Pre-Calculus€¦ · NEPTUNE TOWNSHIP SCHOOL DISTRICT Pre-Calculus Curriculum (College Prep and Honors) Grades 10-12 NEPTUNE TOWNSHIP SCHOOL DISTRICT Office of the Superintendent

NEPTUNE TOWNSHIP SCHOOL DISTRICT

Pre-Calculus Curriculum

(College Prep and Honors) Grades 10-12


Office of the Superintendent

60 Neptune Blvd.

Neptune, NJ 07753-4836

August 23, 2017 Document C1#1

NEPTUNE TOWNSHIP BOARD OF EDUCATION

Laura G. Granelli, President

Monica Kowalski-Lodato, Vice President

Brady M. Connaughton Dorothea L. Fernandez

Nicole M. Green Chanta L. Jackson

Jason A. Jones Michelle A. Moss

Donna L. Puryear

SCHOOL DISTRICT ADMINISTRATION

Tami R. Crader, Ed.D.

Superintendent of Schools

Matthew Gristina, Ed.D.

Assistant Superintendent of Schools

Peter J. Leonard

Business Administrator/Board Secretary

Peter I. Bartlett

Assistant Business Administrator/Assistant Board Secretary

Kathleen M. Skelton

Director of Special Services

Shawanda Beale

Supervisor of School Counseling Services

Cheryl L. Romano

Supervisor for Curriculum, Instruction & Assessment

Kathleen M. Thomsen

Supervisor of Early Childhood Education

Nicholas Pizzulli

Interim Administrator for Athletic & Co-Curricular Activities

ELEMENTARY SCHOOL ADMINISTRATION

Principals

Mark K. Alfone, Ed.D., Midtown Community

Lori B. Burns, Ed.D., Early Childhood Center

Lakeda D. Demery, Shark River Hills

Sally A. Millaway, Ed.D., Gables

James M. Nulle, Green Grove

Jerard L. Terrell, Ed.D., Summerfield

MIDDLE SCHOOL ADMINISTRATION

Arlene M. Rogo, Ed.D., Principal

Thomas Decker, Vice Principal

Michael V. Smurro, Vice Principal

HIGH SCHOOL ADMINISTRATION

Jennifer C. Joseph, Principal

Titania M. Hawkins, Ed.D., Vice Principal

James H. Whitson, Vice Principal

POSEIDON ADMINISTRATION

Richard W. Allen, Principal

DEPARTMENT CHAIRPERSONS

Kelly Baldino

Robert J. Hamm

Charles M. Kolinofsky

Joshua Loveland

Dawn Reinhardt

Nicole Sanyigo

Tara Stephenson

Karen Watt

Hillary L. Wilkins


Office of the Superintendent

60 Neptune Blvd.

Neptune, NJ 07753

An Affirmative Action Equal Opportunity Employer

2017


PRE-CALCULUS

CURRICULUM

Table of Contents

Acknowledgements ............................................................................................................i

District Mission Statement ............................................................................................... ii

District Educational Outcome Goals .............................................................................. iii

Course Description........................................................................................................... iv

Curriculum

Unit Title Page

Functions ........................................................................................................................... 1

Exponential and Logarithmic Functions ........................................................................... 8

Trigonometric Functions ................................................................................................. 13

Analytical Trigonometry ................................................................................................. 18

Applications of Trigonometry Using Vectors................................................................. 23

Systems and Matrices ..................................................................................................... 28

Conics ............................................................................................................................. 33

Sequences, Series, and Limits ......................................................................................... 38

Pacing Guide College-Prep Level ................................................................................... 43

Pacing Guide Honors Level ............................................................................................ 46


Pre-Calculus

Acknowledgements

The Pre-Calculus Curriculum guide was developed for Neptune High School through the efforts

of Erin Seneca, Neptune High School mathematics teacher, in cooperation with Tara Stephenson,

Department Chairperson, and under the guidance of Cheryl Romano, Supervisor for Curriculum,

Instruction and Assessment.

The teacher is to be commended for her dedication in creating this curriculum and formatting it

into UbD and her expertise in the area of mathematics. This curriculum guide prepares students to

topics in calculus. It is our hope that this guide will serve as a valuable resource for the staff

members who teach this course and that they will feel free to make recommendations for its

continued improvement.

The Pre-calculus Curriculum guide was written with related pacing guide in alignment to the 2016

New Jersey Student Learning Standards for Mathematics.

i


DISTRICT MISSION STATEMENT

The primary mission of the Neptune Township School District is to prepare all students for life

in the twenty-first century by encouraging them to recognize that learning is a continuing

process. It is with high expectations that our schools foster:

• A strong foundation in academic areas, modern technologies, life skills and the arts.

• A positive and varied approach to teaching and learning.

• An emphasis on critical thinking skills and problem-solving techniques.

• A respect for and an appreciation of our world, its resources, and its peoples.

• A sense of responsibility, good citizenship, and accountability.

• An involvement by the parents and the community in the learning process.

ii

Neptune Township School District

Educational Outcome Goals

The students in the Neptune Township schools will become life-long learners and will:

Become fluent readers, writers, speakers, listeners, and viewers with comprehension and

critical thinking skills.

Acquire the mathematical skills, understandings, and attitudes that are needed to be

successful in their careers and everyday life.

Understand fundamental scientific principles, develop critical thinking skills, and

demonstrate safe practices, skepticism, and open-mindedness when collecting, analyzing,

and interpreting information.

Become technologically literate.

Demonstrate proficiency in all New Jersey Core Curriculum Content Standards

(NJCCCS), New Jersey Student Learning Standards (NJSLS), and Next Generation

Science Standards (NGSS).

