Standards:
NRN.A.1 Explain how the definition of the meaning
of rational exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for radicals in terms of
rational exponents. For example,
we define 5^{1/3 }to be the cube root of 5 because we want (5^{1/3})^{3}
= 5^{(1/3)3} to hold, so (5^{1/3})^{3} must equal 5.
NRN.A.2
Rewrite expressions involving radicals and
rational exponents using the properties of exponents.
 Standards Clarification:
Including expressions where either base or exponent may contain
variables.
NQ.A.2
Define appropriate quantities for the
purpose of descriptive modeling.
 Standards Clarification:
This standard will be assessed in Algebra II by ensuring that
some modeling tasks (involving Algebra II content or securely held
content from previous grades and courses) require the student to create
a quantity of interest in the situation being described (i.e., this is
not provided in the task). For example, in a situation involving periodic
phenomena, the student might autonomously decide that amplitude is a key
variable in a situation, and then choose to work with peak
amplitude. .
ASSE.B.3c
Choose and produce an equivalent form of an
expression to reveal and explain properties of the quantity represented by
the expression.★
c.
Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15^{t} can be rewritten as
(1.15^{1/12})^{12t }≈ 1.012^{12t} to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.
 Standards Clarification:
Tasks have a realworld context. As described in the standard,
there is an interplay between the mathematical structure of the
expression and the structure of the situation such that choosing and
producing an equivalent form of the expression reveals something about
the situation. In Algebra II, tasks include exponential expressions with
rational or real exponents.
ASSE.B.4
Derive the formula for the sum of a finite
geometric series (when the common ratio is not 1), and use the formula to
solve problems. For example, calculate mortgage payments.★
 Standards Clarification:
This standard includes using the summation notation symbol.
ACED.A.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 exponential
functions.
 Standards Clarification:
Tasks have a realworld context. In Algebra II, tasks include
exponential equations with rational or real exponents, rational
functions, and absolute value functions, but are constrained to simple cases.
AREI.D.11
Explain why the x‐coordinates of the points where the graphs of the equations y =
f(x) and y = g(x) intersect are the
solutions of the equation f(x) = g(x); find the
solutions approximately, e.g., using technology to graph the functions, make
tables of values, or find successive approximations. Include cases where f(x)
and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.★
 Standards Clarification:
In Algebra II, tasks may involve any of the function types
mentioned in the standard.
FIF.A.3
Recognize that sequences are functions,
sometimes defined recursively, whose domain is a subset of the integers. For
example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1,
f(n+1) = f(n) + f(n‐1) for n ≥ 1.
 Standards Clarification:
This standard is Supporting Content in Algebra II. This standard
should support the Major Content in FBF.2 for coherence.
FIF.B.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.★
 Standards Clarification:
Tasks have a realworld context. In Algebra II, tasks may involve
polynomial, exponential, logarithmic, and trigonometric functions. Emphasize the selection of a model
function based on behavior of data and context.
FIF.B.6
Calculate and interpret the average rate of
change of a function (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph.★
 Standards Clarification:
Tasks have a realworld context. In Algebra II, tasks may involve
polynomial, exponential, logarithmic, and trigonometric functions.
Emphasize the selection of a model function based on
behavior of data and context.
FIF.C.7e
Graph functions expressed symbolically and
show key features of the graph, by hand in simple cases and using technology
for more complicated cases.★
e. Graph exponential and logarithmic
functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
 Standards Clarification:
Focus on application and
using key features to guide the selection of the appropriate type of
model function.
FIF.C.8b
Write a function defined by an expression in
different but equivalent forms to reveal and explain different properties of
the function.
b.
Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent rate of change in functions such
as y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t},
y = (1.2)^{t/10}, and classify them as representing exponential
growth or decay.
 Standards Clarification:
Tasks include knowing and applying A = Pe^{rt} and A
= P (1 + r/n)^{nt}^{ }.
FIF.C.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.
 Standards Clarification:
In Algebra II, tasks may involve polynomial, exponential,
logarithmic, and trigonometric functions.
FBF.A.1a
Write a function that describes a relationship
between two quantities.★
a.
Determine an explicit expression, a recursive process, or steps for
calculation from a context.
 Standards Clarification:
Tasks have a realworld context. In Algebra II, tasks may involve
polynomial, exponential, logarithmic, and trigonometric functions.
FBF.A.1b
Write a function that describes a
relationship between two quantities.★
b.
Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
 Standards Clarification:
Combining functions also includes composition of functions.
Include all types of functions studied. Develop models for more complex or
sophisticated situations than in previous courses.
