors; signature of current math teacher

The Pre-Calculus course provides a transition between high school and college texts. Intermediate algebra, analytic geometry, and trigonometry are integrated with other important topics in mathematics by an approach that stresses functions. Separate chapters place special emphasis on trigonometry functions, polynomial functions, and

transcendental functions. A discussion of rational and irrational numbers provides an early introduction to limits. This concept reappears in sections on graphing functions, upper and lower bounds, and sequences. In the latter part of the course the now familiar concept of limit is treated again using a more formal, precise definition. This is an intuitive introduction to differential calculus. A graphing calculator is used in this course.

1. Course Goal :

To prepare the student for a calculus program.

1. Course Objectives :

The student will:

1. Solve problems using the trigonometric functions.

1. The student will use deductive logic to prove trigonometric identities.

1. The student will show the relationship between radians and degrees.

1. The student will classify conics of the form

Ax + Bxy + Cy + Dx + Ey + F = 0.

1. The student will apply the remainder theorem, fundamental theorem of algebra and the rational roots theorem in solving

polynomial equations.

1. The student will add, subtract, multiply, and divide complex numbers.

1. The student will solve equations having complex roots.

1. The student will solve transcendental functions.

1. The student will prove statements using Mathematical Induction.

10. The student will determine if a sequence is infinite or finite.

10. The student will solve matrices.

10. The student will define a limit.

10. The student will use the delta method to differentiate a polynomial expression.

3. Unit One: Circular Functions and Trigonometry:

1. Unit Content:

1. Circular Functions

1. Graphs of Circular Functions

3. Other Circular Functions

4. Sum and Difference Formulas

5. Double and Half Number Formulas

6. Equating Products and Sums

7. Inverse Relations

8. Equations with Circular Functions

9. Using Tables and Calculators to Find Trigonometric Values

10. Solving Right Triangles

11. General Triangles

12. Areas

13. Velocity

14. Complex Numbers

15. DeMoivre's Theorem

2. Unit Objectives:

The student will:

1. define the wrapping functions.

1. map trigonometric functions using the wrapping function.

3. develop the real number values for the trigonometric functions.

3. graph the trigonometric functions.

3. prove the trigonometric identities.

3. find the value of a trigonometric function using the trigonometric identities.

3. convert sums and differences of functions into products.

3. solve for the general and principal values of inverse relations.

3. solve equations involving circular functions.

3. convert angular measure in radians to degrees, and degrees to radians.

3. solve problems using polar coordinates.

3. learn to read trigonometric tables and use a calculator to find trigonometric values.

3. use linear interpolation to estimate angular measures.

3. solve for missing measures involving plane triangles.

3. find the areas of geometric shapes in polygons, circles.

3. solve problems of angular velocity.

3. multiply and divide complex numbers expressed in trigonometric form.

3. evaluate complex expressions using DeMoivre's Theorem.

3. Approximate Time:

55 class periods

Pre-Calculus Mathematics , Crosswhite, Hawkinson, Sachs

Chapters 6 and 7

Teacher's Resource Package by Merrill

E. Suggested Activities:

3. The Wrapping Function Cylinder

3. Review

3. Tests

F. Evaluation:

2. Tests

3. Unit Two: Intermediate Algebra and Graphing

1. Unit Content:

3. Points on a Number Line

3. The Number Line Postulate

3. The Field Properties

4. The Order Properties

4. The Completeness Property

4. The Cartesian Coordinate System

4. Distance

4. Slope

4. Equations of a Line

4. Parallel and Perpendicular Lines

4. Distance Between a Point and a Line

4. Loci

4. Complex Numbers

4. Relations and Functions

4. Compositions of Functions

4. Inverse Functions

4. Increasing and Decreasing Functions

4. Symmetry

4. Intercepts and Excluded Regions

4. Asymptotes

4. Translations and Rotations

4. Continuity

3. Unit Objectives:

The student will:

4. graph the set of real numbers by construction.

4. review the Field, Order, and Completeness Properties.

4. find the distance between two points and the slope of a line.

4. review equations of a line.

4. determine if two lines are parallel or perpendicular.

4. solve for the distance between a point and a line.

4. write the cartesian equations of a loci of points.

4. graph complex numbers on a coordinate plane.

4. solve composite functions.

4. find the inverses of functions.

4. distinguish between different types of functions.

4. identify the symmetry, intercepts, and excluded regions of a relation.

4. derive the asymptotes of a relation.

4. translate and rotate the axes.

4. employ transformations when graphing relation.

4. determine the continuity of a function.

3. Approximate Time:

25 class periods

3. Major Resource:

Pre-Calculus Mathematics , Crosswhite, Hawkinson, Sachs

Chapters 1, 2, 3, 4

Teacher's Resource Package by Merrill

E. Suggested Activities:

4. Homework

4. Worksheets

4. Review

4. Tests

6. Evaluation:

4. Homework

4. Tests

5. Unit Three: Analytic Geometry

5. Unit Content:

4. The Circle

4. The Parabola

4. The Ellipse

4. The Hyperbola

4. General Second-Degree Equations

4. Quadric Surfaces

2. Unit Objectives:

The student will:

4. derive the equation, in general or standard form, of a given circle, parabola, ellipse, hyperbola.

4. transform second-degree equations to their conic form by hand or by graphing calculator.

