PRE-CALCULUS - HONORS - Edison · PRE-CALCULUS - HONORS 3 COURSE OBJECTIVES The student will...

PUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION

PRE-CALCULUS - HONORS Length of Course: Term Elective/Required: Elective School: High Schools Student Eligibility: Grades 10 -11 Credit Value: 5 Credits Date Approved: 9/26/05

PRE-CALCULUS - HONORS

TABLE OF CONTENTS Foreword ------------------------------------------------------------------------------------- 1 Statement of Purpose ------------------------------------------------------------------- 2 Course Objectives ---------------------------------------------------------------------- 3 Suggested Time Schedule --------------------------------------------------------------- 4 Course Content Unit 1: Algebraic Functions --------------------------------------------------- 5 Unit 2: Exponential and Logarithmic Functions ---------------------------- 6 Unit 3: Circular Functions ------------------------------------------------------- 7 Unit 4: Polynomial Functions -------------------------------------------------- 9 Unit 5: Limits ------------------------------------------------------------------------- 10 Unit 6: Graphing Rational Functions ------------------------------------------ 11 Unit 7: Conic Sections ------------------------------------------------------------ 12 Unit 8: Probability, Combinations and Permutations ---------------------- 16 Unit 9: Sequences and Series -------------------- ----------------------------- 18 Unit 10: Polar Coordinates and Complex Numbers ----------------------- 20 Basic Text / References ------------------------------------------------------------------ 22 Course Requirements --------------------------------------------------------------------- 24 Edison Essential Instructional Behaviors (EIB's-Draft 12) ------------------------ 26 Modifications will be made to accommodate IEP mandates for classified students.


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FOREWORD

Mathematics is both cumulative and cyclic. As the student grows more mathematically aware, certain topics should be reexamined from a new level of sophistication. Teachers of the honors course should keep levels of communication open so that each may plan with an awareness of what has been taught in previous courses. Students will be assessed at mid year and at the end of the year. It is expected that the first three units will have been covered by mid year. The final assessment will cover the remaining units. Topics covered in each unit have been written in terms of instructional objectives. “Strategies” and “Notes to Teacher” have been included to indicate potential areas of difficulty and to offer suggestions, explanations and references (including computer and calculator applications). Teachers must realize that this course of study represents the minimum number of concepts to be covered.


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STATEMENT OF PURPOSE Pre-Calculus is an honors-level course which has been designed to prepare students for the Advanced Placement Calculus course offered in grade 12. In this pre-calculus course, emphasis is given to the areas of trigonometry and analytic geometry. The other topics are included to insure that students will enter a calculus course with all of the prerequisite and concepts. This curriculum guide was revised / updated by: Steven Levinson – EHS David Peters – JPS Coordinated by: Don Jobbins, Supervisor of Mathematics Vincent Ciraulo, Supervisor of Mathematics


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COURSE OBJECTIVES The student will demonstrate proficiency in: 1. recognizing and using vocabulary and symbols of Pre-Calculus. 2. applying mathematical functions learned in previous courses. 3. analyzing and graphing the following elementary functions. a. algebraic b. polynomial c. circular d. exponential e. logarithmic 4. applying algebraic and graphing techniques to the study of rectangular and polar

coordinates. 5. applying algebraic and graphing techniques to the study of conic sections,

sequences and limits. 6. applying concepts of basic statistics to the study of permutations, combinations,

and probability.


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SUGGESTED TIME SCHEDULE (Suggested Instructional Time)

UNIT

Class Periods

1. Algebraic Functions

15

2.

Exponents and Logarithms

15

3.

Circular Functions a. Trigonometric Functions b. Triangle Trigonometry c. Advanced Graphing d. Trigonometric Formulas

15 15 14 11

4.

Polynomial Functions

12

5.

Limits

10

6.

Graphing Rational Functions

8

7.

Conic Sections

18

8.

Probability, Combinations, and Permutations

16

9.

Sequences and Series

11

10.

Polar Coordinates and Complex

10

Total Class Periods

170


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UNIT 1: FUNCTIONS Objective: To combine functions in a variety of ways, explore relationships between graphs and algebraic rules, and solve real world function problems.

