Download Alliance for College-Ready Public Schools

Pre-Calculus Instructional Guide 2011-2012
1
Subject: Pre-Calculus/Pre-Calculus Honors
Benchmark Assessments and Instructional Guide
Instructional Guides are provided as resource for Alliance classroom teachers. They identify high priority grade-level standards to be taught during each quarter of
instruction in the context of proposed units with a suggested amount of time. High priority standards are assessed on quarterly benchmark exams.
Pre-Calculus begins with a study of different number systems. The real and complex number systems are explored. Basic set theory is introduced and used to
make logical arguments about number systems and their subsets. The concept of sets are connected to the mapping of set A, domain, to set B, range, through
the use of functions. A library of basic functions is established and transformations and compositions are used to graph and analyze these functions. Math modeling is introduced and connected to equations in one and two variables, and functions. This leads to a discussion of the relationship between the function and its
graph to include the ability to predict behavior. In contrast, an analysis of the general equation of the second degree leads to a thorough study of circles, parabolas, ellipses, and hyperbolas, which are not necessarily functions. Polynomial and rational functions, and their graphs, are then studied in depth. Exponential and
logarithmic functions are further explored, including a study of the logistic growth function. The study of trigonometry is introduced through a review of right triangle
trigonometry and applications of the law of sines and law of cosines in an applied context. Trigonometry functions are defined on the unit circle. Graphs of the
trigonometric functions are investigated and plane transformations are applied. The model for harmonic motion is discussed as an application of trigonometric
functions of a real number. Trigonometric equations are introduced, and methods of verifying trigonometric identities are explored. Trigonometric equations are
solved and their solutions are connected to the graph of a function and the unit circle. Infinite series are reviewed, and summation notation is introduced to write
partial sums. Methods of probability and statistics are reviewed. Pre-calculus ends with a discussion of vectors, polar equations, and parametric equations.
Unit
High Priority Standards
& Learning Targets
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
Algebra 2:
5.0 Students demonstrate
knowledge of how real
and complex numbers
are related both arithmetically and graphically. In particular, they can
plot complex numbers
as points in the plane.
# CST
Items*
Textbook
Prentice Hall**
Unit 1: Mathematical Terminology
Algebra 2:
A67 #13,19,27,39,41
& Notation
15.0 Students determine
1
1
whether a specific alPre-Calculus begins with a formal
A45 #73,79A94 #53,81
gebraic statement indiscussion of terminology and notavolving rational extion that has been used in past
A94 #53,81
pressions, radical exmathematics courses. This begins
pressions, or logawith an introduction to the concept of
A11 #9,11
rithmic or exponential
a set and set theory. The history of
functions is someset theory, specifically the influence
A85-86 #57, 73,83
times true, always
of Georg Cantor, is investigated, and
true, or never true.
the importance of set theory in mathA94 #71,77,83,85
Learning Targets
ematics is highlighted, namely that
1F Use mathematical properties to
the language of set theory (in its
explain how to simplify rasimplest form) provides the foundational and radical expressions
tion for studying all areas of mathe1G Use the conjugate to rationalize the numerator of a commatics. The basic notion of a set is
plex radical expression.
defined, as well as the union and
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
intersection of two sets. Commonly
used sets are defined and given
names, including the integers , the
rational numbers , and the irrational
numbers . The union of the rational
and irrational numbers is defined as
real numbers . A purely imaginary
number is defined, and the complex
numbers are defined as . Properties
associated with these sets are discussed, and used to justify logical
statements. Subsets of these numbers are explored, as well as complements and closure. This naturally
leads to a discussion of the size (or
cardinality) of a set, as well as the
notions of countable and uncountable, which sets up a discussion of
intervals of the real number line, and
interval notation. Quantifier notation
is also developed and used throughout the unit, including (for all, for
every), (there exists), (therefore),
(element of), (implies), and (if and
only if).
Honors Level: With the discussion of
the irrational numbers, honors students prove the irrationality of
High Priority Standards
& Learning Targets
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
# CST
Items*
Textbook
Prentice Hall**
1J Use the distributive property
and exponent rules to simplify
complex rational expressions.
25.0 Students use properties from number systems to justify steps
in combining and
simplifying functions.
Learning Targets
1A Justify mathematical statements using the properties of
real and complex numbers.
1B Write subsets of the Rational,
Real, and Complex number
systems using set notation.
1C Explain relationships between
sets within the Complex
Number System and evaluate
the connections.
1D Explain how the elements in
the real number system and
imaginaries connect to form
the standard form of a complex number.
1H Find the union and intersection
of two sets and justify the solution.
1I Compare and contrast set notation and interval notation and
write mathematical statements using quantifier notation.
1K Explain the closure property
and how a set of numbers
can be closed under addition
or multiplication, using a
counterexample to justify
your reasoning
Math Analysis:
2.0 Students are adept at
the arithmetic of
complex numbers.
Learning Targets
1E Add, subtract, multiply, divide,
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
2
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
& Learning Targets
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
# CST
Items*
Textbook
Prentice Hall**
simplify, and graph complex
numbers.
3.0 Students can give
proofs of various
formulas by using the
technique of mathematical induction.
(honors only)
Learning Targets
1L Prove the irrationality of the
square root of 2 and explain each
step. (Honors)
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
3
Pre-Calculus Instructional Guide 2011-2012
Unit
Unit 2: Domain
We begin this unit by studying the
concept of a function and its properties. The definition of a function, and
the domain and range of a function,
is explored, and multiple representations of functions are used. The domain or pre-image of a function is the
set of allowable inputs, and the range
or image of a function is the set of
outputs. The graphs of a variety of
basic functions are studied including
linear functions (), power functions
(), root functions (), reciprocal functions , and exponential & logarithmic
functions ( and ). This group of functions is often referred to as the parent functions, or the library of functions. Shifting techniques are then
applied to graphs in the library of
functions.
Piecewise functions are discussed in
greater depth, featuring the greatest
integer function and the absolute
value function. The domain of a
piecewise function is carefully analyzed and the behavior of the graph
in each interval of the domain is discussed. Shifting techniques are also
applied to piecewise functions. Once
functions are graphically analyzed,
properties of functions are analyzed,
including even and odd properties.
The average rate of change of a
function is defined and used to de-
High Priority Standards
& Learning Targets
Algebra 2:
1.0 Students solve equations and inequalities
involving absolute
value.
