Download Algebra II Sample Scope and Sequence

Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
In Algebra I, students have already begun their study of algebraic concepts. They have used equations, tables, and graphs to describe
relationships between quantities, with a particular focus on linear, quadratic, and exponential functions and equations.
The Algebra II course outlined in this scope and sequence document begins with connections back to that earlier work, efficiently reviewing
algebraic and statistical concepts that students have already studied while at the same time moving students forward into new concepts.
Students expand their library of functions to include polynomial functions, logarithmic functions, rational functions, and trigonometric
functions. With a larger library of functions, students increase their ability to model situations, make predictions and answer questions about
the situation.
This scope and sequence assumes 160 days for instruction, divided among 13 units. The units are sequenced in a way that we believe best
develops and connects the mathematical content described in the Common Core State Standards for Mathematics; however, the order of
the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times
during the course; others may be only partially addressed in different units, depending on the mathematical focus of the unit.
Throughout Algebra II, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
These practices should become the natural way in which students come to understand and to do mathematics. While, depending on the
content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful
than others. Opportunities for highlighting certain practices are indicated in different units of study in this sample scope and sequence, but
this highlighting should not be interpreted to mean that other practices should be neglected in those units.
This scope and sequence reflects our current thinking related to the intent of the CCSS for Mathematics, but it is an evolving document. We
expect to make refinements to this scope and sequence in the coming months in response to new learnings about the standards. In
planning your district's instructional program, you should be prepared to have similar flexibility in implementing your district's own scope
and sequence for the next 2 to 3 years, as you transition from your state's current standards to full implementation of the CCSS for
Mathematics.
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
1
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for
Mathematical
Practice
Days Comments
Quadratic
functions and
the complex
number system
F--‐IF.8.a (Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.)
N--‐CN.1 (Know there is a complex number i such that i2 = –1, and every complex number has the
form a + bi with a and b real.)
N--‐CN.2 (Use the relation i2 = –1 and the commutative, associative, and distributive properties
to add, subtract, and multiply complex numbers.)
N--‐CN.7 (Solve quadratic equations with real coefficients that have complex solutions.)
N--‐CN.8 ((+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4
as (x + 2i)(x – 2i).)
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
7
Builds on students' learning in Algebra I around
quadratic functions and equations and the
existence of the complex number system. They
understand the complex number system and
perform operations on complex numbers. They
solve quadratic equations with complex
solutions.
Building new
functions
A--‐APR.1 (Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.)
A--‐SSE.1.a (Interpret parts of an expression, such as terms, factors, and
coefficients.★)
A--‐SSE.1.b (Interpret complicated expressions by viewing one or more of their parts as a single
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.★)
reasoning of others.
F--‐BF.1.b (Combine standard function types using arithmetic operations. For example, build a
4. Model with mathematics.
function that models the temperature of a cooling body by adding a constant function to a
5. Use appropriate tools
★
decaying exponential, and relate these functions to the model. )
strategically.
F--‐IF.6 (Calculate and interpret the average rate of change of a function (presented symbolically
6. Attend to precision.
★
or as a table) over a specified interval. Estimate the rate of change from a graph. )
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
8
This topic links the basic functions to which
students were introduced in Algebra I to the new
functions they will master in Algebra II. This topic
builds on previous work with linear and quadratic
functions to help students make sense of the
behavior they see in the larger family of
polynomial functions.The topic leads students to
understand that polynomials form a system
analogous to the integers, namely, they are
closed under the operations of addition,
subtraction, and multiplication.
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
2
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for
Mathematical
Practice
Days Comments
Characteristics
of polynomial
functions
N--‐CN.9 ((+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.)
A--‐APR.2 (Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).)
A--‐APR.3 (Identify zeros of polynomials when suitable factorizations are available, and use the
zeros to construct a rough graph of the function defined by the polynomial.)
F--‐IF.4 (For a function that models a relationship between two quantities, interpret key features
of graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.★)
F--‐IF.5 (Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives the number of person--‐hours it
takes to assemble n engines in a factory, then the positive integers would be an appropriate
domain for the function.★)
F--‐IF.7.c (Graph polynomial functions, identifying zeros when suitable factorizations are
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
available, and showing end behavior.★)
F--‐IF.8.a
F--‐IF.9 (Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger
maximum.)
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
Students learn how polynomials model some
behaviors with varying rates of change, and they
see how the degree of the polynomial relates to
the number of real zeros and the number of local
extreme values of the polynomial function.
Students then apply this knowledge to choose
appropriate models for situations based on how
quantities in the situation vary, with particular
emphasis on short term and end behavior.
Students will also interpret parts of an
expression, such as terms, factors, and
coefficients as well as interpret complicated
expressions by viewing one or more of their
parts as a single entity.
3
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Applying
polynomials
A--‐SSE.2 (Use the structure of an expression to identify ways to rewrite it. For example, see x4 – 1. Make sense of problems
y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 and persevere in solving
them.
+ y2).)
2. Reason abstractly and
A--‐APR.3
quantitatively.
A--‐APR.4 (Prove polynomial identities and use them to describe numerical relationships. For
3. Construct viable
2
2
2
2 2
2
example, the polynomial identity (x + y )2 = (x – y ) + (2xy) can be used to generate
arguments and critique the
Pythagorean triples.)
reasoning of others.
n
A--‐APR.5 ((+) Know and apply the Binomial Theorem for the expansion of (x + y) in powers of x
4. Model with mathematics.
and y for a positive integer n, where x and y are any numbers, with coefficients determined for
5. Use appropriate tools
1 1
example by Pascal’s Triangle. ) The binomial Theorem can be proved by mathematical induction or by a
strategically.
combinatorial argument.
6. Attend to precision.
A--‐REI.11 (Explain why the x--‐coordinates of the points where the graphs of the equations y = f(x)
7. Look for and make use of
and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions
structure.
approximately, e.g., using technology to graph the functions, make tables of values, or find
8. Look for and express
successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational,
regularity in repeated
★
absolute value, exponential, and logarithmic functions. )
reasoning.
A--‐CED.1 (Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and
Standards for
Mathematical
Practice
Days Comments
10
Students solve equations that arise from
situations that can be modeled using polynomial
functions. Students also explore the structure of
polynomial expressions and investigate and
apply polynomial identities.
exponential functions.★)
A--‐CED.2 (Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.★)
A--‐CED.3 (Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or non--‐ viable options in a modeling
context. For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.★)
F--‐IF.5
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
4
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for
Mathematical
Practice
Days Comments
Modeling with
rational
functions
A--‐APR.6 (Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x)
+ r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the
degree of b(x), using inspection, long division, or, for the more complicated examples, a
computer algebra system.)
A--‐APR.7 ((+) Understand that rational expressions form a system analogous to the rational
numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions.)
A--‐REI.2 (Solve simple rational and radical equations in one variable, and give examples showing
how extraneous solutions may arise.)
A--‐REI.11; F--‐IF.4; F--‐IF.5; A--‐CED.1;A--‐CED.2; A--‐CED.3
A--‐CED.4 (Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.)
F--‐BF.4.a (Solve an equation of the form f(x) = c for a simple function f that has an inverse and
write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.)
F--‐BF.1.b
A--‐SSE.4 (Derive the formula for the sum of a finite geometric series (when the common ratio is
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
Students use rational functions to model and
investigate situations, and solve equations
arising from those situations. Through this work,
they learn about the general characteristics and
behavior of rational functions and create and
understand graphs of rational functions. They
also develop the formula for the sum of a finite
geometric series and connect that formula to the
family of rational functions.
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
This topic builds on students' understanding of
simple polynomial functions and inverses to
develop simple radical functions. Students
investigate the characteristics of these functions
and solve equations that arise from situations
that can be modeled by these functions.
not 1), and use the formula to solve problems. For example, calculate mortgage payments.★)
Modeling with
radical functions
A--‐REI.2; A--‐REI.11; A--‐CED.1; A--‐CED.2; A--‐CED.3; A--‐CED.4; F--‐IF.4; F--‐IF.5; F--‐IF.6
F--‐IF.7.b (Graph square root, cube root, and piecewise--‐defined functions, including step
functions and absolute value functions.★)
F--‐IF.9
F--‐BF.3 (Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.)
F--‐BF.4.a
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
5
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Exponential and
logarithmic
models
F--‐LE.4 (For exponential models, express as a logarithm the solution to abct =d where a, c, and d 1. Make sense of problems
and persevere in solving
are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. ★)
them.
F--‐IF.7.e (Graph exponential and logarithmic functions, showing intercepts and end behavior,
2. Reason abstractly and
★
and trigonometric functions, showing period, midline, and amplitude. )
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
10
This topic builds on students' understanding of
exponential functions and inverses to develop
logarithmic functions. Students investigate the
characteristics of these functions and solve
equations that arise from situations that can be
modeled by these functions.
Symmetry and
transformations
in functions
F--‐BF.3
10
In this topic, students generalize the basic
function transformations. They also explore
symmetry of the graphs of functions and learn to
recognize odd and even functions from their
graphs and symbolic representations. Finally,
they begin to investigate situations that exhibit
periodic behavior and how those situations
motivate the need to extend their understanding
of transformations to function models of the
form f(kx).
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
Standards for
Mathematical
Practice
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
10/28/11
Days Comments
6
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for
Mathematical
Practice
Days Comments
Modeling with
trigonometric
functions
F--‐TF.1 (Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.)
F--‐TF.2 (Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.)
F--‐TF.5 (Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.★)
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
F--‐TF.8 (Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or
tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.)
Choosing a
function model
F--‐IF.4; F--‐IF.5; F--‐IF.6; F--‐IF.9; F--‐BF.1.b; F--‐BF.3; A--‐CED.3
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
In this topic, students define three
trigonometric functions: y = sin α, y = cos α and y
= tan α based on the unit circle.
Students learn to use transformations to model
periodic situations. They also develop the
Pythagorean identity from the unit circle and
derive related identities.
In this topic, students draw upon their
knowledge of different parent functions,
including linear, polynomial, power, exponential,
logarithmic and trigonometric functions, to
choose the appropriate model for a given
situation. Students write and solve equations,
inequalities, and systems of equations and
inequalities to answer questions that arise from
these situations.
7
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for
Mathematical
Practice
Days Comments
The design of
statistical
studies
S--‐IC.1 (Understand statistics as a process for making inferences about population parameters
based on a random sample from that population.)
S--‐IC.3 (Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.)
S--‐IC.6 (Evaluate reports based on data.)
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
Drawing correct conclusions from data is highly
dependent on how the data are collected. In
particular, "cause and effect" conclusions can
only arise from properly conducted experiments,
in which the researcher actively imposes a
treatment. Students learn the purposes of and
differences among, surveys, experiments, and
observational studies and explain how
randomization is used in each case.
Normal
distribution as a
model for data
S.--‐ID.4 (Use the mean and standard deviation of a data set to fit it to a normal distribution and
to estimate population percentages. Recognize that there are data sets for which such a
procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under
the normal curve.)
S--‐IC.2 (Decide if a specified model is consistent with results from a given data--‐generating
process, e.g., using simulation. For example, a model says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row cause you to question the model?)
S--‐IC.4 (Use data from a sample survey to estimate a population mean or proportion; develop a
margin of error through the use of simulation models for random sampling.)
S--‐MD.6 ((+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random
number generator).)
S--‐MD.7 ((+) Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).)
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
10
Students investigate the normal distribution and
estimate percentages based on the normal
curve. Students also learn to evaluate whether
the normal distribution is an appropriate model
for a set of data.
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
8
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for
Mathematical
Practice
Days Comments
Drawing
conclusions
from data
S--‐IC.1; S--‐IC.4
S--‐IC.5 (Use data from a randomized experiment to compare two treatments; use simulations to
decide if differences between parameters are significant.)
S--‐IC.6
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
15
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
In this topic students use simulation as a tool to
estimate population parameters and margins of
error, and to compare two treatments.
9