Develop the ability to understand their world and to have an appreciation for the heritage

of America with a high degree of literacy in civics, history, economics and

geography. Develop a respect for different cultures and demonstrate trustworthiness,

responsibility, fairness, caring, and citizenship.

Become culturally literate by being aware of the historical, societal, and multicultural

aspects and implications of the arts.

Demonstrate skills in decision-making, goal setting, and effective communication, with a

focus on character development.

Understand and practice the skills of family living, health, wellness and safety for their

physical, mental, emotional, and social development.

Develop consumer, family, and life skills necessary to be a functioning member of

society.

Develop the ability to be creative, inventive decision-makers with skills in

communicating ideas, thoughts and feelings.

Develop career awareness and essential technical and workplace readiness skills, which

are significant to many aspects of life and work.

iii

PRE-CALCULUS

CURRICULUM

COURSE DESCRIPTION

(5 credits) This course contains the background information that enables students to study advanced

mathematics. In this course the students will be studying extensions of Algebra II along with

advanced trigonometry and analytic geometry. All major areas of Algebra II are covered in greater

depth and application. Students will analyze functions and curve sketching, advanced trigonometry

including radian measures and trigonometric graphing, along with selected topics from discrete

mathematics including sequences, series, and probability models. This course is designed to cover

all topics that are necessary for the student to be successful in a college calculus course.

Appropriate use of technology is integrated throughout the course.

Prerequisites: successful completion of Algebra 2.

iv

1

Unit 1

Functions

Suggested Time

Frame

24 Days (Honors) / 26 Days (CP)

Overview / Rationale

This unit will serve as a review of the various functions that students have studied throughout

Algebra 1 and Algebra 2, as well as offering students an introduction to the various functions that

they will be utilizing throughout their study of Pre-Calculus. Students will study the graphs of

functions, including finding real and complex zeroes. Students will be able to compare the various

types of functions and discuss applications including inverses.

Stage 1 – Desired Results

2016 New Jersey Student Learning Standards

Standards for Mathematical Practice

MP1. Make sense of problems and persevere in solving them.

MP2. Reason abstractly and quantitatively.

MP3. Construct viable arguments and critique the reasoning of others.

MP4. Model with mathematics.

MP5. Use appropriate tools strategically.

MP6. Attend to precision.

MP7. Look for and make use of structure.

MP8. Look for and express regularity in repeated reasoning

Standards for Mathematical Content

N-CN.A.1. Know there is a complex number i such that i2 = –1, and every complex number

has the form a + bi with a and b real.

N-CN.A.2. Use the relation i2 = –1 and the commutative, associative, and distributive

properties to add, subtract, and multiply complex numbers.

N-CN.A.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and

quotients of complex numbers.

N-CN.C. Use complex numbers in polynomial identities and equations.

N-CN.C.7. Solve quadratic equations with real coefficients that have complex solutions.

N-CN.C.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite

x2 + 4 as (x + 2i)(x – 2i).

N-CN.C.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for

quadratic polynomials

A-APR.A. Perform arithmetic operations on polynomials

A-APR.A.1. Understand that polynomials form a system analogous to the integers, namely,

they are closed under the operations of addition, subtraction, and multiplication; add,

subtract, and multiply polynomials.

A-APR.B. Understand the relationship between zeros and factors of polynomials

2

A-APR.B.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number

a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of

p(x).

A-APR.B.3. Identify zeros of polynomials when suitable factorizations are available, and

use the zeros to construct a rough graph of the function defined by the polynomial.

A-APR.D. Rewrite rational expressions

A-APR.D.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the

form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)

less than the degree of b(x), using inspection, long division, or, for the more complicated

examples, a computer algebra system.

A-APR.D.7. (+) Understand that rational expressions form a system analogous to the

rational numbers, closed under addition, subtraction, multiplication, and division by a

nonzero rational expression; add, subtract, multiply, and divide rational expressions.

A-REI.D. Represent and solve equations and inequalities graphically

A-REI.D.11. Explain why the x-coordinates of the points where the graphs of the equations

y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions

approximately, e.g., using technology to graph the functions, make tables of values, or find

successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,

rational, absolute value, exponential, and logarithmic functions.

F-IF.A. Understand the concept of a function and use function notation

F-IF.A.1. Understand that a function from one set (called the domain) to another set (called

the range) assigns to each element of the domain exactly one element of the range. If f is a

function and x is an element of its domain, then f(x) denotes the output of f corresponding to

the input x. The graph of f is the graph of the equation y = f(x).

F-IF.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret

statements that use function notation in terms of a context.

F-IF.A.3. Recognize that sequences are functions, sometimes defined recursively, whose

domain is a subset of the integers. For example, the Fibonacci sequence is defined

recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F-IF.B. Interpret functions that arise in applications in terms of the context

F-IF.B.4. For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing key

features given a verbal description of the relationship. Key features include: intercepts;

intervals where the function is increasing, decreasing, positive, or negative; relative

maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5. Relate the domain of a function to its graph and, where applicable, to the

quantitative relationship it describes. For example, if the function h(n) gives the number of

person-hours it takes to assemble n engines in a factory, then the positive integers would be

an appropriate domain for the function.

F-IF.B.6. Calculate and interpret the average rate of change of a function (presented

symbolically or as a table) over a specified interval. Estimate the rate of change from a

graph.

F-IF.C. Analyze functions using different representations

3

F-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by

hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-defined functions, including step functions

and absolute value functions.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available,

and showing end behavior.

d. (+) Graph rational functions, identifying zeros and asymptotes when suitable

factorizations are available, and showing end behavior.

F-IF.C.8. Write a function defined by an expression in different but equivalent forms to

reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show

zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Essential Questions: 1. How do the algebraic properties of a

function relate to its graph?