Learning Targets
2J Explain the conditions for solving linear absolute value
problems.
2K Solve and graph the solution
sets of absolute value inequalities, and explain the solution set.
24.0 Students solve problems involving functional concepts, such
as composition, defining the inverse function and performing
arithmetic operations
on functions.
Learning Targets
2A Explain the definition of a function.
2B Represent a function in a theoretical and applied context
and justify using a counterexample
2E Composite functions and then
simplify; justify each step in
the simplification process and
connect to concept of domain
2G Reconstruct a function involving composition.
2H Write translations as function
compositions.
Calculus (foundation only):
4.0 Students demonstrate
an understanding of
the formal definition
# CST
Items*
# Q1
Items
1
1
Supporting Medium/Low
Priority Standards
& Learning Targets
Calculus (foundation only):
2.0 Students demonstrate
knowledge of both the
formal definition and the
graphical interpretation
of continuity of a function.
Learning Targets
2C Analyze the domain of a function
and explain the similarities and
differences between domains that
produce different images- some
continuous and others not continuous
2D Find the domain for functions such
as f(x)=Ö(polynomial, degree
greater than or equal to 2)
2P Explain how the domain of a
piecewise function is determined,
and how this relates to evaluating
the function. Support with an example.
# CST
Items*
4
Textbook
Prentice Hall **
Notes-definition
p.67-68 #15,19,33
p.68 #51,55p.75 #13,15p.200
#13
p.166 #9,13p.218 #21,27,33
p.253-254 #11,21,33,35
p.68-69 #39h,77,80
p.254 #53
Notes
p.97 #9-16
A58 #53,57,63
A58 #53,57,63A86 #89,93,95
Libraryp.108 #7-18
p.109-110
#27,39,43,45,47,51,61p.152
#21
p.87 #21,33,39,43
p.24-25#21,33,35,51,59p.75
#13,15
p.97 #25,27
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
termine whether a function is increasing or decreasing at a number.
Given functions are also evaluated
for the difference quotient: . Function
operations, including compositions,
are then explored graphically.
Honors Level: When students are
applying the difference quotient, they
need to justify each step mathematically using a property.
High Priority Standards
& Learning Targets
of the derivative of a
function at a point
and the notion of differentiability:
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
# CST
Items*
Textbook
Prentice Hall **
p.97 #31,35,37
p.68-69 #39h,77,80
Learning Targets
2F Evaluate and simplify functions
involving the difference quotient, explaining the connection between domains and
functions
2R When applying the difference
quotient, justify each step
mathematically using a property. (Honors)
9.0 Students use differentiation to sketch, by
hand, graphs of functions. They can identify maxima, minima,
inflection points, and
intervals in which the
function is increasing
and decreasing.
Learning Targets
2I Illustrate the library of functions
from memory, identify the
domain and range of each
function, and explain the reasoning behind memorizing
these particular functions.
2L Illustrate and describe all possible shifting techniques for
one function in the library of
functions.
2M Graph functions using transformations and describe shifting.
2N Determine whether a function
is even, odd, or neither, and
explain symmetry.
2O Describe and illustrate symmetry about the origin and
give specific cases where it
exists.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
5
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
& Learning Targets
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
# CST
Items*
Textbook
Prentice Hall **
2Q Illustrate a piecewise function
involving shifting techniques,
the greatest integer function,
and the library of functions
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
6
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 3: Introduction to Mathematical Modeling
After formally defining the number
systems and establishing the foundation for functions, it is then natural to
study how the numbers and properties and concept of input and output
discussed in the previous units are
used in a real world context. The
focus of this unit is to introduce the
concept of a mathematical model,
which is studied throughout this
course. Mathematical models are
created using previous knowledge of
a variety of different types of mathematical concepts, with a focus on
linear systems. The discussion of
modeling begins with linear models
in two variables. Linear equations
and inequalities are used to model
real life situations and contrasted to
models involving absolute value
equations and inequalities. Linear
systems in two and three variables
are categorized by their solution sets,
and interpreted graphically. Certain
types of non-linear systems are discussed (i.e. involving and ), and the
u-v method of substitution is used to
solve such systems. Another type of
non-linear mathematical model is
introduced and analyzed, namely a
quadratic model. The quadratic formula is proven using completing the
square, and used to solve a variety
of quadratic equations and quadratic
type equations used in mathematical
Algebra 2:
8.0 Students solve and
graph quadratic equations by factoring,
completing the
square, or using the
quadratic formula.
Students apply these
techniques in solving
word problems. They
also solve quadratic
equations in the complex number system.
Learning Targets
3E Prove the quadratic formula
using completing the square
and explain the connection to
domain and solutions
Linear Algebra:
6.0 Students demonstrate
an understanding that
linear systems are inconsistent (have no
solutions), have exactly one solution, or
have infinitely many
solutions.
Learning Targets
3B Design a model using a system
of linear equations & inequalities (mixture, linear programming, piecewise functions), justify each step within
the analysis and explain the
validity of the solution.
3C Categorize and connect the
solutions to mathematical linear models to types of linear
systems and explain the importance of slope
# CST
Items*
3
# Q1
Items
3
Supporting Medium/Low
Priority Standards
Calculus (foundation only):
2.0 Students demonstrate
knowledge of both the
formal definition and the
graphical interpretation
of continuity of a function.
14.0 Students apply the definition of the integral to
model problems in physics, economics, and so
forth, obtaining results
in terms of integrals.
Learning Targets
3F Design and use a model involving a
quadratic function
3H Apply a quadratic function model
(equations and inequalities) to
projectile motion and explain the
difference between solutions
found theoretically versus within
an applied context (includes inequalities)
# CST
Items*
7
Textbook
Prentice Hall **
p.42-43 #119,120p. 134
#37,38A75-77 #7-53A86-87
#111- 119
p.718-720 #55-68,71-81p.734
#77-88p.788 #57-61, p.794795 #19-31
p.40-41
#11,17,23,33,37,45,59,65,71,7
7,91p.132 #13p.717 #19,25,29
p.717 #35,36,39,40
Notes, Handout: Derive
p.152 #29,35,39,51 (10
steps)p.167 #33,34
A59 #81-86
17.0 Students compute, by
hand, the integrals of a
wide variety of functions
by using techniques of
integration, such as
substitution, integration
by parts, and trigonometric substitution. They
can also combine these
techniques when appropriate.
p.161 #11,12
Learning Targets
3D Use u-v substitution to solve certain non-linear systems of equations and justify each step (i.e.
equations with 1/x).
