Bound Brook Public School District High School PreCalculus Curriculum
2007-2008
Dr. Edward Hoffman, Superintendent
2006-2007 Board of Education Members
Martin Gleason, President
Steve Clouser, Vice – President
Peter Allen – South Bound Brook Representative
Hal Dietrich
Terrence Hoben
Robyn Ann Jeskie
Carol Ann Koupiaris
Robert Murray
Kenneth Sella
Rae Siebel
Carole Deddy, Board Secretary
Administration
Mr. Dan Gallagher, Principal Mr. Mario Bernardo, Vice Principal Mr. Robert Nixon, Vice Principal
Ms. Dianne Ianniello, Director of Pupil Services
Curriculum Revision By:
Richard R. Selander, Mathematics Teacher DISTRICT EDUCATIONAL PHILOSOPHY MISSION STATEMENT Bound Brook High School is a supportive multicultural community that provides an innovative and academically challenging educational program while offering a variety of extra-curricular and social opportunities that encourage life-long learning and citizenship. VISION STATEMENT
The vision of the Bound Brook Public School community is to provide a comprehensive educational environment that will:
• Develop tolerant citizens • Prepare graduates for their educational and vocational choices in life • Develop life long learners • Allow students to be users of technology
• Develop finders and users of data • Provide educational opportunities both within and outside the classroom • Challenge students educationally • Provide a positive learning environment • Make students aware of their strengths and weaknesses • Recognize student successes.
BELIEFS We believe that the Bound Brook community will provide a supportive environment for academic and personal growth that will:
• Foster independence, self-reliance, and self-worth
• Prepare students for a diverse and ever-changing society. • Encourage the development of programs that promote good character in the school community.
• Enable everyone to feel physically, emotionally, and intellectually safe (free to verbally express opinions and ideas). • Value all for their unique qualities.
• Encourage all to pursue their individual goals in a challenging, supportive, and safe environment. • Provide a positive learning environment where mutual respect and opportunity exist for the exchange of ideas among teachers,
students, parents, and community members. • Deliver an instructional program that encompasses a variety of learning styles, interests, and levels of readiness for all students
in all disciplines • Demonstrate honesty, integrity, and trustworthiness in academic pursuits and social interactions.
• Respect all people and cultures • Encourage participation in one’s community as a social, civic, and personal responsibility.
• Promote learning as a life-long process.
ACADEMIC GOALS/EXPECTATION LEARNING GOALS
1. Students are able to use basic communication and mathematics skills for purposes and situations they will encounter throughout their lives. 2. Students shall develop their abilities to apply core concepts and principles from mathematics, the sciences, the arts, the humanities, social studies, practical living studies, and vocational studies to what they will encounter throughout their lives. 3. Students shall develop their abilities to become self-sufficient individuals. 4. Students shall develop their abilities to become responsible members of a family, work group, or community, including demonstrating effectiveness in community service. 5. Students shall develop their abilities to think and solve problems in school situations and in a variety of situations they will encounter in life. 6. Students shall develop their abilities to connect and integrate experiences and new knowledge from all subject matter fields with what they have previously learned and build on past learning experiences to acquire new information through various media sources.
Academic Expectations
• Students will use reference tools such as dictionaries, almanacs, encyclopedias, and computer reference programs and research tools such as interviews and surveys to find the information they need to meet specific demands, explore interests, or solve specific problems.
• Students will make sense of the variety of materials they read, observe, and hear.
• Students will use mathematical concepts and procedures to communicate, reason, and solve problems. • Students will organize and classify information through an understanding of terms defined in this course
• Students will use appropriate conventions and styles in their written work to communicate ideas and information to different
audiences and for different purposes. • Students’ oral communication will incorporate appropriate forms, conventions, and styles to communicate ideas and
information to different audiences and different purposes.
• Students will use of technology to collect, organize, and communicate information and ideas. • Students will understand scientific ways of thinking and working and use those methods to solve real-life problems.
• Students will identify, analyze, and use patterns such as cycles and trends to understand past and present events and predict
possible future events. • Students will identify and analyze systems and understand how their components work together or affect each other.
• Students will use and scientific models and scales to explain the organization and functioning of living and non-living entities
and predict other characteristics that might be observed.
• Students will understand that under certain conditions nature tends to remain the same or move toward a balance.
• Students will understand how living and nonliving things change over time and the factors that influence the changes.
• Students will understand number concepts and use numbers appropriately and accurately.
• Students will understand various mathematical procedures and use them appropriately and accurately.
• Students will understand space and dimensionality concepts and use them appropriately and accurately.
• Students will understand measurement concepts and use measurement appropriately and accurately.
• Students will understand mathematical change concepts and use them appropriately and accurately. • Students will understand mathematical structure concepts including the properties and logic of various mathematical systems.
• Students will understand probability and use statistics appropriately.
• Students will understand the democratic principles of justice, equality, responsibility, and freedom and apply them to real-life
situations.
• Students will accurately describe various forms of government and analyze issues that relate to the rights and responsibilities
of citizens in a democracy.
• Students will observe, analyze, and interpret human behaviors, social groupings, and institutions to better understand people and the relationships among individuals and among groups.
• Students will interact effectively and work cooperatively with the many ethnic and cultural groups of our nation and world.
• Students will understand economic principles and are able to make economic decisions that have consequences in daily living.
• Students will understand, analyze, and interpret historical events, conditions, trends, and issues to develop historical
perspective. • Students will recognize and understand the relationship between people and geography and apply their knowledge in real-life
situations.
• Students will present works of art convey a point of view. • Students will analyze and reflect on their own and others' artistic products and performances using accepted standards.