2. What is the importance of the domain

in studying a function?

Enduring Understandings: 1. The algebraic properties of a graph,

including the calculation of zeros, can

help us describe the behavior of a

function including its intercepts.

2. The domain of a function is needed to

determine any extraneous solutions and

to properly study the graph of the

function.

Knowledge: Students will know…

● Methods to determine end behavior of

the graph of polynomial functions.

● How to apply synthetic division to

divide polynomials and find zeros of

the graph of the polynomial.

● Properties of complex numbers.

● Methods for combining functions.

● Methods for finding and graphing

inverse functions.

Skills: Students will be able to…

● Graph polynomial functions and

describe end behavior.

● Perform polynomial and synthetic

division given polynomials.

● Perform operations with complex

numbers.

● Find real and complex zeros of

polynomial functions.

● Simplify and evaluate rational

functions

● Perform various combinations of two or

more functions.

● Determine if a function has an inverse

and find the inverse if it exists.

4

In this unit plan, the following 21st Century Life and Careers skills are addressed:

Check ALL that apply –

21st Century Themes

Indicate whether these skills are:

● E – encouraged

● T – taught

● A – assessed

Career Ready Practices

9.1 Personal Financial Literacy E CRP1. Act as a responsible and

contributing citizen and employee.

Income and Careers E CRP2. Apply appropriate

academic and technical skills.

Money Management CRP3. Attend to personal health

and financial well-being.

Credit and Debt Management ET CRP4. Communicate clearly and

effectively and with reason.

Planning, Saving, and Investing CRP5. Consider the

environmental, social and economic

impacts of decisions.

Becoming a Critical Consumer E CRP6. Demonstrate creativity and

innovation.

Civic Financial Responsibility ET CRP7. Employ valid and reliable

research strategies.

Insuring and Protecting ETA CRP8. Utilize critical thinking to

make sense of problems and

persevere in solving them.

9.2 Career Awareness, Exploration,

and Preparation

E CRP9. Model integrity, ethical

leadership and effective

management.

Career Awareness E CRP10. Plan education and career

paths aligned to personal goals.

Career Exploration ETA CRP11. Use technology to enhance

productivity.

Career Preparation ET CRP12. Work productively in

teams while using cultural global

competence.

Student Resources

Primary Source

Readings

Demana, Franklin D., Bert K. Waits, Gregory D. Foley, Daniel

Kennedy, and David E. Bock. Precalculus: graphical, numerical,

algebraic. Boston: Pearson, 2015. Print.

Secondary Source

Readings

Supporting Text pages

5

Teacher Resources

Texts:

Demana, Franklin D., Bert K. Waits, Gregory D. Foley, Daniel Kennedy, and David E. Bock.

Precalculus: graphical, numerical, algebraic. Boston: Pearson, 2015. Print.

Larson, Ron, Robert P. Hostetler, and David C. Falvo. Precalculus with Limits. Boston,

Masachusets: Houghton Mifflin Co., 2007. Print.

Carter, John A. Glencoe Precalculus. Bothall, WA: McGraw-Hill Education, 2014. Print.

Supplemental Workbooks: Carter, John A. Glencoe Precalculus Workbook. Bothall, WA: McGraw-Hill Education, 2014.

Print.

Websites: http://www.larsonprecalculus.com/precalc9e/content/interactive-activities/

Worksheets: Teacher created worksheets

www.kutasoftware.com

Videos: Teacher created videos

www.khanacademy.com

http://patrickjmt.com/

Stage 2 – Assessment Evidence

Performance Task(s):

Transformations Project

- Students will create a drawing on the

coordinate plane and use

transformations - translations, and

reflections using function notation to

describe them.

Unit Assessment

- Differentiated for College Prep and

Honors levels

Other Evidence:

Do Now/Anticipatory assignments

Class Notes

Teacher Observations

Participation in class discussions

Peer/Self Assessments.

Guided practice individually, in pairs and in

group

Classroom assignments

Homework assignments

Lesson Quizzes

Closure activities/exit slips

Stage 3 – Learning Plan

Instructional

Strategies

Descriptions

http://www.larsonprecalculus.com/precalc9e/content/interactive-activities/

http://www.kutasoftware.com/

http://www.khanacademy.com/


6

Suggested

Learning

Activities

Teacher directed lessons

● Course introduction(grading, syllabus, rules)

● Functions and a Library of Parent Functions

● Polynomial Functions of Higher Degree

● Polynomial and Synthetic Division

● Complex Numbers

● Zeros of Polynomial Functions

● Rational Functions

● Combinations of Functions: Composite Functions

● Inverse Functions

Modifications Special Education Students: (These are just suggested ideas to modify

instruction. All modifications and accommodations should be specific to

each individual child’s IEP) ● Multi-sensory instruction.

● Differentiated instruction.

● Additional Vocabulary Activities.

● Provide hands-on manipulatives with format skeletons to groups of

students.

● Draw and label diagrams to represent the data for visual learners.

● Provide time for revision of work when students show need.

● Facilitate group discussions to assess understanding among varying

ability levels of students.

● Scaffolding content.

● Graphic organizers.

English Language Learners:

● Identify key phrases or new vocabulary to pre-teach.

● Additional Vocabulary Activities: to support the ELL students to

build mathematical understanding


● Provide visual cues.




● Encourage students to offer bilingual assistance to each other.

Students at Risk of Failure:

● Reteach to Build Understanding: for struggling learners to revisit

and practice the lesson concept or skill modeling.




● Mneumonics.

7

Gifted Students: ● Enrichment Activities: to challenge the advanced-proficient student.

● Provide extension assignments and activities.