3G Solve quadratic type problems,
Notes
p.117 #25,26(a-c)p.160 #7,8(ab)p.778-779 #85,87A76 #31,32
Notes
p.117 #25,26(d)p.160-162
#3,7c,8c,
9,10,11,13,17,27p.169 #39
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
models, including projectile motion.
Solutions found theoretically are analyzed within an applied context for
quadratic models. Next models for
“fence”, “box” , and “garden” problems and designed and used to establish the foundation for optimization
problems in Calculus. Models of direct, indirect, and joint variation are
discussed, and used to solve problems related to physics.
Honors Level: Students solve applied
minimum and maximum problems,
focusing on the box, garden, and
fence problems
Calculus (foundation only):
11.0 Students use differentiation to solve optimization (maximumminimum problems)
in a variety of pure
and applied contexts.
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
Textbook
Prentice Hall **
justify each step, and explain the
purpose of u- substitution.
Learning Targets
3A Explain how the four step process (given, want, know,
analysis) for problem solving
supports finding solutions to a
variety of complex problems
3I Design and use a model for
fence, garden, and box problems
3J Explain the similarities and
differences in solving a "box"
problem vs. a "garden" problem vs. a fence problem vs a
projectile motion problem
3K Write a direct, indirect, or joint
variation model, and explain
the differences between the
three different types of variation.
3L Design and use a model to
maximize/minimize solutions
to “projectile”, “box”, “garden”
problems.
3M Describe the purpose of mathematical modeling and give
examples.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
8
Pre-Calculus Instructional Guide 2011-2012
Unit
Unit 4: The General Equation of
the Second Degree in Two Variables
The history of the conic sections begins in ancient Greece with the
mathematician Apollonius. His role
in mathematics and the study of conic sections is investigated. The general quadratic equation of second
degree in two variables is then defined as . The coefficient of represents the rotation of the graph of the
equation, which cannot be studied
without trigonometry. Therefore, by
setting, the following conditions are
developed for and to determine
which conic sections are produced
from the equation: parabolas [ or ,
not both], ellipses [], circles [], and
hyperbolas []. Using completing the
square, each type of conic section is
written in its standard form, and key
information about the graph is identified. Using this information, graphs
of each conic section are drawn from
the standard forms and equations in
standard form are written given certain information. Continuing the
theme of mathematical modeling,
conic sections are used to solve
problems in a real world context, including those about physics and astronomy.
Honors Level: While the discussing
the history of the conic sections,
honors students investigate the intersection of a plane and a double-
High Priority Standards
Math Analysis:
5.0 Students are familiar
with conic sections,
both analytically and
geometrically:
Learning Targets
3A Know the general equation of
the second degree in two variables.
3B Describe the conditions of the
coefficients of the general
second-degree equation that
produce each conic section
(circle, parabola, ellipse, hyperbola).
3C Describe how each conic section is obtained from the intersection of a plane and a
cone, including the degenerate forms.
3D Describe how completing the
square is involved in the
study of conic sections.
3N Design and use a mathematical model involving conic sections.
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
Math Analysis:
5.2 Students can take a geometric description of a
conic section - for example, the locus of
points whose sum of its
distances from (1, 0) and
(-1, 0) is 6 - and derive a
quadratic equation representing it.
# CST
Items*
Textbook
Prentice Hall **
notes
notes
Learning Targets
3K Given the focus and directrix of a
parabola find the standard form
3L Write equations for conic sections
given key information (foci, vertices, etc.)
3M Write equations for conic sections
from a graph.
5.1 Students can take a
quadratic equation in
two variables; put it in
standard form by
completing the
square and using rotations and translations, if necessary;
determine what type
of conic section the
equation represents;
and determine its geometric components
(foci, asymptotes, and
so forth).
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
9
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
napped cone, including the degenerate forms.
Learning Targets
3E Rewrite a circle from the general form into the standard
form and identify the center,
radius, and sketch
3F Rewrite a parabola from the
general form into the standard form and identify the vertex, focus, directrix, and
sketch
3G Rewrite an ellipse from the
general form into the standard form and identify the center, vertices, foci, co-vertices,
eccentricity, and sketch
3J Rewrite a hyperbola from the
general form into the standard form and identify the center, vertices, foci, asymptotes,
and sketch
# CST
Items*
# Q1
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
10
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
& Learning Targets
Unit 5: Polynomial and Rational
Functions
The focus of the next few units is to
use the relationships developed in
the previous unit between functions
and graphs, to analyze specific types
of functions, and then to compare
and contrast the different types of
functions studied. This discussion
begins with polynomial functions.
Polynomial functions in standard
form are defined, and the graphs of
polynomial functions are studied, as
well as the domain (pre-image) and
range (image). Graphs of polynomials and non-polynomials are compared and contrasted in order to develop an intuitive understanding of
the smooth and continuous nature of
the graph of polynomial functions.
The end behavior of graphs of polynomial functions is then explored.
Math Analysis:
4.0 Students know the
statement of, and can
apply, the fundamental theorem of algebra.
The focus of the unit shifts from the
graphs of polynomial functions to the
roots/zeros/solutions of the function.
The roots/zeros/solutions of a polynomial function are defined, and the
values of any real
roots/zeros/solutions are estimated
from the graph of the function. Connections are then made between the
Binomial Theorem and
roots/zeros/solutions of multiplicity,
and this information is used to review
the binomial expansion to a certain
8.0 Students are familiar
with the notion of the
limit of a sequence
and the limit of a
function as the independent variable approaches a number or
infinity. They determine whether certain
sequences converge
or diverge.
Learning Targets
5J Explain how the division algorithm, the remainder theorem,
and the factor theorem are
related.
5L Explain how to find the equation of a polynomial if given
the degree of the polynomial
and one of the conjugate
pairs: , as well as information
about multiplicity.
5M Describe the behavior of the
graph of a polynomial function near a root of multiplicity.
5N Classify the real roots of a
polynomial function using
Descartes Rule of Signs and
FTA.
5O Find the roots of a polynomial
function using the Rational
Roots Theorem
Learning Targets
5U Explain how rational functions
behave close to vertical as-
# CST
Items*
# Q2
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
Math Analysis:
6.0 Students find the roots
and poles of a rational
function and can graph
the function and locate
its asymptotes.