FBF.A.2
Write arithmetic and geometric sequences
both recursively and with an explicit formula, use them to model situations,
and translate between the two forms.★
FBF.B.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.
 Standards Clarification:
In Algebra II, tasks may involve simple radical, rational,
polynomial, exponential, logarithmic, and trigonometric functions. Tasks
may involve recognizing even and odd functions. Note the effect of
multiple transformations on a single graph and emphasize the common
effect of each transformation across function types.
FBFB.4a
Find inverse functions.
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) =2x^{3} or f(x) = (x+1)/(x‐1) for x ≠ 1.
 Standards Clarification:
Extend to simple
rational, simple radical, and simple exponential functions.
FLE.A.2
Construct linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a description of
a relationship, or two input‐output pairs (include reading these from a
table).
 Standards Clarification:
In Algebra II, tasks will include solving multistep problems by
constructing linear and exponential functions.
FLE.A.4
For exponential models, express as a
logarithm the solution to ab^{ct} = d where a, c,
and d are numbers and the base b is 2, 10, or e;
evaluate the logarithm using technology.
 Standards Clarification:
Students learn terminology that logarithm without a base
specified is base 10 and that natural logarithm always refers to base e.
Recognize and use logarithms as solutions for exponentials.
FLE.B.5
Interpret the parameters in a linear,
quadratic, or exponential function in terms of a context.
 Standards Clarification:
Tasks have a realworld context. In Algebra II, tasks include
exponential functions with domains not in the integers.
Focus Standards of Mathematical Practice:
MP.1 Make
sense of problems and persevere in solving them.
MP.2 Reason
abstractly and quantitatively.
MP.3
Construct viable arguments and critique the reasoning of others.
MP.4 Model
with mathematics.
MP.5 Use
appropriate tools strategically.
MP.6 Attend
to precision.
MP.7 Look for
and make use of structure.
MP.8 Look for
and express regularity in repeated reasoning.
Instructional Outcomes:
Full
Development of the Major Clusters, Supporting Clusters, Additional Clusters
and Mathematical Practices for this unit could include the following
instructional outcomes:
NRN.A.1
·
I can define radical
notation as a convention used to represent rational exponents.
·
I can explain the
properties of operations of rational exponents as an extension of the
properties of integer exponents.
·
I can explain how
radical notation, rational exponents, and properties of integer exponents
relate to one another.
NRN.A.2
·
I can, using the
properties of exponents, rewrite a radical expression as an expression with a
rational exponent.
·
I can, using the
properties of exponents, rewrite an expression with a rational exponent as a
radical expression.
NQ.A.2
·
I can define
descriptive modeling.
·
I can determine
appropriate quantities for the purpose of descriptive modeling.
ASSE.B.3c
·
I can use the
properties of exponents to transform simple expressions for exponential
functions.
·
I can use the
properties of exponents to transform expressions for exponential functions.
·
I can choose and
produce an equivalent form of an exponential expression to reveal and explain
properties of the quantity represented by the original expression.
·
I can explain the
properties of the quantity or quantities represented by the transformed exponential
expression.
ASSE.B.4
·
I can find the first
term in a geometric sequence given at least two other terms.
·
I can define a
geometric series as a series with a constant ratio between successive terms.
·
I can use the formula
S = ((a (1r^{n}))/ (1r)) to solve problems.
ACED.A.1
·
I can solve equations
specified in this standard in one variable.
·
I can solve
inequalities specified in this standard in one variable.
·
I can describe the
relationships between the quantities in the problem (for example, how the
quantities are changing or growing with respect to each other); express these
relationships using mathematical operations to create an appropriate equation
or inequality to solve.
·
I can create
equations and inequalities specified in this standard in one variable and use
them to solve problems.
·
I can create
equations and inequalities specified in this standard in one variable to
model realworld situations.
·
I can compare and
contrast problems that can be solved by the different types of equations specified
in this standard.
AREI.D.11
·
I can approximate or
find the solutions to a system involving any of the function types listed in
this standard.
·
I can explain why the
solution to a system involving any of the function types listed in this
standard will occur at the point(s) of intersection.
FIF.A.3
·
I can recognize that
sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers. For example, the Fibonacci sequence is defined by
f(0)=f(1)=1, f(n+1) = f(n) + f(n1) for n≥1.
FIF.B.4
·
I can identify the
key features from a table or graph of polynomial, exponential, logarithmic,
and trigonometric functions.
·
I can sketch
functions that model key feature behavior.
FIF.B.6
·
I can calculate the
average rate of change of a given interval for polynomial, exponential,
logarithmic, and trigonometric functions.
