Standards:
AAPR.A.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.
AAPR.B.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.
o Standards Clarification: In Algebra I, tasks
are limited to quadratic and cubic polynomials, in which linear and quadratic
factors are available. For example, find the zeros of (x – 2)(x^{2} –
9).
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: In Algebra I,
tasks are limited to linear, quadratic, or exponential equations with
integer exponents.
ACED.A.2 Create
equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.^{★}
AREI.B.4 Solve
quadratic equations in one variable.
 Use the method of completing the square
to transform any quadratic equation in x into an equation of the form (x
– p)^{2} = q that has the same solutions. Derive the quadratic
formula from this form.
 Solve quadratic equations by inspection
(e.g., for ), taking square roots, completing the
square, the quadratic formula and factoring, as appropriate to the
initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real
numbers a and b.
o Standards Clarification: Tasks do not
require students to write solutions for quadratic equations that have roots
with nonzero imaginary parts. However, tasks can require that students
recognize cases in which a quadratic equation has no real solutions.
AREI.D.11 Explain
why the 𝑥coordinates 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 (𝑥) and/or (𝑥) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.^{★}
o Standards Clarification: In Algebra I, tasks
that assess conceptual understanding of the indicated concept may involve any
of the function types mentioned in the standard except exponential and
logarithmic functions. Finding the solutions approximately is limited to
cases where f(x) and g(x) are polynomial functions.
ASSE.A.1 Interpret
expressions that represent a quantity in terms of its
context.^{★}
 Interpret parts of an expression, such
as terms, factors, and coefficients.
 Interpret complicated expressions by
viewing one or more of their parts as a single entity. For example,
interpret P(1 + r)^{n} as the product of P and a factor not
depending on P.
ASSE.A.2 Use
the structure of an expression to identify ways to rewrite it. For example,
see x^{4} – y^{4} as (x^{2})^{2} – (y^{2})^{2},
thus recognizing it as a difference of squares that can be factored as (x^{2}
– y^{2})(x^{2} + y^{2}).
o Standards Clarification: In Algebra I, tasks
are limited to numerical expressions and polynomial expressions in one
variable. Examples: Recognize that 53^{2} – 47^{2} is the
difference of squares and see an opportunity to rewrite it in the
easiertoevaluate form (53 – 47)(53 + 47). See an opportunity to rewrite a^{2}
+ 9a + 14 as (a + 7)(a + 2). Can include the sum or difference of cubes (in
one variable), and factoring by grouping.
ASSE.B.3 Choose
and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression.^{★}
 Factor a quadratic expression to reveal
the zeros of the function it defines.
 Complete the square in a quadratic
expression to reveal the maximum or minimum value of the function it
defines.
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.
o Standards Clarification: In Algebra I,
identifying 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) is
limited to linear and quadratic functions. Experimenting with cases and
illustrating an explanation of the effects on the graph using technology is
limited to linear functions, quadratic functions, square root functions, cube
root functions, piecewisedefined functions (including step functions and
absolute value functions), and exponential functions with domains in the
integers. Tasks do not involve recognizing even and odd functions.
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.^{★}
o Standards Clarification: Tasks have a
realworld context. In Algebra I, tasks are limited to linear functions,
quadratic functions, square root functions, cube root functions,
piecewisedefined functions (including step functions and absolute value
functions), and exponential functions with domains in the integers.
FIF.B.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 personhours it takes to assemble 𝑛 engines in a
factory, then the positive integers would be an appropriate domain for the
function.^{★}
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.^{★}
o Standards Clarification: Tasks have a
realworld context. In Algebra I, tasks are limited to linear functions,
quadratic functions, square root functions, cube root functions,
piecewisedefined functions (including step functions and absolute value
functions), and exponential functions with domains in the integers.
FIF.C.7 Graph
functions expressed symbolically and show key features of the graph, by hand
in simple cases and using technology for more complicated cases.^{★}
 Graph linear and quadratic functions
and show intercepts, maxima, and minima.
 Graph square root, cube root, and
piecewisedefined functions, including step functions and absolute value
functions.
 Standards
Clarification: Tasks
have a realworld context. In Algebra I, tasks are limited to linear
functions, quadratic functions, square root functions, cube root
functions, piecewisedefined functions (including step functions,
radical functions, and absolute value functions), and exponential
functions with domains in the integers.
FIF.C.8 Write
a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
 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.
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.
o Standards Clarification: In Algebra I, tasks
are limited to linear functions, quadratic functions, square root functions,
cube root functions, piecewisedefined functions (including step functions
and absolute value functions), and exponential functions with domains in the
integers.
NRN.B.3 Explain
why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and that the product
of a nonzero rational number and an irrational number is irrational.
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:
AAPR.A.1
· I can identify that the sum, difference, or
product of two polynomials will always be a polynomial, which means that
polynomials are closed under the operations of addition, subtraction, and
multiplication
· I can define “closure”
· I can apply arithmetic operations of
addition, subtraction, and multiplication to polynomials
AAPR.B.3
· I can factor polynomials using any method.
· I can sketch graphs of polynomials using
zeroes and a sign chart.