Algebra 2:
3.0 Students are adept at
operations on polynomials, including long division.
# CST
Items*
11
Textbook
Prentice Hall **
1
Learning Targets
5H Use the division algorithm, remainder theorem, and factor theorem.
4.0 Students factor polynomials representing the
difference of squares,
perfect square trinomials, and the sum and difference of two cubes.
1
Learning Targets
G Describe the relationship between
roots and factors.
5P Compare and contrast linear factors and quadratic factors.
7.0 Students add, subtract,
multiply, divide, reduce,
and evaluate rational expressions with monomial and polynomial denominators and simplify
complicated rational expressions, including
those with negative exponents in the denominator.
2
Learning Targets
5K Demonstrate how to use long division to find the roots of a poly-
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
whole number power. The remainder and factor theorems are reviewed, and used to find
roots/zeros/solutions of polynomial
functions given certain information.
The connection between a
root/zero/solution of a polynomial
function and a factor of the function
is emphasized. The Fundamental
Theorem of Algebra is reviewed, as
well as the Complex Conjugate Theorem. Descartes Rule of Signs is
used to classify the
roots/zeros/solutions of a polynomial
function and the Rational Roots Theorem is used to find the rational
roots/zeros/solutions of a polynomial
function.
The focus of the unit shifts to the
study of rational functions, with an
emphasis on the domain (pre-image)
of a rational function. Asymptotic
behavior is reviewed, and the concept of a limit is discussed using limit
notation. Conditions for the existence of different asymptotes of a
rational function are discussed. The
process of long division is introduced
in order to find the oblique asymptotes of a rational function. Rational
functions are analyzed (find domain/range, intercepts, asymptotes,
symmetry) and graphed. Both polynomial and rational functions are
used to develop mathematical models representing a real world situa-
High Priority Standards
& Learning Targets
ymptotes and find the vertical
asymptotes if they exist
5V Explain how rational functions
behave close to horizontal
asymptotes and find the horizontal asymptotes if they exist using the limit of the function as x approaches infinity
# CST
Items*
# Q2
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
# CST
Items*
12
Textbook
Prentice Hall **
nomial function.
Calculus (foundation only):
9.0 Students use differentiation to sketch, by
hand, graphs of functions. They can identify maxima, minima,
inflection points, and
intervals in which the
function is increasing
and decreasing.
Learning Targets
5A Describe a polynomial function
in standard form and classify
special cases of polynomials
5B Apply the Fundamental Theorem of Algebra.
5C Determine the end behavior of
a graph a polynomial function.
5D Explain how symmetry affects
the behavior of the polynomial graph
5E Explain how symmetry affects
the behavior of the graph
5F Find the maximum(s) and minimum(s) of a function.
5Q Determine whether a polynomial function is increasing or
decreasing over an interval.
5R Analyze a polynomial that involves multiplicity and nonreal roots
5S Explain the difference between
a polynomial function and rational function
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
tion.
Honors Level: Honors students use
limit notation to describe the end behavior of the graphs of polynomial
functions..
High Priority Standards
& Learning Targets
# CST
Items*
# Q2
Items
Supporting Medium/Low
Priority Standards
& Learning Targets
# CST
Items*
13
Textbook
Prentice Hall **
5X Graph rational functions.
5Y Defend the purpose for analyzing functions.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 6: Functions, Graphs, and
their Inverses
Composite functions are re-visited
within the context of solving radical
equations, rational equations and
defining an inverse. Namely, f and g
are inverses if and only if and . The
concept of a one to one function is
then explored algebraically and
graphically (via symmetry), and an
alternative definition is developed for
an inverse function: and are inverses if and only if is a one-to-one function and . The method for finding
inverses by switching variables is
used to find inverse functions. Finally, the action of taking an inverse is
linked with the concept of reflection
about the line and connected to odd
functions
Honors Level: Honors students analyze the proof of solving a depressed
general cubic equation, to include the
history behind the solution to the depressed cubic.
Algebra 2:
24.0 Students solve problems involving functional concepts, such
as composition, defining the inverse function and performing
arithmetic operations
on functions.
Learning Targets
6C Explain how to justify that a
function is one-to-one.
6D Explain how composition is
used to determine if two functions are inverses
6E Describe the connection between symmetry and one-toone functions.
6F Find inverse functions for algebraic and transcendental
functions
# CST
Items*
# Q2
Items
1
1
Supporting Medium/Low
Priority Standards
Algebra 2:
15.0 Students determine
whether a specific algebraic statement involving rational expressions,
radical expressions, or
logarithmic or exponential functions is sometimes true, always true,
or never true.
# CST
Items*
14
Textbook
Prentice Hall **
1
Learning Targets
6A Simplify radical expressions and
solve radical equations and explain extraneous solutions
6B Solve rational equations, justify
each step, and connect to analyzing rational functions
Trigonometry (foundational):
8.0 Students know the
definitions of the inverse trigonometric
functions and can
graph the functions.
Learning Targets
6F Find inverse functions for algebraic and transcendental
functions
6G Explain the relationship between domain and range for
inverses.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 7: The Relationship Between
Exponential and Logarithmic
Functions
The focus of the previous unit was
the study of certain algebraic functions. Algebraic functions are functions that can be produced using
basic operations. The focus of this
unit, on the other hand, is to study
certain type of non-algebraic, or transcendental, functions. The discussion begins with analyzing exponential and logarithmic functions and
their graphs. These functions are
analyzed and graphed in base and
arbitrary , and key information about
these functions is identified. This
analysis includes the study of logistics growth functions. The unit begins with an emphasis on the inverse
relationship between exponential and
logarithmic functions. Exponential
equations and functions are analyzed
both algebraically and graphically,
followed by an analysis of logarithmic
equations and functions. The properties of logarithms are proven and
used to expand or condense expressions containing logarithms. Methods of estimating the value of a logarithm are discussed, including mental
estimation and approximating using
the change of base formulas. The
focus of the unit then shifts to explore
mathematical modeling and equations involving exponentials and logarithms. The methods of solving cer-
Algebra 2:
11.1 Students understand
the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and exponents.
# CST
Items*
# Q2
Items
1
1
Learning Targets
7A Explain the relationship between exponents and logarithms.
7C Explain the definition of a logarithm and use to simplify and
solve.