Objective

Materials

Strategies

Notes To Teacher

Reference to Standards

The student will be able to: 1. identify functions. 2. determine domain, range, and zeros

of a function. 3. graph a function. 4. perform operations on functions. 5. reflect graphs and to use symmetry. 6. determine periodicity and amplitude

from graphs. 7. stretch, shrink, and translate graphs. 8. determine the equation and sketch the graph of the inverse of a function. 9. use a graphing calculator to graph functions and interpret data. 10. form functions from verbal descriptions.

Teacher text: Advanced Mathematics Chapter 4 pp. 118-167 Graphing Calculator

1. The use of the graphing calculator throughout this unit facilitates understanding.

2. The overhead graphing calculator makes it easier for students to follow along.

3. Many students will own their own graphing calculator. Using a classroom set of T1-82 or T1-83 makes it easier for everyone to stay together.

4. Cooperative learning activities work well throughout this unit and particularly with section 4.7.

1. Functions are a common

thread to many topics in Pre-Calculus. Periodicity, odd and even, increasing and decreasing, symmetry and transformation occur throughout the course.

2. Graphing calculators are

required. By the end of this unit, students should be familiar with the “y=”, window, table, graph, and zoom buttons. They should be able to use the calc button to find values and roots.

3. www.ti.com provides many wonderful lesson ideas for use with this unit.

4.1 A 4.2 ABCE 4.3 BCD 4.5 ABCDEF

http://www.ti.com/


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UNIT 2: EXPONENTS AND LOGARITHMS Objective: To study the laws of exponents and logarithms and apply them to real-life situations.

Objective

Materials

Strategies

Notes To Teacher


The student will be able to: 1. define and apply integral exponents. 2. define and apply rational exponents. 3. define and use exponential

functions and their inverse logarithmic functions.

4. define and apply the natural

exponential function (the number e). 5. apply the laws of logarithms. 6. solve exponential equations and to

change logs from one base to another.

7. solve problems involving exponential growth and decay and half-life problems.

Teacher text: Advanced Mathematics Chapter 5 pp. 168-211 Graphing Calculators CBL

1. A cooperative learning review works well in reviewing the laws of exponents in section 5.1.

2. Using the graphing calculator helps students see the patterns in the graphs of exponential functions.

3. Use and discuss the application problems throughout this unit. Some lend themselves to mini-projects which are more real-life relevant than the simplifying of given expressions.

4. Using the CBL, explore Newton’s Law of Coding.

1. Students need to be reminded

that the laws of exponents are good for products of expressions, not sums.

2. Some scientific calculators will not

calculate the power of a negative number directly. Remind students to “know their calculator”.

3. Confer with the science department to determine the possibility of some interdisciplinary lessons involving growth and decay. 4. Section 5.4 (natural exponential functions) lends itself to an historical discussion of Leonhard Euler (1707-1783)

4.1 ABC 4.2 ACD 4.3 BCD 4.5 ABCDEF


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UNIT 3: CIRCULAR FUNCTIONS

Objective: To explore the trigonometric functions, discover relationships between graphs and algebraic rules, and to solve real world problems.

Objective

Materials

Strategies

Notes To Teacher


The student will be able to:

1. find the measure of an angle in degrees or radians and to find cuterminal angles.

2. find the arc length and area of a sector of a circle and to solve apparent size problems.

3. find the value of trig functions by definition, calculators, and tables.

4. use reference angles and sketch graphs of the trigonometric functions.

5. find values of the inverse trigonometric functions.

6. solve and apply trigonometric equations.

7. find the inclination of a line.

8. graph sine and cosine curves and to interpret sine and cosine graphs.

9. use trig functions to model periodic behavior.

Teacher text: Advanced Mathematics Chapter 7-10 pp. 256-393 Graphing Calculators CBL Computers

1. Teacher Edition pp. 256B-25CC describe an excellent apparent size cooperative learning activity.

2. Have students make an index card of sin/cos/tan values. They need to memorize. The trace feature of the T1-82 confirms entries and also shows repetition of values.

3. A CBL and a tuning fork is an excellent way to show sine/cosine curves.

4. Use the T1-82 to graph the trig functions. Graph reciprocal functions on the same screen. Graph co-functions on the same screen. Graph y = 3sinx and y=sin3x on the same screen. Students will have a vivid display of relationship among curves.