• Students will gain knowledge of major works of art, music, and literature and appreciate creativity and the contributions of the
arts and humanities.
• In the products they make and the performances they present, students will show that they understand how time, place, and society influence the Arts and Humanities such as languages, literature, and history.
• Students will demonstrate skills that promote individual well-being and healthy family relationships.
• Students will evaluate consumer products and services and make effective consumer decisions.
• Students will demonstrate the knowledge and skills they need to remain physically healthy and to accept responsibility for their
own physical well-being.
• Students will demonstrate strategies for becoming and remaining mentally and emotionally healthy.
• Students will demonstrate the skills to evaluate and use services and expectation resources available in their community.
• Students will perform physical movement skills effectively in a variety of settings.
• Students will demonstrate knowledge and skills that promote physical activity and involvement in physical activity throughout their lives.
• Students will use strategies for choosing and preparing for a career.
• Students will demonstrate skills and work habits that lead to success in future schooling and work.
• Students will demonstrate skills such as interviewing, writing resumes, and completing applications that are needed to be
accepted into college or other postsecondary training or to get a job.
• Students will use critical thinking skills such as analyzing, prioritizing, categorizing, evaluating, and comparing to solve a variety of problems in real-life situations.
• Students will use creative thinking skills to develop or invent novel, constructive ideas or products.
• Students will organize information to develop or change their understanding of a concept. • Students will use a decision-making process to make informed decisions. • Students will use problem-solving processes to develop solutions to complex problems.
• Students will connect knowledge and experiences from different subject areas.
• Students will use scaffolding to acquire new knowledge, develop new skills, or interpret new experiences. • Students will expand their understanding by making connections to new paradigms, skills, and experiences
PRECALCULUS COURSE DESCRIPTION This course covers all of the necessary topics of a PreCalculus course. Students coming into the course should have completed one year of geometry and two years of algebra. This class is designed for students who are either honors students or are preparing for college, as it will give them a solid foundation and knowledge of what they will study in college mathematics courses. It will also help students prepare for college entrance exams such as the SAT or ACT.
PREREQUISITES Completion of Algebra II or IMP III. OUTCOMES Participation in the district’s mathematics programs will enable students to develop mathematical proficiency. This program is designed to help students gain an ability to use and understand mathematics in a variety of contexts in order to prepare them to succeed both socially and academically. NEW JERSEY CORE CURRICULUM CONTENT STANDARDS AND STRANDS FOR MATHEMATICS There are five standards altogether, each of which has a number of lettered strands. These standards, and their associated strands, are enumerated below: 4.1. Number and Numerical Operations A. Number Sense B. Numerical Operations C. Estimation 4.2. Geometry and Measurement A. Geometric Properties B. Transforming Shapes C. Coordinate Geometry D. Units of Measurement E. Measuring Geometric Objects
4.3. Patterns and Algebra A. Patterns and Relationships B. Functions C. Modeling D. Procedures 4.4. Data Analysis, Probability, and Discrete Mathematics A. Data Analysis (Statistics) B. Probability C. Discrete Mathematics--Systematic Listing and Counting D. Discrete Mathematics--Vertex-Edge Graphs and Algorithms 4.5. Mathematical Processes A. Problem Solving B. Communication C. Connections D. Reasoning E. Representations F. Technology
The first four of these "standards" also serve as what have been called "content clusters" in the current state assessments; the lettered strands replace what have been called "macros" in the directories of test specifications. The fifth standard will continue to provide the "power base" of the assessments. It is anticipated that the expectations presented here will be used as the basis for test specifications for the next version of the statewide assessments.
WRITTEN CURRICULUM Instructional Methods and Strategies This course will be taught using a variety of instructional methods. Students will be taught directly via an auditory and visual approach through lecture and note-taking. Indirect instruction via an auditory and visual approach will be presented through the use of multimedia presentations as well as technological presentations. Teachers will utilize technology and manipulatives to assist students who require a more kinesthetic teaching approach for success. Students will be given homework and projects so they can work at home independently. Students will not only be expected to work individually, but cooperatively as well. Students will be expected to interact with one another and work cooperatively in groups, as well as create their own group multimedia presentations. Group oral/visual representations will also be required of students during the school year. Academic Expectations All students should be able to work both independently and cooperatively to develop their mathematical skills. Students should be able to prepare for standardized tests by focusing on expressing mathematical ideas and concepts clearly. Students should also become mathematically proficient by mastering a variety of problem-solving strategies. Finally, students should be able to present their learnings via the use of oral/visual presentations using classroom technology. Student Assessment Students will be assessed both formally and informally. Informally, teachers will note how students have comprehended the material via classroom and discussions and activities. Teachers will informally assess students orally by asking students questions during lectures, as well as visually by looking over the students’ classwork. Formally, teachers will assess students via homework, quizzes and tests. Homework will be checked regularly, and the homework average will account for 20% of each marking period grade. Quizzes, projects, and oral/visual presentations will be given at different intervals during each chapter of the book, and the quiz/project/presentation average will account for 30% of each marking period grade. Finally, tests will be given at the end of every chapter, and the test average will account for the remaining 50% of each marking period grade. The midterm and final exams will each count for 10% of the final grade for the course.
District Grading Philosophy and Policy The Board of Education recognizes that a system of measuring, recording, and reporting the achievements of individual pupils is important to the continuing process of learning. The Board, therefore directs the instructional program of this school district include a system of grading that measures progress toward the New Jersey Core Curriculum Content Standard and the educational goals of the district. Pupils shall be informed at the outset of any course of study of the behaviors and achievements that are expected of them and shall be kept informed of their progress during the course of study. As a rule, grading should reward pupils for positive efforts and minimize failure, and pupils should be encouraged to evaluate their own achievements. The Superintendent shall develop and continually review in consultation with teaching staff members, parent(s) or legal guardian(s), and pupils, a grading program appropriate to the course of study and maturity of pupils. The final decision on any contested grade will be the responsibility of the Principal. A pupil classified as disabled will be graded in accordance with his/her Individualized Educational Program (IEP) or the Section 504 Plan. Suggested Materials and Textbooks Notebook (to keep notes, homework, quizzes, and tests in) Writing utensil TI-83Plus Graphing Calculator Honors Textbook (Redlin, Lothar, James Stewart, and Saleem Watson. Precalculus: Mathematics for Calculus. 3rd ed. Pacific
Grove, CA: Brooks/Cole Publishing Company, 1998.) CP Textbook (Hostetler, Robert P., and Roland E. Larson. Precalculus. 3rd ed. Lexington, Ma: D.C. Heath and Company, 1993.)
COURSE MAP: PRECALCULUS
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-Curricular Support
Activities/ Experiences
4.1A,. 4.1B, 4.2A, 4.2C, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5A, 4.5C, 4.5D,
4.5E, 4.5F
SWBAT review the topics they covered in
Algebra II. These topics include the following
subjects: sets of numbers, real numbers
and operations, algebraic expressions, properties of numbers, properties of exponents
and radicals, solving equations and
inequalities, writing equations and
inequalities, using equations and
inequalities, graphing equations and
inequalities, testing for symmetry, rational
expressions, scientific notation, using the
graphing calculator, and linear equations. They
will learn to graph a circle, and be able to
write the equation of a circle in standard form.
Students will review what they
covered in Algebra II to
prepare them for the material to
be presented to them in
PreCalculus.
What are the different sets of numbers, and how do
we distinguish among them? How do we add, subtract, multiply, and
divide real numbers, and scientific notation? What are the different numerical properties, and how do we
use them? What are exponents and radicals,
and what are the different rules that apply to them? What are equations and inequalities, how do write them, how do we graph them, and how do we
solve them? How do we use equations to solve
word problems? How do we perform the basic operations on rational
expressions? What is the Cartesian Coordinate
Plane, and how can we use it to graph lines,
curves, and circles? What are the main parts of the
graph of an equation, and how do we test an
1) Summer Assignment on concepts covered in Algebra II. 2) Test on concepts
covered in Summer
Assignment.
1) Science: Scientific
Notation and its uses.
2) Language Arts: Being able
to read and properly interpret a word problem.
3) Physical Education/Health: Using quadratic
functions to model scientific,
physical, or nutritional situations.
equation for symmetry? How can we use a
graphing calculator to view the graphs of equations?
What are the main characteristics of linear
equations?
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.2A, 4.2B, 4.2C, 4.3A, 4.3B, 4.3C, 4.3D, 4.5A, 4.5C, 4.5D,
4.5E, 4.5F
SWBAT learn that a function is a subset of
equations, and that they are treated the same as
equations. They will learn how to graph
functions, be able to define and explain the
terms domain and range, and find the critical points of any graph. They will
continue to see the correlation between a function and the graph
that represents it, and be able to see how changing
one will change the other. Students will revisit the topic of
variation, and be able to apply it to different word problems. Students will
be introduced to the concept of one-to-one
functions, and how one must have a one-to-one function in order to be able to find an inverse
function.
Students will learn how to
logically apply their
mathematical knowledge to
situations involving linear relationships.
What is a function? What are different types of
functions, and how are they used? What are the
domain and range of a function, and what does it stand for? What are the
direct and inverse variation equations, and where are
they used outside the classroom? What happens to a graph when we change its algebraic equation, and vice-versa? How do we
find the extreme values of functions? How are
combinations of functions solved? What are one-to-one functions, and how do
we find and graph their inverses?
1) Quiz on functions, graphs of functions, applied
functions, and transformations
of functions. 2) Quiz on
extreme values of functions, combining
functions, and one-to-one
functions and their inverses.
3) Test on chapter
concepts mentioned
above.
1) Science or Social
Studies: Students will
be able to take any situation
involving a starting point
and a constant
slope (such as spending money or a
increasing or declining
population or growth of
bacteria), and be able to
write a linear equation to represent
that situation.
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.2A, 4.2C, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5C, 4.5D, 4.5E,
4.5F
SWBAT continue to learn polynomials and
polynomial operations. They will solve polynomial equations, and be able to find the critical points of
the graph of any polynomial. They will be
able to find the zeroes/roots of a
polynomial not only by using the graphing calculator, but also
algebraically by using a set of different rules and theorems. Students will review complex numbers
and complex number operations, as well as
rational expressions and rational expression
operations. They will also review how to find the
asymptotes of a polynomial, and how they
affect that polynomial's graph.
Students will demonstrate the
mathematical thinking and processes
required of them in the fields of mathematics and science. Students will learn what it means to "do
Algebra" as they learn to isolate and solve for
variables.
What are polynomials, and how are they graphed? How
do we add, subtract, multiply, and divide
polynomials? How do we factor polynomials? How do
we solve an equation by factoring, and can you solve word problems that involve
solving equations by factoring? What are the
critical points of the graph of a polynomial, and how do
we find them? What are the zeroes/roots of a polynomial,
and how do we find them algebraically and
graphically? What are complex numbers, and how do we perform the four basic
operations on this set of numbers? What is a rational expression, and how do we simplify one? How do we
add, subtract, multiply, and divide rational expressions? What are the asymptotes of an equation, and what do
they represent?
1) Quiz on polynomial functions, graphs of
polynomial functions, and real zeros of
polynomials. 2) Quiz on
complex numbers, complex
roots, The Fundamental Theorem of Algebra, and
rational functions. 3) Test on
chapter concepts
mentioned above.
1) Social Studies:
Finding the constant of variation or
finding different
values given different
populational or
economical situations that
involve variation.
2) Science: Being able to isolate and solve for a variable in different scientific formulas.
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.1C, 4.2A, 4.2B, 4.2C, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5A, 4.5C, 4.5D, 4.5E,
4.5F
SWBAT learn about exponential equations and how to graph them. They will also learn about the
natural exponential equation, how to graph it,
and how it is used to determine compound
interest and exponential growth of populations. Students will also be
introduced to logarithmic equations, how to graph
them, and the three basic laws of logarithms. They
will also be able to see the relationship between
exponential and logarithmic equations, and
be able to take an equation in one form and rewrite it using the other. This will enable them to solve for a variable in
different equations. They will also be able to use the fact that exponential and logarithmic functions are inverses in order to solve more difficult equations. Students will also apply
the rules of transformations of graphs
Students will learn different ways to invest
their money, and how to become more financially
responsible.
What are exponential functions, and how are they
graphed? What is the natural exponential function, how is it graphed, and where
is it applied outside of the classroom? What are
logarithmic functions, how are they solved, and how
are they graphed? What are the different laws that
govern logarithmic functions? How are
exponential and logarithmic functions related to one
another? How do we solve exponential and logarithmic
equations?
1) Quiz on exponential functions, the natural exponential
function, and logarithmic functions. 2) Quiz on
laws of logarithms, exponential equations, logarithmic equations,
and applications
of exponential
and logarithmic functions. 3) Test on
chapter concepts
mentioned above.
1) Social Studies/
Economics: Computing the interest
gained using different
investment strategies. 2) Science: Finding the
half-life given a rate of
decay for any substance.
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.1C, 4.2A, 4.2B, 4.2C, 4.2D, 4.2E, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5E, 4.5F
SWBAT review trigonometric functions in terms of the unit circle.
Students will learn different identities and
relationships among the trigonometric functions.
They will also learn how to graph trigonometric functions and their
transformations, and find the distinctive points and behavior of each type of
function. Students will be able to use the graphing
calculator to evaluate and graph these functions.
Given a situation involving
trigonometric relationships,
students will be able to create an
equation or formula to match
the situation.
What is the unit circle, and how do we find the
trigonometric functions for an angle in the unit circle?
What is the domain and range of the basic
trigonometric functions? What are some properties
and fundamental identities of trigonometric functions?
How can we find trigonometric function values using a graphing calculator?
How do we graph trigonometric functions and their transformations? What
are some relationships among the trigonometric
functions?
1) Quiz on the Unit
Circle and trigonometric functions of
real numbers. 2) Quiz on
trigonometric graphs.
3) Test on chapter
concepts mentioned
above.
1) Science: Using
trigonometry to solve
problems in physics involving
finding the distance
between two objects given the distance between one of the objects
and a third object, and the angle
from one of the objects to
the third.
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.1C, 4.2A, 4.2B, 4.2C, 4.2D, 4.2E, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5A, 4.5C, 4.5D, 4.5E, 4.5F
SWBAT see the relationship between
degrees and radians, and be able to convert
between the two. They will be able to take
information about the measure of an interior
angle of a circle and be able to find certain pieces of information regarding the section of the circle
that interior angle creates. They will explore
trigonometric functions in terms of right triangles.
Students will learn different identities and
relationships among the trigonometric functions.
They will also learn about the Law of Sines and Law of Cosines, and be able to use this information to find the side lengths and area
of any triangle.
Students will be able to apply a
given or created trigonometric equation or
formula in order to make
discoveries.
How do we convert from degrees to radians? How do
we find the length of a circular arc and the area of a circular sector? How do we
find the trigonometric functions for an angle of a right triangle? Given an
acute angle measure of a right triangle, how can we
find the lengths of the sides of that triangle? What are some of the fundamental trigonometric identities?
What are the Law of Sines and Law of Cosines and
how can they help us find the dimensions of any
triangle?
1) Quiz on angle
measure, trigonometry
of right triangles,
and trigonometric functions of
angles. 2) Quiz on
Law of Sines and Law of Cosines.
3) Test on chapter
concepts mentioned
above.
1) Science: Using
trigonometry to solve
problems in physics that
involve vectors, such
as motion and force.
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-Curricular
Support Activities/
Experiences
4.1A, 4.1B, 4.1C, 4.2A, 4.2B, 4.2C, 4.2D, 4.2E, 4.3A, 4.3B, 4.3C, 4.3D,
4.5E, 4.5F
SWBAT further explore trigonometric identities, relationships, and laws.
Students will learn to simplify trigonometric
identities. They will learn to prove identities, step-by-step, from beginning
to end. Students will also review inverse
trigonometric functions and how to graph them.
They will continue learning how to solve
equations that mix both algebraic and
trigonometric concepts. Students will learn to solve trigonometric
equations. They will be able to write a complex
number using trigonometric notation, use this new form to
perform the four basic operations and further investigate complex
numbers. Students will be introduced to the
concept of vectors, how to manipulate them, and
why they are an important mathematical concept.
Students will demonstrate the
mathematical thinking and processes
required of them in the fields of mathematics and science.
What are some more fundamental trigonometric
identities, and how can they help us find trigonometric function values? How do we simplify trigonometric expressions? How do we
prove trigonometric identities? What are the trigonometric functions of
sums and differences, and what identities describe
them? What are the double-angle and half-angle
identities and product-to-sum and sum-to-product formulas, and how do we
use them? How do we find the inverse of a
trigonometric function, and how do we graph it? How do we solve trigonometric equations? How can we
algebraically manipulate an equation that contains
trigonometric functions? How do we write complex
numbers using trigonometric notation?
How can we now multiply, divide, and apply powers to
complex numbers? How can we find the roots of
1) Quiz on trigonometric
identities, addition and subtraction formulas, double-
angle, half-angle, and
product-sum formulas. 2) Quiz on
inverse trigonometric
functions, trigonometric
equations, trigonometric
form of complex numbers,
DeMoivre’s Theorem,
and vectors. 3) Test on
chapter concepts
mentioned above.
1) Science: Using different trigonometric identities and
relationships to simplify scientific
problems that involve angles and vectors.
2) Social Studies/
Technology: Research a
famous mathematician,
and then prepare and
present a PowerPoint
presentation to the class.
complex numbers? What are vectors, and where are they applied outside of the
classroom?
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.1C, 4.2A, 4.2C, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5C, 4.5D,
4.5E, 4.5F
SWBAT review what they learned in Algebra II of
systems of equations and inequalities. They will be introduced to the concept of matrices, and how they can help solve a system of equations. They will further explore matrices algebraically. Students
will also learn how to take one large fraction and split it up into rational
factors known as partial fractions.
Given a situation involving
mathematical relationships,
students will be able to create an
equation, inequality, or
formula to match the situation,
and then apply that equation, inequality, or
formula to make discoveries.
What is a system of equations/inequalities, and
how do we solve them graphically and
algebraically? What kinds of system of equations are
there? How does a graphing calculator work,
and can you use it to solve a system of equations? What are matrices, and how can
we use them to solve a system of equations? What
is the algebra behind matrices? How can we find
the inverse of a matrix? How can we solve matrix equations? What is the
determinant of a matrix, and how can we use it to find the
solution of a system of equations? How can we split a single fraction into
partial fractions?
1) Quiz on systems of equations,
pairs of lines, and systems
of linear equations. 2) Quiz on the algebra of matrices, inverses of matrices,
matrix equations,
determinants, Cramer’s
Rule, systems of inequalities, and partial fractions. 3) Test on
chapter concepts
mentioned above.
1) Social Studies/
Economics/ Technology:
Using a graphing
calculator or computer software,
students will be able to
take a situation involving selling
multiple items at the same time (with
restrictions on the total number of items that
can be sold), and be able to determine
the best number of
each item to sell to make the biggest
profit. Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.2A, 4.2B, 4.2C, 4.2D, 4.2E, 4.3A, 4.3B, 4.3C, 4.3D, 4.4A, 4.5C, 4.5E, 4.5F
SWBAT learn about these conic sections: parabolas, ellipses, and hyperbolas.
They will learn how to recognize the equation of a conic, how to graph it,
and what the major points of each graph are. They
will also learn to transform the graph of a conic.
Students will learn about polar coordinates, how to graph them, and how they
are used in equations. They will learn how to write polar equations. They will learn about
parametric equations, how to graph them, and how they are used to rewrite
polar equations.
Students will also continue to
explore the relationship
between Algebra and Geometry. Students will
also be able to see how
different objects that are circular and elliptical in
shape are created by
understanding their equations.
What is a parabola, and what are its major points?
What is an ellipse, and what are its major points? What
is a hyperbola, and what are its major points? How do we shift and transform the graph of a conic? What are polar coordinates, and how are
they related to trigonometry? How do we graph polar coordinates and polar
equations on paper and on the graphing calculator?
How do we write the equation for a conic using polar coordinates? What are parametric equations,
how do we graph them, and how are they used? How do
we put polar equations in parametric form?
1) Quiz on parabolas, ellipses,
hyperbolas, and shifted
conics. 2) Quiz on rotation of axes, polar
coordinates, polar
equations of conics, and parametric equations. 3) Test on
chapter concepts
mentioned above.
1) Science: Using
second-degree
equations to model
different scientific
situations. Also, using a
graphing calculator or
computer software,
students will be able to
take multiple different scientific situations
involving the same number
of scientific variables, and find
when each situation has
an equal amount of
each variable.
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-
Curricular Support
Activities/ Experiences
4.1A, 4.1B, 4.1C, 4.3A, 4.3B, 4.3D, 4.4A, 4.4C, 4.5A, 4.5C, 4.5D, 4.5E,
4.5F
SWBAT build and recognize arithmetic and
geometric sequences. They will be able to find
different sums of a series through the use of
summation notation. Students will be introduced to the
important investment strategy of annuities, how they work, and how they grow and are calculated over time. An important buying strategy they will
be introduced to is that of installment buying, where students will learn how to
calculate monthly payments. Students will
learn about how to attempt proving an identity
through the use of induction. They will also
be introduced to both Pascal's Triangle and the Binomial Theorem, and how they are related to
each other.
Students will become more
adept in finding patterns given a set of data, and
be apply to apply those patterns to
make future predictions.
What a sequence? How can we find the sum of a certain number of terms of a given sequence? What is sigma
notation, and how is it used? What is the difference
between arithmetic and geometric sequences, and
how are they used? How do we find the partial sums of
these sequences, as well as the sum of an infinite
geometric series? What is an annuity, and how do we
find the amount of an annuity? What is installment
buying, and how can you calculate monthly
payments? What is the Principle of Mathematical Induction, and how does it
help us prove mathematical conjectures? What is the Binomial Theorem, how does it relate to Pascal's
Triangle, and how do we use it to expand a binomial?
1) Quiz on sequences, summation notation, arithmetic
sequences, and
geometric sequences. 2) Quiz on
annuities and installment
buying, mathematical
induction, and the Binomial Theorem. 3) Test on
chapter concepts
mentioned above.
1) Social Studies: Using
patterns among sets
of economical or
populational data to make predictions about future
results. 2) Science: Finding a pattern or sequence
among sets of biological, chemical, or physical data
in order to make
predictions about future
results.
Unit Reflections:
NJCCCS/Content Content Topics/Key Skills
Enduring Understandings Essential Questions Assessment
Connected Co-Curricular
Support Activities/
Experiences
4.1A, 4.1B, 4.1C, 4.4B, 4.5A, 4.5F
SWBAT refresh their probability skills. They
will find the probability of individual as well as
multiple events, through use of the Fundamental
Counting Principle, permutations,
combinations, and factorials. Students will
also learn about expected value, and its applications in the world
of games.
Students will become more
adept at solving logical problems
that involves discovering and utilizing patterns, such as games and the stock
market.
What is the Fundamental Counting Principle, and
how is it used in probability situations? How do we
perform permutations and combinations? How do we
find the probability of a single event or multiple
events? What is expected value, and how is it used in
probability situations?
1) Quiz on counting
principles, permutations,
and combinations.
2) Quiz on probability and
expected value.
3) Test on chapter
concepts mentioned
above. 4) PowerPoint presentation
on the life of a famous
mathematician.
1) Social Studies: Using historical data to predict the possibility of
history repeating itself
or the probability of
an event happening. 2) Science: Using past
scientific data to predict the probability of
an event occurring. 3) Social Studies/
Technology: PowerPoint presentation
on the life of a famous
mathematician.
Unit Reflections:
Subject: PRECALCULUS
National & State
Standards Guiding Program
Essential Question Skills Assessment
Unit 1: Numbers, Equations, and
Graphing (Days to complete: 16
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.2A - Geometric Properties 4.2C - Coordinate Geometry 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5A - Problem Solving 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F - Technology
What are the different sets of numbers, and how do we distinguish among them? How do we add, subtract, multiply, and divide real numbers, and scientific notation? What are the different numerical properties, and how do we use them? What are exponents and radicals, and what are the different rules that apply to them? What are equations and inequalities, how do write them, how do we graph them, and how do we solve them? How do we use equations to solve word problems? How do we perform the basic operations on rational expressions? What is the Cartesian Coordinate Plane, and how can we use it to graph lines, curves, and circles? What are the main parts of the graph of an equation, and how do we test an equation for symmetry? How can we use a graphing calculator to view the graphs of equations? What are the main characteristics of linear equations?
SWBAT review the topics they covered in Algebra II. These topics include the following subjects: sets of numbers, real numbers and operations, algebraic expressions, properties of numbers, properties of exponents and radicals, solving equations and inequalities, writing equations and inequalities, using equations and inequalities, graphing equations and inequalities, testing for symmetry, rational expressions, scientific notation, using the graphing calculator, and linear equations. They will learn to graph a circle, and be able to write the equation of a circle in standard form.
1) Summer Assignment on
concepts covered in Algebra II. 2) Test on concepts
covered in Summer
Assignment.
Unit 2: Functions and Graphs (Days to complete: 21
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.2A - Geometric Properties 4.2B - Transforming Shapes 4.2C - Coordinate Geometry 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.5A - Problem Solving 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F - Technology
What is a function? What are different types of functions, and how are they used? What are the domain and range of a function, and what does it stand for? What are the direct and inverse variation equations, and where are they used outside the classroom? What happens to a graph when we change its algebraic equation, and vice-versa? How do we find the extreme values of functions? How are combinations of functions solved? What are one-to-one functions, and how do we find and graph their inverses?
SWBAT learn that a function is a subset of equations, and that they are treated the same as equations. They will learn how to graph functions, be able to define and explain the terms domain and range, and find the critical points of any graph. They will continue to see the correlation between a function and the graph that represents it, and be able to see how changing one will change the other. Students will revisit the topic of variation, and be able to apply it to different word problems. Students will be introduced to the concept of one-to-one functions, and how one must have a one-to-one function in order to be able to find an inverse function.
1) Quiz on functions, graphs of functions, applied
functions, and transformations
of functions. 2) Quiz on
extreme values of functions, combining
functions, and one-to-one
functions and their inverses.
3) Test on chapter
concepts mentioned
above.
Unit 3: Polynomials and
Rational Functions (Days to complete: 27
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.2A - Geometric Properties 4.2C - Coordinate Geometry 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F - Technology
What are polynomials, and how are they graphed? How do we add, subtract, multiply, and divide polynomials? How do we factor polynomials? How do we solve an equation by factoring, and can you solve word problems that involve solving equations by factoring? What are the critical points of the graph of a polynomial, and how do we find them? What are the zeroes/roots of a polynomial, and how do we find them algebraically and graphically? What are complex numbers, and how do we perform the four basic operations on this set of numbers? What is a rational expression, and how do we simplify one? How do we add, subtract, multiply, and divide rational expressions? What are the asymptotes of an equation, and what do they represent?