# CST
Items*
Textbook
Prentice Hall **
Calculus (foundational):
4.2 Students demonstrate an
understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of
problems from physics,
chemistry, economics,
and so forth that involve
the rate of change of a
function.
Learning Targets
7H Graph and analyze logistics growth
functions.
7J Solve mathematical modeling problems using exponential functions
involving compound interest,
growth and decay. and logarithmic functions to include growth
and decay and justify each step
within the analysis.
7K Solve mathematical modeling problems using logarithmic functions
11.2 Students judge the
validity of an argument according to
whether the properties of real numbers,
exponents, and logarithms have been applied correctly at each
step.
Learning Targets
7I Solve exponential and logarithmic equations, justifying each
step.
12.0 Students know the
laws of fractional exponents, understand
exponential functions,
and use these functions in problems involving exponential
growth and decay.
14.0 Students understand
Supporting Medium/Low
Priority Standards
15
2
1
2
1
9.0 Students use differentiation to sketch, by hand,
graphs of functions.
They can identify maxima, minima, inflection
points, and intervals in
which the function is increasing and decreasing.
Learning Targets
7B Analyze an exponential function,
including finding the domain,
range, horizontal asymptote, intercepts, and inverse- sketch
7G Analyze a logarithmic function,
including finding the domain,
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
tain [accessible] equations that involve exponentials and logarithms
are explored, and are applied to
solve problems involving exponential
growth & decay, logistic growth, and
logarithmic models.
Honors Level: The “rule of 72” is analyzed in terms of logarithmic and exponential functions. Furthermore,
honors students develop more complex mathematical models involving
growth and decay, including logistic
growth.
High Priority Standards
and use the properties of logarithms to
simplify logarithmic
numeric expressions
and to identify their
approximate values.
# CST
Items*
# Q2
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
16
Textbook
Prentice Hall **
range, vertical asymptote, intercepts, and inverse- sketch
Learning Targets
7D Prove the multiplication and
division property of logarithms.
7E Expand and condense logarithmic expressions using
properties of logarithms.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
Unit 8: Applied Trigonometry
The study of non-algebraic functions
continues as the trigonometric functions and their properties are developed. The trigonometric ratios of a
right triangle, learned in previous
courses, are reviewed and used to
solve right triangles for all sides and
angles. Applications of right triangle
trigonometry are studied, including
problems related to indirect measurement.
Up to this point, the focus of the
study of trigonometry has been strictly related to right triangle trigonometry. Concepts of right triangle trigonometry are the foundation for studying the trigonometry of acute or obtuse triangles. The Law of Sines and
Law of Cosines are introduced and
proven in order to establish trigonometric relationships of these triangles. Any triangle (acute, right, or
obtuse) is solved using these laws
and the inverse trigonometric functions, and the different conditions for
which each law is used are explored.
Applications of the Law of Sines and
Law of Cosines are explored, including problems related to navigation
and finding area.
Honor’s Level: While discussing the
Law of Sines, honors students explore the SSA ambiguous case for
the Law of Sines and the reasons
why two there are two cases. Fur-
High Priority Standards
Trigonometry:
13.0 Students know the
law of sines and the
law of cosines and
apply those laws to
solve problems.
Learning Targets
8C Derive the Law of Sines and
the different forms of the Law
of Cosines.
8D Solve triangles using the Law
of Sines
8E Explain why the SSA case
yields more than one triangle.
(challenge)
8F Describe conditions when an
SSA case yields two solutions, one solution, and no
solutions. (challenge)
8G Solve triangles using the Law
of Cosines
19.0 Students are adept at
using trigonometry in
a variety of applications and word problems.
# CST
Items*
# Q2
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
17
Textbook
Prentice Hall **
Trigonometry:
12.0 Students use trigonometry to determine unknown sides or angles in
right triangles.
Learning Targets
8A Define the six trigonometric ratios
of a right triangle, and solve a
right triangle for all sides and angles
8D Solve triangles using the Law of
Sines
8G Solve triangles using the Law of
Cosines
14.0 Students determine the
area of a triangle, given
one angle and the two
adjacent sides.
Learning Targets
8J Find the area of any triangle
8K Write a formal proof of Heron's
formula using the Law of Cosines. (challenge)
8L Prove Heron's formula for finding
the area of a triangle. (challenge)
Learning Targets
8B Use right triangle trigonometry
to solve problems of indirect
measurement.
8H Explain the difference between
and angle of depression and
angle of elevation.
8I Design and solve problems
related to navigation using
right triangle trigonometry,
the Law of Sines and the Law
of Cosines and explain the
analysis leading to the solutions
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
# CST
Items*
# Q2
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
18
Textbook
Prentice Hall **
thermore, honors students study
formulas for the area of a triangle,
and prove Heron’s formula using either Law of Sines or Law of Cosines.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 9: Analyzing Trigonometric
Functions
This unit begins with a discussion of
how angles are measured. Radian
measure is defined in terms of arc
length on the unit circle. Trigonometric functions of angles are defined in
terms of the unit circle. This definition is used to determine which quadrants the trigonometric functions are
positive and negative. The concept
of a reference angle is explored and
references angles are then used to
find the value of the trigonometric
functions of any angle.
Trigonometry
4.0 Students graph functions of the form f(t) =
A sin (Bt + C) or f(t) =
A cos (Bt + C) and interpret A, B, and C in
terms of amplitude,
frequency, period,
and phase shift.
The unit circle is then discussed in
more detail. Terminal points on a
unit circle are defined, as well as reference numbers. The trigonometric
functions of a real number are then
defined, using terminal points of the
unit circle. Comparisons are made
between the concept of the trigonometric functions as functions of an
angle, and functions of a real number. Special values of the trigonometric functions are derived. The
domain and range of the trigonometric functions are discussed and the
graphs of the sine and cosine are
built. Connections between the sine
and cosine graphs and the unit circle
are illustrated, including properties of
periodicity. Transformations are applied to the graphs of the sine and
cosine, and the general forms of
Learning Targets
9J Explain the relationship between the graph of a sine and
cosine function, and the unit
circle.
Learning Targets
9K Use transformations to graph
functions of the form or ,
and identify key information
about the function (i.e. period,
amplitude, phase shift, etc.).
5.0 Students know the
definitions of the tangent and cotangent
functions and can
graph them.
Learning Targets
9L Graph the tangent, cotangent,
secant, cosecant functions,
and identify the domain,
range, and period for each.
8.0 Students know the
definitions of the inverse trigonometric
functions and can
graph the functions.
