Download Pre Calculus PreAP Scope and Sequence

Pre-Calculus Pre-AP – Scope and Sequence - Year at a Glance
Three Weeks
Topics/
Concepts
Pre-Calculus Pre-AP - First Semester – Pre-calculus with Limits; Larson/Hostetler
1st 3 weeks
2nd 3 weeks
3rd 3 weeks
4th 3 weeks
Rectangular
Coordinates
Linear and NonLinear Systems of
Equation
Graphs of Equations
Linear Equations in
Two Variables
Two-Variable Linear
Systems
Polynomials and
Rational Functions
Quadratic
Equations and
Inequalities
Functions
Multivariable Linear
Systems
Remainder and
Factor Theorems
Parent functions
Partial Fractions
Rational Root
Theorem
Transformations
Systems of
Inequalities
Inverse Functions
Analyzing graphs
Matrices and Systems
of Equations
5th 3 weeks
6th 3 weeks
Sequences and Series
Circle
Exponential and
Logarithmic
Functions
Parabola
Rational Exponents
Ellipse
Number e
Hyperbola
Common
Logarithms
Conics
Transformations
Natural Logarithms
Locating Zeros of a
Function
Systems of Second
Degree Equations
and Inequalities
Rational Equations
and Partial
Fractions
Tangents and
Normals to Conic
Sections
Exponential
Growth
Chapter 1, Appendix A
Chapter 16(f)
Teks – P.1A, P.1B,
P.1C, P.1D, P.1E,
P.2A, P.2B, P.2C,
P.3A
Chapter 7, Chapter 8
P.1B, P.2B, P.3A,
P.3B, P.3C,
Chapter 2
Chapter 16(f)
Teks – P.1C, P.1D,
P.1E, P.3A
Geometric
Infinite
Convergent and
Divergent
Sigma Notation and
the nth Term
Binomial Theorem
Permutations
Probability and
Odds
Combinations
Radical Equations
and Inequalities
Resource:
Precalculus with
Limits;
Larson/Hostetler,
Foerster (f)
Arithmetic
Chapter 10
Chapter 12 & 13 (f)
Teks – P.1D, P.2A,
P.5A, P.5B, P.5C
Chapter 3
Teks – P.1A, P.1B,
P.1D, P.1E, P.2C,
P.3A
Chapter 9
Teks – P.1B, P.3A,
P.3B, P.3C, P.4A,
P.4B, P.4C, P.4D
Three Weeks
Topics/
Concepts
Pre-Calculus Pre-AP - Second Semester – Pre-calculus with Limits; Larson/Hostetler
1st 3 weeks
2nd 3 weeks
3rd 3 weeks
4th 3 weeks
Trigonometric
Functions
Angles and Their
Measure
Central Angles and
Arcs
Graphs and Inverses of
Trigonometric
Functions
Amplitude, Period and
Phase Shift
Principal Values of
Inverses
Circular Functions
Special Angles
Simple Harmonic
Motion
Trigonometric
Identities and
Equations
Sum and
Difference
Identities
Double Angle and
Half-Angle
Identities
Solving
Trigonometric
Equations
Right Triangles
Vectors and
Parametric Equations
Geometric and
Algebraic Vectors
Perpendicular
Vectors
Applications with
Vectors
Resource:
Precalculus with
Limits;
Larson/Hostetler
Foerster (f)
Chapters 4, 5, 6
Teks – P.1C, P.2B,
P.3A, P.3C, P.3D,
P.3E
Chapters 4, 5, 6
Teks – P.2C, P.3A,
P.3C, P.3D
Applied Problems
Connections to
Calculus
Polar Graphs
Polar-Rectangular
Coordinates
Derivatives
Polar Form of
Linear Function
Polar Form of
Complex Numbers
Products and
Quotients of
Complex Numbers
Distance form a
Point to a Line
Chapters 4, 5, 6
Teks – P.1A, P.1B ,
P.1C, P.1D, P.2A,
P.2B, P.2C, P.3A, P.3B,
P.3C
Polar Coordinates
and Complex
Numbers
Motion Modeling
Normal Form of a
Linear Equation
Area of Triangles
6th 3 weeks
Limits
3D Space
Law of Sines
Law of Cosines
5th 3 weeks
Chapter 6
Teks – P.5A, P.5C,
P.5D, P.6A, P.6B
Powers and Roots
of Complex
Numbers
Chapter 10
Teks – P.1A, P.1B,
P.2A, P.3D
Chapter 12
Chapter 16(f)
*Teks 111.54b
Calculus AB
* These Teks, 111.54b Calculus AB, have been included for cross reference due to the lack of in-depth explanation with-in the PreCalculus Teks.
Functions and Their Graphs (1st 3 weeks)
Essential Learning Outcomes
TEKS
111.35
The Student Will:
P.1A, P.1B,
P.1C, P.1D,
 Identify functions by definition and
graph and state their domain and range P.1E, P.2A,
 Discuss continuity; determine intervals P.2B, P.2C, P.3A
of increase, decrease or constant;
describe end behavior
 Find the sum, difference, product, and
quotient of two functions
 Find the composition of two functions
given their equations and be able to
determine functions given their
composition.
 Be able to reflect graphs of functions
over x-axis, y-axis and line y = x.
 Be able to test a function for symmetry
and determine if it is odd, even or
neither algebraically and from the
graph.
 Given the graph of y = f(x), be able to
graph a f(x), f(x) + d, f(x - c), f(bx),
f( x ) , and f(|x|).




Given a graph that has a
transformation, write the equation for
the graph.
(Optional to address this in this unit)
Determine if a function is periodic
and, if so, give its period and
amplitude and graph it using shifts and
stretches.
Find the inverse of a function and,
given two functions, determine if they
are inverses of each other algebraically
and graphically.
Solve application problems using
composite functions.
Topics (not in sequential order)
Polynomials and Rational Functions
Quadratic Equations and Inequalities
Remainder and Factor Theorems
Rational Root Theorem
Locating Zeros of a Function
Rational Equations and Partial Fractions
Radical Equations and Inequalities
Suggested
Resources
Chapter 1,
Appendix A
Assessments
TAKS
Objectives
1,2,3,4
Conics (2nd 3 weeks)
Essential Learning Outcomes
The Student Will:







solve 2 X 2 systems
 linear combination
 substitution
 graphing (with a graphing
calculator)

intersecting lines

parallel lines

lines that coincide
 table (graphing calculator)
 matrices

define a matrix and use a
matrix to represent data

inverse matrices by hand (2X2
only)

inverse matrices with calculator

use operations of matrices to
solve problems
systems of inequalities
 graphing calculator
 graph by hand
 Test a point
Solve 3 X 3 systems
 Matrices (calculator)
Perform matrix multiplication to show
multiplication is not commutative
 By hand (demo only)
 With technology
determine what a solution to a system of
equations/inequalities means in
relationship to the problem
determine if the solution to a system of
equations/inequalities is reasonable for
given contexts
connect algebraic solutions to graphical
and tabular solutions
TEKS
111.35
P.1B, P.2B, P.3A,
P.3B, P.3C,
Topics (not in sequential
order)
Circle
Parabola
Ellipse
Hyperbola
Transformations
Systems of Second Degree
Equations and Inequalities
Tangents and Normals to
Conic Sections
Suggested
Resources
Chapter 7, Chapter 8
Assessments
TAKS
Objectives
1,2,3,4,5,6,7
Polynomials and Rational Functions (3rd 3 weeks)
Essential Learning Outcomes
The Student Will:

Identify and classify polynomials by degree

Find zeros (roots) and factors of polynomial functions (equations)

Use synthetic substitution/division

Apply Theorems: Remainder, Factor, Rational Root, Complex
Conjugate, and Fundamental Theorem of Algebra

Graph a polynomial function given a factorable equation

Write the equation for a polynomial function given its graph or a set of
data

Find maximum and minimum values with or without a graphing
calculator and apply to situations (area, volume, cost, distance, etc.)

Solve and graph (on a number line) polynomial inequalities with
answer in interval notation

Sketch the graph of a polynomial function using properties of end
behavior, zeros, and multiplicity of zeros

Graph rational functions, including those containing more than one
vertical asymptote, an oblique asymptote, removable discontinuities,
and a crossover of the horizontal or oblique asymptote by locating the
asymptotes, “holes”, and intercepts as transformations of one of the
parent functions

f  x 
1
1
or f  x   2 .
x
x
Manipulate the function rule algebraically to obtain graphing form.
(Example: Use long division to rewrite the





1
x3
as 1 
x2
x2
Describe the continuity and limits of the function.
State the domain and range of the function and write the equations of
its asymptotes.
Write a rational function given the graph, or description of
characteristics such as a verbal description of the transformations of
the parent function, the asymptote equations, or function notation
showing the transformations.
Make connections between the graphical, tabular, verbal, and
symbolic representations of the function.
Describe the symmetry of the function by identifying it as odd or
even.
TEKS 111.35
P.1C, P.1D,
P.1E, P.3A
Topics (not in sequential
order)
Polynomials and Rational
Functions
Quadratic Equations and
Inequalities
Remainder and Factor Theorems
Rational Root Theorem
Locating Zeros of a Function
Rational Equations and Partial
Fractions
Radical Equations and
Inequalities
Suggested
Resources
Chapter 2
Assessments
TAKS
Objectives
1,2,3,4,5
Conics (4th 3 weeks)
Essential Learning Outcomes
The Student Will:

Know the geometric and algebraic
descriptions of circle, parabola, hyperbola,
and ellipse

Represent conic sections in standard form
(graphing form) and in parametric form
and graph

Determine the equation of a conic given
the graph

Determine the equation and graph of a
conic from given information

Apply properties of conic sections to solve
real-life problems (both as parametric and
rectangular)
 orbits of planets
 LORAN

Convert from general to standard form

Describe how properties of conics are used
in the physical world, such as
 reflective properties (flashlight,
whispering room, lithotripsy. etc.)
 elliptipool
 cooling towers for nuclear reactors

(optional) graphs of rotated conics
TEKS
111.35
P.5A,
P.5C,
P.5D,
P.6A,
P.6B
Topics (not in sequential order)
Conics
Circle
Parabola
Ellipse
Hyperbola
Transformations
Systems of Second Degree Equations
and Inequalities
Tangents and Normals to Conic Sections
Suggested
Resources
Chapter 10,
Chapter 12 & 13(f)
Assessments
TAKS
Objectives
1,2,3,4,5,6,10
Exponential and Logarithmic Functions (5th 3 weeks)
Essential Learning Outcomes
TEKS
Topics (not in sequential order)
The Student Will:

Know the definition of an exponential
function, identify restrictions and graph

Define the logarithmic function as the
inverse of an exponential function and
graph

Know and apply laws of integral and
rational exponents

Apply transformations to exponential
and logarithmic functions

Know restrictions on the log function

Identify intervals where graph of an
exponential function is increasing or
decreasing, as well as its domain, range,
and zeros

Use exponential growth and decay
formulas to solve application problems:
A  A0 1  r 
t
t
A  A0b k
r

A  A0  1  
n

nt
A  A0ert



Use rational exponents to simplify
expressions and solve equations
Write an exponential function from a set
of data
Know the definition of e and the
function f(x) = ex and use it in
application problem.
P.1A,
P.1B,
P.1D,
P.1E,
P.2C,
P.3A
Exponential and Logarithmic Functions
Rational Exponents
Number e
Common Logarithms
Natural Logarithms
Exponential Growth
Suggested Resources
Chapter 3
Assessments
TAKS
Objectives
1,2,3,4,5,8
Sequences and Series (6th 3 weeks)
Essential Learning Outcomes
TEKS
The Student Will:

Identify a sequence as arithmetic
(linear), geometric (exponential
when r > 0), or other

Define a sequence both explicitly
and recursively

Explore representations of a
sequence on a graph and table

Use sequences defined recursively
to solve problems

Find the sum of the first n terms
and the value of a specific term of
an arithmetic and geometric series

Find or estimate the limit of an
infinite sequence or determine that
the limit does not exist

Find the sum of an infinite
geometric series when possible

Use sigma notation to represent
series

Expand a binomial of the form
a  b 

n
and find the nth term
of a binomial expansion
Use sequences and series to solve
real-life problems
P.1B,
P.3A,
P.3B,
P.3C,
P.4A,
P.4B,
P.4C,
P.4D
Topics (not in sequential order)
Suggested Resources
Sequences and Series
Chapter 9
Arithmetic
Geometric
Infinite
Convergent and Divergent
Sigma Notation and the nth Term
Binomial Theorem
Permutations
Probability and Odds
Combinations
Assessments
TAKS
Objectives
1,2,3,4,5,8
Trigonometric Functions (1st 3 weeks, 2nd Semester)
Essential Learning Outcomes
The Student Will:
 Unit Circle: know definition of radians, degree
measure, standard position, coterminal angles,
quadrantal angles, coordinates, and the six trig
functions
 Understand the concept of radian measure as a
location on the Unit Circle)
 Define sine, cosine and tangent in terms of x, y
and r
 Convert between radian measure and degree
measure of special angles (multiples
   

, , , )
6 4 3 2
Know the sine, cosine and tangent values of
, , , )
6 4 3 2
Find the values of cotangent, secant, and cosecant
from sine, cosine, and tangent
Find the reference angle of any given angle
Recognize equivalent fractions created by
rationalized denominators
Find arc length and area of a sector (may be
taught in another unit)
Solve right triangles using sin, cos, and tan
Find area of triangles
Use Law of Sines and Law of Cosines
Solve application problems involving navigation
and surveying, including angles of elevation and
depression, area of parallelogram using area of
triangle.
special angles (multiples








   
TEKS
111.35
P.1C,
P.2B,
P.3A,
P.3C,
P.3D,
P.3E
Topics (not in sequential order)
Suggested Resources
Trigonometric Functions
Chapter 4, 5, 6
Angles and Their Measure
Central Angles and Arcs
Circular Functions
Special Angles
Right Triangles
Law of Sines
Law of Cosines
Area of Triangles
Assessments
TAKS
Objectives
1,2,3,4
Graphs and Inverses of Trigonometric Functions (2nd 3 weeks, 2 nd Semester)
Essential Learning Outcomes
TEKS Topics (not in sequential
111.35 order)
The Student Will:
P.1A,
Graphs and Inverses of
P.1B , Trigonometric Functions
 Graph all six trig functions and identify their
P.1C,
period, amplitude, domain, range and zeros
P.1D,
Amplitude, Period and Phase
 Graph all six trig functions with parameter
P.2A,
Shift
changes (i.e.
P.2B,
y = a sin b(x – c) + d; f( x ) , and f(|x|) are
P.2C,
Principal Values of Inverses
optional)
P.3A,
 Write/identify the equation of a trig function
P.3B,
Simple Harmonic Motion
from its graph
P.3C
 State whether a trig function graph is odd or
even
 Investigate continuity, end behavior, vertical
asymptotes, increasing/decreasing (especially
tan, cot, sec, and csc)
 Model real-life data using sine and cosine
functions with and without a regression
equation
 Define and graph principle trig functions (1
to 1 functions) and their inverses (sine,
cosine, and tangent).
 Make connections between graphs of inverse
trig functions and the Unit Circle).
 Find values of inverse trig functions with and
without the calculator (sine, cosine, and
tangent)

Sketch the graphs of the functions Sin-1, Cos1
, Tan-1 and state the domain and range.
(Emphasize the range of inverse sine and
tangent)
Suggested Resources
Chapter 4, 5, 6
Assessments
TAKS
Objectives
1,2,3,4,5
Trigonometric Identities and Equations (3rd 3 weeks, 2nd Semester)
Essential Learning Outcomes
TEKS Topics (not in sequential order)
111.35
The Student Will:
P.2C,
Trigonometric Identities and Equations
P.3A,
 Know and apply the following
P.3C,
Sum and Difference Identities
identities:
P.3D
 Pythagorean
Double Angle and Half-Angle Identities
 Reciprocal
 Quotient
Solving Trigonometric Equations
 Negative angle
Normal Form of a Linear Equation
 Cofunctions
 Sum and difference
Distance form a Point to a Line
 Double angle
 Half angle (optional)
 Prove trig identities
 Simplify and expand trig
expressions
 Solve trig equations
Suggested Resources
Chapter 4, 5, 6
Assessments
TAKS
Objectives
1,2,3,4,5,6
Vectors and Parametric Equations (4th 3 weeks, 2 nd Semester)
Essential Learning Outcomes
TEKS Topics (not in sequential order)
111.35
The Student Will:
P.5A,
Vectors and Parametric Equations
P.5C,
 Using geometric and algebraic
P.5D,
Geometric and Algebraic Vectors
representation of vectors, add,
P.6A,
subtract, and multiply by a scalar
P.6B
3D Space
and find the magnitude (length)
and direction
Perpendicular Vectors
 Show, analyze, and solve
navigation and force problems
Applications with Vectors
using vectors and trig
 Use vectors, vector equations, and
Motion Modeling
parametric equations to show an
object’s motion.
 Use polar coordinates to plot
points on a graph
 Determine if two vectors are
parallel or perpendicular using dot
product
 Calculate the angle between two
vectors
 Work with 3-dimensional vectors
to find their magnitude and
midpoint
 Find equations of planes parallel
and perpendicular to another
plane
 Write complex numbers and find
products in polar form
 Find powers and roots of complex
numbers
 Convert polar coordinates to
rectangular coordinates and vice
versa
Suggested Resources
Chapter 6
Assessments
TAKS
Objectives
1,2,3,4,5,6
Polar Coordinates and Complex Numbers (5th 3 weeks, 2 nd Semester)
Essential Learning Outcomes
TEKS Topics (not in sequential order)
111.35
The Student Will:
P.1A,
Polar Coordinates and Complex Numbers
P.1B,
 Determine if two vectors are
P.2A,
Polar Graphs
parallel or perpendicular using
P.3D
dot product
Polar-Rectangular Coordinates
 Calculate the angle between two
vectors
Polar Form of Linear Function
 Work with 3-dimensional
vectors to find their magnitude
Polar Form of Complex Numbers
and midpoint
 Find equations of planes parallel
Products and Quotients of Complex Numbers
and perpendicular to another
plane
Powers and Roots of Complex Numbers
 Write complex numbers and
find products in polar form
 Find powers and roots of
complex numbers
 Convert polar coordinates to
rectangular coordinates and vice
versa
Suggested Resources
Chapter 10
Assessments
TAKS
Objectives
2, 5
Limits and an introduction to Calculus (16th 3 weeks, 2nd Semester)
Essential Learning Outcomes
TEKS
Topics (not in sequential order)
The student will
 Estimate limits using graphical and
numerical approaches
 Learn different ways a limit can fail to
exist
 Evaluate limits using properties of
limits
 Develop and use a strategy for finding
limits
 Evaluate a limit using division and
rationalization techniques
 Use properties of continuity
 Determine infinite limits from the left
and right
 Find and sketch the vertical
asymptotes of the graph of a function
 Use limits to justify vertical
asymptotes
 Find limits as x approaches infinity
 Find and sketch the horizontal
asymptotes of the graph of a function
 Find the slope of the tangent line to a
curve at a point
 Use the limit definition to find the
derivative of a function
 Understand the relationship between
differentiability and continuity
 Find the derivative of a function using
the basic differentiation rules
All 111.35
Applied Problems
*111.54b
Calculus AB
Connections to Calculus
Limits
Derivatives
Suggested Resources
Chapter 12
Chapter 16(f)
Assessments
TAKS
Objectives
N/A