1. The use of the graphing calculator is a must throughout this unit. By the end of this unit, students should be familiar with all trig function buttons, mode, trace, and calc.

2. Students will frequently forget to set

the mode to radians before graphing. A discussion of why this needs to be done tends to help them remember.

3. There are many real world application problems throughout these 4 chapters that warrant discussion. Use as many as time permits. 4. Students using calculators will want to give approximate answers such as 4525.48 as opposed to

the exact answer of 3200√2. Both

answers should be acceptable, however, students need to be informed that if taking the traditional midterm, answers will be in exact form.

4.1 ABC 4.2 ACDE 4.3 BCD 4.5 ABCDEF


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UNIT 3: CIRCULAR FUNCTIONS - CONTINUED

Objective: To explore the trigonometric functions, discover relationships between graphs and algebraic rules, and to solve real world problems.

Objective

Materials

Strategies

Notes To Teacher


10. simplify trigonometric expressions.

11. prove trigonometric identities.

12. use identifies and technology to solve more difficult trigonometric equations.

13. use trigonometry to find missing sides or angles of a right triangle.

14. find the area of a triangle given two sides and an included angle.

15. find the area of a triangle given three sides (Hero’s Formula)

16. use the Law of Sines and the Law of Cosines to solve triangles.

17. use the trigonometric addition, double-angle, and half-angle formulas to simplify and evaluate expressions, prove identities, and solve equations.

18. find the angle between two intersecting lines.

5. Using trig ratios, take the students outside to find the height of a telephone pole or tree. 6. Geometer Sketchpad can be used to explore the ambiguous case for Law of Sines. If a classroom set of computers is not possible, then a demonstration lesson using the Proxima can be just as effective. 7. For assessment purposes, this unit can be divided into four parts – trig functions, triangle trig, graphing, and formulas.

5. There are programs available for the graphing calculators that will compute law of sines, cosines, area formulas, etc. If you do not want these formulas available to them during tests or quizzes, issue a classroom set on that day. (The new T1-83 plus has an archive feature so that even if you clear a student’s calculator memory, any program that’s been archived will not be cleared.)

6. Students can check their

solutions of trig equations by using the graphing calculator. Review the use of the window feature at this time. Students may also use the test feature on the graphing calculator to check their solutions.

7. Students are sometimes resistant to the use of radian measure. Advise them that angles are most frequently expressed in radians in AP Calculus.


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UNIT 4: POLYNOMIAL FUNCTIONS

Objective: The student will demonstrate the ability to graph polynomial functions.

Objective

Materials

Strategies

Notes To Teacher


The student will be able to demonstrate the ability to:

1. use remainder and factor theorems and synthetic division to find factors of the polynomial function P(x) = a1x

n + a2x

n-1 +….+an

2. sketch cubic and quantic polynomial functions. 3. find possible rational roots of a polynomial function by using the rational root theorem. 4. to use the location principal to locate roots during the synthetic division process. 5. to use Descartes Rule of signs and upper and lower bounds for the real roots of P(x) = 0. 6. to graph a polynomial function P(x) = a1x

n + a2x

n-1 +…. ….+an by

using objectives (1-5).

Text Graphing Calculator

1. Spend time for lesson on use of graphing calculator.

1. Introduce methods to approximate irrational roots. 2. See Appendix 3, pp. 835-837 in text.

4.1 A 4.2 AC 4.3 BCD 4.5 ABCDEF


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UNIT 5: LIMITS

Objective: The student will demonstrate the ability to find the limit of a function.

Objective

Materials

Strategies

Notes To Teacher


The student will demonstrate: 1. the ability to find lim f(x) by using x → c

a. left and right hand limits. b. the properties of limits

(1) Lim f(x) ● g(x) = Lim f(x) ● Lim g(x)

x → c x → c x → c

(2) Lim K f(x) = K Lim f(x) x → c x → c

(3) Lim K - K x → c

(4) Lim f(x) = x → c, Lim g(x) ≠ 0 9(x) Lim g(x) x → c

x → c

C. the graph of the function.

2. the ability to find limits as x → ∞

3. the ability to find points of discontinuity and removable discontinuity. *Optional topics: 4. The student will demonstrate the ability to find limits using the property.