Learning Targets
9M Compare the domain and
range of the trigonometric
functions and their inverses
and explain the relationship
between the two.
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
19
Textbook
Prentice Hall **
Trigonometry
1.0 Students understand the
notion of angle and how
to measure it, in both
degrees and radians.
They can convert between degrees and radians.
Learning Targets
9A Explain how to find arc length, and
how this concept is used in the
definition of radian measure.
9B Identify angles in degree and radian measure, and convert between the measures.
9C Find exact values of trig functions
2.0 Students know the definition of sine and cosine
as y-and x-coordinates
of points on the unit circle and are familiar with
the graphs of the sine
and cosine functions.
Learning Targets
9D Define a reference angle and describe the reference angle in
each quadrant.
9E Describe a method of finding reference angles.
9F Calculate the exact trig function
values and explain the purpose of
the angle
9G Use the unit circle to evaluate trig
expressions and explain the process
19.0 Students are adept at
using trigonometry in a
variety of applications
and word problems.
Learning Targets
9N Describe simple harmonic motion
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
these functions are built in the following order: , , , and . The same order
is used for the cosine function. Other graphs of trigonometric functions
are studied including the tangent, the
reciprocal functions, and the inverse
functions. The difference between
reciprocal functions and inverse functions is emphasized. The domain
and range of trigonometric functions
and their inverses are also compared.
9.0 Students compute, by
hand, the values of
the trigonometric
functions and the inverse trigonometric
functions at various
standard points.
Trigonometric functions are then
used as mathematical models, specifically modeling harmonic motion.
The general form of the sine and cosine is applied to simple harmonic
motion, and used to find amplitude,
frequency, and period. These components of the graph of a harmonic
motion model are all interpreted in a
real world context, and used to solve
problems within that context.
Honors Level: Honors students extend the study of harmonic motion, to
include damped harmonic motion.
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
20
Textbook
Prentice Hall **
and identify the amplitude, frequency, and period of harmonic
motion model, and explain their
real world significance.
9O Describe damped harmonic motion
verbally and algebraically.
9P Compare and contrast simple and
damped harmonic motion.
Learning Targets
Use the concept of inverse functions and the unit circle to find
exact values
9I Explain how to use the unit
circle to evaluate trig functions as well as find locations
of ordered pairs (detail how to
“work” all aspects of the unit
circle).
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 10: Trigonometric Equations
The trigonometric functions have
now been study in depth, as well as
their application. The focus of this
unit is on trigonometric equations.
The discussion begins with a review
of basic identities (e.g. ). The fundamental trigonometric identities are
introduced and illustrated. These
include the reciprocal identities, the
Pythagorean identities, the even and
odd identities, and the cofunction
identities. Basic trigonometric identities are used to simplify trigonometric
expressions, and different methods
of simplifying these expressions are
explored.
Trigonometry:
3.1 Students prove that
this identity is equivalent to the Pythagorean theorem (i.e., students can prove this
identity by using the
Pythagorean theorem
and, conversely, they
can prove the Pythagorean theorem as
a consequence of this
identity).
The basic trigonometric identities are
then used to prove more complex
trigonometric identities. A variety of
methods of proving trigonometric
identities are used. More advanced
identities are introduced, including
the following: addition and subtraction identities, double angle and half
angle identities, and product and
sum identities. All of these are used
to prove more complex trigonometric
identities.
The focus of the unit shifts to a study
of solving trigonometric equations.
The inverse trigonometric functions
are used to solve basic trigonometric
equations on a restricted domain
(e.g. on ). Once this domain re-
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
21
Textbook
Prentice Hall **
Trigonometry:
3.0 Students know the iden2
2
tity cos (x) + sin (x) = 1:
10C Know the Pythagorean identities
Learning Targets
10B Derive the Pythagorean Trig.
identities.
10L Verify more complex trigonometric identities using the
basic trigonometric identities,
Pythagorean identities, cofunction identities, odd/even
identities, sum & difference
identities, double angle identities, half-angle identities and
justify each step in the verification process.
3.2 Students prove other
trigonometric identities and simplify others by using the iden2
2
tity cos (x) + sin (x) =
1. For example, students use this identity
2
to prove that sec (x)
2
= tan (x) + 1.
Learning Targets
10A Know the basic trigonometric
identities, including the reciprocal identities
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
striction is removed, the periodic nature of the solution sets of trigonometric equations is explored. Connections are made between trigonometric equations, the unit circle, and
graphs of trigonometric functions.
Various methods of solving trigonometric equations are studied, and
used to solve mathematical models
involving the trigonometric functions.
Honors Level: While solving trigonometric equations, honors students
manipulate the known trigonometric
identities before finding the solution
set. Honors students also prove the
multiple angle identities, and the addition and subtraction identities.
10D Simplify trigonometric expressions using the basic trigonometric identities.
10E Know the co-function identities using the subtraction
identities.
10F Know and explain the
odd/even identities for sine &
cosine
10G Know the sum and difference
identities
10I Prove the co-function identities
using the subtraction identities.
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
22
Textbook
Prentice Hall **
9.0 Students compute, by
hand, the values of
the trigonometric
functions and the inverse trigonometric
functions at various
standard points.
Learning Targets
10O Explain the role of the inverse
trigonometric functions in
solving trigonometric expressions and equations.
10.0 Students demonstrate an understanding of the addition
formulas for sines
and cosines and their
proofs and can use
those formulas to
prove and/or simplify
other trigonometric
identities.
Learning Targets
0H Use the sum and difference
identities to find the exact
value of trigonometric functions at a given number.
10L Verify more complex trigonometric identities using the
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
23
Textbook
Prentice Hall **
basic trigonometric identities,
Pythagorean identities, cofunction identities, odd/even
identities, sum & difference
identities, double angle identities, half-angle identities and
justify each step in the verification process.
10M Solve trigonometric equations
on a restricted domain and
justify each step
10N Use the periodicity of the
trigonometric functions to find
the general solutions of a
trigonometric equation.
11.0 Students demonstrate an understanding of half-angle and
double-angle formulas for sines and cosines and can use
those formulas to
prove and/or simplify
other trigonometric
identities.
Learning Targets
10J Know the double angle identities
10K Know the half-angle identities
10L Verify more complex trigonometric identities using the
basic trigonometric identities,
Pythagorean identities, cofunction identities, odd/even
identities, sum & difference
identities, double angle identities, half-angle identities and
justify each step in the verification process.