ACED.A.1
· I can solve linear and exponential equations
in one variable
· I can solve inequalities 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 (linear and
exponential) and inequalities in one variable and use them to solve problems
· I can create equations and inequalities in
one variable to model realworld situations
· I can compare and contrast problems that can
be solved by different types of equations (linear and exponential)
ACED.A.2
· I can identify the quantities in a
mathematical problem or realworld situation that should be represented by
distinct variables and describe what quantities the variables represent
· I can create at least two equations in two
or more variables to represent relationships between quantities
· I can justify which quantities in a
mathematical problem or realworld situation are dependent and independent of
one another and which operations represent those relationships
· I can determine appropriate units for the
labels and scale of a graph depicting the relationship between equations
created in two or more variables
· I can graph one or more created equation on
a coordinate axes with appropriate labels and scales
AREI.B.4
· I can use the method of completing the
square to transform any quadratic equation in x into an equation of the form
(xp)^{2} = q that has the same solutions
· I can solve quadratic equations in one
variable
· I can derive the quadratic formula by
completing the square on a quadratic equation in x
· I can solve quadratic equations by
inspection (e.g., for x^{2} = 49), taking square roots, completing
the square, the quadratic formula and factoring
· I can determine appropriate strategies (see
first knowledge target listed) to solve problems involving quadratic
equations, as appropriate to the initial form of the equation
· I can recognize when the quadratic formula
gives complex solutions
AREI.D.11
·
I
can recognize and use function notation to represent linear and exponential
equations
·
I
can recognize that if (x_{1} , y_{1}) and (x_{2} , y_{2})
share the same location in the coordinate plane that x_{1} = x_{2}
and y_{1} = y_{2}
·
I
can recognize that f(x) = g(x) means that there may be particular inputs of f
and g for which the outputs of f and g are equal
·
I
can explain why the xcoordinates of the points where the graph of the
equations y=f(x) and y=g(x) intersect are the solutions of the equations
f(x) = g(x). (Include cases where f(x)
and/or g(x) are linear and exponential equations)
·
I
can approximate/find the solution(s) using an appropriate method for example,
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
and exponential equations)
ASSE.A.1
· I can, for expressions that represent a
contextual quantity, define and recognize parts of an expression, such as
terms, factors, and coefficients
· I can, for expressions that represent a
contextual quantity, interpret parts of an expression, such as terms,
factors, and coefficients in terms of the context
· I can, for expressions that represent a
contextual quantity, interpret complicated expressions, in terms of the
context, by viewing one or more of their parts as a single entity.
ASSE.A.2
· I can identify ways to rewrite expressions,
such as difference of squares, factoring out a common monomial, regrouping,
etc
· I can identify ways to rewrite expressions
based on the structure of the expression
· I can use the structure of an expression to
identify ways to rewrite it.
· I can classify expression by structure and
develop strategies to assist in classification
ASSE.B.3
· I can factor a quadratic expression to
produce an equivalent form of the original expression
· I can explain the connection between the
factored form of a quadratic expression and the zeros of the function it
defines
· I can explain the properties of the quantity
represented by the quadratic expression
· I can choose and produce an equivalent form
of a quadratic expression to reveal and explain properties of the quantity
represented by the original expression
· I can complete the square on a quadratic
expression to produce an equivalent form of an expression
· I can explain the connection between the
completed square form of a quadratic expression and the maximum or minimum
value of the function it define
· I can explain the properties of the quantity
represented by the expression
· I can choose and produce an equivalent form
of a quadratic expression to reveal and explain properties of the quantity
represented by the original expression
FBF.B.3
·
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
FIF.B.4
·
I
can define and recognize the key features in tables and graphs of linear and
exponential functions: intercepts; intervals where the function is
increasing, decreasing, positive, or negative, and end behavior
·
I
can identify whether the function is linear or exponential, given its table
or graph
·
I
can interpret key features of graphs and tables of function in the terms of
the contextual quantities the function represents
·
I
can sketch graphs showing key features of a function that models a
relationship between two quantities from a given verbal description of the
relationship
FIF.B.5
·
I
can, given the graph or a verbal/written description of a function, identify
and describe the domain of the function
·
I
can identify an appropriate domain based on the unit, quantity , and type of
the function it describes
·
I
can relate the domain of the function to its graph and, where applicable, to
the quantitative relationship it describes
·
I
can explain why a domain is appropriate for a given situation
FIF.B.6
·
I
can recognize slope as an average rate of change
·
I
can calculate the average rate of change of a function (presented
symbolically or as a table) over a specified interval
·
I
can estimate the rate of change from a linear or exponential graph
·
I
can interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval
FIF.C.7
· I can graph linear functions by hand in
simple cases or using technology for more complicated cases and show/label
intercepts of the graph
· I can graph square root, cube root, and
piecewisedefined functions, including step functions and absolute value
functions, by hand in simple cases or using technology for more complicated
cases, and show/label key features of the graph
· I can determine the difference between
simple and complicated linear, quadratic, square root, cube root, and
piecewisedefined functions, including step functions and absolute value
functions and know when the use of technology is appropriate
· I can compare and contrast the domain and
range of absolute vale, step and piecewise defined functions with linear,
quadratic, and exponential
FIF.C.8a
· I can identify different forms of a
quadratic expression
· I can write functions in equivalent forms
using the process of factoring
· I can identify zeros, extreme values, and
symmetry of the graph of a quadratic function
· I can interpret different but equivalent
forms of a function defined by an expression in terms of context
· I can use the process of factoring and
completing the square in a quadratic function to show zeros, extreme values,
and symmetry of the graph, and intercept these in terms of a context
FIF.C.9
·
I
can identify types of functions based on verbal, numerical, algebraic, and
graphical descriptions and state key properties (e.g. intercepts, growth
rates, average rates of change, and end behaviors)
·
I
can differentiate between exponential and linear functions using a variety of
descriptors (graphically, verbally, numerically, and algebraically)
·
I
can use a variety of function representations algebraically, graphically,
numerically in tables, or by verbal descriptions) to compare and contrast
properties of two functions
NRN.B.3
· I can find the sums and products of rational
and irrational numbers
· I can recognize that the sum of a rational
number and an irrational number is irrational
· I can recognize that the product of a
nonzero rational number and irrational number is irrational
· I can explain why rational numbers are
closed under addition or multiplication