Lim sinθ = 1

θ→0 θ

Text Graphing Calculator

1. Use graphing calculator as a tool to investigate what is happening as x approaches a certain number. 2. Students need additional worksheets in addition to text problems.

1. Students should have ability to solve limit problem by using Algebraic and pencil graphing first then check results using graphing calculator. 2. Students need additional worksheets in addition to text problems.

4.1 A 4.2 AC 4.3 ABD 4.5 ABCDEF


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UNIT 6: GRAPHING RATIONAL FUNCTIONS

f(x) Objective: The students will demonstrate the ability to graph functions of the type y=g(x).

Objective

Materials

Strategies

Notes To Teacher


The student will demonstrate the ability:

1. to determine if a function is symmetric to the y axis, x axis, origin or the line y = x.

2. to find x and y intercepts of the graph of y = f(x). g(x)

3. to find vertical and horizontal asymptotes of the graph of y = f(x). g(x)

4. to find holes in the graph of y = f(x). g(x)

5. to find the equation of a slant asymptote of the graph of y=f(x). g(x)

6. to find crossing pt(s) on horizontal and slant asymptotes.

*Optional topics: The student will demonstrate the ability to graph functions of the form

y= f(x) g(x)

7. to graph functions of the form y=f(x) by using objectives (1-6) g(x)

Text Also Chapter 7 Algebraic Curves Analytic Geometry Gordon Fuller Graphing Calculator

1. Students should have the ability to graph functions using a combination of graphing calculators, pencil and paper.

4.1 4.5 A1 4.2 4.4.6 A3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4, 4.5F6 4.6 4.3.8A1, 4.3.12A2 4.8 4.1.12B2,4.1.12B4

4.10 4.4.12C1 4.11 4.3.12B2 4.13 4.3.1D2 4.15 4.3.12A2 4.16 Encouraging all students to go beyond the standards in the introduction, under “The Vision –Equity and Excellence.”


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UNIT 7: CONIC SECTIONS

Objective: The students will demonstrate the ability to graph conic sections of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

Objective

Materials

Strategies

Notes To Teacher


The student will demonstrate: 1. the ability to: a. use distance between two points

d = √(X2 – X1)2 + (y2 – Y1)

2

b. directed distance from a point (X1, Y1) to a line Ax + By + C = 0.

d = Ax1 + By1 + C ______

±√A2 + B

2

(Use same sign as B)

2. the ability to graph a conic of the form Ax

2 + Cy

2 + Dx +EY + F = 0 by

completing the square and writing it in one of the following forms:

a. (x – h )

2 + (y – K)

2 = y

2 (circle) b. (x – h)

2 = 4a (y – k) (parabola)

(y – k)2 = 4a (x – h)

c. (x – h)

2 + (y – K)

2 = 1

a2

b2

(ellipse)

(y – k)2 + (x – h)

2 = 1

a2 b

2

Text Chapter 3 Conic Analytic Geometry Gordon Fuller Chapter 4 Simplifications of Equations Analytic Geometry Gordon Fuller Graphing Calculator

1. Use formal definitions of conic sections to derive formulas. (optional) 2. Show examples of real world applications for conic sections. The planets have elliptic paths with the sun at a focus. A projectile, for example, travels in a path which approximates a parabola.

4.1 4.5 A1 4.2 4.4.6 A3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4, 4.5F6 4.6 4.3.8A1, 4.3.12A2 4.7 4.2.6 A2 4.8 4.4.8C1 4.9 4.2.12E1 4.11 4.3.12B2 4.13 4.3.1D2 4.15 4.31D2 4.16


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UNIT 7: CONIC SECTIONS - CONTINUED


Objective

Materials

Strategies

Notes To Teacher


c. (x – h)

2 - (y – K)

2 = 1

a2

b2

(hyperbola)

(y – k)2 - (x – h)

2 = 1

a2 b

2

3. the ability to find the equation of a circle given a. center and radius. b. center and a point on the circumference.

c. endpoints of a diameter. d. center and the equation at a line tangent to the circle. e. 3 points on the circle. 4. the ability to find the equation of a parabola given

a. vertex and focus.

b. vertex and directrix.

c. vertex, axis of symmetry and a point on the parabola.

4.1 4.5 A1 4.2 4.4.6 A3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4, 4.5F6 4.7 4.2.6 A2 4.9 4.2.12E1 4.10 4.4.12C1 4.11 4.3.12B2 4.13 4.3.1D2 4.15 4.3.12A2 4.16


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Objective

Materials

Strategies

Notes To Teacher


d. endpoints of Latus Rectum and vertex.

e. 3 points on the parabola.

5. the ability to find the equation of a parabola given

a. foci and vertices.

b. endpoints of transverse and conjugate axis.

c. equations of asymptotes and vertices.

6. the ability to find the equation of an ellipse given

a. endpoints major and minor axis.

b. foci and endpoints of major or minor axis.

c. length of Latus Rectum and endpoints of major axis.

7. the ability to compute eccentricity c/a = e and understand it’s value affects the hyperbola and ellipse.


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Objective

Materials

Strategies

Notes To Teacher


8. the ability to find the intersection points of a system of conic sections. *Optional – if time permits 9. the ability to graph a non-degenerate conic of the form Ax

2 + Cy

2 Dx + EY + F = 0

by find the angle of rotation necessary to eliminate the xy term and using tan 2θ = B and x = x’cos θ – y’sin θ A – C y = x’sin θ + y’sin θ

10. the ability to find the degenerate cases of Ax

2 + Cy

2 + Dx + EY + F = 0.

4.1 4.5 A1 4.2 4.4.6 A3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4, 4.5F6 4.7 4.2.6 A2 4.9 4.2.12E1 4.10 4.412C1 4.11 4.3.12B2 4.13 4.3.1D2 4.15 4.312A2 4.16 Encouraging all students to go beyond the standards in the introduction, under “The Vision –Equity and Excellence.”


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UNIT 8: PROBABILITY, COMBINATIONS AND PERMUTATIONS

Objective: The students will demonstrate the ability to solve Probability, counting, combinations and permutations problems.

Objective

Materials

Strategies

Notes To Teacher


The student will demonstrate the ability: 1. to use Venn diagrams to illustrate the union and intersection of sets. 2. to use the multiplication and addition principle to solve counting problems. 3. to use the Tree diagrams to count the number of ways. 4. to use P = __n!__ a. Permutations n r (n – r)! C = __n!__ b. Combinations n r (n – r) !n!

5. to use the binomial theorem (a+b)

n = c + c a

n-1b + ca

n-2b

2…+c a

0b

n

n0

n1 n2 nn

6. to find the probability of an event or either of two events by using a. P(Event) = #of outcomes for event__

total # of possible outcomes

Decks of cards Dice Textbook Calculator Sets of different colored marbles

1. Students should be able to solve permutation and combination problems both algebraically and by using the calculator. 2. Introduce with sample spaces then do problems algebraically.

4.1 4.5 A1 4.2 4.4.6 A3, 4.5B3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4 4.6 4.3.8A1 4.8 4.4.8C1 4.10 4.412B5 4.11 4.3.12C1 4.13 4.3.12D2 4.14 4.4.12C1 4.16


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UNIT 8: PROBABILITY, COMBINATIONS AND PERMUTATIONS - CONTINUED


Objective

Materials

Strategies

Notes To Teacher


b. for intersecting events

P(A or B) = P(A) + P(B) – P(A and B)

c. mutually exclusive events P(A or B) = P(A) + P(B)

7. to find the probability of events occurring together and to determine whether the two events are independent? (Events A and B are independent if and only if P(B/A) = P(B) 8. to solve probability problems by using combinations.


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UNIT 9: SEQUENCES AND SERIES


Objective

Materials

Strategies

Notes To Teacher


The student will demonstrate the ability: 1. to classify a sequence as arithmetic, geometric or neitez. 2. to find a formula for the n

th term of an

arithmetic orageometric sequence by using tn = t1 + (n-1)d (arithmetic) tn = t1 (r)

n-1 (geometric)

3. to use sequences defined recursively to solve problems. 4. to find the sum of the first n terms of an arithmetic or a geometric series by using Sn = n(t1 + tn) (arithmetic) 2

Sn = t1 (1 - rn) (geometric)

1 - r 5. to find the sum of an infinite geometric series using S = t1 1 – r

if r < 1

Textbook Graphing Calculator

1. You will find that students have little experience with other than direct proof. Mathematical induction proofs call for a great deal of creativity; indicate this to the students.