10M Solve trigonometric equations
on a restricted domain and
justify each step
10N Use the periodicity of the
trigonometric functions to find
the general solutions of a
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
24
Textbook
Prentice Hall **
trigonometric equation.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 11: Statistics and Probability
This unit begins with a discussion of
how most distributions encountered
in statistics give a normal (“bell”
curve) distribution, which corresponds to the graph of the function .
In this graph, we note that 67% of all
values lie within one standard deviation of the mean, 95% of all values
within two standard deviations of the
mean, and 99.7% of all values within
three standard deviations of the
mean. Examples from standardized
tests such as the SAT are given and
explored. The binomial distribution
and its relation to Pascal’s triangle
are discussed, as well as how the
binomial distribution approximates
the normal distribution. The definition of the mean using sigma notation, is explored. The difference
between a subscript and a value is
emphasized. This leads into the
equation for the standard deviation .
These concepts are then explored
through examples and problems.
Probability and Statistics
7.0 Students compute the
variance and the
standard deviation of
a distribution of data.
Learning Targets
11C Use formulas to find variation
and standard deviation of a
distribution of data.
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
25
Textbook
Prentice Hall **
Probability and Statistics
4.0 Students are familiar
with the standard distributions (normal, binomial, and exponential) and
can use them to solve
for events in problems in
which the distribution
belongs to those families.
Learning Targets
11A Identify and explain the normal
distribution.
11D Explain the binomial distribution,
and its relationship to the normal
distribution.
5.0 Students determine the
mean and the standard
deviation of a normally
distributed random variable.
Learning Targets
11B Explain the concept of standard
deviation and connect to the concept of mean
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 12: Sequences and Series
Properties of sequences and series
are reviewed, with an emphasis on
the algebraic derivation of the summation formulas for the arithmetic
and geometric series. For the arithmetic series, mean of the terms is
used to find the sum (the pairing
method: pair the 1st and nth terms,
the 2nd and st terms, etc.). Geometric series and the factorization
Algebra 2:
20.0 Students know the
binomial theorem and
use it to expand binomial expressions
that are raised to positive integer powers.
Learning Targets
12D Explain the relationship between Pascal's triangle and
the Binomial theorem.
Calculus (foundational):
1  x  (1  x  x  ...  x )(1  x)13.0 Students know the
definition of the defiare discussed. Summation notation
nite integral by using
is then used to simplify summation
Riemann sums. They
calculations and do slightly more
use this definition to
complex sums. The relationship beapproximate intetween Pascal’s triangle, the binomial
grals.
coefficients, and the binomial theoLearning Targets
rem is also discussed and the bino12A Know the summation formulas
mial theorem and Pascal’s triangle is
for k^1, k^2, and k^3
used to expand powers of binomials.
Honors Level: The derivation of the
summation properties using Pascal’s
theorem is discussed. This includes
the “hockey stick” formula: .
n 1
2
n
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
Algebra 2:
4.0 Students are familiar
with the standard distributions (normal, binomial, and exponential) and
can use them to solve
for events in problems in
which the distribution
belongs to those families.
5.0 Students determine the
mean and the standard
deviation of a normally
distributed random variable.
22.0 Students find the general term and the sums
of arithmetic series and
of both finite and infinite
geometric series.
# CST
Items*
26
Textbook
Prentice Hall **
1
1
Learning Targets
12B Explain the derivations of the
summation formulas for the
arithmetic and geometric series.
12C Find the sum of an arithmetic and
geometric series.
23.0 Students derive the
summation formulas for
arithmetic series and for
both finite and infinite
geometric series.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 13: CST Review
The CST unit in Pre-Calculus is a
review of essential topics from Algebra I, Geometry, and Algebra II in
preparation for students to take the
summative mathematics exam. The
essential standards below are from
the blueprint for the STAR Summative exam.
Algebra 2:
1.0 Students solve equations and inequalities
involving absolute
value.
2.0 Students solve systems of linear equations and inequalities
(in two or three variables) by substitution,
with graphs, or with
matrices.
3.0 Students are adept at
operations on polynomials, including
long division.
4.0 Students factor polynomials representing
the difference of
squares, perfect
square trinomials,
and the sum and difference of two cubes.
6.0 Students add, subtract, multiply, and divide complex numbers.
7.0 Students add, subtract, multiply, divide,
reduce, and evaluate
rational expressions
with monomial and
polynomial denominators and simplify
complicated rational
expressions, including those with nega-
# CST
Items*
# Q3
Items
1
1
3
3
1
1
1
1
1
1
2
2
Supporting Medium/Low
Priority Standards
# CST
Items*
27
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
tive exponents in the
denominator.
8.0 Students solve and
graph quadratic equations by factoring,
completing the
square, or using the
quadratic formula.
Students apply these
techniques in solving
word problems. They
also solve quadratic
equations in the complex number system.
10.0 Students graph
quadratic functions
and determine the
maxima, minima, and
zeros of the function.
11.1 Students understand
the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and exponents.
12.0 Students know the
laws of fractional exponents, understand
exponential functions,
and use these functions in problems involving exponential
growth and decay.
14.0 Students understand
and use the proper-
# CST
Items*
# Q3
Items
3
3
2
2
1
1
2
2
1
1
Supporting Medium/Low
Priority Standards
# CST
Items*
28
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
ties of logarithms to
simplify logarithmic
numeric expressions
and to identify their
approximate values.
15.0 Students determine
whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential
functions is sometimes true, always
true, or never true.
18.0 Students use fundamental counting principles to compute
combinations and
permutations.
19.0 Students use combinations and permutations to compute
probabilities.
22.0 Students find the
general term and the
sums of arithmetic
series and of both finite and infinite geometric series.
24.0 Students solve problems involving functional concepts, such
as composition, defining the inverse function and performing
arithmetic operations
on functions.
# CST
Items*
# Q3
Items
1
1
1
1
1
1
1
1
1
1
Supporting Medium/Low
Priority Standards
# CST
Items*
29
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Geometry
3.0 Students construct
and judge the validity
of a logical argument
and give counterexamples to disprove a
statement.
4.0 Students prove basic
theorems involving
congruence and similarity.
5.0 Students prove that
triangles are congruent or similar, and
they are able to use
the concept of corresponding parts of
congruent triangles.