SWBAT continue to learn polynomials and polynomial operations. They will solve polynomial equations, and be able to find the critical points of the graph of any polynomial. They will be able to find the zeroes/roots of a polynomial not only by using the graphing calculator, but also algebraically by using a set of different rules and theorems. Students will review complex numbers and complex number operations, as well as rational expressions and rational expression operations. They will also review how to find the asymptotes of a polynomial, and how they affect that polynomial's graph.
1) Quiz on polynomial functions, graphs of
polynomial functions, and real zeros of polynomials. 2) Quiz on
complex numbers,
complex roots, The
Fundamental Theorem of
Algebra, and rational
functions. 3) Test on
chapter concepts
mentioned above.
Unit 4: Exponential and
Logarithmic Functions (Days to complete: 23
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.2A - Geometric Properties 4.2B - Transforming Shapes 4.2C - Coordinate Geometry 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5A - Problem Solving 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F - Technology
What are exponential functions, and how are they graphed? What is the natural exponential function, how is it graphed, and where is it applied outside of the classroom? What are logarithmic functions, how are they solved, and how are they graphed? What are the different laws that govern logarithmic functions? How are exponential and logarithmic functions related to one another? How do we solve exponential and logarithmic equations?
SWBAT learn about exponential equations and how to graph them. They will also learn about the natural exponential equation, how to graph it, and how it is used to determine compound interest and exponential growth of populations. Students will also be introduced to logarithmic equations, how to graph them, and the three basic laws of logarithms. They will also be able to see the relationship between exponential and logarithmic equations, and be able to take an equation in one form and rewrite it using the other. This will enable them to solve for a variable in different equations. They will also be able to use the fact that exponential and logarithmic functions are inverses in order to solve more difficult equations. Students will also apply the rules of transformations of graphs on both exponential and logarithmic graphs and equations.
1) Quiz on exponential
functions, the natural
exponential function, and logarithmic functions.
2) Quiz on laws of logarithms, exponential equations, logarithmic
equations, and applications of
exponential and logarithmic
functions. 3) Test on
chapter concepts
mentioned above.
Unit 5: Trigonometric Functions
of Real Numbers (Days to complete: 9
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.2A - Geometric Properties 4.2B - Transforming Shapes 4.2C - Coordinate Geometry 4.2D - Units of Measurement 4.2E - Measuring Geometric Objects 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5E - Representations 4.5F - Technology
What is the unit circle, and how do we find the trigonometric functions for an angle in the unit circle? What is the domain and range of the basic trigonometric functions? What are some properties and fundamental identities of trigonometric functions? How can we find trigonometric function values using a graphing calculator? How do we graph trigonometric functions and their transformations? What are some relationships among the trigonometric functions?
SWBAT review trigonometric functions in terms of the unit circle. Students will learn different identities and relationships among the trigonometric functions. They will also learn how to graph trigonometric functions and their transformations, and find the distinctive points and behavior of each type of function. Students will be able to use the graphing calculator to evaluate and graph these functions.
1) Quiz on the Unit Circle and trigonometric functions of
real numbers. 2) Quiz on
trigonometric graphs.
3) Test on chapter
concepts mentioned
above.
Unit 6: Trigonometric Functions
of Angles (Days to complete: 10
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.2A - Geometric Properties 4.2B - Transforming Shapes 4.2C - Coordinate Geometry 4.2D - Units of Measurement 4.2E - Measuring Geometric Objects 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5A - Problem Solving 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F - Technology
How do we convert from degrees to radians? How do we find the length of a circular arc and the area of a circular sector? How do we find the trigonometric functions for an angle of a right triangle? Given an acute angle measure of a right triangle, how can we find the lengths of the sides of that triangle? What are some of the fundamental trigonometric identities? What are the Law of Sines and Law of Cosines and how can they help us find the dimensions of any triangle?
SWBAT see the relationship between degrees and radians, and be able to convert between the two. They will be able to take information about the measure of an interior angle of a circle and be able to find certain pieces of information regarding the section of the circle that interior angle creates. They will explore trigonometric functions in terms of right triangles. Students will learn different identities and relationships among the trigonometric functions. They will also learn about the Law of Sines and Law of Cosines, and be able to use this information to find the side lengths and area of any triangle.
1) Quiz on angle measure, trigonometry of right triangles,
and trigonometric functions of
angles. 2) Quiz on Law
of Sines and Law of
Cosines. 3) Test on
chapter concepts
mentioned above.
Unit 7: Analytic Trigonometry (Days to complete: 15
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.2A - Geometric Properties 4.2B - Transforming Shapes 4.2C - Coordinate Geometry 4.2D - Units of Measurement 4.2E - Measuring Geometric Objects 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.5E - Representations 4.5F - Technology
What are some more fundamental trigonometric identities, and how can they help us find trigonometric function values? How do we simplify trigonometric expressions? How do we prove trigonometric identities? What are the trigonometric functions of sums and differences, and what identities describe them? What are the double-angle and half-angle identities and product-to-sum and sum-to-product formulas, and how do we use them? How do we find the inverse of a trigonometric function, and how do we graph it? How do we solve trigonometric equations? How can we algebraically manipulate an equation that contains trigonometric functions? How do we write complex numbers using trigonometric notation? How can we now multiply, divide, and apply powers to complex numbers? How can we find the roots of complex numbers? What are vectors, and where are they applied outside of the classroom?
SWBAT further explore trigonometric identities, relationships, and laws. Students will learn to simplify trigonometric identities. They will learn to prove identities, step-by-step, from beginning to end. Students will also review inverse trigonometric functions and how to graph them. They will continue learning how to solve equations that mix both algebraic and trigonometric concepts. Students will learn to solve trigonometric equations. They will be able to write a complex number using trigonometric notation, use this new form to perform the four basic operations and further investigate complex numbers. Students will be introduced to the concept of vectors, how to manipulate them, and why they are an important mathematical concept.
1) Quiz on trigonometric
identities, addition and subtraction formulas,
double-angle, half-angle, and product-sum
formulas. 2) Quiz on
inverse trigonometric
functions, trigonometric
equations, trigonometric
form of complex numbers,
DeMoivre’s Theorem, and
vectors. 3) Test on
chapter concepts
mentioned above.
Unit 8: Systems of Equations, Systems of Inequalities (Days to complete: 15
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.2A - Geometric Properties 4.2C - Coordinate Geometry 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F – Technology
What is a system of equations/inequalities, and how do we solve them graphically and algebraically? What kinds of system of equations are there? How does a graphing calculator work, and can you use it to solve a system of equations? What are matrices, and how can we use them to solve a system of equations? What is the algebra behind matrices? How can we find the inverse of a matrix? How can we solve matrix equations? What is the determinant of a matrix, and how can we use it to find the solution of a system of equations? How can we split a single fraction into partial fractions?
SWBAT review what they learned in Algebra II of systems of equations and inequalities. They will be introduced to the concept of matrices, and how they can help solve a system of equations. They will further explore matrices algebraically. Students will also learn how to take one large fraction and split it up into rational factors known as partial fractions.
1) Quiz on systems of equations,
pairs of lines, and systems of
linear equations.
2) Quiz on the algebra of matrices,
inverses of matrices,
matrix equations,
determinants, Cramer’s Rule,
systems of inequalities, and partial fractions. 3) Test on
chapter concepts
mentioned above.
Unit 9: Topics in Analytic
Geometry (Days to complete: 14
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.2A - Geometric Properties 4.2B - Transforming Shapes 4.2C - Coordinate Geometry 4.2D - Units of Measurement 4.2E - Measuring Geometric Objects 4.3A - Patterns and Relationships 4.3B - Functions 4.3C - Modeling 4.3D - Procedures 4.4A - Data Analysis 4.5C - Connections 4.5E - Representations 4.5F – Technology
What is a parabola, and what is its major points? What is an ellipse, and what is its major points? What is a hyperbola, and what is its major points? How do we shift and transform the graph of a conic? What are polar coordinates, and how are they related to trigonometry? How do we graph polar coordinates and polar equations on paper and on the graphing calculator? How do we write the equation for a conic using polar coordinates? What are parametric equations, how do we graph them, and how are they used? How do we put polar equations in parametric form?
SWBAT learn about these conic sections: parabolas, ellipses, and hyperbolas. They will learn how to recognize the equation of a conic, how to graph it, and what the major points of each graph are. They will also learn to transform the graph of a conic. Students will learn about polar coordinates, how to graph them, and how they are used in equations. They will learn how to write polar equations. They will learn about parametric equations, how to graph them, and how they are used to rewrite polar equations.
1) Quiz on parabolas, ellipses,
hyperbolas, and shifted
conics. 2) Quiz on rotation of axes, polar
coordinates, polar equations of conics, and
parametric equations. 3) Test on
chapter concepts
mentioned above.
Unit 10: Sequences and Series (Days to complete: 14
Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.3A - Patterns and Relationships 4.3B - Functions 4.3D - Procedures 4.4A - Data Analysis 4.4C - Discrete Mathematics - Systematic Listing and Counting 4.5A - Problem Solving 4.5C - Connections 4.5D - Reasoning 4.5E - Representations 4.5F – Technology
What a sequence? How can we find the sum of a certain number of terms of a given sequence? What is sigma notation, and how is it used? What is the difference between arithmetic and geometric sequences, and how are they used? How do we find the partial sums of these sequences, as well as the sum of an infinite geometric series? What is an annuity, and how do we find the amount of an annuity? What is installment buying, and how can you calculate monthly payments? What is the Principle of Mathematical Induction, and how does it help us prove mathematical conjectures? What is the Binomial Theorem, how does it relate to Pascal's Triangle, and how do we use it to expand a binomial?
SWBAT build and recognize arithmetic and geometric sequences. They will be able to find different sums of a series through the use of summation notation. Students will be introduced to the important investment strategy of annuities, how they work, and how they grow and are calculated over time. An important buying strategy they will be introduced to is that of installment buying, where students will learn how to calculate monthly payments. Students will learn about how to attempt proving an identity through the use of induction. They will also be introduced to both Pascal's Triangle and the Binomial Theorem, and how they are related to each other.
1) Quiz on sequences, summation notation, arithmetic
sequences, and geometric
sequences. 2) Quiz on
annuities and installment
buying, mathematical induction, and the Binomial
Theorem. 3) Test on
chapter concepts
mentioned above.
Unit 11: Counting and Probability
(Days to complete: 8 Days)
4.1A - Number Sense 4.1B - Numerical Operations 4.1C - Estimation 4.4B - Probability 4.5A - Problem Solving 4.5F – Technology
What is the Fundamental Counting Principle, and how is it used in probability situations? How do we perform permutations and combinations? How do we find the probability of a single event or multiple events? What is expected value, and how is it used in probability situations?
SWBAT refresh their probability skills. They will find the probability of individual as well as multiple events, through use of the Fundamental Counting Principle, permutations, combinations, and factorials. Students will also learn about expected value, and its applications in the world of games.
1) Quiz on counting
principles, permutations,
and combinations.
2) Quiz on probability and
expected value.
3) Test on chapter
concepts mentioned
above. 4) PowerPoint
presentation on the life of a
famous
Top Related