4. use the discriminant to solve for a conic figure or degenerate.

4. identify a quadric surface by using traces.

3. Approximate Time:

17 class periods

3. Major Resource:

Pre-Calculus Mathematics , Crosswhite, Hawkinson, Sachs

Chapter 8

Teacher's Resource Package by Merrill

3. Suggested Activities:

4. Homework

4. Worksheets

4. Cone

4. Review

4. Tests

6. Evaluation :

2. Tests

6. Unit Four: Advanced Algebra

3. Unit Content:

2. Polynomials

2. Graphing Polynomials

2. The Division Algorithm and the Remainder Theorem

2. The Factor Theorem

2. The Fundamental Theorem of Algebra

2. Locating Zeros

2. Rational Zero Theorem

2. Complex Zeros

2. Irrational Zeros

2. Rational Functions

2. The Exponential Function

2. The Number e

2. The Logarithmic Function

2. Exponential and Logarithmic Equations

2. Growth and Decay Functions

16. Definition of Matrix and Determinate

16. Addition and Multiplication of Matrices

16. Identity and inverse Matrices

16. Matrix Solutions of Systems of Equations

16. Homogeneous Systems and Systems Without Unique Solutions

16. Matrices Using a Graphing Calculator

2. Unit Objectives:

The student will:

2. identify polynomial expressions.

2. analyze the properties of polynomials and check with a graphing calculator.

2. evaluate polynomial expressions using the Division

Algorithm and Remainder Theorem.

2. factor polynomials using the Factor Theorem.

2. the real, rational, and complex roots of a polynomial.

2. approximate the irrational roots of a polynomial

expression.

2. graph rational functions.

2. identify transcendental functions.

2. define the number e and expand it using the binomial theorem.

2. develop the logarithmic functions.

2. solve equations containing transcendental expressions.

2. apply transcendental expressions in solving problems.

2. define and calculate determinants.

2. define addition, scalar multiplication, and multiplication of matrices.

2. define multiplication identity and find inverse matrices.

2. solve systems of equations using matrices.

2. introduce systems that have only the trivial solution (0, 0, 0) and systems that have an infinite number of solutions.

2. Use a graphing calculator to solve matrices.

2. Approximate Time:

33 class periods

2. Major Resource:

Pre-Calculus Mathematics , Crosswhite, Hawkinson, Sachs

Teacher's Resource Package by Merrill

2. Suggested Activities:

2. Homework

2. Worksheets

2. Review

2. Tests

2. Homework

2. Tests

7. Unit Five: Limits

6. Unit Content:

2. Sequences

2. Summations

2. Arithmetic Sequences and Sums

2. Geometric Sequences and Sums

2. Mathematical Induction

2. Neighborhoods

2. Limit of a Sequence

2. Series

2. Limit of a Function

2. Limit Theorems

2. Continuity

2. An Application of Continuity

2. Finding Other Limits

2. Limits of Trigonometric Functions

6. Unit Objectives:

The student will:

2. derive a formula for the nth term of a sequence.

2. compute the terms and evaluate the sum of an

arithmetic sequence and a geometric sequence.

3. expand and calculate summations.

3. use Mathematical Induction to prove a theorem.

3. decide which terms are inside or outside a given

neighborhood.

3. identify a specific neighborhood of a sequence.

3. determine the limit of a sequence.

3. calculate the sum of a series.

3. solve problems concerning two-dimensional

neighborhoods.

3. use the limit theorems to determine specific limits.

3. use a false position to derive or approximate a zero.

3. derive the value of trigonometric limits.

3. Approximate Time:

28 class periods

3. Major Resource:

Pre-Calculus Mathematics , Crosswhite, Hawkinson, Sachs

Chapters 13, 14

Teacher's Resource Package by Merrill

3. Suggested Activities:

3. Homework

3. Worksheets

3. Review

3. Tests

3. Evaluation :

1. Homework

1. Tests

6. Unit Content:

3. Differentiation

3. Applications of Differentiation

3. Geometric Interpretation

3. Increasing and Decreasing Functions

3. Stationary Values

3. Curve Sketching

3. Maxima and Minima Problems

3. Integration

3. Applications of Integration

3. Negative Integrals

3. The Fundamental Theorem of Calculus

3. Indefinite Integrals

6. Unit Objectives:

The student will:

3. calculate the average speed for the given time intervals.

3. evaluate the derivative of a function.

3. interpret the derivative geometrically.

3. develop and apply differentiation techniques for nonpolynomial functions.

3. locate and interpret stationary values of a function.

3. apply differentiation techniques to curve sketching.

3. apply differentiation techniques to maximum and minimum solutions.

3. approximate the area under a curve using upper andlower rectangular approximations.

3. compute the area under a curve as the limit of a series.

3. interpret the fundamental theorem of calculus.

3. determine the antiderivatives of a function.

3. will apply indefinite integration as a technique for problem solving.
3. will evaluate definite integrals.

3. compute the area bounded by curves in the plane.

6. Approximate Time:

29 class periods

6. Major Resource:

Pre-Calculus Mathematics , Crosswhite, Hawkinson, Sachs

Chapters 15,16

Teacher's Resource Package by Merrill

6. Suggested Activities:

3. Homework

3. Worksheets

3. Review

3. Tests

6. Evaluation :

3. Homework

3. Tests