● Projects in multiple tasks.

● Grouping.

● Honors-level courses should feature activities and assessments

that challenge students beyond the general education class

requirements.

8

Unit 2

Exponential and Logarithmic Functions

Suggested Time

Frame

8 Days


This unit will serve as a review of the Exponential and Logarithmic functions that students have

studied throughout Algebra 2, as well as offering students an introduction to using the function to

model real life phenomena that they will be utilizing throughout their study of Pre-Calculus.

Students will be able to compare the exponential and log functions and discuss applications

including inverses.













F-IF.B. Interpret functions that arise in applications in terms of the context

F-IF.B.4. For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing key

features given a verbal description of the relationship. Key features include: intercepts;

intervals where the function is increasing, decreasing, positive, or negative; relative

maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5. Relate the domain of a function to its graph and, where applicable, to the

quantitative relationship it describes. For example, if the function h(n) gives the number of

person-hours it takes to assemble n engines in a factory, then the positive integers would be

an appropriate domain for the function.

F-IF.C. Analyze functions using different representations

F-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by

hand in simple cases and using technology for more complicated cases.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and

trigonometric functions, showing period, midline, and amplitude.

F-IF.C.8. Write a function defined by an expression in different but equivalent forms to

reveal and explain different properties of the function.

9

b. Use the properties of exponents to interpret expressions for exponential functions. For

example, identify percent rate of change in functions such as y = (1.02)t , y = (0.97)t , y =

(1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay

Essential Questions: 1. How can exponential and logarithmic

functions be used to model real life

phenomena?

Enduring Understandings: 1. Functions can be used to represent

situations including the study of the

growth or decline of populations and

the maximization of volume for an

object, among other phenomena.


● Properties of logarithmic and

exponential functions.

● Methods of rewriting and evaluating

logarithmic expressions.

● Methods of solving logarithmic and

exponential equations.


● Graph and evaluate exponential and

logarithmic functions.

● Apply the properties of logarithms to

simplify logarithmic expressions.

● Solve exponential and logarithmic

equations.



21st Century Themes



● T – taught

● A – assessed


9.1 Personal Financial Literacy E CRP1. Act as a responsible and


Income and Careers E CRP2. Apply appropriate academic

and technical skills.





Planning, Saving, and Investing CRP5. Consider the environmental,

social and economic impacts of

decisions.


innovation.

Civic Financial Responsibility E CRP7. Employ valid and reliable






and Preparation

CRP9. Model integrity, ethical


10

management.

X Career Awareness E CRP10. Plan education and career


X Career Exploration ET CRP11. Use technology to enhance

productivity.

X Career Preparation ET CRP12. Work productively in teams

while using cultural global

competence.

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:







Print.





www.khanacademy.com






11



Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group



Lesson Quizzes



Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Exponential Functions and Their Graphs

● Logarithmic Functions and Their Graphs

● Properties of Logarithms

● Exponential and Logarithmic Equations

● Exponential and Logarithmic Models







students.













12











● Mneumonics.




● Grouping.



requirements.

13

Unit 3

Trigonometric Functions

Suggested Time

Frame



This unit will build on previous studies of trigonometry done in Geometry and Algebra 2.

Students will build on their understanding of the unit circle to evaluate trigonometric functions.

They will expand their understanding to all triangles by using the law of sines and law of cosines

to be able to solve any triangle. Students will explore the graphs of trigonometric functions,

building an understanding of periodicity and amplitude.













F-TF.A. Extend the domain of trigonometric functions using the unit circle

F-TF.A.1. Understand radian measure of an angle as the length of the arc on the unit circle

subtended by the angle.

F-TF.A.2. Explain how the unit circle in the coordinate plane enables the extension of

trigonometric functions to all real numbers, interpreted as radian measures of angles

traversed counterclockwise around the unit circle.

F-TF.A.3. (+) Use special triangles to determine geometrically the values of sine, cosine,

tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and

tangent for π-x, π+x, and 2π–x in terms of their values for x, where x is any real number.

F-TF.A.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of

trigonometric functions.

F-TF.B. Model periodic phenomena with trigonometric functions

F-TF.B.5. Choose trigonometric functions to model periodic phenomena with specified

amplitude, frequency, and midline. F-TF.B.6. (+) Understand that restricting a trigonometric function to a domain on which it is

always increasing or always decreasing allows its inverse to be constructed.

F-TF.B.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling

contexts; evaluate the solutions using technology, and interpret them in terms of the context. G-SRT.C. Define trigonometric ratios and solve problems involving right triangles

14

G-SRT.C.6. Understand that by similarity, side ratios in right triangles are properties of the

angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT.C.7. Explain and use the relationship between the sine and cosine of complementary

angles.

G-SRT.C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles

in applied problems.«

G-SRT.D. Apply trigonometry to general triangles

G-SRT.D.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing

an auxiliary line from a vertex perpendicular to the opposite side.

G-SRT.D.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.

G-SRT.D.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find

unknown measurements in right and non-right triangles (e.g., surveying problems, resultant

forces).

Essential Questions: 1. How can trigonometric functions be

applied to triangles other than right

triangles?

2. How do trigonometric functions model

real world problems and their

solutions?

Enduring Understandings: 1. Trigonometric functions can be used to

find the missing sides and angles for

any triangle and to find the area of any

triangle.

2. Trigonometric functions can model real

world problems through their use in

finding various hard to find

measurements and through modeling of

phenomena such as sound waves.


● How the unit circle can be used to find

trigonometric values.

● Properties of special right triangles.

● How to graph a trigonometric function

using technology.

● How to describe a trigonometric graph

using its equation.

● Use the laws of sines and cosines to

solve triangles.


