Algebra II Scope and Sequence 2014-15 (edited May 2014)
HOLT Algebra 2
Algebra II - Unit Outline
First Semester
Page
Unit
Abbreviated
Name
TEKS
2-4
Linear and Absolute
Value Functions
LINABS
2A.1A, 2A.1B, 2A.4B
2A.2A, 2A.4A,
20
5-8
Quadratics
QUADS
2A.1B, 2A.4B, 2A.6A, 2A.6B, 2A.7A,
2A.8A, 2A.8D
2A.2A, 2A.2B, 2A.4A, 2A.6C,
2A.7B, 2A.8B, 2A.8C
40
9
Systems
SYS
2A.3A, 2A.3B, 2A.3C
---
8 (+10 for
end of sem
testing)
Readiness
Supporting
Total First Semester
Time
(days)
78
Second Semester
10-12
13-17
Square Root Functions
and Inverses
Exponential and
Logarithmic Functions
SQRTINV
2A.4B, 2A.9F
2A.4A, 2A.4C, 2A.9A, 2A.9B,
2A.9C, 2A.9D, 2A.9E, 2A.9G
21
EXPLOG
2A.4B, 2A.11A, 2A.11F
2A.2A, 2A.4A, 2A.4C, 2A.11B,
2A.11C, 2A.11D, 2A.11E,
29
18-20
Rational Functions
RATFUN
2A.1B, 2A.4B, 2A.10F,
2A.4A, 2A.10A, 2A.10B, 2A.10C,
2A.10D, 2A.10E, 2A.10G
26
21
Conics
CONICS
---
2A.5A, 2A.5B, 2A.5C, 2A.5D,
2A.5E
6
22
Cube and Cube Root
Functions
CUBRT
New TEKS 2A.6A, 2A.6B
23
Polynomials
POLY
New TEKS 2A.7A, 2A.7B, 2A.7C, 2A.7D, 2A.7E
Total Second Semester
7 (+10 for
end of year
testing)
99
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
The following “process standards” from the new Algebra II TEKS are consistent across all high school math courses and
should be embedded in instruction throughout the scope and sequence.
(1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding.
The student is expected to:
(A) apply mathematics to problems arising in everyday life, society, and the workplace;
(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a
solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including
mental math, estimation, and number sense as appropriate, to solve problems;
(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams,
graphs, and language as appropriate;
(E) create and use representations to organize, record, and communicate mathematical ideas;
(F) analyze mathematical relationships to connect and communicate mathematical ideas; and
(G) display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral
communication.
2
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Linear and Absolute Value Functions (20 days)
Enduring Understandings
The student understands that a function represents a dependence of one quantity on another and can
be described in a variety of ways.
The student understands that linear and absolute value functions have unique properties and attributes,
including domain and range.
The student understands that changes to a function rule affect the function’s graph and associated
ordered pairs.
The student understands the concept of absolute value and how it is represented in a function.
The student understands that data representing real-world situations can be collected, organized, and
interpreted in order to solve problems.
The student understands that linear and absolute value functions can be used to model real-world
situations.
The student understands that an equation and inequality can be solved using a variety of methods.
Vocabulary
parent function, linear, point-slope form, slope-intercept form, standard form, absolute value, piece-wise
function, parameter, transformation, translation, reflection, expansion, compression, dilation, horizontal,
vertical, domain, range, set notation, interval notation, discrete, continuous, solution
2A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions.
The student is expected to identify and sketch graphs of parent functions, including linear f ( x )  x , [quadratic
f ( x )  x 2 , exponential f ( x )  a x , and logarithmic functions f ( x )  loga x ,] absolute value of x f ( x )  x , [square
1
].
x
2A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions.
a
The student is expected to extend parent functions with parameters such as a in f ( x ) 
and describe the effects
x
of the parameter changes on the graph of parent functions.
The student will know…
The student will be able to…
 The parent function is the simplest  Identify and sketch the graph of the parent function for linear and
function with the defining
absolute value.
characteristics of the family.
 Predict the effects of parameter changes on the graphs of parent
functions
 That by transforming the graph of
a parent function, you can create
 Vertical/horizontal shifts.
infinitely many new functions.
 Vertical expansion(stretch) and compression
 That the absolute value function is
 Reflections across the x-axis and y-axis
related to a linear function and its
 Horizontal expansion and compression (K)
reflection across a vertical line
 Describe the effects of parameter changes on the graph of the linear
through the vertex of the absolute
parent function.
value function.
 y = mx + b
 Changes in m
 Changes in b
 Connect changes in m and b to changes in a problem situation
 Define the absolute value function as a piece-wise function.
 Compare parameter changes on the absolute value function to that of the
linear function.
 Make connections between the point-slope formula of a linear equation
and the transformations of the parent function. (Ex. y  2  4( x  1) is
root of x f ( x ) 
x , and reciprocal of x f ( x ) 
the same as y  4( x  1)  2 and is the translation of the parent
function y = x with a vertical expansion of 4 which effects the slope and a
translation 1 unit left and 2 units down.
 Record/describe parameter changes of parent functions using function
notation (Ex. y = f(x + 2) is a transformation of y = f(x) 2 units to the left.).
3
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Linear and Absolute Value Functions, continued
2A.1A(R) Foundations for functions. The student uses properties and attributes of functions and applied functions
to problem situations. The student is expected to identify the mathematical domains and ranges of functions and
determine reasonable domain and range values for continuous and discrete situations.
The student will know…
The student will be able to…
 The domain represents the
 Identify domain and range of linear and absolute value functions.
independent values (x-values) in a
 From a graph
linear function.
 From a verbal situation
 The range represents the
 From a table
dependent values (y-values) in a
 Determine reasonable domain and range values from a given situation.
linear function.
 Continuous situations
Interval and set-builder notation will
 The difference between
 Discrete situations
not be assessed on district
continuous data and discrete data.
 From a graph
questions in this unit, including
 The domain and range of a
 From a verbal description
DPM2; campuses may assess on
function may be different from the
 From a table
their assessments.
domain and range for a situation
 From an equation (linear)
represented by that function.
 Distinguish between the domain and range of a situation and the domain
and range of the function modeling the situation.
 Determine if a situation is represented by continuous data or discrete
data.
 Express domain and range using different forms.
 Example: for x ≥ 0, write as:
 Informal set notation {x ≥ 0}
 Interval notation [0 , ∞)
 Set-builder notation, {x | x ≥ 0}
2A.1B(R) Foundations for functions. The student uses properties and attributes of functions and applied functions
to problem situations. The student is expected to collect and organize data, make and interpret scatterplots, fit the
graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical
judgments.
The student will know…
The student will be able to…
 There are different ways to
 Identify independent and dependent quantities from a situation.
organize data (table, scatterplot,
 Describe the relationship between two quantities.
etc.).
 Verbal
 The attributes of a scatterplot
 Equation
(horizontal axis represents the
 Graph
independent variable and vertical
 Collect and use real world data to create scatterplots.
axis represents the dependent
 Make predictions from scatterplots.
variable).
 Using parameter changes, write the equation of a trend line that best
 Information can be gathered and
models linear data.
interpreted from a scatterplot.
 Determine that a set of data has a positive, negative, or no correlation.
 Predictions can be made based on  Use the regression feature of the graphing calculator to find the line of
trends in data.
best fit that models the data.
 Compare/contrast the trend line vs. the line of best fit for a set of data.
 Which is a better fit for the situation? (y-intercept, slope)
 Which is a better fit mathematically based on the data?
4
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Linear and Absolute Value Functions, continued
2A.2A(S) Foundations for functions. The student understands the importance of the skills required to manipulate
symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic
expressions and solve equations and inequalities in problem situations. The student is expected to use tools
[including factoring and properties of exponents] to simplify expressions and to transform and solve equations.
The student will know…
 That a variety of methods can be
used to solve linear equations and
inequalities.
 That linear functions can be
written in a variety of forms.
 That equations of lines can be
determined from various
information sets.
 How to graph a line from various
information sets.
 There are various, equivalent
forms of linear equations that can
represent the same line.
The student will be able to…
 Simplify algebraic expressions using order of operations.
 Connect to concrete models such as algebra tiles
 Evaluate algebraic expressions.
 Formulate linear equations and inequalities from problem situations.
 Use a variety of strategies to solve linear equations and inequalities.
 Graph
 Table
 Graphing calculator (Y1 = left side; Y2 = right side)
 Simplify and solve linear equations containing fractions (multiply by
LCD) and decimals (multiply by power of 10)
 Transform linear equations from one form to another and graph.
 From standard to slope-intercept form
 From point-slope form (new instruction) to slope-intercept form and
vice versa
 Solve absolute value equations (L level stay with basic equations such as
|2x – 5|=7).
 Solve absolute value inequalities (L level stay with basic inequalities).
Linear and Absolute Value Functions: Looking forward to 2015-16…
What’s new?


Write absolute value equations to solve problems
Solve absolute value linear inequalities
What’s going away?


Simplifying algebraic expressions
Domain/range of linear functions
How will this affect my teaching this year?
5
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Quadratics (40 days)
Enduring Understandings
The student understands that symbols can be manipulated algebraically in order to simplify algebraic
expressions and solve equations and inequalities in problem situations.
The student understands that quadratic equations and inequalities can be solved in a variety of ways.
The student understands the relationships between various components and representations of a
quadratic function.
The student understands that changes to the equation of a function will cause related changes to the
table and graph representing the function.
The student understands that quadratic functions have unique properties and attributes, including
domain and range.
The student understands that quadratic functions can be used to model real-world situations.
The student understands that an equation and inequality can be solved using a variety of methods.
Vocabulary
polynomial, degree, exponent, factor, monomial, binomial, trinomial, coefficient, constant, leading term,
product, quotient, perfect square trinomial, vertex, axis of symmetry, intercepts, maximum/minimum,
parabola, transformations, translation, reflection, dilation, expansion, compression, domain, range, xand y-intercepts, curve of best fit, solutions, quadratic formula, complex number, discriminant, real,
imaginary, rational, irrational, roots, zeros
2A.2A(S) Foundations for functions. The student understands the importance of the skills required to manipulate
symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic
expressions and solve equations and inequalities in problem situations. The student is expected to use tools
including factoring and properties of exponents to simplify expressions [and to transform and solve equations].
The student will know…
 There are rules for simplifying
expressions that involve
exponents.
 The difference between
simplifying and multiplying
polynomials and when each is
appropriate.
 Polynomials can be simplified
and multiplied, using various
strategies.
 Polynomials may have different,
but equivalent forms.
 Not all polynomials have factors
other than 1 (some are prime).
The student will be able to…
 Use laws of exponents to simplify, multiply and divide polynomial
expressions.
 Product rule
 Quotient rule
 These rules repeated in Exponential unit but needed here for GCF
factoring. Other exponent rules taught in Exponential unit.
 Simplify polynomials (quadratics only).
 Algebraically
 Verify on graphing calculator
 Multiply polynomials (quadratics only).
 With concrete models such as algebra tiles
 Algebraically
 Verify on graphing calculator
 Multiply polynomials in quadratic problem situations.
 Factor polynomial expressions.
 GCF, including variables so that the remaining polynomial is degree 2.
 Trinomial factoring (include “a” not equal to 1)
 Factor by grouping
 Connect to area models (Ex. algebra tiles)
Recognize and apply special patterns in factoring.
 Difference of two squares (include multiple variables. Ex: a2b2 – c2)
 Perfect square trinomials
 Note: Sum and difference of cubes moved to Polynomials unit – all
levels.
Note: Fractional, negative, and higher degree exponents are taught for all
levels in Exponential unit.
6
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Quadratics, continued
2A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to identify and sketch graphs of parent functions, including [ linear f ( x )  x ,] quadratic
f ( x )  x 2 , [ exponential f ( x )  a x , and logarithmic functions f ( x )  loga x , absolute value of x f ( x )  x , square
root of x f ( x ) 
x , and reciprocal of x f ( x ) 
1
].
x
2A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions.
a
The student is expected to extend parent functions with parameters such as a in f ( x ) 
and describe the effects
x
of the parameter changes on the graph of parent functions.
2A.7B(S) Quadratic and square root functions. The student interprets and describes the effects of changes in the
parameters of quadratic functions in applied and mathematical situations. The student is expected to use the
parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of
y  a( x  h )2  k form of a function in applied and purely mathematical situations.
The student will know…
The student will be able to…
 Quadratic functions can be
 Identify the graph of the quadratic parent function.
written in a variety of forms.
 Sketch the graph of the quadratic parent function.
 The parent function of a
 Recognize the attributes of the quadratic parent function. See 2A.7A
quadratic function is a parabola.
 Goes through origin
 The parent function is the
 Symmetric about line x = 0
simplest function with the
 Table of values has a 2nd difference of 2
defining characteristics of the
 Use transformations to sketch y  a( x  h )2  k from the parent function.
family.
 Identify the vertex of the graph from y  a( x  h )2  k .
 Changing the values in the
 Predict changes to the graph when a, h, or k are changed.
vertex form of a quadratic
function will change the graph in  Connect the effects of changing a, h, or k to a problem situation.
a predictable manner.
2A.7A(R) Quadratic and square root functions. The student interprets and describes the effects of changes in the
parameters of quadratic functions in applied and mathematical situations. The student is expected to use
characteristics of the quadratic parent function to sketch the related graphs and connect between the
y  ax 2  bx  c and the y  a( x  h )2  k symbolic representations of quadratic functions.
2A.6A(R) Quadratic and square root functions. The student understands that quadratic functions can be
represented in different ways and translates among their various representations. The student is expected to
determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the
reasonableness of solutions to quadratic equations and inequalities.
The student will know…
The student will be able to…
 The form of a quadratic function
 Use completing the square to change a quadratic function from standard
is based on the situation.
form to vertex form.
 The domain and range of a
 Identify y-intercept from standard (polynomial) form of a quadratic function.
quadratic function can be found
 Choose the appropriate form (standard or vertex form) of a quadratic
from a variety of
function based on the situation or information needed.
representations.
 Determine the axis of symmetry from the graph of a quadratic function.
 The difference between the
 Sketch the graph of a quadratic function using key attributes (vertex,
domain and range of a quadratic
intercepts, direction).
function and the domain and
 Determine domain and range of a quadratic function.
range of a situation modeled by
 Given a graph.
a quadratic function.
 Given a table.
 Express domain and range of a quadratic function.
 Example: for x ≥ 0, write as:
 Informal set notation {x ≥ 0}
 Interval notation [0 , ∞)
 Set-builder notation, {x | x ≥ 0}
 Compare the domain and range of a quadratic function and the domain and
range of a situation that can be modeled by the same quadratic function.
7
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Quadratics, continued
2A.6B(R) Quadratic and square root functions. The student understands that quadratic functions can be
represented in different ways and translates among their various representations. The student is expected to relate
representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions.
2A.8A(R) Quadratic and square root functions. The student formulates equations and inequalities based on
quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
The student is expected to analyze situations involving quadratic functions and formulate quadratic equations or
inequalities to solve problems.
The student will know…
The student will be able to…
 Quadratic functions can be
 Create the other representations of quadratic functions when given a verbal
represented in a variety of ways.
description, equation, graph, or table.
 Some situations can be
 Determine which form of a quadratic function is appropriate when solving
represented by a quadratic
problems (i.e. vertex or general form).
function.
 Write a quadratic equation/inequality to solve application problems.
 Use a quadratic function to answer questions and make predictions in a
given situation.
 Determine if a situation can be modeled by a quadratic function.
2A.6A(R) Quadratic and square root functions. The student understands that quadratic functions can be
represented in different ways and translates among their various representations. The student is expected to
determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the
reasonableness of solutions to quadratic equations and inequalities.
2A.8D(R) Quadratic and square root functions. The student formulates equations and inequalities based on
quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
The student is expected to solve quadratic equations and inequalities using graphs, tables, and algebraic methods.
The student will know…
The student will be able to…
 There are various ways to solve  Determine domain and range of a quadratic function.
a quadratic equation/inequality.
 Given an equation
 Given a situation that can be modeled with a quadratic function
 Interpret solution of a quadratic equation/inequality in context of the situation.
 Determine if the solution to a quadratic equation/inequality is reasonable in
context of the situation.
 Solve quadratic inequalities in problem situations and in purely mathematical
situations.
 Graph (graphing calculator)
 Table (graphing calculator)
 Y1 = left side; Y2 = right side
2A.1B(R) Foundations for functions. The student uses properties and attributes of functions and applied functions
to problem situations. The student is expected to collect and organize data, make and interpret scatterplots, fit the
graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical
judgments.
The student will know…
The student will be able to…
 There are different ways to
 Collect data in a problem situation that can be described using a quadratic
organize data (table,
function in tabular form (optional).
scatterplot. etc.).
 Use the table to generate a graph and a function rule.
 The attributes of a scatterplot
 Use curve-fitting techniques and transformations to make a curve of the
(horizontal axis represents the
parent function y = x2 to model the data in a scatterplot.
independent variable and
 Use a graphing calculator to find the regression equation (curve of best fit) of
vertical axis represents the
a set of data.
dependent variable).
 Use a graphing calculator to confirm the function rule over a scatterplot of
 Information can be gathered
the data.
and interpreted from a
 Use the function rule to make predictions and judgments.
scatterplot.
 Interpret the meaning of the maximum/minimum values from a graph or
 Predictions can be made based
table to the situation.
on trends in data.
8
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Quadratics, continued
2A.2B(S) Foundations for functions. The student understands the importance of the skills required to manipulate
symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic
expressions and solve equations and inequalities in problem situations. The student is expected to use complex
numbers to describe the solutions of quadratic equations.
2A.8C(S) Quadratic and square root functions. The student formulates equations and inequalities based on
quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
The student is expected to compare and translate between algebraic and graphical solutions of quadratic
equations.
The student will know…
The student will be able to…
 There is a relationship between  Solve quadratic equations in problem situations and in purely mathematical
situations.
the graph of a quadratic
function and its zeros.
 Factoring
 The x-intercept of a graph is the
 Graph
ordered pair that describes the
 Table
point where the graph crosses
 Quadratic Formula
the x-axis.
 Connect the solution (root) to a quadratic equation.
 The zeros of a function are the
 x-intercepts of the related function from a table and a graph
value(s) of the independent
 Zeros of the related function
variable, x, that makes the
 Simplify solutions from quadratic formula involving radicals.
function equal to zero.
6  12
Ex.
x
 The zeros of a quadratic
2
function can be determined


x
3
3
from either a graph, a table, or
an algebraic representation.
{3  3, 3  3 }
 The value of i is 1 .
 Express the solution of a quadratic equation in terms of a complex number.
Ex.
4  20
2
x  2  2i 5
x
2  2i
5, 2  2i 5

 Note: Simplifying complex numbers (inc powers of i) and operations on
complex numbers will be taught in the Polynomials unit after STAAR.
2A.6C(S) Quadratic and square root functions. The student understands that quadratic functions can be
represented in different ways and translates among their various representations. The student is expected to
determine a quadratic function from its roots or a graph.
2A.8B(S) Quadratic and square root functions. The student formulates equations and inequalities based on
quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
The student is expected to analyze and interpret the solutions of quadratic equations using discriminants and solve
quadratic equations using the quadratic formula.
The student will know…
The student will be able to…
 The discriminant of a quadratic
 Calculate and use the discriminant in determining types of solutions of
equation determines the
quadratic equations, including real, imaginary, rational, irrational.
number and type of roots of the  Perform basic calculations with radical expressions in order to use sum and
equation.
product of roots when determining a quadratic function.
 Solutions to some quadratic
 Write a quadratic function.
equations are nonreal complex
 Given two roots
numbers.
 Write in factored form
 The quadratic formula can be
 Use sum and product of roots (real roots only)
used to solve quadratic
 Given a graph
equations.
 Given three points
 Quadratic equations may have
 Use a graphing calculator to find the regression equation.
2, 1, or no real solutions.
 Optional: Use matrices to find A, B, and C.
9
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Quadratics: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
10
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Systems (18 days)
Enduring Understandings
The student understands that a system of two functions can have a varying number of solutions
depending on the type of functions.
The student understands that matrices can be used to represent and solve real-life problems.
The student understands that systems of equations and inequalities can be used to model real-world
situations.
The student understands that systems of equations and inequalities can be solved using a variety of
methods.
Vocabulary
system, solution, intersection, linear combination (elimination), substitution, consistent, inconsistent,
dependent, independent, matrix/matrices, inverse matrix, matrix multiplication, test point, region
2A.3A(R) Foundations for functions. The student formulates systems of equations and inequalities from problem
situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The
student is expected to analyze situations and formulate systems of equations in two or more unknowns or
inequalities in two unknowns to solve problems.
2A.3B(R) Foundations for functions. The student formulates systems of equations and inequalities from problem
situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The
student is expected to use algebraic methods, graphs, tables, or matrices, to solve systems of equations or
inequalities.
2A.3C(R) Foundations for functions. The student formulates systems of equations and inequalities from problem
situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The
student is expected to interpret and determine the reasonableness of solutions to systems of equations or
inequalities for given contexts.
The student will know…
The student will be able to…
 Situations that involve two or more  Represent 2  2 systems of equations, a graph and table, including:
unknowns can be expressed by a
linear/linear and linear/quadratic.
system of equations or
 Write a 2  2 system of equations or inequalities from a problem situation.
inequalities.
 Solve 2  2 systems, including linear/quadratic
 That systems of equations or
 Algebraically (Linear combination and Substitution)
inequalities can be solved using a
 Graphing (with and without a graphing calculator)
variety of methods, including
technology.
 Table (with and without graphing calculator)
 The solution to systems of
 Include parallel lines and coincidental lines
equations should be reasonable to  Connect algebraic solutions to graphical and tabular solutions.
the situation it describes.
 Solve systems of inequalities.
 Matrix multiplication is not
 Include 2 or more linear inequalities
commutative.
 Graphing calculator
 Graph by hand (include using test point)
 Determine what a solution to a system of equations/inequalities means in
relationship to the problem and determine if it is reasonable.
 Introduce matrices with a 2  2 system.
 Define a matrix and use a matrix to represent data
 Inverse matrices by hand as demo (optional)
 Inverse matrices with calculator (method to be determined by campus
teams)
 Inverse matrices by hand (K optional)
 Perform matrix multiplication to show multiplication is not commutative.
 By hand (demo only)
 With technology
 Write a 3  3 system of equations from a problem situation.
 Solve 3  3 systems using inverse matrix equation on calculator.
 Solve 3  3 systems by hand (optional for L).
11
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Systems: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
12
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Square Root Functions and Inverses (21 days)
Enduring Understandings
The student understands the relationship between quadratic and square root functions and equations.
The student understands that square root functions have unique properties and attributes, including
domain and range.
The student understands that square root functions can be used to model real-world situations.
The student understands that square root equations and inequalities can be solved using a variety of
methods.
Vocabulary
inverse, reflection, one-to-one correspondence, composition of functions, restricted domain, interchange,
square root, radical, radicand
2A.4C(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to describe and analyze the relationship between a function and its inverse.
2A.9G(S) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to connect inverses of square root functions with quadratic functions.
The student will know…
The student will be able to…
 The inverse of a function is
 Determine compositions of functions algebraically
when the independent variable  Graph a function and its inverse.
is exchanged with the
 By hand using table or reflection about y = x
dependent variable.
 On calculator (using y1 and y2 or using L1 and L2)
 The inverse of a function is
 Use composition of functions to verify an inverse. f f 1  x    f 1 f  x    x
always a relation, but not
 Discuss restricted domain for 1 to 1 correspondence.
always a function.
 Develop inverse relations and functions.
 How the domain and range of
 From a situation (i.e. the area of an equilateral triangle in terms of its base
a function and its inverse are
length) by reversing the independent and dependent variables
related.
 Algebraically (i.e. interchange x and y, then solve for y)

 Calculator (table and graph)
 Patty paper (optional)
 From a graph (using key points)
 Compare and contrast the domain and range of a relation or function and its
inverse.
 Discover graph of square root function as inverse of quadratic function with
restrictions.
 Explore and formalize the relationship between the parent functions y = x2
and y  x .
13
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Square Root Functions and Inverses, continued
2A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to identify and sketch graphs of parent functions, including [ linear f ( x )  x , quadratic
f ( x )  x 2 , exponential f ( x )  a x , and logarithmic functions f ( x )  loga x , absolute value of x f ( x )  x ,] square
1
].
x
2A.9B(S) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to relate representations of square root functions, such as algebraic, tabular, graphical, and
verbal descriptions.
The student will know…
The student will be able to…
 The parent function is the
 Identify and sketch the graph of the square root parent function.
simplest function with the
 Create the other representations of square root functions when given one of
defining characteristics of the
the following: verbal description, equation, graph, or table.
family.
 Determine which representation of a square root function is appropriate when
 A square root function can be
solving problems.
represented in a variety of
ways.
root of x f ( x )  x ,[ and reciprocal of x f ( x ) 
2A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
a
student is expected to extend parent functions with parameters such as a in f ( x ) 
and describe the effects of the
x
parameter changes on the graph of parent functions.
2A.9A(S) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to use the parent function to investigate, describe, and predict the effects of parameter changes
on the graphs of square root functions and limitations on the domains and ranges.
The student will know…
The student will be able to…
 By transforming the graph of a
 Use transformations to sketch y  a x  c  d from the parent function.
parent function, you can create
 Predict changes to the graph when a, c, or d are changed.
infinitely many new functions.
 Connect the effects of changing a, c, or d to a problem situation.
 Specific changes to a function
 Verify parameter changes on graphing calculator.
equation will result in a specific
 Horizontal expansion and compression (K)
change to the graph of that

Determine the equation of square root function from a graph.
function.
14
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Square Root Functions and Inverses, continued
2A.9D(S) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to determine solutions of square root equations using graphs, tables, and algebraic methods.
2A.9F(R) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to analyze situations modeled by square root functions, formulate equations or inequalities,
select a method, and solve problems.
2A.9E(S) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to determine solutions of square root inequalities using graphs and tables.
2A.9C(S) Quadratic and square root functions. The student formulates equations and inequalities based on square
root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The
student is expected to determine the reasonable domain and range values of square root functions, as well as
interpret and determine the reasonableness of solutions to square root equations and inequalities.
The student will know…
The student will be able to…
 Solutions to square root
 Solve square root equations in real-life or purely mathematical situations
equations can be found in a
algebraically and on a calculator (by graph or table).
variety of ways.
 Write and solve a square root equation from a given situation.
 The solution of a square root
 Determine the independent and dependent variables of the situation
equation or inequality does not
 Select an appropriate method for solving the equation (algebraically,
always match the solution for a
graphically, tabular)
scenario using that same
 Solve the equation and relate the solution to the situation
equation or inequality.
 Use the graphing calculator to determine the solution to a square root
 The domain represents the set
inequality from a graph and table (Ex. ( x  3)  3 ).
of all input values in a function.
 Y1  ( x  3)
 The range represents the set
of all output values in a
 Y2 = 3
function.
 Look for >3 values
 The domain and range of a

Graph
solutions to a square root inequality on a number line.
function representing a

Interpret
solution of a radical equation/inequality in context of the situation.
situation may vary from the

Determine
if the solution to a radical equation/inequality is reasonable in
domain and range of the
context
of
the
situation.
situation.

Determine
domain
and range of a square root function when given a graph, a
 The domain and range of a
table,
or
a
situation
that can be modeled with a square root function.
function can be represented in

Express
limitations
on
the domain and range of square root functions.
a variety of ways.

Compare
the
domain
and
range of a square root function and the domain and

range of a situation that can be modeled by the same quadratic function.
 Express domain and range of a square root function.
 Example: for x ≥ 0, write as:
 Informal set notation {x ≥ 0}
 Interval notation [0 , ∞)
 Set-builder notation, {x | x ≥ 0}
15
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Square Root Functions and Inverses: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
16
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Exponential and Logarithmic Functions (29 days)
Enduring Understandings
The student understands the relationship between exponential and logarithmic functions and equations.
The student understands that exponential and logarithmic functions have unique properties and
attributes, including domain and range.
The student understands that exponential and logarithmic expressions can be written in a variety of
equivalent forms.
The student understands that exponential and logarithmic functions can be used to model real-world
situations.
The student understands that exponential and logarithmic equations and inequalities can be solved
using a variety of methods.
Vocabulary
exponential, rate of growth/decay, asymptote, base (of exponent), power, logarithm, base (of logarithm),
argument, common log, natural log, antilog (optional)
2A.2A(S) Foundations for functions. The student understands the importance of the skills required to manipulate
symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic
expressions and solve equations and inequalities in problem situations. The student is expected to use tools
including [factoring and] properties of exponents to simplify expressions and to transform and solve equations.
The student will know…
The student will be able to…
 Use laws of exponents to simplify expressions, including negative and
 When to apply specific
rational exponents.
exponent rules to problems.

Product rule
 The value of i is 1 .

Quotient rule
 Powers of i repeat after every

*Product
and quotient rule were taught in Quadratics unit but are
3
7
11
4 powers (Ex. i  i  i ).
extended in this unit.
 Power to a power rule
 Power of the product rule
 Power of the quotient rule
 Connect laws of exponents to expanded form of expression
x2
xx
(Ex. 3 
).
xxx
x
 Use a graphing calculator to verify answers.
 Simplify powers of i (Ex. i 23  i 3  i ) (Optional here – will be taught in
Polynomials unit)
 Convert an exponential expression to a radical expression (i.e:
1



3 2  3 ) and vice versa.
verify on graphing calculator
number bases
variable bases
17
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Exponential and Logarithmic Functions, continued
2A.11F(R) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in
terms of the situation. The student is expected to analyze a situation modeled by an exponential function, formulate
an equation or inequality, and solve the problem.
The student will know…
The student will be able to…
 That exponential equations
 Write and solve an exponential equation from a given situation, such as
can be solved in a variety of
bacterial growth and decay, population growth and decay, and finances.
ways.
 Determine the independent and dependent variables of the situation.
 Select an appropriate method for solving the equation (algebraically,
graphically, tabular).
 Solve the equation and relate the solution to the situation.
 Write and solve an exponential inequality from a given situation such as
bacterial growth and decay, population growth and decay, and finances.
 Select an appropriate method for solving the inequality (graphical or
tabular).
 Solve the inequality and relate the solution to the situation.
2A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to identify and sketch graphs of parent functions, including [ linear f ( x )  x , quadratic
f ( x )  x 2 ,] exponential f ( x )  a x [, and logarithmic functions f ( x )  loga x , absolute value of x f ( x )  x , square
root of x f ( x ) 
x , and reciprocal of x f ( x ) 
1
].
x
2A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
a
student is expected to extend parent functions with parameters such as a in f ( x ) 
and describe the effects of the
x
parameter changes on the graph of parent functions.
2A.11B(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in
terms of the situation. The student is expected to use the parent functions to investigate, describe, and predict the
effects of parameter changes on the graphs of exponential [and logarithmic] functions, describe limitations on the
domains and ranges, and examine asymptotic behavior.
The student will know…
The student will be able to…
 That the parent function is the
 Identify the graph of the exponential parent function.
simplest function with the
 Sketch the graph of the exponential parent function.
defining characteristics of the
 Use transformations to sketch y  a  b( x  c )  d from the parent function.
family.
 Predict changes to the graph when a, c, or d are changed.
 By transforming the graph of a
 Connect the effects of changing a, c, or d to a problem situation.
parent function, you can create
 Discover changes to the domain, range, and asymptote.
infinitely many new functions.
 Write equations of asymptotes.
 Specific changes to a function
equation will result in a specific  Determine if the graph is increasing or decreasing.
change to the graph of that
function.
18
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Exponential and Logarithmic Functions, continued
2A.11C(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in terms
of the situation. The student is expected to determine the reasonable domain and range values of exponential [and
logarithmic] functions, as well as interpret and determine the reasonableness of solutions to exponential [and
logarithmic] equations and inequalities.
The student will know…
The student will be able to…
 The domain represents the set  Determine domain and range of an exponential function when given a graph,
of all input values in a function.
a table, a situation that can be modeled with an exponential function.
 The range represents the set
 Example: for x ≥ 0, write as:
 Informal set notation {x ≥ 0}
of all output values in a
 Interval notation [0 , ∞)
function.
 Set-builder notation, {x | x ≥ 0}
 The domain and range of a
 Interpret solution of an exponential equation/inequality in context of the
function representing a
situation and determine if it is reasonable.
situation may vary from the
domain and range of the
 Compare the domain and range of an exponential function and the domain
situation.
and range of a situation that can be modeled by the same functions.
 The domain and range of a
function can be represented in
a variety of ways.
2A.11D(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in
terms of the situation. The student is expected to determine solutions of exponential [and logarithmic] equations
using graphs, tables, and algebraic methods.
2A.11E(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in
terms of the situation. The student is expected to determine solutions of exponential [and logarithmic] inequalities
using graphs and tables.
The student will know…
The student will be able to…
 Some exponential equations
 Solve exponential equations.
can be written so that both
 By reducing both sides to a common base
exponential expressions have
 From a graph (calculator)
the same base.
 From a table (calculator)
 Equivalent exponential
 From a given situation
expressions that have the
 Solve exponential inequalities from a problem-solving situation and a purely
same base have equivalent
mathematical situation from a graph or table.
exponents.
 Graph solutions to an exponential inequality on a number line (optional).
 Exponential inequalities can be  Use the graphing calculator to represent the solution to an exponential
solved in a variety of ways.
inequality.
2A.4C(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to describe and analyze the relationship between a function and its inverse.
2A.11A(R) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms
of the situation. The student is expected to develop the definition of logarithms by exploring and describing the
relationship between exponential functions and their inverses.
The student will know…
The student will be able to…
 The relationship between a
 Define the inverse of y  2 x as a logarithmic function y  log2 x .
function and its inverse.
 Compare the domain, range, and asymptotes of y  2 x to y  log2 x .
 Connect exponential notation and logarithmic notation.
19
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Exponential and Logarithmic Functions, continued
2A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to identify and sketch graphs of parent functions, including [ linear f ( x )  x , quadratic
f ( x )  x 2 , exponential f ( x )  a x , and] logarithmic functions f ( x )  loga x [, absolute value of x f ( x )  x , square
root of x f ( x ) 
x , and reciprocal of x f ( x ) 
1
].
x
2A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
a
student is expected to extend parent functions with parameters such as a in f ( x ) 
and describe the effects of the
x
parameter changes on the graph of parent functions.
2A.11B(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
[exponential and] logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms
of the situation. The student is expected to use the parent functions to investigate, describe, and predict the effects
of parameter changes on the graphs of [exponential and] logarithmic functions, describe limitations on the domains
and ranges, and examine asymptotic behavior.
The student will know…
The student will be able to…
 That the parent function is the
 Identify the graph of the logarithmic parent function.
simplest function with the
 Sketch the graph of the logarithmic parent function.
defining characteristics of the
 Use transformations to sketch y  a logb  x  c   d from the parent function.
family.
 Predict changes to the graph when a, c, or d are changed.
 By transforming the graph of a
parent function, you can create  Connect the effects of changing a, c, or d to a problem situation.
 Discover changes to the domain, range, and asymptote
infinitely many new functions.
 Write equations of asymptotes
 Specific changes to a function
equation will result in a specific
change to the graph of that
function.
2A.11C(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
[exponential and] logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms
of the situation. The student is expected to determine the reasonable domain and range values of [exponential and]
logarithmic functions, as well as interpret and determine the reasonableness of solutions to [exponential and]
logarithmic equations and inequalities.
The student will know…
The student will be able to…
 The domain represents the set
 Determine domain and range of a logarithmic function when given a graph, a
of all input values in a function.
table, a situation that can be modeled with a logarithmic function.
 The range represents the set of
 Example: for x ≥ 0, write as:
 Informal set notation {x ≥ 0}
all output values in a function.
 Interval notation [0 , ∞)
 The domain and range of a
 Set-builder notation, {x | x ≥ 0}
function representing a
 Interpret solution of a logarithmic equation/inequality in context of the situation.
situation may vary from the
domain and range of the
 Determine if the solution to a logarithmic equation/inequality is reasonable in
situation.
context of the situation.
 The domain and range of a
 Compare the domain and range of a logarithmic function and the domain and
function can be represented in
range of a situation that can be modeled by the same functions.
a variety of ways.
20
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Exponential and Logarithmic Functions, continued
2A.11D(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of
the situation. The student is expected to determine solutions of exponential and logarithmic equations using graphs,
tables, and algebraic methods.
2A.11E(S) Exponential and logarithmic functions. The student formulates equations and inequalities based on
exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of
the situation. The student is expected to determine solutions of exponential and logarithmic inequalities using graphs
and tables.
2A.11F(R) Exponential and logarithmic functions. The student formulates equations and inequalities based on
[exponential] and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms
of the situation. The student is expected to analyze a situation modeled by an exponential function, formulate an
equation or inequality, and solve the problem.
The student will know…
The student will be able to…
 Exponential equations can be
 Write exponential equations in logarithmic form and vice versa.
written in logarithmic form.
 Include f(x) notation.
 Logarithmic equations can be
 Solve exponential equations using logarithms.
written in exponential form.
 Use the change of base formula for logarithms.
 The connection between
 Simplify logarithmic expressions using logarithmic properties.
exponents and logarithms.
 Find common and natural logarithms and antilogarithms on the graphing
 Logarithmic inequalities can be
calculator.
solved in a variety of ways.
 Solve logarithmic equations.

 With like bases (by rewriting them as exponential equations and finding a
common base)
 With calculator (graph or table)
 From a given situation
 Solve logarithmic inequalities (integer base) from a problem-solving situation
and a purely mathematical situation.
 From a graph (calculator)
 From a table (calculator)
 Graph solutions to a logarithmic inequality on a number line (optional).
 Use the graphing calculator to represent the solution to a logarithmic
inequality.
 Write and solve a logarithmic equation from a given situation.
 Determine the independent and dependent variables of the situation
 Select an appropriate method for solving the equation (algebraically,
graphically, tabular)
 Solve the equation and relate the solution to the situation
 Write and solve a logarithmic inequality from a given situation.
 Select an appropriate method for solving the inequality (graphical or
tabular)
 Solve the inequality and relate the solution to the situation
21
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Exponential and Logarithmic Functions: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
22
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Rational Functions (26 days)
Enduring Understandings
The student understands the relationship between the numerator and denominator of a rational function
determines the function’s behavior.
The student understands that rational functions have unique properties and attributes, including domain
and range.
The student understands that rational functions can be used to model real-world situations.
The student understands that rational equations and inequalities can be solved using a variety of
methods.
Vocabulary
rational, proportion, constant of proportionality, inverse variation, direct variation, joint
variation (K optional), combined variation (K optional), asymptotes (horizontal/
vertical), discontinuity, removable discontinuity, converge, infinity, end behavior, oblique
asymptote (K optional),
2A.10G(S) Rational functions. The student formulates equations and inequalities based on rational functions, uses
a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to
use a function to model and make predictions in problem situations involving direct and inverse variation.
The student will know…
The student will be able to…
 The characteristics of direct
y
 List attributes of proportional relationships, including constant ratio of ,
variation (proportional
x
relationships) given in a table,
graph passing through the origin, and equation of the form y = kx.
graph, or algebraic
 Introduce rational functions with inverse variation scenario.
representation.
 Make predictions in problem situations involving direct variation and inverse
 The differences between
variation.
proportional and non Make predictions in problem situations involving joint and combined
proportional situations.
variation (optional for L and K).
2A.1B(R) Foundations for functions. The student uses properties and attributes of functions and applied functions
to problem situations. The student is expected to collect and organize data, make and interpret scatterplots, fit the
graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical
judgments.
2A.10B(S) Rational functions. The student formulates equations and inequalities based on rational functions,
uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is
expected to analyze various representations of rational functions with respect to problem situations.
2A.10F(R) Rational functions. The student formulates equations and inequalities based on rational functions,
uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is
expected to analyze a situation modeled by a rational function, formulate an equation or inequality composed of a
linear or quadratic function and solve the problem.
The student will know…
The student will be able to…
 Data in specific problem
 Represent a real-world situation using a table, appropriate symbolic
situations can be modeled by
representation, and a graph.
a rational function.
 Use transformations to fit a rational function to model a data set.
 Rational equations can be
 Formulate and solve rational equations using tables or graphs in a problemsolved in a variety of ways.
solving context.
23
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Rational Functions, continued
2A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The
student is expected to identify and sketch graphs of parent functions, including [linear f ( x )  x , quadratic f ( x )  x 2 ,
exponential f ( x )  a x , and logarithmic functions f ( x )  loga x , absolute value of x f ( x )  x , square root of x
1
.
x
2A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions.
a
The student is expected to extend parent functions with parameters such as a in f ( x ) 
and describe the effects
x
of the parameter changes on the graph of parent functions.
2A.10A(S) Rational functions. The student formulates equations and inequalities based on rational functions, uses
a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to
use quotients of polynomials to describe the graphs of rational functions, predict the effects of parameter changes,
describe limitations on the domains and ranges, and examine asymptotic behavior.
The student will know…
The student will be able to…
 The parent function is the
 Predict and describe the effect of parameter changes in the graph of a
simplest function with the
a
rational function given in the form y 
 d , use graphing calculators to
defining characteristics of the
x c
family.
check.
 By transforming the graph of a  Using tables and graphs, describe end behavior and behavior near
parent function, you can
asymptotes.
create infinitely many new
a
a
functions.
 Use tables and graphs to compare the functions y  and y  2 .
x
x
 Specific changes to a function

Write
the
equation
of
a
rational
function
from
the
graph.
equation will result in a
specific change to the graph of  Use factoring to determine discontinuities (vertical asymptote(s), holes),
 Use tables and graphs to identify discontinuities and asymptotes of functions
that function.
f (x) 
x , and] reciprocal of x f ( x ) 
in the form of
y
p( x )
q( x )
.
 Write equations of horizontal and vertical asymptotes.
 Sketch the graphs of rational functions using transformations and long
division (synthetic division, K optional).
 Using tables and graphs, describe the domain and range of rational functions
and how they change given certain parameter changes.
 Graph (only – no domain and range) rational functions with oblique
asymptotes. (K optional).
 Graph and write equations of oblique asymptotes (K optional) .
2A.10C(S) Rational functions. The student formulates equations and inequalities based on rational functions, uses
a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to
determine the reasonable domain and range values of rational functions, as well as interpret and determine the
reasonableness of solutions to rational equations and inequalities.
The student will know…
The student will be able to…
 The domain represents the set  Describe the physical meaning of the asymptote in a problem-solving context.
of all input values in a function.  Find and describe locations of removable discontinuities and explain why
 The range represents the set
there is a discontinuity at this location in a problem-solving context.
of all output values in a
 Determine reasonable domain and range values of rational functions.
function.
 Determine if a solution to a rational equation or inequality is reasonable in a
 The domain and range of a
problem-solving context.
function representing a
situation may vary from the
domain and range of the
situation.
 The domain and range of a
function can be represented in
a variety of ways.
24
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Rational Functions, continued
2A.10D(S) Rational functions. The student formulates equations and inequalities based on rational functions, uses
a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to
determine the solutions of rational equations using graphs, tables, and algebraic methods.
2A.10E(S) Rational functions. The student formulates equations and inequalities based on rational functions, uses
a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to
determine solutions of rational inequalities using graphs and tables.
The student will know…
The student will be able to…
 That rational equations can be  Solve rational equations algebraically and with graphing calculator (graph and
solved in a variety of ways.
table), including equations with more than two terms
 That rational inequalities can
2
5
24 

 Ex. x  1  x  1  x 2  1  . - New to L level.
be solved in a variety of ways.


 Simplify rational expressions involving addition, subtraction, multiplication,
and division – new to L level.
 Decompose a rational expression into partial fractions (K).
 Solve rational inequalities with graphing calculator (graph and table).
2


 4  Looking at the part of the graph from y=4 and
Add example:  Ex.

x
1



upward, the solution is  1, 

1
.
2 
Rational Functions: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
25
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Conics (6 days)
Enduring Understandings
The student understands how a conic section is related to a cone.
The student understands that the geometric shape of a conic is related to its algebraic
representation.
The student understands that equations of conics represent the relationship between two
variables but not all conics are functions.
The student understands that equations of conics can be written in a variety of ways.
Vocabulary
cone, conic section, nappe, focus(foci), vertex(vertices), directrix, axis of symmetry, circle,
ellipse, major/minor axis, parabola, latus rectum (K level), hyperbola, transverse axis,
conjugate axis, asymptotes, standard form, vertex form
2A.5A(S) Algebra and Geometry. The student knows the relationship between the geometric and algebraic
descriptions of conic sections. The student is expected to describe a conic section as the intersection of a plane
and a cone.
2A.5C(S) Algebra and Geometry. The student knows the relationship between the geometric and algebraic
descriptions of conic sections. The student is expected to identify symmetries from graphs of conic sections.
The student will know…
The student will be able to…
 A cone is formed by rotating  Describe each conic section as the set of points resulting from a double
a line about a vertical axis at
napped cone being intersected by a plane at different angles to the edge of
a point called the vertex so
the cone.
that two nappes are formed.  Define and model a parabola geometrically as the set of points equidistant
 Conic sections are the result
from a point, the focus, and a line, the directrix.
of a plane intersecting a
 Define and model a circle geometrically as the set of all points that are
cone.
equidistant (the radius) from a given point (the center of the circle).
 Define and model an ellipse geometrically as the set of points such that the
sum of their distances from each of two fixed points (foci) is a constant.
 Define and model a hyperbola geometrically as the set of points such that the
absolute value of the difference of their distances from each of two fixed
points (foci) is a constant.
2A.5B(S) Algebra and Geometry. The student knows the relationship between the geometric and algebraic
descriptions of conic sections. The student is expected to sketch graphs of conic sections to relate simple
parameter changes in the equation to corresponding changes in the graph.
2A.5D(S) Algebra and Geometry. The student knows the relationship between the geometric and algebraic
descriptions of conic sections. The student is expected to identify conic sections from a given equation.
2A.5E(S) Algebra and Geometry. The student knows the relationship between the geometric and algebraic
descriptions of conic sections. The student is expected to use the method of completing the square.
The student will know…
The student will be able to…
 Each conic has attributes
 Graph each conic section given the equation in graphing form (from a
which are described by its
transformations perspective).
relation equation and related
 Circles: Identify center and radius
measures.
 Ellipses: Identify center, foci, lengths of major/minor axes, lines of
 The characteristics of each
symmetry, pairs of symmetric points
conic.
 Hyperbolas: Identify center, lengths of transverse and conjugate axes,
 The process of completing the
direction, foci, vertices, slopes/equations of asymptotes
square for a quadratic
 Parabolas: Identify vertex, focus, directrix, axis of symmetry, pairs of
polynomial.
symmetric points, and latus rectum (K)
 Correctly identify a conic section from a given equation.
 From the standard form of the equation
 From the graphing form of the equation
 Complete the square to convert a conic equation from standard (polynomial)
form to graphing form.
26
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Conics: Looking forward to 2015-16…
What’s new?
What’s going away?
All of conics is moving to Precalculus.
How will this affect my teaching this year?
27
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Cube and Cube Root Functions (? days)
Enduring Understandings
The student understands cube and cube root functions have unique properties and attributes, including domain and
range.
The student understands that cube and cube root functions can be used to model real-world situations.
The student understands that cube and cube root equations and inequalities can be solved using a variety of
methods.
Vocabulary
cube, cube root, inverse operation
2A.6A Cubic, Cube Root, Absolute Value and Rational Functions, Equations, and Inequalities. The student applies
mathematical processes to understand that cubic, cube root, [rational, and absolute value functions] and
inequalities can be used to model situations, solve problems, and make predictions. The student is expected to
analyze the effect on the graphs of f  x   x3 and f  x   3 x when f (x) is replaced by a·f (x), f(bx), f(x - c), and f
(x) + d for specific positive and negative values of a, b, c, and d.
2A.6B Cubic, Cube Root, Absolute Value and Rational Functions, Equations, and Inequalities. The student applies
mathematical processes to understand that cubic, cube root, [rational, and absolute value functions] and
inequalities can be used to model situations, solve problems, and make predictions. The student is expected to solve
cube root equations.
The student will know…
The student will be able to…
 Parameter changes for cube and cube
 Graph the parent functions f  x   x3 and f  x   3 x .
root functions follow the same rules as for  Make connections between the graph, table, and equations for
other functions.
cube and cube root functions.
 The cube root of a negative number is a
 Graph and describe parameter changes of the cube and cube
real number.
root functions.
 The cube and cube root of a number are
 Solve equations involving cube roots.
inverse operations.
Cube and Cube Root Functions: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
28
Algebra II Scope and Sequence 2014-15
HOLT Algebra 2
Polynomials (? days)
Enduring Understandings
The student understands that mathematical patterns can be used to simplify expressions and solve equations.
The student understands that mathematical processes and skills used with linear, quadratic, and exponential
expressions can be extended to polynomial expressions of higher degree.
Vocabulary
complex number, imaginary unit, polynomial, degree (of term and of polynomial), leading term, constant,
coefficient, sum, difference, like terms, product, quotient, remainder, factor
2A.7A Number and Algebraic Methods. The student applies mathematical processes to simplify and perform
operations on expressions and to solve equations. The student is expected to add, subtract, and multiply complex
numbers.
2A.7B Number and Algebraic Methods. The student applies mathematical processes to simplify and perform
operations on expressions and to solve equations. The student is expected to add, subtract, and multiply
polynomials.
2A.7C Number and Algebraic Methods. The student applies mathematical processes to simplify and perform
operations on expressions and to solve equations. The student is expected to determine the quotient of a polynomial
of degree three and of degree four when divided by a polynomial of degree one and of degree two.
2A.7D Number and Algebraic Methods. The student applies mathematical processes to simplify and perform
operations on expressions and to solve equations. The student is expected to determine the linear factors of a
polynomial function of degree three and of degree four using algebraic methods such as the Remainder Theorem.
2A.7E Number and Algebraic Methods. The student applies mathematical processes to simplify and perform
operations on expressions and to solve equations. The student is expected to determine linear and quadratic factors
of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two
cubes and factoring by grouping.





The student will know…
The value of i is 1 .
Powers of i repeat after every 4 powers (Ex.
i 3  i 7  i11 ).
A factor of a polynomial will result in a
remainder of 0 when divided into the
polynomial.
The distributive property can be used to
multiply polynomials of any degree.
The patterns involved in the factors of the sum
or difference of two cubes.
The student will be able to…
Interpret powers of i.
Simplify and perform operations on complex numbers.
Add and subtract polynomials by adding/subtracting like terms.
Multiply polynomials using the distributive property.
Divide polynomials using factor/cancel, long division, and synthetic
division (optional) – Note: degree 3 or 4 for the dividend and degree 1
or 2 for divisor.
 Find the factors of a polynomial of degree greater than 2 with the
Remainder Theorem.
 Factor the sum and difference of two cubes using formulas.






 a  b a

 ab  b 
 a 3  b3   a  b  a 2  ab  b 2
 a 3  b3
2
2
Polynomials: Looking forward to 2015-16…
What’s new?
What’s going away?
How will this affect my teaching this year?
29