PARS I PPANY - TR OY HI LLS TOW N SHI P SC HOOLS CO URS E ...sharepoint.pthsd.k12.nj.us/ci/Approved Curriculum... · technology-dependent lessons. These methodologies are the expected

PARS I PPANY - TR OY HI LLS TOWN SHI P SC HOOLS

CO URS E O F S T UDY

FO R

PRE CAL CUL US

M TH 3 1 3

APPROVED BY THE BOARD OF EDUCATION

January 24, 2013

Approved: June 1990 Revised: July 1996 November 2012

MTH 313 – Precalculus 2

STA TEM EN T OF P U R P OSE

Precalculus is the fourth course in a sequence of four college-preparatory courses (Algebra I, Geometry, Algebra II and Precalculus) that develop the foundation for success in the future study of mathematics. This course provides continuing experiences with technology, advanced analysis, mathematical modeling to real-world situations, discovery techniques in problem-solving and intuitive foundations to develop an understanding of the conceptual building blocks of calculus.

The advancement to this high level of mathematics is based upon the assumption that prior learned concepts have been retained by the student. Thus, the sophisticated analysis inherent in Algebra II must be further developed using the graphic and visual techniques to enhance and justify the analytical and algebraic strategies in problem solving. Consistent with this philosophy will be student accountability for knowledge acquired in previous mathematics courses. Separately we assess students to gauge progress and inform instruction. Benchmark assessments for students in grades 9 through 12 are administered in the form of a midterm and final exam for full year courses. *Special Note: Only final exams are administered at the end of quarter courses and semester courses. This revision was undertaken to align with the New Jersey Student Learning Standards, the New Jersey Student Learning Standards for Technology, and the College Board AP Calculus BC course outline.

M ETHOD S A N D OU TC OM ES

The Common Core State Standards involve a variety of skills and problem-solving techniques and an extensive range of topics. Precalculus is an area that can tie these strands together, bridging algebra, geometry, and discrete branches of mathematics.

The National Council of Teachers of Mathematics Standards endorses the use of discovery-based, hands-on, non-routine problem solving and technology-dependent lessons. These methodologies are the expected mode of instruction that must be used on a consistent basis throughout the course. Graphing calculators and advanced technology allow mathematics problems to become concrete representations of concepts. Work with these sophisticated tools emphasizes the appropriateness of the many varied techniques of successful problem solving.

Traditional methods of “watch and do” mathematics must be de-emphasized. Whenever possible, students will be given the opportunity to observe and collect information, explore relationships, analyze, draw inferences, make conjectures, design models, synthesize, and reach new conclusions about concepts. Students will be encouraged to communicate their mathematical understanding and to validate their conclusions in classroom discussions and in written explanations such as those required on the New Jersey High School Proficiency Assessment (HSPA) and the Scholastic Aptitude Test (SAT).

A variety of different techniques are to be used throughout the course to enhance student discovery, broaden student participation, increase learner self-confidence, and enable students to demonstrate their competence through traditional and alternate forms of assessment.

MTH 313 – Precalculus 3 This course offers students the opportunity to:

GOA LS

1. expand the understanding of functions, developing the concepts of inverse relationship, periodicity, continuity and limits. 2. demonstrate the ability to use the advanced capabilities of the graphing calculator and recognize which approach: graphic, numeric or algebraic,

is the most advantageous or efficient. 3. explore intuitively and examine graphically various functions including polynomial, rational, exponential, and logarithmic and their appropriate

applications. 4. increase familiarity with the complex number system, gaining comprehension of relationships among the subsets and developing advanced skills

with imaginary numbers. 5. acquire an understanding of trigonometry through a variety of perspectives and apply a range of techniques to model real-world situations. 6. examine the relationships and patterns that exist in sequences and series using appropriate notation to extended pattern-based thinking as a

problem-solving technique. 7. explore the geometric and algebraic properties of the conic sections. 8. use self-assessment to identify their mathematical strengths and weaknesses and to help foster a better understanding of the concepts being taught.


A SSESSM EN T C OM P ON EN T “Assessment must be more than testing: it must be a continuous, dynamic and often informal process,” according to National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics.

The assessment tools for the Precalculus course should include:

� open-ended questions � problem-solving experiences using real-life data � labs: individual/partner/group � projects: individual/group � concepts and application of technology � partner quizzes/tests � notebook quizzes � unit test/quizzes

Utilizing a variety of assessment techniques enables the teacher to understand more fully the mathematical strengths and weaknesses of the student and the program while also allowing the student to understand his/her mathematical strengths and weaknesses.


T H E L I VI NG CURRI CUL UM Curriculum guides are designed to be working documents. Teachers are encouraged to make notes in the document. Written comments can serve as the basis for future revisions. In addition, the teachers and administrators are invited to discuss elements of the guides as implemented in the classroom and to work collaboratively to develop recommendations for curriculum reforms as needed.