4.1 4.5A1 4.2 4.4.6A3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4, 4.5F6 4.6 4.3.8A1 4.8 4.1.12B2, 4.1.12B4 4.9 4.2.12E1 4.10 4.4.12C1 4.11 4.3.12C2 4.13 4.3.12D2 4.14 4.3.12D2 4.15 4.3.12A1 4.16


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UNIT 9: SEQUENCES AND SERIES - CONTINUED


Objective

Materials

Strategies

Notes To Teacher


6. To represent series using Sigma Notation. *Optional topics 7. to estimate the limit of an infinite sequence or to determine that the limit does not exist.

8. to prove that a statement is true by using mathematics induction. n

Example: Prove ∑ 2i = n2 +n i = 1

9. to find individual terms of a power series. Example: find the 3 terms of ∞ ∑ (-1)

n x

2n+1

n = o (2n)!


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UNIT 10: POLAR COORDINATES AND COMPLEX NUMBERS

Objective:

Objective

Materials

Strategies

Notes To Teacher


The student will demonstrate the ability: 1. to plot points on a polar graph. 2. to change from rectangular coordinates to polar coordinates

using r =±√x2 + y

2 and Tan θ = y

x 3. to change from polar coordinates to rectangular coordinates using

x = rcosθ and y = rsinθ 4. to graph equations of the form

a. r = asinθ or r = acosθ (rose curves)

b. r = a + bsinθ or r = a + bcosθ 5. to add, subtract, multiply and divide complex numbers and simplify powers of i. 6. to plot complex numbers on the complex plane. 7. to find the absolute value of a complex number using

a+bi =±√a2 + b

2

Polar graph board Polar graph paper Textbook Graphing Calculator Chapter 5 Polar Coordinates Analytic Geometry Gordon Fuller

1. Use cooperative learning and graphing calculator. To explore patterns in polar graphs. 2. Treat this section of complex number operations with only a short review.

1. Demonstrate that certain equations are easier to graph in polar form than in rectangular forma and conversely. 2. Possible short lesson might be given on using graphing calculator to graph polar equations. 3. This is a good spot to introduce vectors.

4.1 4.5A1 4.2 4.4.6A3 4.3 4.5C5 4.4 4.5D5 4.5 4.5F4, 4.5F6 4.6 4.1.12A2 4.7 4.2.12A2 4.8 4.1.12B2, 4.1.12B4 4.9 4.2.12E1 4.11 4.31C2 4.13 4.31D2 4.16


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UNIT 10: POLAR COORDINATES AND COMPLEX NUMBERS-CONTINUED

Objective:

Objective

Materials

Strategies

Notes To Teacher


8. to change a complex number a+bi

(rectangular form) to (r, θ) or

rcosθ + isinθ (polar form) and vice

versa.

9. to multiply two complex numbers in polar form by using

Z1Z2 = (r1, θ1) (r2, θ2) = (r1•r2, θ1 + θ2)

Z1Z2 = (r, cis A)(S, cis B) = rs cis (A + B)

10. to use DeMoivre’s Theorem If Z = (r, θ) Then Z

n = (r, θ)

n = (r

n, nθ)

OR If Z = rcisθ Then Z

n = r

ncisnθ

To find power of (a + bi)n or to find

the roots of a complex number


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BASIC TEXTS/REFERENCES

BASIC TEXTS:

Brown, Richard G., et al. Advanced Mathematics. Boston: Houghton Mifflin Co., 1994. RERERENCES:

Buchanan, O. Lexington, Jr. Limits: A Transition to Calculus. Boston: Houghton Mifflin Co., 1985. Coxford, Arthur F., et al. HJB Advanced Mathematics – A Preparation for Calculus. Orlando: Harcourt Brace Jovanovich, 1988. Crabtree, Douglas E., et al. Precalculus. Glenview, IL: Scott, Foresman and Co., 1987. Crosswhite, F. Joe, et al. Precalculus Mathematics. Columbus, OH: Chas E. Merrill Pub. Co., 1988. Dolciani, Mary P., et al. Introductory Analysis. Boston: Houghton Mifflin Co., 1988. Foerster, Paul A. Precalculus with Trigonometry: Functions and Applications. Menlo Park, CA: Addison-Wesley Pub. Co., 1986. Fuller, Gordon. Analytic Geometry, 4th Edition. Reading, MA: Addison-Wesley Pub. Co., 1973. Gaughan, Edward D. Sequences and Limits. Glenview, IL: Scott, Foresman and Co., 1972. Hungerford, Thomas W., et al. Precalculus Mathematics. Philadelphia: Saunders College Pub., 1985. Larson, Roland E., et al. Precalculus. Lexington, MA: D.C. Heath and Co., 1985. Leithold, Louis. Before Calculus Functions, Graphs and Analytic Geometry. New York: Harper and Row, 1985. Meserve, Bruce E., et al. Principles of Advanced Mathematics. New York: L.W. Singer Co., Inc., 1970. Myers, David L. Precalculus Computer Activities. Boston: Houghton Mifflin Co., 1987. O’Brien, Katherine E. Sequences. Boston: Houghton Mifflin Co., 1966. Rosenbaum, Louise J. Induction in Mathematics. Boston: Houghton Mifflin Co., 1966. Selby, Peter H. Analytic Geometry. Orlando: Harcourt Brace Jovanoich, 1986. Shanks, Merrill E. Pre-Calculus Mathematics. Menlo Park, CA: Addison-Wesley Pub. Co., 1981.


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Steen, Fred H., et al. Analytic Geometry. 3rd Edition. Lexington, MA: Ginn and Co., 1963. Yaglom, I.M. Introduction in Geometry. Lexington, MA: D.C. Heath and Co., 1963. Yates, Robert C. Curves and Their Properties. National Council of Teachers of Mathematics, 1974.


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PUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION

COURSE REQUIREMENTS


Grades: 10-11 Length of Course: Term

________________________________________________________________ 1. Course Content – This course will consist of the following units of study:

A. Algebraic Functions: relation, function, range, domain, increasing,

decreasing, odd, even, symmetry, basic operations involving algebraic functions, composite mappings, group properties

B. Exponential and Logarithmic Functions: graphs and properties of

exponential and logarithmic functions, natural and common logarithms, equations, growth and decay problems

C. Circular Functions: periodic function, unit circle, six circular functions,

graphing, formulas, identities, inverse of a circular function, equations, trigonometric functions, radians, formulas

D. Polar Coordinates and Complex Numbers: polar coordinate plane, polar

equations, graphing properties of families of curves, complex numbers, operations with complex numbers, DeMoivre’s Theorem

E. Polynomial Functions: properties and graphs of polynomial functions,

remainder Theorem, factor Theorem, synthetic division, location principle, zeros of a function, Descartes Rule of Signs, upper and Lower bounds for roots

F. Limits: right and left hand limits, intervals, continuity limit theorems, limits as

x approaches G. Graphing Rational Functions: symmetry, holes in the graph, horizontal

asymptotes, vertical asymptotes, slant asymptotes, crossing points H. Conic Sections: distance formula, distance at a point to a line, circle,

parabola, ellipse, hyperbola, translations, rotations, degenerate forms, applications

I. Probability Combinations and Permutations: counting methods, tree

diagrams, counting principles, permutations, combinations, binomial theorem, simple probability, probability of intersecting and mutually exclusive events, independent events, probability using combinations

J. Sequenced and Series: Arithmetic sequence, geometric sequence,

recursion formulas, arithmetic and geometric series, power series, mathematical induction, summation notation

K. Polar Coordinates and Complex Numbers: polar coordinate plane, polar

equations, graphing, converting from polar to rectangular and vice versa, operations with complex numbers, DeMoivre’s Theorem

(Additionally, career-related topics and information will be presented/reviewed.)

________________________________________________________________


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Course Requirements – Pre-Calculus (page 2)

II. Course Requirements – To complete this course successfully, students will be

required to demonstrate a satisfactory (or higher) level of proficiency in:

A. recognizing and using vocabulary and symbols of Pre-Calculus. B. applying mathematical functions learned in previous courses. C. analyzing and graphing the following elementary functions: algebraic,

polynomial, circular, exponential, logarithmic, D. applying algebraic and graphing techniques to the study of rectangular and

polar coordinates. E. applying algebraic and graphing techniques to the study of conic sections,

sequences and limits, probability, permutations and combinations.

________________________________________________________________ III. Evaluation Process – Throughout the length of this course, students will be

evaluated on the basis of:

A. Test/quizzes B. Homework assignments C. Class participation D. Notebook

Note: Both midterm and final examinations will be administered.

________________________________________________________________


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Public Schools of Edison Township Division of Curriculum and Instruction

DRAFT 12

Framework For Essential Instructional Behaviors, K-12

Common Threads

Edison’s Framework for Essential Instructional Behaviors, K-12, represents a collaboratively developed statement of effective

teaching. The lettered indicators are designed to explain each common thread, but do not denote order of importance. This

concise statement of instructional expectation is intended as a framework; its use as an observation checklist would be

inappropriate.

1. Planning Which Sets The Stage For Learning & Assessment Does the instruction show evidence of: a. long-range and specific short-range planning which shows a direct relationship between student learning needs, the written curriculum

and assessments. b. communicating a clearly defined, curriculum-based learning objective based on observation and diagnosis of student needs.

c. setting high standards and expectations and deliberately selecting instructional strategies and materials to help students achieve them. d. sequencing lesson design. e. using thematic/interdisciplinary units of fundamental importance to connect science, social studies, language arts, and/or math, which result in integrated learning. f. integrating reading, writing, listening, speaking, viewing. g. using available materials, including technology and outside resources, effectively. h. demonstrating accurate knowledge of subject matter.

i. planning to enable co-teaching in shared teaching situations. j. allocating nearly all of the available time to learning and activities to accomplish learning objective(s). k. improving teaching practice by frequent reflection, individually and with others, on past student learning and instructional experiences. l. designing lessons that gradually and deliberately model and give over time increasing student independence and responsibility for learning. m. developing assessment tasks that realistically measure what is taught. n. developing multiple means of assessments including performance (students participating in self assessment, projects, demonstrations)

o. using appropriate homework assignments that extend learning, coupled with efficient assessment of student performance.

2. Productive Learning Climate & Classroom Management Does the student-teacher interaction show evidence of: a. establishing an environment which is learner-centered (content rich and reflective of children’s efforts). b. creating a climate of mutual respect, one that is considerate of and addresses differences in culture, race, gender. c. providing opportunities for student choice. d. using proactive rules, routines (which students have internalized) and reactive management strategies effectively. e. creating a safe, positive and open classroom environment in which children and teachers take risks, use trial and error.

3. Teaching & Learning Does the instruction show evidence of: a. demonstrating a belief that all children can learn and learn in different ways. b. deliberately selecting a variety of grouping strategies (individual conferences, whole class, flexible, small groups, partners, cooperative learning structures).

c. addressing varied learning modalities (visual, auditory, kinesthetic/tactile). d. deliberately selecting and using graphic organizers, auditory tapes, and manipulatives. e. intentionally modifying content, strategies, materials and/or assessment during a lesson. f. helping students interact with concepts/skills/content knowledge through a variety of planned and, when appropriate, unplanned

learning strategies. g. focusing on strategies for concept building rather than on isolated facts, information, and skills (teacher with students exploring/discovering, not teacher assigning) h. accessing prior knowledge to build background information (e.g., anticipatory set, K-W-L, I wonder list..)

i. building in ample connections, including students’ home cultures, to real life. j. providing students the opportunity to actively process the learning through closure. k. using questioning strategies that promote discussion, problem solving, divergent thinking, multiple responses, and higher levels of thinking (analysis, evaluation, synthesis). l. integrating active student participation, individually and collaboratively. m. using varied writing activities (journals, learning logs, creative pieces, letters, charts, notes, reports) that connect to and extend learning in all subjects. n. using assessment to monitor and adjust instruction continually. o. assessing student learning before, during, and after the lesson and providing timely feedback to students.

-over-


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4. Professional Responsibilities & Characteristics Does the teacher show evidence of: a. reflecting upon teaching to inform instruction. b. maintaining accurate records and completes forms/reports in a timely manner. c. communicating with parents to keep them informed of child’s progress and instructional process. d. treating learners with care, fairness and respect. e. working collaboratively and cooperatively with colleagues.

f. being flexible and open to suggestions from supervisors. g. presenting a professional appearance.

/jv cui/frameworks12

PRE-CALCULUS - HONORS - Edison · PRE-CALCULUS - HONORS 3 COURSE OBJECTIVES The student will...

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