·
I can demonstrate
that the average rate of change of a nonlinear function is different for
different intervals.
FIF.C.7e
·
I can graph
polynomial, exponential, logarithmic, and trigonometric functions.
·
I can describe key
features of polynomial, exponential, logarithmic, and trigonometric
functions.
FIF.C.8b
·
I can identify how
key features of an exponential function relate to characteristics in a
realworld context.
·
I can write and
classify realworld problems as an exponential growth or decay.
·
I can use and
applying A = Pe^{rt} and A = P (1 + r/n)^{nt}^{ }.
FIF.C.9
·
I can compare key
features of two representations of functions (polynomial, exponential,
logarithmic, and trigonometric).
FBF.A.1a
·
I can define
“explicit function” and “recursive process”.
·
I can write a
function that describes a relationship between two quantities by determining
an explicit expression, a recursive process, or steps for calculation from a
context (involving polynomial, exponential, logarithmic, and trigonometric
functions).
FBF.A.1b
·
I can combine two
functions (all functions studied) using the operations of addition,
subtraction, multiplication, and division.
·
I can evaluate the
domain of the combined function (all functions studied).
·
I can build standard
functions (all functions studied) to represent relevant
relationships/quantities given a realworld situation or mathematical
process.
·
I can determine which
arithmetic operation should be performed to build the appropriate combined
function (all functions studied) given a realworld situation or mathematical
process.
·
I can relate the
combined function (all functions studied) to the context of the problem.
FBF.A.2
·
I can identify
arithmetic and geometric patterns in given sequences.
·
I can generate
arithmetic and geometric sequences from recursive and explicit formulas.
·
I can, given an
arithmetic or geometric sequence in recursive form, translate into the
explicit formula.
·
I can, given an
arithmetic or geometric sequence as an explicit formula, translate into the
recursive form.
·
I can use given and
constructed arithmetic and geometric sequences, expresses both recursively
and with explicit formulas, to model reallife situations.
·
I can determine the
recursive rule given arithmetic and geometric sequences.
·
I can determine the
explicit formula given arithmetic and geometric sequences.
·
I can justify the
translation between the recursive form and explicit formula for arithmetic and
geometric sequences.
FBF.B.3
·
I can perform the
tasks below which may involve simple radical, rational, polynomial,
exponential, logarithmic, and trigonometric functions.
·
I can identify the
effect a single transformation will have on the function (symbolic or
graphic).
·
I can use technology
to identify effects of single transformations on graphs of functions.
·
I can graph a given
function by replacing f(x) with f(x)+k, kf(x), f(kx), or f(x+k) for specific
values of k (both positive and negative).
·
I can describe the
differences and similarities between a parent function and the transformed
function.
·
I can find the value
of k, given the graphs of a parent function, f(x), and the transformed
function: f(x)+k, kf(x), f(kx), or f(x+k).
·
I can recognize even
and odd functions from their graphs and from their equations.
·
I can experiment with
cases and illustrate an explanation of the effects on the graph using
technology.
·
I can identify
transformations of a function on a graph.
·
I can describe the
effects of transformations on parent functions.
FBFB.4a
·
I can define inverse
function (this extends to simple
rational, simple radical, and simple exponential functions).
·
I can 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 (this extends to simple rational, simple radical, and simple exponential functions).
FLE.A.2
·
I can recognize
arithmetic sequences can be expressed as linear functions.
·
I can recognize
geometric sequences can be expressed as exponential functions.
·
I can construct
linear functions, including arithmetic sequences, given a graph, a
description of a relationship, or two inputoutput pairs (include reading
these from a table).
·
I can construct
exponential functions, including geometric sequences, given a graph, a
description of relationship, or two inputoutput pairs (include reading these
from a table).
·
I can determine when
a graph, a description of a relationship, or two inputoutput pairs (include
reading these from a table) represents a linear or exponential function in
order to solve problems.
FLE.A.4
·
I can use the
properties of logs.
·
I can describe the
key features of logs.
·
I can use logarithmic
form to solve exponential models.
·
I can recognize that
logarithm without a base specified is base 10 and that natural logarithm
always refers to base e.
·
I can recognize and use logarithms as solutions
for exponentials.
FLE.B.5
·
I can recognize the
parameters in a linear or exponential function including: vertical and
horizontal shifts, vertical and horizontal dilations.
·
I can recognize rate
of change and intercept as “parameters” in linear or exponential functions.
·
I can interpret the
parameters in a linear or exponential function in terms of a context.
·
I can interpret the
parameters of exponential functions with domains not in the integers.