7.0 Students prove and
use theorems involving the properties of
parallel lines cut by a
transversal, the properties of quadrilaterals, and the properties of circles.
8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume,
lateral area, and surface area of common
geometric figures.
9.0 Students compute the
volumes and surface
areas of prisms, pyr-
# CST
Items*
# Q3
Items
1
1
3
3
2
2
2
2
1
1
1
1
Supporting Medium/Low
Priority Standards
# CST
Items*
30
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
amids, cylinders,
cones, and spheres;
and students commit
to memory the formulas for prisms, pyramids, and cylinders.
10.0 Students compute
areas of polygons, including rectangles,
scalene triangles,
equilateral triangles,
rhombi, parallelograms, and trapezoids.
11.0 Students determine
how changes in dimensions affect the
perimeter, area, and
volume of common
geometric figures and
solids.
15.0 Students use the Pythagorean theorem to
determine distance
and find missing
lengths of sides of
right triangles.
18.0 Students know the
definitions of the
basic trigonometric
functions defined by
the angles of a right
triangle. They also
know and are able to
use elementary relationships between
them. For example,
tan(x) = sin(x)/cos(x),
# CST
Items*
# Q3
Items
1
1
1
1
2
2
2
2
Supporting Medium/Low
Priority Standards
# CST
Items*
31
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
(sin(x))2 + (cos(x)) 2 =
1.
19.0 Students use trigonometric functions to
solve for an unknown
length of a side of a
right triangle, given
an angle and a length
of a side.
21.0 Students prove and
solve problems regarding relationships
among chords, secants, tangents, inscribed angles, and
inscribed and circumscribed polygons of
circles.
Algebra 1:
4.0 Students simplify expressions before
solving linear equations and inequalities
in one variable, such
as 3(2x-5) + 4(x-2) =
12.
5.0 Students solve multistep problems, including word problems, involving linear
equations and linear
inequalities in one
variable and provide
justification for each
step.
6.0 Students graph a linear equation and compute the x- and y-
# CST
Items*
# Q3
Items
1
1
2
2
1
1
3
3
2
2
Supporting Medium/Low
Priority Standards
# CST
Items*
32
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
intercepts (e.g., graph
2x + 6y = 4). They are
also able to sketch
the region defined by
linear inequality (e.g.,
they sketch the region
defined by 2x + 6y <
4).
7.0 Students verify that a
point lies on a line,
given an equation of
the line. Students are
able to derive linear
equations by using
the point-slope formula.
8.0 Students understand
the concepts of parallel lines and perpendicular lines and how
those slopes are related. Students are
able to find the equation of a line perpendicular to a given line
that passes through a
given point.
10.0 Students add, subtract, multiply, and divide monomials and
polynomials. Students solve multistep
problems, including
word problems, by
using these techniques.
11.0 Students apply basic
factoring techniques
# CST
Items*
# Q3
Items
1
1
1
1
3
3
1
1
Supporting Medium/Low
Priority Standards
# CST
Items*
33
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
to second- and simple
third-degree polynomials. These techniques include finding
a common factor for
all terms in a polynomial, recognizing the
difference of two
squares, and recognizing perfect squares
of binomials.
12.0 Students simplify
fractions with polynomials in the numerator and denominator
by factoring both and
reducing them to the
lowest terms.
14.0 Students solve a
quadratic equation by
factoring or completing the square.
15.0 Students apply algebraic techniques to
solve rate problems,
work problems, and
percent mixture problems.
20.0 Students use the
quadratic formula to
find the roots of a
second-degree polynomial and to solve
quadratic equations.
23.0 Students apply
quadratic equations
to physical problems,
such as the motion of
# CST
Items*
# Q3
Items
1
1
1
1
2
2
1
1
1
1
Supporting Medium/Low
Priority Standards
# CST
Items*
34
Textbook
Prentice Hall **
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
35
Textbook
Prentice Hall **
an object under the
force of gravity.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
Unit 14: Introduction to Vectors,
Polars, Parametrics
This unit begins with the introduction
of a vector space. Vectors are introduced algebraically and geometrically in two and three dimensions. Vector addition and scalar multiplication
are introduced. The polar coordinate
system is also introduced, and connections are made between polar
and rectangular coordinates. Parametric equations are then introduced,
and connected to the rectangular
coordinate system by the method of
eliminating the parameter.
Math Analysis:
1.0 Students are familiar
with, and can apply,
polar coordinates and
vectors in the plane.
In particular, they can
translate between polar and rectangular
coordinates and can
interpret polar coordinates and vectors
graphically.
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
36
Textbook
Prentice Hall **
Trigonometry:
16.0 Students represent
equations given in rectangular coordinates in
terms of polar coordinates.
Learning Targets
14A Explain the meaning of a
vector within the context of
vector space
14B Write and draw vectors in two
and three dimensions.
14C Perform vector addition and
scalar multiplication in two
dimensions, and explain their
representation in R2.
14D Describe and categorize polar
graphs
14E Convert between the polar
coordinate system and the
rectangular coordinate system.
7.0 Students demonstrate
an understanding of
functions and equations defined parametrically and can
graph them.
Learning Targets
14F Explain the parametric coordinate system, and eliminate
the parameter in order to
convert between the polar
coordinate system and the
rectangular coordinate system.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall
Pre-Calculus Instructional Guide 2011-2012
Unit
High Priority Standards
# CST
Items*
# Q3
Items
Supporting Medium/Low
Priority Standards
# CST
Items*
37
Textbook
Prentice Hall **
14G Draw a graph of a parametric
equation.
Unit 15: Extension Project
Honors Level: This final unit is in two
parts. Individual students choose
from a list of certain topics. Research is conducted on the chosen
topic, and results are presented to
the class. The following topics are
used: proof by induction, DeMoivre’s
theorem, the Chinese remainder theorem, modular arithmetic, Gaussian
Elimination, introduction to group
theory, and an overview of string
theory. As a group, the concept of
infinity is debated in the style of
Kroneker and Cantor.
Non-Honors Level: This final unit is in
two parts. Individual students
choose from a list of certain topics.
Research is conducted on the chosen topic, and results are presented
to the class. The following topics are
used: Simpson’s paradox, Boolean
algebra, chaos theory (fractals), angular and linear speed, history and
mathematics of 0, history and mathematics of , and Brahmagupta theorem.
Instruction Continues After CST- Q4
Math Culminating Projects
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
*CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13)
**Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall