©2010 Austin Independent School District
Austin ISD Instructional Planning Guide – Mathematics
5th Six Weeks IPG-February 22 to April 16 – (34 days; 2 days for 6 weeks review/test, 1 day ELA TAKS)
Major Concept #1: Polynomial Function
Overarching
Idea
Teacher
Guiding
Questions
240A
242
262
271
274
240
240A
242
What is the degree of the polynomial function?

How does the degree of the function relate to the solutions of the function?
TEKS
Knowledge & Skill
Attributes of Functions
240

Equations, Functions,
and Function Models
271
7 DAYS
Finding the solutions to polynomial functions.
The student defines
functions, describes
characteristics of
functions, and translates
among verbal, numerical,
graphical, and symbolic
representations of
functions, including
polynomial, rational, power
(including radical),
exponential, logarithmic,
trigonometric, and
piecewise-defined
functions. (P.1)
The student uses functions
and their properties, tools
and technology, to model
and solve meaningful
problems. (P.3)
Attributes of
Functions
262
Matrix
Strand
Equations,
Functions,
and Function
Models
Matrix
#
Pre-Calculus
Student Expectation
Describe parent functions
symbolically and graphically,
including f(x)=xn, f(x)=ln x,
f(x)=logax, f(x)=1/x, f(x)=ex,
f(x)=|x|, f(x)=ax, f(x)=sin x,
f(x)=arcsin x, etc. (P.1A)
Recognize and use
connections among significant
values of a function (zeros,
maximum values, and
minimum values, etc.), points
on the graph of a function, and
the symbolic representation of
a function. (P.1D)
Investigate properties of
trigonometric and polynomial
functions. (P.3A)
Use functions such as
logarithmic, exponential,
trigonometric, polynomial, etc.
to model real-life data. (P.3B)
Use properties of functions to
analyze and solve problems
and make predictions. (P.3D)
(P.1A)
TAKS
Obj
Resource
LH
(Section 2-1)
Quadratic
Functions and
Models
Time/
Pace
Teacher Tools
1 day
Vocabulary: axis of symmetry, degree of
polynomial, polynomial, quadratic function,
standard form, vertex
Teacher Notes: Review the standard form of a
quadratic function and discuss the definitions of
polynomial and quadratic functions (p. 128). Draw
the graph of y = x2 and identify the vertex (at the
origin) and the axis.
Allow students 5-10 minutes to work the
Exploration (box in margin) on page 129 of the
text.
Review the material previously covered in Section
1.7 to produce the following.
The quadratic function f(x) = a(x – h)2 + k, a ≠ 0 is
in standard form. The graph of f is a parabola with
vertical axis x = h and with vertex at (h, k). If a >
0, the parabola opens upward, and if a < 0, the
parabola opens downward.
Have students graph y = −(x – 4)2 + 5. Ask the
class, “What is the minimum value of y?” Ask
several students to explain how they know the
minimum value? How are the minimum and
maximum similar and different? How are the max
and min related to the vertex?
2 day
Vocabulary: Continuous, Repeated zero, Multiplicity,
Intermediate Value Theorem
Teacher Notes: Discuss these characteristics of
graphs of polynomial functions.
1. Polynomial functions are continuous. This means
that the graphs of polynomial functions have no
breaks, holes, or gaps.
2. The graphs of polynomial functions have only
nice, smooth turns and bends. There are no sharp
turns as in the graph of y = |x|.
• Look first at the simplest polynomials, f(x) = xn.
These are called power functions. We can break
these into two cases, n is even and n is odd.
Have students work in pairs or small groups to
complete the Exploration in the margin of page 141
before discussing the leading coefficient test.
(P.1D)
(P.1)
(P.3)
Investigate the concepts of
continuity, end behavior,
asymptotes, and limits and
connect these characteristics
to functions represented
graphically and numerically.
(P.1E)
(P.3A)
(P.3B)
(P.3D)
LH: Pre-Calculus with Limits, Larson-Hostetler, Houghton-Mifflin, 2007
PTII: Precalculus TEXTEAMS Institute Part II
LH
(Section 2-2)
Polynomial
Functions of Higher
Degree
Pre-Calculus Page 1 of 6
LTF/A2: Laying the Foundation, Connecting Algebra 2 LTF/PC: Laying the Foundation, Connecting Pre-Calculus
APTI: Algebra II/Precalculus TEXTEAMS Part I
©2010 Austin Independent School District
Austin ISD Instructional Planning Guide – Mathematics
5th Six Weeks IPG-February 22 to April 16 – (34 days; 2 days for 6 weeks review/test, 1 day ELA TAKS)
Major Concept #1: Polynomial Function (Continued)
Overarching
Idea
Teacher
Guiding
Questions
271
240
240A
242
Attributes of Functions
271
Matrix
Strand
Equations, Functions, and Function
Models
Matrix
#
Pre-Calculus
7 DAYS (Continued)
Finding the solutions to polynomial functions.

What is the degree of the polynomial function?

How does the degree of the function relate to the solutions of the function?
TEKS
Knowledge & Skill
(P.1)
(P.1)
Student Expectation
(P.1D)
TAKS
Obj
Resource
LH
(Section 2-3)
Polynomial and
Synthetic Division
Time/
Pace
Teacher Tools
2 days
Vocabulary: Division Algorithm, proper, improper,
synthetic division, Remainder Theorem, Factor
Theorem
Teacher Notes: Students should be able to relate
long division of polynomials with previous
experiences with the division algorithm.
(P.1D)
(P.3A)
(P.3B)
LH
(Section 2-5)
Zeros of
Polynomial
Functions
1 day
(P.3)
(P.3D)
Review and Test
LH: Pre-Calculus with Limits, Larson-Hostetler, Houghton-Mifflin, 2007
PTII: Precalculus TEXTEAMS Institute Part II
Vocabulary: Conjugates, Irreducible over the
reals, variation in signs, upper bound, lower
bound, Fundamental Theorem of Algebra, Linear
Factorization Theorem, rational zero test
Teacher Notes: When discussing the
Fundamental Theorem of Algebra, it should be
pointed out that the zeros might not be distinct.
Ask the students how they would solve x3 + 6x –
7 = 0. Then ask them how they would solve the
same equation if they knew that, if there were
any, the rational zeros would have to be in the list
±1, ±2, ±3, ±6. Then discuss the Rational Zero
Test.
Note that in Examples 1(c) and 1(d) of the text,
the two complex zeros were conjugates. State
that if f is a polynomial function with real
coefficients, then whenever a + bi is a zero of f, a
– bi is also a zero of f.
Discuss the multiple ways of dealing with a very
large list generated by the Rational Zero Test
2 days
Pre-Calculus Page 2 of 6
LTF/A2: Laying the Foundation, Connecting Algebra 2 LTF/PC: Laying the Foundation, Connecting Pre-Calculus
APTI: Algebra II/Precalculus TEXTEAMS Part I
©2010 Austin Independent School District
Austin ISD Instructional Planning Guide – Mathematics
5th Six Weeks IPG-February 22 to April 16 – (34 days; 2 days for 6 weeks review/test, 1 day ELA TAKS)
Major Concept #2: Rational Functions
Overarching
Idea
Teacher
Guiding
Questions
Matrix
#
Matrix
Strand
Attributes of
Functions
271

How are rational expressions, fractions, and rational numbers related?

What are the differences between adding/subtracting and multiplying/dividing rational expressions?

When can rational functions be used to find solutions to specific problems?
TEKS
Knowledge & Skill
(P.1)
Attributes of
Functions
(P.1)
2 days
(P.1B)
Activity 1: Should
You See a
Discontinuity?
Apply basic
transformations, including
af(x), f(x)+d, f(x-c) f(bx),
and compositions with
absolute value functions,
including |f(x)|, and f(|x|), to
the parent functions. (P.2A)
LH: Pre-Calculus with Limits, Larson-Hostetler, Houghton-Mifflin, 2007
PTII: Precalculus TEXTEAMS Institute Part II
Vocabulary: Rational function, vertical asymptote,
horizontal asymptote, slant (or oblique)
asymptote
Teacher Notes: Review the basics of rational
functions. Draw attention to the Guidelines for
Analyzing Graphs of Rational Functions and the
Technology feature on page 187 of the text.
Discuss the concept of slant asymptotes and how
they are related to vertical and horizontal
asymptotes.
Applications of rational functions are explored on
pages 192-193 of the text.
Vocabulary: discontinuity, removable
PTII: II.2.2.1
Seeing Too Much
(P.1E)
The student interprets the
meaning of the symbolic
representations of
functions and operations
on functions to solve
meaningful problems. (P.2)
Teacher Tools
As a campus the Precalculus teachers might
determine that the TexTEAMS activities below
should precede these text resources.
(P.3D)
(P.1A)
Graphing and
Transformations
Time/
Pace
(P.3)
274
321
Resource
LH
(Section 2-6)
Rational
Functions
(P.3B)
262
260
TAKS
OBJ
(P.1A)
Determine the domain and
range of functions using
graphs, tables, and symbols.
(P.1B)
(P.1E)
Equations,
Functions, and
Function Models
242
Student Expectation
(P.1D)
274
240A
7 DAYS
Manipulations of Rational expressions and solutions of functions with fractions.
262
260
Pre-Calculus
Activity 2: How Big
is a Hole?
Reflect and Apply
2 days
3 days
discontinuity, pixel
Teacher Notes: The transformation of the parent
function is investigated first with paper and pencil
before turning to the graphing calculator
Sometimes the exact nature of technology can
make graphs appear incorrect.
This can be used to our advantage to help teach
topics like discontinuities. The Zoom Decimal
window on the calculator will allow students to
“see” the hole in the graph. A dilation of the
original decimal window settings will also result in
a hole. Also, keeping the original YMIN and
YMAX at -3.1 and 3.1 respectively will also
“show” the hole or maintain the integrity of the
graphs.
Pre-Calculus Page 3 of 6
LTF/A2: Laying the Foundation, Connecting Algebra 2 LTF/PC: Laying the Foundation, Connecting Pre-Calculus
APTI: Algebra II/Precalculus TEXTEAMS Part I
©2010 Austin Independent School District
Austin ISD Instructional Planning Guide – Mathematics
5th Six Weeks IPG-February 22 to April 16 – (34 days; 2 days for 6 weeks review/test, 1 day ELA TAKS)
Major Concept #3: Exponential and Logarithmic Functions
Overarching
Idea
Teacher
Guiding
Questions
260
274
240A
242
Attributes
of
Functions
262
Matrix
Strand
Equations, Functions,
and Function Models
Matrix
#

What is the relationship between exponential and logarithmic functions?

What is the relationship between the rules of exponents and the rules of logarithms?

How are the rules used in growth and decay curves?
TEKS
Knowledge & Skill
240A
242
Attributes of Functions
275
Student Expectation
TAKS
OBJ
Resource
Time/
Pace
Teacher Tools
3 days
Vocabulary: algebraic functions, transcendental
functions, natural base e, continuous
compounding
Teacher Notes: Consider beginning this lesson
with a “Think, Pair, Share” where students tell you
(record student responses publicly) everything
they know about exponential functions (student
should have encountered exponential and
logarithmic functions in Algebra II). Review the
natural exponential function. Determine how
much prior knowledge students have retained
about exponential functions and consider allowing
cooperative groups to work the three applications
examples and present them.
3 days
Vocabulary: common logarithmic function, natural
logarithmic function
Teacher Notes: Students should have prior
knowledge of logarithmic functions as the inverse
of exponential functions. Review the
characteristics and transformations of the graphs
of logarithmic functions.
State that the logarithmic function with base e is
called the natural logarithmic function and is
denoted by f(x) = ln x.
Tip: Students will have trouble remembering that
ln x = logex; so extra emphasis may be required.
An interesting application concerning human
memory is the last example of the lesson.
(P.1A)
(P.1)
(P.1B)
(P.1E)
LH
(Section 3-1)
Exponential
Functions and
Their Graphs
(P.3B)
(P.3)
(P.3D)
(P.1A)
Equations,
Functions, and
Function Models
274
17 DAYS
The relationship between exponential and logarithmic functions.
262
260
Pre-Calculus
(P.1)
(P.1B)
(P.1E)
The student interprets the
meaning of the symbolic
representations of
functions and operations
on functions to solve
meaningful problems.
(P.2)
Investigate identities
graphically and verify them
symbolically, including
logarithmic properties,
trigonometric identities, and
exponential properties. (P.2C)
(P.3B)
(P.3)
(P.3D)
LH: Pre-Calculus with Limits, Larson-Hostetler, Houghton-Mifflin, 2007
PTII: Precalculus TEXTEAMS Institute Part II
LH
(Section 3-2)
Logarithmic
Functions and
Their Graphs
Pre-Calculus Page 4 of 6
LTF/A2: Laying the Foundation, Connecting Algebra 2 LTF/PC: Laying the Foundation, Connecting Pre-Calculus
APTI: Algebra II/Precalculus TEXTEAMS Part I
©2010 Austin Independent School District
Austin ISD Instructional Planning Guide – Mathematics
5th Six Weeks IPG-February 22 to April 16 – (34 days; 2 days for 6 weeks review/test, 1 day ELA TAKS)
Major Concept #3: Exponential and Logarithmic Functions (Continued)
Overarching
Idea
Teacher
Guiding
Questions
242
275
240A
242
Attributes
of
Functions
Equations,
Functions,
and
Function
Models
240A
Attributes
of
Functions
275
Matrix
Strand
Equations, Functions, and Function
Models
Matrix
#
Pre-Calculus
17 DAYS (Continued)
The relationship between exponential and logarithmic functions.

What is the relationship between exponential and logarithmic functions?

What is the relationship between the rules of exponents and the rules of logarithms?

How are the rules used in growth and decay curves?
TEKS
Knowledge & Skill
(P.2)
Student Expectation
(P.2C)
(P.3B)
TAKS
OBJ
Resource
LH
(Section 3-3)
Properties of
Logarithms
Time/
Pace
Teacher Tools
3 days
Vocabulary: change-of-base formula
Teacher Notes: Review the change-of-base
formula and properties of logarithms. The
exploration in the margin of page 241 uses the
graphing calculator as a tool for students to think
critically about logarithmic expressions. This
exploration introduces the concept of expanding
and condensing logarithmic expressions. An
application problem is the final example.
3 days
Vocabulary:
Teacher Notes: Review the following one-to-one
and inverse properties, which will be key in
solving exponential and logarithmic equations.
One-to-One Properties
1. a x = a y if and only if x = y.
2. logax = logay if and only if x = y.
Inverse Properties
1. logaax = x
2. a log a x = x
Two very general keys to solving exponential
equations are:
1. Isolate the exponential expression.
2. Use the second one-to-one property from
above.
Two basic ways of solving logarithmic equations.
1. Isolate the logarithmic expression and then
write the equation in equivalent exponential
form.
2. Get a single logarithmic expression with the
same base on each side of the equation; then
use the one-to-one property.
(P.3)
(P.3D)
(P.2)
(P.2C)
(P.3B)
LH
(Section 3-4)
Exponential and
Logarithmic
Equations
(P.3)
(P.3D)
LH: Pre-Calculus with Limits, Larson-Hostetler, Houghton-Mifflin, 2007
PTII: Precalculus TEXTEAMS Institute Part II
Pre-Calculus Page 5 of 6
LTF/A2: Laying the Foundation, Connecting Algebra 2 LTF/PC: Laying the Foundation, Connecting Pre-Calculus
APTI: Algebra II/Precalculus TEXTEAMS Part I
©2010 Austin Independent School District
240A
242
Matrix
Strand
Equations, Functions, and
Function Models
Matrix
#
Austin ISD Instructional Planning Guide – Mathematics
5th Six Weeks IPG-February 22 to April 16 – (34 days; 2 days for 6 weeks review/test, 1 day ELA TAKS)
TEKS
Knowledge & Skill
Student Expectation
TAKS
OBJ
Teacher Tools
APTI: II.1.1.1
Exponential Growth
& Decay –
Activities 1 and 4
3 days
Vocabulary: Exponential function, exponential
equation, logarithmic equation, mathematical
model, Logistic Growth Curve, ExponentialGrowth Stage, Dampened-Growth Stage,
Equilibrium-Growth Stage
Teacher Notes: These activities are for groups of
3-4 students. Several transparencies are included
to facilitate intermittent discussions during the
cooperative activities. Please read all included
Leader’s Notes carefully and consider working
through this activity with other teachers prior to
implementing it in the classroom with students.
Review and Test
2 days
Resource
(P.3B)
(P.3)
(P.3D)
LH: Pre-Calculus with Limits, Larson-Hostetler, Houghton-Mifflin, 2007
PTII: Precalculus TEXTEAMS Institute Part II
Pre-Calculus
Time/
Pace
Pre-Calculus Page 6 of 6
LTF/A2: Laying the Foundation, Connecting Algebra 2 LTF/PC: Laying the Foundation, Connecting Pre-Calculus
APTI: Algebra II/Precalculus TEXTEAMS Part I