AFFI RMAT I VE ACT I O N During the development of this course of study, particular attention was paid to material which might discriminate on the basis of sex, race, religion, national origin, or creed. Every effort has been made to uphold both the letter and spirit of affirmative action mandates as applied to the content, the texts and the instruction inherent in this course.

MODIFICATIONS AND ADAPTATIONS

For guidelines on how to modify and adapt curricula to best meet the needs of all students, instructional staff should refer to the Curriculum Modifications and Adaptations included as an Appendix in this curriculum. Instructional staff of students with Individualized Education Plans (IEPs) must adhere to the recommended modifications outlined in each individual plan.


PARS I PPANY - TR OY HI LLS TOWN SHI P SC HOOLS

CO URS E PRO FI CI E NCI E S AND G RADI NG PRO CE DURE S

COURSE #: MTH 313 TITLE: PRECALCULUS

IN ACCORDANCE WITH DISTRICT POLICY AS MANDATED BY THE NEW JERSEY ADMINISTRATIVE CODE AND THE NEW JERSEY STUDENT LEARNING STANDARDS, THE FOLLOWING ARE PROFICIENCIES REQUIRED FOR THE SUCCESSFUL COMPLETION OF THE ABOVE NAMED COURSE.

The student will:

1. sketch many functions intuitively , including the identify function, the quadratic function, the square root function, the cubic function, and the

cube root function. 2. recognize in these basic functions and in unfamiliar functions the domain, range, and even or odd symmetry. 3. apply to both familiar and unfamiliar functions all basic transformations including vertical or horizontal shifts, reflections through x-axis and y-

axis, and vertical stretching or shrinking. 4. find the inverse of a function, prove an inverse relationship using composites, and describe the relationship of the graphs of inverse functions. 5. solve quadratic equations, rational equations, and equations containing radical expressions using a range of algebraic skills; exclude extraneous

roots. 6. identify functions as continuous or discontinuous. 7. sketch polynomial functions, including end behavior, roots, and y-intercept. 8. analyze a polynomial function using the Fundamental Theorem of Algebra, the Remainder Theorem, the Rational Root Theorem, synthetic

division, Descartes’ Rule of Signs, and the upper and lower bounds of real zeroes. 9. use a graphing calculator to find the real roots of a polynomial function. 10. find a polynomial equation given its roots. 11. identify the intercepts, the vertical asymptote of a rational function. 12. write the equation of the slant asymptote of a rational function. 13. analyze the behavior of a rational function as it approaches a vertical asymptote; analyze the end behavior with respect to horizontal or slant

asymptotes. 14. define and graph exponential and logarithmic functions; recognize their inverse relationship. 15. apply the properties of logarithms to expand or condense expressions. 16. solve exponential and logarithmic equations.

MTH 313 – Precalculus 7 Proficiencies (continued)

17. apply exponential and logarithmic functions to model situations and solve problems. 18. using a graphing calculator to fit a line or curve to a set of data. 19. define the trigonometric functions through the unit circle. 20. find the argument of a trigonometric expression whose value is known.

� � �

21. find the values of trigonometric functions of quadrantal angles and any angle related to , , . 6 4 3

22. use a calculator to evaluate expressions containing trigonometric functions of any angle. 23. graph the six basic trigonometric functions and variations by changing parameters. 24. evaluate both simple and composite inverse trigonometric expressions. 25. define, describe, and graph the inverses of the sine, cosine, and tangent functions. 26. solve triangles and apply the trigonometric functions to solve verbal problems. 27. define and apply to basic trigonometric identities. 28. verify trigonometric identities. 29. solve trigonometric equations. 30. apply the trigonometric functions of the sum and difference of two angles. 31. derive and apply the double angle and half-angle formulas. 32. derive and apply the Laws of Sines and Cosines. 33. find the area of a triangle using the oblique triangle formula or Heron’s formula. 34. define an geometric sequence, find the nth term, insert means, find the sum. 35. define an arithmetic sequence, find the nth term, find a finite sum, find an infinite sum, or determine that it does not exist. 36. explain a series written in sigma notation, and rewrite an expanded series in sigma form. 37. use the Binomial Theorem to expand a binomial or to find a specific term. 38. identify the conic sections in standard or general form. 39. given the general form of a conic section, convert to standard form and graph. 40. use a variety of techniques to evaluate a limit. 41. (optional) develop the slope of a tangent to a curve and the concept of the derivative.


G RADI NG PRO CE DURE S

Marking Period Grades:

Long- and Short-Term Assessments 90% Publisher prepared tests, quizzes and/or worksheets Teacher prepared tests, quizzes and/or worksheets Authentic Assessments Technology applications Projects Reports Labs

Daily Assessments 10%

Homework Do Now / Exit Questions Class participation Journal Writing Notebook - checks and open notebook assessments Explorations

Final Grade:

Final Grade – Full Year Course

Full Year Course • Each marking period shall count as

20% of the final grade (80% total).

The midterm assessment will count as 10% of the final grade, and the final assessment will count as 10% of the final grade.


Content Outline Numbers in parentheses indicate coordination

with the Course Proficiencies

Standards

Suggested Activities

Evaluation/Assessment

Teacher Notes

Students will: Students will: I. Functions and Graphs

A. Functions (Review) (2) � function notation � determine whether a rela-

tion is a function � domain and range � evaluating functions

B. Graphs of Functions (Review) (2) � increasing, decreasing,

relative, maximum and minimum.

� even and odd functions

C. Shifting, Reflecting and Stretching Graphs (Review) (3)

F.1F.1,2

F.1F.4,7a,c,d F.BF.3 8.1.12.A.1

F-BF.3 8.1.12.A.1

� recall various techniques to

determine if a relation is a function.

� use the graphing calculator to determine the intervals where functions are increasing, decreasing or constant.

� use the graphing calculator to discover transformations.

� Find the domain and

range of: 1. y = x − 3 2. y = x2 + 2

3. y = 1 x

� Determine whether the indicated functions are even, odd or neither:

1. y = x2 + 2 2. y = x3

3. y = x − 3

� Compare y = x2 , y = x2 + 2,

y = ( x + 2)2 , y = 2x2 .




Standards



Teacher Notes

I. Functions and Graphs (continued) Students will: Students will: D. Functions Operations (Review)

� add, subtract, multiply, divide and determine the composition

E. Inverse Functions (4) � sketch the graph of the

inverse � find the inverse algebraically

A-APR.1 F-BF.1b,1c

F-BF.4a,4b,4c,4d 8.1.12.A.1

� recall and explain techniques for adding, subtracting and multi- plying polynomials.

� draw reflections of the graphs

f ( x ) = 3x + 1

f ( x ) = x3

over the line y = x.

� verify, explain and

demonstrate that a function and its inverse are reflections over the line y = x.

� Given: f ( x) = x2 + 2x − 3 g ( x) = x − 5

Find: 1. f ( x) + g ( x) 2. f ( x) − g ( x) 3. f ( x) ⋅ g ( x) 4. f ( x) g (

x) 5. f ( g ( x)) + g ( f ( x))

� Given f ( x) = 2x − 4 , find f −1 ( x) and graph both

f ( x )and f −1 ( x ) on the same coordinate plane.

Emphasize that inverses are reflections over the line, y = x




Standards



Teacher Notes

Students will: Students will: II. Polynomial Functions

A. Quadratics (5) � graphs of quadratics � write a quadratic in standard

form and find its minimum and maximum values

� applications

B. Polynomial Functions of Higher Degree (6, 7, 9) � use the Leading Coefficient

Test to determine end behavior.

� find zeros both algebraically and graphically.

� sketch graphs of polynomials � use Intermediate Value

Theorem

F-1F.7a,9 A-SSE.3b F-1F.4

A-RE1.4b A-SSE.3a F-1F.8a 8.1.12.A.1

� compare the benefits of a

quadratic written in general form (ax2 + bx + c = y) with a quadratic written in standard form ( y = a( x − h)2 + k ) .

� model end behavior using their arms. (Mathletics!)

� write y = x2 + 10x + 14 in

standard form and sketch its graph.

� determine the end behavior, find the zeros and sketch the graph of: f ( x) = x4 − x3 − 20x2 .




Standards



Teacher Notes

II. Polynomial Functions (continued) Students will: Students will: C. Real Zeros of Polynomial

Functions (8) � divide polynomials using long

and synthetic division. � use remainder and factor

theorems � use Descartes Rule of Signs � use Rational Zero Test � determine lower and upper

bounds

D. Complex Numbers (Review) � operations with complex

numbers

E. The Fundamental Theorem of Algebra (10) � complex zeros in conjugate

pairs. � write the equation given the

roots.

A-APR.2,3,6 N-CN.8,9 8.1.12.A.1

N-CN.5

N-CN.8,9 A-CED.1

� use a graphing calculator to eliminate possible rational zeros and help determine zeros of a polynomial function.

� determine: i2, i3, i4

and verify i20 = 1 recalling that i = −1

� review rational, irrational, real and complex numbers.

� find the real zeros of f ( x) = x3 + x2 − 4x − 4

using Descartes Rule of Signs and the Rational Zero Test.

� given g = 4 + i and h = 7 − 2i , find: 1. g + h 2. g − h 3. g ⋅ h

� find all zeros of: f ( x) = x4 − 8x3 + 17 x2 − 8x + 16




Standards



Teacher Notes

Students will: Students will: III. Rational Functions

A. Graphs (11, 12, 13) � vertical asymptotes � horizontal asymptotes � slant asymptotes � intercepts � domain

A-APR.6 F-1F.7d,5 8.1.12.A.1

� review methods for

finding horizontal and vertical asymptotes and explain what type of functions have horizontal totes.

� sketch the graph of:

x2 + 2x f ( x) = 2 x2 − x

by finding all asymptotes and intercepts.

IV.

Exponential and Logarithmic Functions A. Exponential Functions and Their

Graphs (14) � graph exponential functions

F-1F.7e 8.1.12.A.1

� use a graphing calculator to:

a. evaluate (1 + 1 ) for x

increasing large values of x by viewing the table as x goes from 0 – 10,000 in increments of 100.

b. graph = (1 + 1 ) x

y for 1 x

x > 0 and y2 = e and compare.

� graph and identify the domain and range of y = 2x −1 + 4 .

B. Logarithmic Functions and Their Graphs (14) � graph logarithmic functions

F-1F.7e 8.1.12.A.1

� review inverse functions.

� show the relationship between: y = a x and loga y = x .

� graph and identify the domain and range of y = log2 ( x − 3) .

x




Standards



Teacher Notes

IV. Exponential and Logarithmic Functions (continued)

Students will: Students will:

B. (Continued)

C. Properties of Logarithms (15) � expand and condense loga-

rithmic expressions.

D. Solving Exponential and Loga- rithmic Equations (16, 17)

E. Applications (17)

� compound interest � population growth � half-life decay

F-1F.8

A-RE1.11 F-BF.4a F-LE.4 8.2.12.C.4 8.2.12.E.3

F-LE.4,1c 8.2.12.C.4 8.2.12.E.3

� graph y = ex and y = ln x y = 10x and y = log x

and compare the graphs.

� review and use properties of logarithms to simplify and evaluate logarithmic expressions and explain why logb xy = logb x + logb y .

� review and use change of base formula to evaluate expressions like log2 5 .

� compare interest earned by different compound- ings.

A = Pert

A = P(1 + r )rt n

� explain when com-

pounding more often is beneficial.

� expand: x y 4

logb z 4

� condense: ln x − 3 ln( x + 1)

� solve for x:

1. 2 log5 3x = 4 2. 4e2 x − 3 = 2

� find the number of years it would take for money to double compounding con- tinuously.

Use a calculator to determine the amount




Standards



Teacher Notes

Students will: Students will: V. Exploring Data and Curve Fitting

A. Scatter Plots and Linear Models (18)

B. Scatter Plots and Non-linear models (18)

VI. Trigonometric Functions

A. Radian and Degree Measure (Review)

B. Trigonometric Functions: The Unit Circle (19)

C. Right Triangle Trigonometry (Review) � trigonometric identities � applications

S-1D.1,6 8.1.12.A.1

F-TF.1

F-TF.2,3,4

G-SRT.6,7,8 F-TF.8

� collect data, create a scatter plot, and determine best-fit model.

� explain how to convert from degree ⇔ radian measure.

� create a unit circle and determine the coordi- nates of the points where the circle intersects a � and 5� angle. 6 3

� explain the definitions of the trigonometric functions.

� given a set of data, use the

calculator to determine the best-fit model.

� convert 75o into a radian measure in terms of � .

� evaluate the Six Trigono-

metric Functions at t = 3� . 4

� find :

Students should memorize � =180o




Standards



Teacher Notes

VI. Trigonometric Functions (continued)


D. Trigonometric Functions of Any Angle (21, 22) � find reference angles � evaluate trigonometric func-

tions of any angle

E. Graphs of Sine and Cosine Functions (23) � graph y = sin x and y = cos x

with transformations

F. Graphs of Other Trigonometric Functions (23)

Graph y = tanx

y = cotx y = secx y = cscx

with transformations

F-TF.3 F-TF.5 F-1F.7e F-BF.3 8.1.12.A.1

F-TF.5 F-1F.7e F-BF.3 8.1.12.A.1

� determine the sin, cos, and tan of an angle

5� which measures

6 S A T C

� discuss amplitude, period,

vertical and horizontal shifts.

� create a cos x equation

with an amplitude of 3, a period of � , no vertical shift and a horizontal

shift of � . 3

� use the graphs of

y = sin x and y = cos x and facts about recip- rocals to graph y = sec x and y = csc x . Explain the difference between the graphs of the tanx and the cotx.

� find the reference angle for 4� and evaluate the Six 3

Trigonometric Functions.

� graph:

y = −3sin 1 ( x − � ) + 4 2 2

� graph: y = −3 tan 2 x

Students need to know where the trigonometric functions are positive




Standards



Teacher Notes

VI. Trigonometric Functions (continued)


G. Inverse Trigonometric Functions (24, 25) � definitions � composites � graphs of y = sin −1 x ,

y = cos−1 x and y = tan −1 x

F-TF.6 F-BF.4a,4b,4c,4d 8.1.12.A.1 8.2.12.C.4 8.2.12.E.3

� define inverse trigonometric functions and explain the domain and range of the arc sin x, arc cos x and arc tan x.

� graph: y = arc sin x

H. Applications of Trigonometry

(26)

F-TF.5 G-SRT.8

� given word problems,

sketch diagrams to model the problem and use trigonometry to solve.

� A ladder 20 feet long leans

against the side of a house. The angle of elevation is 80o. How far up the side of the house does the ladder reach?

VII.

Analytic Trigonometry A. Using Fundamental Identities

(28) � simplifying trigonometric ex-

pressions using identities.

F-TF.8

� explain why sin x2 + cos2 x = 1 and use this pythagorean identity to derive

1 + tan2 x = sec2 x 1 + cot 2 x = csc2 x

� simplify: sin x sec x. .

Students should memorize the fundamental identities


Content Outline Numbers in parentheses indicate coordination with

the Course Proficiencies

Standards



Teacher Notes

VII. Analytic Trigonometry (continued) Students will: Students will: B. Verify Trigonometric Identities

(28) F-TF.8 � explain how the basic

identities can be used to simplify expressions.

� verify: sec2 −1

sec2 = sin

Students should review guidelines for verifying identities

C. Solving Trigonometric Equations

(29) � solving equations by factoring,

using identities, combining like terms, squaring and extracting square roots.

� solving equations involving multiple angles.

F-TF.7 8.2.12.C.4 8.2.12.E.3

� explain why a trigonometric equation has numerous solutions.

� solve for x: 2 sin x −1 = 0 .

Students should review general solution versus answers given over an interval

D. Sum and Difference Formulas (30) � evaluate trigonometric functions

and expressions � simplify trigonometric expres-

sions

F-TF.9 � use special angle formulas to evaluate and simplify expressions.

� evaluate: sin 75 Students should memorize formulas using song or story

2




Standards



Teacher Notes

VII. Analytic Trigonometry (continued) Students will: Students will: E. Multiple-Angle Formulas (31)

� use double-angle formulas to solve trigonometric equations, analyze graphs and evaluate trigonometric functions.

F. Half-Angle Formulas (31) � use half-angle formulas to

evaluate trigonometric functions.

VIII. Additional Topics in Trigonometry

A. Law of Sines (32) � ASA � SSA (ambiguous case) � AAS

F-TF.5,9 8.1.12.C.4 8.2.12.E.3

F-TF.9

G-SRT.10.11

� explain what formula to utilize to find cos 2 given:

sin = 3 5

� < < �

2

� explain what formula to use to find cos 2 given:

sin 4 = −3 5 .

3� < 4 < 2�

2

� explain the three different possibilities for solutions when given SSA.

� given cos = 5 and 13

3� < < 2�, find:

2 sin (2 ) cos (2 ) tan (2 )

� find: sin105

� given: A = 42 , B = 100

and a = 14, solve: the ∆ .

� given: A = 35o, b = 10 and a = 7 determine the number of solutions for a ∆ .

Students should memorize formulas

Students should memorize formulas

Students should recall ASA, AAS and SSA from Geometry




Standards



Teacher Notes

VIII. Additional Topics in Trigonometry (continued)


B. Law of Cosines (32) � SAS � SSS

G-SRT.10,11 � explain how to determine whether to use Law of Sines or Law of Cosines

� given a=8, b=4, c=10, solve the ∆ .

Students should memorize Law of Cosines

C. Areas of Triangles (33)

� oblique triangle formula � Heron’s Formula

G-SRT.9

� find the area of ∆ ABC

given A = 50o, b = 10 and c = 12.

� given a=8, b=4, and c=10,

find the area

Students should memorize area formulas

D. Applications (32, 33)

G-SRT.9,10,11 � design their own word problems.

� critique and solve their peer’s word problems

IX. Sequences and Series A. Sequences and Series (Review)

(36) � finding the nth term in a

sequence � factorials � summation notation

F-BF.2 F-1F.3

� find the 8th term of the

sequence defined by 2n

an = n2 .

� find the sum:

6 n! ∑ n=1 n +1

Students should review factorial and summation notation

B. Arithmetic Sequences (Review) (34) � definition � finding the nth term � sum � applications

F-BF.2 � read the Story of Gauss. � write the formula for an if: 3, 7, 11, 15,




Standards



Teacher Notes

IX. Sequences and Series (continued) Students will: Students will: C. Geometric Sequences (35)

� definition (review) � finding the nth term (review) � sum of a finite series (review) � sum of an infinite series � applications

D. The Binomial Theorem (37) � combination notation � Pascal’s Triangle � expand a binomial � find a specific term

A-SSE.4 F-BF.2

A-APR.5

� discuss the existence of the sum of an infinite geometric series and determine when a geometric series will have a sum.

� discover and identify patterns in Pascal’s Triangle, and relate these patterns to the binomial expansion.

� find the sum:

4 + 2 + 1 + 1 +

2

� expand: (2 x − y)6




Standards



Teacher Notes

Students will: Students will: X. Conic Sections

A. Definitions of Conic Sections (40, 41) � circle – sketch the graph by

finding the center radius. � parabolas – sketch the graph

by finding the vertex, focus and directrix.

� ellipses – sketch graph by finding vertices, co-vertices, foci, center, axes and eccentricity.

� hyperbolas – sketch graph by finding center, vertices, asymptotes and foci.

B. Recognize Conics in General Form and Write Conics in Standard Form (40)

8.1.12.A.1 G-GPE.1-3

G-GPE.1-3

� cut ice cream sugar

cones to discover different conic sections formed.

classify conics based on squared terms and the coefficients of x2 and y 2 .

� sketch the graph of:

( x − 2)2 ( y + 1)2

4 +

9 = 1

label center, vertices, foci and asymptotes.

� classify the conic: 1. 4 x2 − 9 x + y − 5 = 0 2. 4x2 − y 2 + 8x − 6 y + 4 = 0

The teacher will illustrate, with solids, the different conic sections created by passing a plane through a double-napped cone.

Textbook Appendix: A29 and A30




Standards



Teacher Notes

Students will: Students will: XI. Limits (44)

A. Definition of Limit B. Estimate the Limit by:

� Table of Values � graph � direct substitution

C. Limits That Do Not Exist

D. Techniques for Evaluating Limits � factoring � rationalizing

8.1.12.A.1 (AP Calc.) I.B.1,2,3

(AP Calc.) I.C.1,2

(AP Calc.) I.B.2

� sketch a graph of a

function with a limit of 1 at x = 4.

� determine the

2 lim (1 − x ) x x → 0

� sketch the graph of a function for which the

limit of f ( x ) as x

approaches 3 is 5 but for which f (3) ≠ 5 .

� find lim x + 1 −1

x → 0 x

� use a calculator, a table and

direct substitution to find the limit:

lim ( 3x2 + 2x − 2 )

x → 2

� find the limit: lim x

x → 0 x

� find the limit: lim x2 + x − 6

x → 3 x + 3

emphasize that the limit is a number that the function is approaching, not necessarily the value of the function at that place

Discuss unbounded behavior and one-sided limits

Review algebraic techniques




Standards



Teacher Notes

Students will: Students will: E. Limits at Infinity (AP Calc.) I.C.2 � explain the meaning of a

limit as x → ∞ using

f ( x ) = 1 x

� find the limit: lim x + 1

x → ∞ 2 x

Review horizontal asymptotes

MTH 313 – Precalculus 25 TEXTBOOK:

BI BLI OGR A P HY

Larson, Hostetler, Edwards. Precalculus with Limits, A Graphing Approach. 4th ed. New York, New York: Houghton Mifflin, 2005. RESOURCES:

Barnes-Robinson, Linda, Sue Jeweler and Mary Cay-Ricci. “Potential: Winged Possibilities to Dreams Realized.” Parenting for High Potential

June 2004: 20. (Self-Assessment Rubric for Work Habits)

Curriculum and Evaluation Standards of School Mathematics. Reston, Virginia: National Council of Teachers of Mathematics, Inc., 1989.

Demana, F. and B Waits. Precalculus Mathematic. Reading, Massachusetts: Addison-Wesley Publishing Co., 1990.

Foerster, P. Precalculus with Trigonometry: Functions and Applications. Menlo Park, California: Addison Wesley Publishing Company, 1993.

Gordon, B., L. Yunker, G. Vannatta and F. Merrill Crosswhite. Advanced Mathematical Concepts. Columbus, Ohio: Glencoe/McGraw Hill, 1994.

Larson, R., R. Hostetler and B. Edwards. Calculus with Analytic Geometry, 5th ed. Lexington, Massachusetts: D. C. Heath and Company, 1994.

---Precalculus. Lexington, Massachusetts: D.C. Heath and Company, 1989.

---Precalculus with Limits. 4th ed. Lexington, Massachusetts: D.C. Heath and Company, 1995.

Common Core State Standards for Mathematics. Adopted by the New Jersey State Department of Education, October, 2009.

New Jersey Core Curriculum Content Standards for Technological Literacy. Adopted by the New Jersey State Department of Education, October, 2004.

Purcell, E. and D Varberg. Calculus with Analytic Geometry. 4th ed. Englewood Cliffs, New Jersey: Prentice Hall, 1984. WEBSITES:

www.classzone.com www.domath.org www.enc.org www.forum.swarthmore.edu www.illuminations.nctm.org www.mathforum.com www.mathgoodies.com www.nctm.org

http://www.classzone.com/

http://www.classzone.com/

http://www.domath.org/

http://www.domath.org/

http://www.enc.org/

http://www.enc.org/

http://www.forum.swarthmore.edu/

http://www.forum.swarthmore.edu/

http://www.illuminations.nctm.org/

http://www.illuminations.nctm.org/

http://www.mathforum.com/

http://www.mathforum.com/

http://www.mathgoodies.com/

http://www.mathgoodies.com/

http://www.nctm.org/


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SA M P LE A U THEN TI C A SSESSM EN TS


SA M P LE A U THEN TI C A SSESSM EN TS TASK #1:

You were stopped by a policeman because your “hot rod” car is simply too loud. The officer tells you that he measured the noise level of your car to be 88 decibels and, if you do not want to be ticketed again, you must reduce the noise level to 72 decibels or less. You really like the sound of your car and want to prove to the officer that reduction to 72 decibels is much too high a percentage reduction. How are you going to convince the officer of this so you do not have to install a new muffler on your car?

TASK #2:

The Parsippany Fire Department has called upon the intelligent youth of the town for help. There are some proposals to build high-rise office buildings in town. The department is concerned about being able to rescue people from the top stories of these new buildings. They believe that the maximum safe angle of elevation for the ladder is 72 degrees and the longest ladder they currently have is 100 feet. Based on this information, would you advise the fire company to support the building of these new offices? Suppose the Fire Department asked you to go before the Planning Board with a proposal regarding this building, what would you report?

TASK #3:

Summer is here and you have to get a job, after all, college is just around the corner! You want to try your hand at selling Cutco knives, as many of your older sibling’s friends have made a lot of money doing this. The company offers you a choice. You can earn 6% of your total sales or you can take a salary of $200 a week plus 4% of your sales. Which offer do you decide to take and why?


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RUB RI C FO R S CO RI NG O PE N - E NDE D PRO B L E MS


RUB RI C FO R S CO RI NG O PE N - E NDE D PRO B L E MS

3-POINT RESPONSE

Your response showed a complete understanding of the problem’s essential mathematical concepts. You executed procedures completely and gave relevant responses to all parts of the task. Your response contained a few minor errors, if any. Your response was clear and effective, detailing how the problem was solved so that the reader did not need to infer how and why decisions were made.

2 POINT RESPONSE

Your response showed a nearly complete understanding of the problem’s essential mathematical concepts. You executed nearly all procedures and gave relevant responses to most parts of the task. Your response may have had minor errors. Your explanation detailing how the problem was solved, may not have been clear, causing the reader to make some inferences.

1 POINT RESPONSE

Your response showed a limited understanding of the problem’s essential mathematical concepts. Your response and procedures may have been incomplete and/or may have contained major errors. An incomplete explanation of how the problem was solved may have contributed to questions as to how and why decisions were made.

0 POINT RESPONSE

Your response showed an insufficient understanding of the problem’s essential mathematical concepts. Your procedures, if any, contained major errors. There may have been no explanations of the solution or the reader may not have been able to understand the explanation. The reader may not have been able to understand how and why decisions were made.

SOURCE: NEW JERSEY HSPA


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SELF - A SSESSM EN T


SELF-ASSESSMENT SHEET FOR STUDENT WORKFOLDER Name Course

Date

Assessment

Grade What I did well... How I could improve… Things I need to work on...

Marking Period

MTH 313 – Precalculus 32 NAME:

MI D - YE AR RE FL E CT I O N

After looking over your work folder with all of your assessments from this year, what are your strengths in math? What are your weaknesses?

How can you continue to use your strengths to be successful? Be specific and explain.

How can you improve your areas of weakness? Give yourself at least one goal in order to help you improve.

Which assessment(s) are you most proud of? Explain why.

Which assessment(s) do you think you could have done better on? Explain why and how.

We are now half-way through the school year. What will you continue to strive for in math? How do you plan on doing this?


Work Folder Reflection Look through the various items in your work folder and take a moment to think about this school year. Answer the following questions in the form of a paragraph to reflect on your mathematical progress so far this year.

What were some of your goals in the beginning of this school year? Have you made progress towards achieving them?

What are some goals you have for the rest of this school year?

In what areas did you have the most success? Be specific by indicating the topics in which you feel

most confident.

In what areas did you have difficulty? What are some ways you can improve in those areas?

What can you do to prepare yourself for the final exam?

Now that more than half of the year has passed, what are some things that you have learned that will help you next year? (i.e. study skills, putting more effort in homework, etc.)

What are some things that you enjoy about this class? What are some things you don’t like? Do you

have any suggestions as to what would make the class better?

MTH 313 – Precalculus 34 2

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NEW JERSEY STUDENT LEARNING STANDARDS

MTH 313 – Precalculus

35

NEW JERSEY STUDENT LEARNING STANDARDS

4 - Mathematics

8 - Technology

9 - 21st Century Life and Careers

http://www.state.nj.us/education/cccs/2016/math/

http://www.state.nj.us/education/aps/cccs/tech/

http://www.state.nj.us/education/aps/cccs/career/


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ADVANCE D PL ACE ME NT CAL CUL US B C T O PI C O UT L I NE


ADVANCE D PL ACE ME NT CAL CUL US B C T O PI C O UT L I NE

Topic Outline The topic outline for Calculus BC includes all Calculus AB topics. Additional topics are found in paragraphs that are marked with a plus sign (+) or an asterisk (*). The additional topics can be taught anywhere in the course that the instructor wishes. Some topics will naturally fit immediately after their Calculus AB counterparts. Other topics may fit best after the completion of the Calculus AB topic outline. Although the examination is based on the topics listed here, teachers may wish to enrich their courses with additional topics.

I. Functions, Graphs, and Limits

II. Derivatives

III. Integrals

IV. *Polynomial Approximations and Series

I. Functions, Graphs, and Limits A. Analysis of Graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

B. Limits of Functions (incl. one-sided limits)

An intuitive understanding of the limiting process.

Calculating limits using algebra.

Estimating limits from graphs or tables of data. C. Asymptotic and Unbounded Behavior

Understanding asymptotes in terms of graphical behavior.

Describing asymptotic behavior in terms of limits involving infinity.

Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

D. Continuity as a Property of Functions

An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)

Understanding continuity in terms of limits.

http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc&funcGraph

http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc&derivatives

http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc&integrals

http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc&poly

MTH 313 – Precalculus Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

E. *Parametric, Polar, and Vector Functions 66 The analysis of planar curves includes those given in parametric form, polar form, and vector form.

II. Derivatives

A. Concept of the Derivative

Derivative presented graphically, numerically, and analytically. Derivative interpreted as an instantaneous rate of change. Derivative defined as the limit of the difference quotient. Relationship between differentiability and continuity.

B. Derivative at a Point

Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points

at which there are no tangents. Tangent line to a curve at a point and local linear approximation. Instantaneous rate of change as the limit of average rate of change. Approximate rate of change from graphs and tables of values.

C. Derivative as a Function

Corresponding characteristics of graphs of 'f and f '. Relationship between the increasing and decreasing behavior of f and the sign of f '. The Mean Value Theorem and its geometric consequences. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

D. Second Derivatives

Corresponding characteristics of the graphs of f, f ', and f ". Relationship between the concavity of f and the sign of f ". Points of inflection as places where concavity changes.

E. Applications of Derivatives

Analysis of curves, including the notions of monotonicity and concavity. + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.

MTH 313 – Precalculus Optimization, both absolute (global) and relative (local) extrema. 67

Modeling rates of change, including related rates problems.

Use of implicit differentiation to find the derivative of an inverse function.

Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and

acceleration.

Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.

+ Numerical solution of differential equations using Euler's method.

+ L'Hôpital's Rule, including its use in determining limits and convergence of improper integrals and series.

F. Computation of Derivatives

III. Integrals

Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse

trigonometric functions. Basic rules for the derivative of sums, products, and quotients of functions. Chain rule and implicit differentiation. + Derivatives of parametric, polar, and vector functions.

A. Interpretations and Properties of Definite Integrals

Definite integral as a limit of Riemann sums.

Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

Basic properties of definite integrals. (Examples include additivity and linearity.)

B. *Applications of Integrals Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a

MTH 313 – Precalculus solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and for BC only the length of a curve 68 (including a curve given in parametric form).

C. Fundamental Theorem of Calculus

Use of the Fundamental Theorem to evaluate definite integrals.

Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

D. Techniques of Antidifferentiation

Antiderivatives following directly from derivatives of basic functions.

+ Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).

+ Improper integrals (as limits of definite integrals).

E. Applications of Antidifferentiation

Finding specific antiderivatives using initial conditions, including applications to motion along a line.

Solving separable differential equations and using them in modeling. In particular, studying the equation y ' = ky and exponential growth.

+ Solving logistic differential equations and using them in modeling.

F. Numerical Approximations to Definite Integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

IV. *Polynomial Approximations and Series A. *Concept of Series A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.

B. *Series of constants

+ Motivating examples, including decimal expansion. + Geometric series with applications. + The harmonic series. + Alternating series with error bound. + Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its

use in testing the convergence of p-series.

MTH 313 – Precalculus + The ratio test for convergence and divergence. 69

+ Comparing series to test for convergence or divergence.

C. *Taylor Series

+ Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of

various Taylor polynomials of the sine function approximating the sine curve.) + Maclaurin series and the general Taylor series centered at x = a. + Maclaurin series for the functions ex, sin x, cos x, and 1/(1-x). + Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation,

antidifferentiation, and the formation of new series from known series. + Functions defined by power series. + Radius and interval of convergence of power series. + Lagrange error bound for Taylor polynomials.


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CURRICULUM MODIFICATIONS AND ADAPTATIONS





















84

A P P E N D I X G

S H O W C A S E P O R T F O L I O G U I D E L I N E S

MTH 31MTH313 – Precalculus

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Parsippany – Troy Hills Secondary Math Departments

Showcase Portfolio Guidelines

All secondary math courses showcase portfolios will contain evidence of the following NJ Core Curriculum Content Standards in Mathematics:

1. Problem-Solving 2. Reasoning 3. Tools and Technology 4. Patterns, Relationships and Functions

Specifically the student’s showcase portfolio for each subject will display evidence of the following local standards:

Precalculus:

1. Understanding of transcendental functions and their graphs. 2. Understanding and applications of trigonometric functions. 3. Problem-solving ability.

Documents

PARS I PPANY - TR OY HI LLS TOW N SHI P SC HOOLS CO URS E ...sharepoint.pthsd.k12.nj.us/ci/Approved Curriculum... · technology-dependent lessons. These methodologies are the expected