● Use the unit circle to find trigonometric

values.

● Solve special right triangles using their

properties.

● Graph a trigonometric function using

technology.

● Describe the graph of a trigonometric

function using its period and amplitude.

● Solve triangles using the law of sines

and law of cosines.



21st Century Themes



● T – taught

● A – assessed


9.1 Personal Financial Literacy CRP1. Act as a responsible and


15

Income and Careers CRP2. Apply appropriate academic


Money Management CRP3. Attend to personal health and

financial well-being.

Credit and Debt Management ETA CRP4. Communicate clearly and




decisions.


innovation.

Civic Financial Responsibility CRP7. Employ valid and reliable






and Preparation

ET CRP9. Model integrity, ethical


management.

X Career Awareness CRP10. Plan education and career



productivity.



competence.

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:






16


Print.





www.khanacademy.com




Unit Circle Project

- Students will create a unit circle with

special right triangles and complete unit

circle worksheets using their created

unit circle.

Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group



Lesson Quizzes



Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Radian and Degree Measure

● Trigonometric Functions: The Unit Circle

● Right Triangle Trigonometry

● Trigonometric Functions of Any Angle

● Graphs of Sine and Cosine Functions

● Graphs of Other Trigonometric Functions

● Inverse Trigonometric Functions

● Applications and Models

● Law of Sines

● Law of Cosines





17







students.























● Mneumonics.




● Grouping.



requirements.

18

Unit 4

Analytical Trigonometry

Suggested Time

Frame

9 Days


Students will further explore trigonometric functions throughout this unit. Students will study

trigonometric identities and use the identities to verify that the identities are true. Students will

then make connections between their understanding of trig identities and strategies for solving

equations to solve trigonometric equations and use various related trigonometric formulas.













F-TF.B. Model periodic phenomena with trigonometric functions.

F-TF.B.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling

contexts; evaluate the solutions using technology, and interpret them in terms of the context.

F-TF.C. Prove and apply trigonometric identities

F-TF.C.8. Prove the Pythagorean identity sin2 (θ) + cos2 (θ) = 1 and use it to find sin(θ),

cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

F-TF.C.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and

use them to solve problems.

Essential Questions: 1. How can using a trigonometric identity

be used to verify other identities?

2. How can methods for solving general

equations be used to solve

trigonometric equations?

Enduring Understandings: 1. Manipulating the fundamental

trigonometric identities can create

equivalent expressions that can verify

other trigonometric identities.

2. Strategies for solving linear and

quadratic equations can be applied to

trigonometric equations. Vectors

represent motion



19

● The fundamental trigonometric

identities.

● Strategies for verifying trigonometric

identities.

● How to solve trigonometric equations.

● Sum and difference formulas for


● Multiple-angle and product-to-sum

formulas for trigonometric functions.

● Verify trigonometric identities.

● Solve trigonometric equations.

● Use sum-and difference formulas for


● Use multiple-angle and product-to-sum

formulas for trigonometric functions.



21st Century Themes



● T – taught

● A – assessed




Income and Careers CRP2. Apply appropriate academic






Planning, Saving, and

Investing

CRP5. Consider the environmental,


decisions.

Becoming a Critical

Consumer

ET CRP6. Demonstrate creativity and

innovation.






9.2 Career Awareness,

Exploration, and

Preparation



management.



X Career Exploration CRP11. Use technology to enhance

productivity.



competence.

20

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:







Print.





www.khanacademy.com




Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group



Lesson Quizzes






21


Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Using Fundamental Identities

● Verifying Trigonometric Identities

● Solving Trigonometric Equations

● Sum and Difference Formulas

● Multiple-Angle and Product-to-Sum Formulas







students.























● Mneumonics.

Gifted Students:

22

● Enrichment Activities: to challenge the advanced-proficient student.



● Grouping.



requirements.

23

Unit 5

Applications of Trigonometry Using Vectors

Suggested Time

Frame

8 Days


Students will be introduced to vectors which will tie back to their previous study of complex

numbers. They will use the vectors to model problems of motion. Students will be introduced

to polar coordinates which will lead to new and interesting graphs. At the end of the unit

students will use polar coordinates in the context of complex numbers to explore De-Moive’s

Theorem.













N-CN.B. Represent complex numbers and their operations on the complex plane.

N-CN.B.4. (+) Represent complex numbers on the complex plane in rectangular and polar

form (including real and imaginary numbers), and explain why the rectangular and polar

forms of a given complex number represent the same number.

N-CN.B.5. (+) Represent addition, subtraction, multiplication, and conjugation of complex

numbers geometrically on the complex plane; use properties of this representation for

computation. For example, (-1 + √3i)3 = 8 because (-1+ √3i) has modulus 2 and argument

120°.

N-CN.B.6. (+) Calculate the distance between numbers in the complex plane as the modulus

of the difference, and the midpoint of a segment as the average of the numbers at its

endpoints.

N-VM.A. Represent and model with vector quantities.

N-VM.A.1. (+) Recognize vector quantities as having both magnitude and direction.

Represent vector quantities by directed line segments, and use appropriate symbols for

vectors and their magnitudes (e.g., v, |v|, ||v||, v).

N-VM.A.2. (+) Find the components of a vector by subtracting the coordinates of an initial

point from the coordinates of a terminal point.

N-VM.A.3. (+) Solve problems involving velocity and other quantities that can be

represented by vectors.

24

N-VM.B. Perform operations on vectors.

N-VM.B.4. (+) Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that

the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

b. Given two vectors in magnitude and direction form, determine the magnitude and

direction of their sum.

c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w,

with the same magnitude as w and pointing in the opposite direction. Represent vector

subtraction graphically by connecting the tips in the appropriate order, and perform vector

subtraction component-wise.

N-VM.B.5. (+) Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their

direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of

cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v

(for c < 0).

Essential Questions: 1. How can vectors be used to represent

real life phenomena?

Enduring Understandings: 1. Vectors represent motion involving

both speed and direction which models

real life events.


● How to draw vectors in the coordinate

plane.

● How to perform operations vectors.

● How to write the trigonometric form of

a complex number.


● Draw vectors in a coordinate plane.

● Find the dot product of two vectors.

● Determine the magnitude of a vector.

● Find the angle between two vectors.

● Write the trigonometric form of a

complex number.



21st Century Themes



● T – taught

● A – assessed




Income and Careers ETA CRP2. Apply appropriate academic






25



decisions.


innovation.







and Preparation



management.



X Career Exploration ETA CRP11. Use technology to enhance

productivity.



competence.

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:







Print.


Worksheets:


26

Teacher created worksheets



www.khanacademy.com




Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group



Lesson Quizzes



Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Vectors in the Plane

● Vectors and Dot Products

● Modeling with Vectors

● Trigonometric Form of a Complex Number







students.









27


















● Mneumonics.




● Grouping.



requirements.

28

Unit 6

Systems and Matrices

Suggested Time

Frame

6 Days


Students will review strategies for solving systems of equations and inequalities and expand their

understanding to nonlinear systems. Students will work with systems with more than two variables

to compare how the methods they use are similar to those used with two variables. Students will

then develop skills for performing operations on matrices and to see how they can be applied to

solve systems of equations, including the use of technology.













A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs),

focusing on pairs of linear equations in two variables.

A-REI.7. Solve a simple system consisting of a linear equation and a quadratic equation in

two variables algebraically and graphically. For example, find the points of intersection

between the line y = –3x and the circle x 2 + y 2 = 3.

A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector

variable.

A-REI.9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear

equations (using technology for matrices of dimension 3 × 3 or greater). Represent and solve

equations and inequalities graphically

A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y

= f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions

approximately, e.g., using technology to graph the functions, make tables of values, or find

successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,

rational, absolute value, exponential, and logarithmic functions.

N-VM. Perform operations on matrices and use matrices in applications.

N-VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or

incidence relationships in a network.

29

N-VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the

payoffs in a game are doubled.

N-VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.

N-VM.9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for

square 16 matrices is not a commutative operation, but still satisfies the associative and

distributive properties.

N-VM.10. (+) Understand that the zero and identity matrices play a role in matrix addition

and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a

square matrix is nonzero if and only if the matrix has a multiplicative inverse.

N-VM.11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of

suitable dimensions to produce another vector. Work with matrices as transformations of

vectors.

N-VM.12. (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the

absolute value of the determinant in terms of area.

Essential Questions: 1. How can matrices be used to solve

systems of linear equations?

2. How can systems of equations be used

to represent real life situations?

Enduring Understandings: 1. Matrices can be created from a linear

system and reduced to find the solution

to the system.

2. Systems of equations can represent

many real life situations, such as

finance and chemistry, which require

multiple variables.


● Methods for solving linear systems of

two or more variables algebraically.

● Methods for solving non-linear systems

of two variables algebraically.

● How to write and solve a linear system

using a matrix and technology.

● How to find the inverse of a square

matrix.

● How to calculate the determinant of a

square matrix.

● Applications of the determinant of a

square matrix.


● Solve systems of linear and nonlinear

equations algebraically.

● Perform operations on a matrix.

● Use a matrix to solve a system of

equations.

● Find the inverse of a square matrix

(using technology for a 3x3 or above.)

● Calculate the determinant of a square

matrix (using technology for a 3x3 or

above).

● Use the determinant of a matrix in real

life situations (including finding the

area of a triangle).



21st Century Themes



● T – taught

● A – assessed


30



Income and Careers ET CRP2. Apply appropriate academic








decisions.


innovation.







and Preparation



management.




productivity.



competence.

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:





31



Print.





www.khanacademy.com




Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group



Lesson Quizzes



Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Linear and Nonlinear Systems of Equations

● Two-Variable Linear Systems

● Multivariable Linear Systems

● Systems of Inequalities

● Matrices and Systems of Equations

● Operations with Matrices

● The Inverse of a Square Matrix

● The Determinant of a Square Matrix

● Applications of Matrices and Determinants.





32







students.























● Mneumonics.




● Grouping.



requirements.

33

Unit 7

Conics

Suggested Time

Frame

5 Days


In this unit, students will explore the different conic sections in mathematics. Students will build

on what they have learned previously about quadratic equations to understand parabolas. Students

will be introduced to ellipses and hyperbolas and discuss the various properties of each conic

section.













G-GPE. Translate between the geometric description and the equation for a

conic section

G-GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean

Theorem; complete the square to find the center and radius of a circle given by an equation.

G-GPE.2. Derive the equation of a parabola given a focus and directrix.

G-GPE.3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact

that the sum or difference of distances from the foci is constant.

Essential Questions: 1. How can various conic sections be

created from the same figure?

Enduring Understandings: 1. The different ways that a plane can

intersect a double-napped cone will

create different conic sections based on

the nature of the intersection, each with

its own properties.


● The various conic shapes that can be

created.


● Write the equation of a circle given its

center and radius.

34

● The standard equations of each of the

conic sections.

● How to write an equation in standard

form for each conic section.

● Write the equation of a parabola in

standard form given is focus and

directrix.

● Write the equations of an ellipse given

its foci, vertices and axes.

● Write the equation of a hyperbola given

its foci, vertices, and axes.

● Classify a conic from its general

equation.



21st Century Themes



● T – taught

● A – assessed




Income and Careers ET CRP2. Apply appropriate academic






Planning, Saving, and

Investing

CRP5. Consider the environmental,


decisions.


innovation.






9.2 Career Awareness,

Exploration,

and Preparation



management.

X Career Awareness ET CRP10. Plan education and career



productivity.

35



competence.

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:







Print.





www.khanacademy.com




Conic Sections Project

- Students will create a picture using a

minimum of 4 conic sections using a

graphing application, desmos.

Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group






36


Lesson Quizzes



Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Graphs of Equations (Circles)

● Introduction to Conics: Parabolas

● Ellipses

● Hyperbolas







students.























● Mneumonics.

37




● Grouping.



requirements.

38

Unit 8

Sequences, Series, and Limits

Suggested Time

Frame



In this culminating unit, students will explore the concepts of sequences, series, and limits (Limits

are Honors Only). Students will learn about various concepts that will be needed to move forward

into Calculus.













A-SSE.B. Write expressions in equivalent forms to solve problems

A-SSE.B.4. Derive and/or explain the formula for the sum of a finite geometric series (when

the common ratio is not 1), and use the formula to solve problems. For example, calculate

mortgage payments.

F-BF.A. Build a function that models a relationship between two quantities

F-BF.A.1. Write a function that describes a relationship between two quantities.

F-BF.A.1.a. Determine an explicit expression, a recursive process, or steps for calculation

from a context.

F-BF.A.2. Write arithmetic and geometric sequences both recursively and with an explicit

formula, use them to model situations, and translate between the two forms.

F-LE.A.2. Construct linear and exponential functions, including arithmetic and geometric

sequences, given a graph, a description of a relationship, or two input-output pairs (include

reading these from a table).

A-APR.C.5. (+) Know and apply the Binomial Theorem for the expansion of (x + y) n in

powers of x and y for a positive integer n, where x and y are any numbers, with coefficients

determined for example by Pascal’s Triangle.

Essential Questions: 1. How can sequences and series be used

to model real life phenomena?

2. How do limits help us to study

mathematical functions? (Honors Only)

Enduring Understandings: 1. How can sequences and series be used

to model real life phenomena?

2. How do limits help us to study

mathematical functions? (Honors Only)

39


● How to evaluate arithmetic and

geometric series.

● How to raise a binomial to a power by

using the Binomial Theorem.

● How to evaluate a limit. (Honors Only).


● Evaluate an arithmetic sequence.

● Evaluate a geometric sequence.

● Use the Binomial Theorem to raise a

binomial expression to a given power.

● Evaluate a limit for a given function.

(Honors Only).



21st Century Themes



● T – taught

● A – assessed




Income and Careers ETA CRP2. Apply appropriate academic








decisions.


innovation.

Civic Financial Responsibility CRP7. Employ valid and reliable






and Preparation

ET CRP9. Model integrity, ethical


management.

X Career Awareness E CRP10. Plan education and career



productivity.



competence.

40

Student Resources

Primary Source

Readings




Secondary Source

Readings


Teacher Resources

Texts:







Print.





www.khanacademy.com




Unit Assessment


Honors levels.

Other Evidence:


Class Notes





group



Lesson Quizzes






41


Instructional

Strategies

Descriptions

Suggested

Learning

Activities


● Sequences and Series

● Arithmetic Sequences and Partial Sums

● Geometric Sequences and Series

● The Binomial Theorem

● Introduction to Limits (Honors Only)

● Techniques for Evaluating Limits (Honors Only)







students.























● Mneumonics.


42



● Grouping.



requirements.

43

College-Prep Precalculus Pacing Guide

Day Unit Topic Chapter Day Unit Topic Chapter

1

Intro to class, syllabus, rules,

etc. Algebra 2 Review P 46 3 Inverse Trig Functions 4

2 Algebra 2 Review P 47 3

Solving Problems with Trig

Functions 4

3 Algebra 2 Review P 48 3 Graphs of Sine and Cosine 4


Graphs of Tangent,

Cotangent, Secant and

Cosecant 4

5 1

Functions; Domain and

Range; Increasing and

Decreasing 1 50 3 Unit 3B Test 4

6 1 Extrema and End Behavior 1 51 4 Fundamental Trig Identities 5

7 1 Parent Functions 1 52 4 Fundamental Trig Identities 5

8 1 Transformations 1 53 4 Proving Trig Identities 5

9 1 Transformation Project 1 54 4

Sum and Difference

Identities 5

10 1 Transformation Project 1 55 4 Multiple Angle Identities 5

11 1

Function Combinations and

Compositions 1 56 4 Law of Sines 5

12 1 Inverse Relations 1 57 4 Law of Cosines 5

13 1 Unit 1A Review 1 58 4 Unit 4 Review 5

14 1 Unit 1A Test 1 59 4 Unit 4 Test 5

15 1

Polynomial Functions and

Rate of Change 2 60 5

Vector Operations; Unit

Vectors; Direction Angles 6

16 1 Quadratic Functions 2 61 5

Dot Product; Angle Between

Vectors; Vector Projections 6

17 1 Power Functions 2 62 5 Vector Projections and Work 6

44

18 1

Graphs of Polynomial

Functions; End Behavior;

Zeros 2 63 5 Parametric Equations 6

19 1

Polynomial Division;

Remainder and Factor

Theorems 2 64 5 Polar Coordinates 6

20 1 Complex Zeros 2 65 5

Polar Form of Complex

Numbers 6

21 1 Graphs of Rational Functions 2 66 5 Unit 5 Review 6

22 1 Graphs of Rational Functions 2 67 5 Unit 5 Test 6

23 1 Solving Rational Equations 2 68 6

Solving Systems of

Equations 7

24 1 Solving Rational Equations 2 69 6 Matrix Algebra 7

25 1 Unit 1B Review 2 70 6 Matrix Algebra 7

26 1 Unit 1B Test 2 71 6 Systems of Inequalities 7

27 2 Exponential and Log Graphs 3 72 6 Unit 6 Review 7

28 2 Growth and Decay Models 3 73 6 Unit 6 Test 7

29 2

Exp. Inverses; Properties of

Logs 3 74 7

Intro to Conic Sections;

Parabolas 8

30 2

Properties of Logs/ Solving

Exponentials 3 75 7 Circles and Ellipses 8

31 2 Solving Log Equations 3 76 7 Hyperbolas 8

32 2 Compound Interest 3 77 7 Unit 7 Review 8

33 2 Unit 2 Review 3 78 7 Unit 7 Test 8

34 2 Unit 2 Test 3 79 8

Intro to Sequences and

Series 9

35 3

Degrees and Radians; Arc

Length; Angular and Linear

Motion 4 80 8

Arithmetic Sequences and

Series 9

36 3 Unit Circle 4 81 8

Geometric Sequences and

Series 9

37 3 Unit Circle Project 4 82 8 Pascal's Triangle 9

45

38 3 Unit Circle Project 4 83 8 Binomial Theorem 9

39 3 Right Traingle Trig 4 84 8 Unit 8 Review 9

40 3 Trig Functions of Any Angle 4 85 8 Unit 8 Test 9

41 3 Trig Functions of Any Angle 4 86 Final Review

42 Midterm Review 87 Final Review


44 MIDTERM EXAMS

89 FINAL EXAMS

45 90

46

Honors Pre-Calculus Pacing Guide

Day Unit Topic Chapter Day Unit Topic Chapter

1

Intro to class, syllabus, rules,

etc. Algebra 2 Review P 46 4

Fundamental Trig

Identities 5


Fundamental Trig

Identities 5

3 Algebra 2 Review P 48 4 Proving Trig Identities 5

4 1

Functions; Domain and

Range; Increasing and

Decreasing 1 49 4

Sum and Difference

Identities 5

5 1 Extrema and End Behavior 1 50 4 Multiple Angle Identities 5

6 1 Parent Functions 1 51 4 Law of Sines 5

7 1 Transformations 1 52 4 Law of Cosines 5

8 1 Transformation Project 1 53 4 Unit 4 Review 5

9 1 Transformation Project 1 54 4 Unit 4 Test 5

10 1

Function Combinations and

Compositions 1 55 5

Vector Operations; Unit

Vectors; Direction

Angles 6

11 1 Inverse Relations 1 56 5

Dot Product; Angle

Between Vectors; Vector

Projections 6

12 1 Unit 1A Review 1 57 5

Vector Projections and

Work 6

13 1 Unit 1A Test 1 58 5 Parametric Equations 6

14 1

Polynomial Functions and

Rate of Change 2 59 5 Polar Coordinates 6

15 1 Quadratic Functions 2 60 5

Polar Form of Complex

Numbers 6

16 1 Power Functions 2 61 5 Unit 5 Review 6

47

17 1

Graphs of Polynomial

Functions; End Behavior;

Zeros 2 62 5 Unit 5 Test 6

18 1

Polynomial Division;

Remainder and Factor

Theorems 2 63 6

Solving Systems of

Equations 7

19 1 Complex Zeros 2 64 6 Matrix Algebra 7

20 1 Graphs of Rational Functions 2 65 6 Matrix Algebra 7

21 1 Graphs of Rational Functions 2 66 6 Systems of Inequalities 7

22 1 Solving Rational Equations 2 67 6 Unit 6 Review 7

23 1 Unit 1B Review 2 68 6 Unit 6 Test 7

24 1 Unit 1B Test 2 69 7

Intro to Conic Sections;

Parabolas 8

25 2 Exponential and Log Graphs 3 70 7 Circles and Ellipses 8

26 2 Growth and Decay Models 3 71 7 Hyperbolas 8

27 2

Exp. Inverses; Properties of

Logs 3 72 7 Unit 7 Review 8

28 2

Properties of Logs/ Solving

Exponentials 3 73 7 Unit 7 Test 8

29 2 Solving Log Equations 3 74 8

Intro to Sequences and

Series 9

30 2 Compound Interest 3 75 8

Arithmetic Sequences

and Series 9

31 2 Unit 2 Review 3 76 8

Geometric Sequences and

Series 9

32 2 Unit 2 Test 3 77 8 Pascal's Triangle 9

33 3

Degrees and Radians; Arc

Length; Angular and Linear

Motion 4 78 8 Binomial Theorem 9

34 3 Unit Circle 4 79 8 Unit 8A Review 9

35 3 Unit Circle Project 4 80 8 Unit 8A Test 9

36 3 Right Traingle Trig 4 81 8 Introduction to Limits 11

48

37 3 Trig Functions of Any Angle 4 82 8 Limits Graphically 11

38 3 Inverse Trig Functions 4 83 8 Limits Algebraically 11

39 3

Solving Problems with Trig

Functions 4 84 8 Limits Review 11

40 3 Graphs of Sine and Cosine 4 85 8 Limits Test 11

41 3

Graphs of Tangent,

Cotangent, Secant and

Cosecant 4 86 Final Review



44 MIDTERM EXAMS

89 FINAL EXAMS

45 90

Documents

Pre-Calculus€¦ · NEPTUNE TOWNSHIP SCHOOL DISTRICT Pre-Calculus Curriculum (College Prep and Honors) Grades 10-12 NEPTUNE TOWNSHIP SCHOOL DISTRICT Office of the Superintendent