Download CCRS Standard - the Franklin County Schools Website

ALGI
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
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.
Example: We
define 51/3 to be
the cube root of 5
because we want
(51/3)3 = 5(1/3)3 to
hold, so (51/3)3
must equal 5.
The Real
Number
System
Extend the
properties of
exponents to
rational
exponents.
Number/Quantity
N-RN.1
2. Rewrite
expressions
involving radicals
and rational
exponents using
the properties of
exponents.
The Real
Number
System
Extend the
properties of
exponents to
rational
exponents.
Number/Quantity
N-RN.2
Students:
Given expressions
involving radicals and
rational exponents,
The Real
Number
System
Use properties of
rational and
irrational
numbers.
Number/Quantity
Students:
Given the definition of
rational numbers,
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
Evidence of Student
Attainment
Students:
Teacher
Vocabulary
Exponent
Use the repeated
Root
reasoning from prior
knowledge of properties
of exponents to progress
from, for example,
(23*23*23) = 29 to the
extension what three
equal factors multiplied
together yields 21,
Knowledge
Franklin County Schools
Use
mathematical/logical
reasoning to demonstrate
that various forms of
radicals and roots
actually represent the
same quantity, that is
(23/2)3 = 29/2 = √29 =
16√2.
Show that the sum or
product of two rational
numbers is rational and
Understanding
Students know:
Students are able to: Students understand
that:
Techniques for
applying the
properties of
exponents.
Correctly perform
the manipulations of
exponents which
apply the properties
of exponents,
Students know:
Students are able to: Students understand
that:
Connect this result to
the definition of cube
root and extend their
understanding to other
roots.
Rational
exponent
Skills
Properties of
exponents,
The properties of
exponents are true
regardless of the type Click below to
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of numbers being
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used.
to this standard.
Use mathematical
reasoning and prior
 ALEX
knowledge of the
Resources
meaning of integer
exponents to explain
notation for radicals
with rational
exponents.
Use mathematical
The properties of
reasoning to justify
the equality of various exponents are true
regardless of the type
The meaning of forms of radical
expressions,
of numbers being
algebraic symbols
used.
such as radicals and
rational exponents. Correctly perform
the manipulations of
exponents which
apply properties of
exponents.
Irrational number Students know:
Resources
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resources aligned
to this standard.

ALEX
Resources
Students are able to: Students understand
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The distinction
Convert between
resources aligned
between rational and fraction and decimal The decimal
irrational numbers.
form of rational
approximation of an to this standard.
numbers.
irrational will never
produce an exact or
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infinite repeating
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
irrational; and that N-RN.3
the product of a
nonzero rational
number and an
irrational number is
irrational.
4. Use units as a
way to understand
problems and to
guide the solution
of multistep
problems; choose
and interpret units
consistently in
formulas; choose
and interpret the
scale and the origin
in graphs and data
displays. {THIS IS
A TEST}
Quantities★
Reason
quantitatively
and use units to
solve problems.
(Foundation for
Evidence of Student
Attainment
Knowledge
Skills
Understanding
explain their reasoning,
decimal,
Use repeated
reasoning from examples
to demonstrate that the
sum (or product) of a
rational number and an
irrational number
appears to always result
in an infinite nonrepeating decimal, for
example 2 + π = 2 +
3.14159... = 5.14159...
Making the same
change to every digit
in an infinite nonrepeating decimal will
not make the decimal
repeat,
Students:
Interpret and make
sense of problems
through analyzing units
for agreement using
work with
dimensional analysis
expressions,
(e.g., knowing that there
equations, and
are 5,280 feet in a mile
functions.)
we might change 20
Number/Quantity miles/hour to inches per
N-Q.1
second by 20 miles/hour
x 5280 feet/mile x 12
inches/foot x 1 hour/60
minutes x 1 minute/60
seconds to yield just over
352 inches per second),
Model contextual
problem situations with
appropriately chosen
units or derived units,
analyze the data using
those units, and interpret
Franklin County Schools
Teacher
Vocabulary
Resources
Resources
Reasoning from
repeated examples is
not a proof but can
lend credibility to a
statement when there
is reasoning to why
the regularity
continues.
Students know:
Students are able to: Students understand
that:
Techniques for
Choose the
dimensional analysis, appropriate known
conversions to
perform dimensional
Uses of
analysis to convert
technology in
producing graphs of units,
data,
Criteria for
selecting different
displays for data
(e.g., knowing how
to select the window
on a graphing
calculator to be able
to see the important
parts of the graph).
The relationships
of units to each other
and how using a
chain of conversions
allows one to reach a
desired unit or rate. Click below to
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Correctly use
resources aligned
graphing window and
to this standard.
other technology
features to precisely
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determine features of
Resources
interest in a graph.
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
the solution (e.g.,
problems involving per
capita income, person
hours, heat degree days,
or currency conversions),
Interpret and
evaluate, with and
without appropriate
technology, graphical and
tabular data displays for
consistency with the data
and precisely determine
and interpret a scale and
origin that is useful in
examining the problem of
interest.
5. Define
appropriate
quantities for the
purpose of
descriptive
modeling.
6. Choose a level of
accuracy
appropriate to
limitations on
measurement when
Quantities★
Reason
quantitatively
and use units to
solve problems.
(Foundation for
Students know:
Students are able to: Students understand
that:
Choose appropriate
Descriptive
quantities for
modeling
descriptively modeling
important features of the
work with
phenomenon being
expressions,
investigated (e.g., find a
equations, and
good measure of overall
functions.)
highway safety: propose
Number/Quantity and debate such
N-Q.2
measures as fatalities per
year, fatalities per driver
per year, or fatalities per
vehicle mile driven).
descriptive
models
Determine when
a descriptive model
accurately portrays
the phenomenon it
was chosen to model,
Quantities★
Reason
quantitatively
and use units to
solve problems.
Students know:
Franklin County Schools
Students:
Students:
Given contextual
situations involving
measurements,
Quantities
Accuracy
Justify their
selection of model
and choice of
quantities in the
context of the
situation modeled and
critique the
arguments of others
concerning the same
situation.
Attributes of
measurements
including precision
Different models
reveal different
Click below to
features of the
phenomenon that is access all ALEX
resources aligned
being modeled.
to this standard.

ALEX
Resources
Students are able to: Students understand
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that:
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Determine and
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Calculations
distinguish the
to this standard.
accuracy and
involving
ALGI
CCRS Standard
reporting
quantities.
Mathematics CCRS Standards and Alabama COS
Standard ID
Teacher
Vocabulary
(Foundation for
Report direct
measurements and
measurements gained by
combining direct
Number/Quantity measurements to levels
N-Q.3
of accuracy allowed by
the units on the
quantities and will not
report combined or
converted results with
accuracy beyond that of
the original
measurements (e.g., if
one side of a rectangle is
measured to the nearest
meter, and the other side
to the nearest
centimeter, the perimeter
can only be accurate to
the nearest meter).
exponential,
quadratic.)
Algebra
A-SSE.1A
b. Interpret
complicated
expressions by
viewing one or
more of their parts
as a single entity.
Example: Interpret
P(1+r)n as the
Franklin County Schools
Students:
Given a contextual
situation and an
expression that does
model it,
Connect each part of
the expression to the
corresponding piece of
the situation,
Interpret parts of the
expression such as
terms, factors, and
coefficients.
Knowledge
and accuracy and
techniques for
determining each.
work with
expressions,
equations, and
functions.)
7. Interpret
Seeing
expressions that
Structure in
represent a
Expressions
quantity in terms of Interpret the
its context.★
structure of
expressions.
(Linear,
a. Interpret parts
of an expression
such as terms,
factors, and
coefficients.
Evidence of Student
Attainment
Skills
precision of
measurements.
Understanding
measurements can't
produce results more
accurate than the
least accuracy in the
original
measurements,
Resources

ALEX
Resources
The margin of
error in a
measurement, (often
expressed as a
tolerance limit), varies
according to the
measurement, tool
used, and problem
context.
Terms
Students know:
Students are able to: Students understand
that:
Factors
Interpretations of
parts of algebraic
expressions such as
terms, factors, and
coefficients.
Produce
Physical situations
mathematical
expressions that
can be represented
model given contexts, by algebraic
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expressions which
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combine
numbers
Provide a context
resources aligned
from
the
context,
that a given
variables representing to this standard.
mathematical
expression accurately unknown quantities,
and operations
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fits,
indicated by the
Resources
context,
Explain the
reasoning for
selecting a particular Different but
algebraic expression equivalent algebraic
expressions can be
by connecting the
formed by
quantities in the
approaching the
expression to the
Coefficients
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
product of P and a
factor not
depending on P.
Understanding
Resources
physical situation that context from a
produced them, (e.g., different perspective.
the formula for the
area of a trapezoid
can be explained as
the average of the
two bases multiplied
by height).
8. Use the
structure of an
expression to
identify ways to
rewrite it.
Example: See x4 –
y4 as (x2)2 – (y2)2,
thus recognizing it
as a difference of
squares that can be
factored as (x2 –
y2)(x2 + y2).
Seeing
Structure in
Expressions
Interpret the
structure of
expressions.
(Linear,
9. Choose and
produce an
equivalent form of
an expression to
reveal and explain
properties of the
quantity
represented by the
expression.★
Seeing
Structure in
Expressions
Write
expressions in
equivalent forms
to solve
problems.
(Quadratic and
exponential.)
Algebra
A-SSE.3
a. Factor a
quadratic
expression to
reveal the zeros of
the function it
defines.
Skills
exponential,
quadratic.)
Students:
Students know:
Students are able to: Students understand
that:
Make sense of
algebraic expressions by
identifying structures
within the expression
which allow them to
rewrite it in useful ways.
Algebraic
properties (including
those in Tables 3, 4,
and 5),
Use algebraic
properties to produce
equivalent forms of
the same expression
by recognizing
underlying
mathematical
structures.
When one form
of an algebraic
expression is more
useful than an
equivalent form of
that same
expression.
Algebra
A-SSE.2A
b. Complete the
Franklin County Schools
Students:
Quadratic
expression
Make sense of
algebraic expressions by Zeros
identifying structures
within the expression
Complete the
which allow them to
square
rewrite it in useful ways
that assist in the solution
of given problems,
Produce the useful
equivalent forms of
expressions, in particular,
factor a quadratic
expression to reveal the
zeros of the function it
defines and complete the
Generating
simpler, but
equivalent, algebraic
expressions facilitates
the investigation of
more complex
algebraic expressions.
Students know:
Students are able to: Students understand
that:
Techniques for
generating
equivalent forms of
an algebraic
expression including
factoring and
completing the
square for quadratic
expressions and
using properties of
exponents,
Use algebraic
properties including
properties of
exponents to produce
equivalent forms of
the same expression
by recognizing
underlying
mathematical
structures,
When one form
of an algebraic
expression is more
useful than an
Factor quadratic
expressions,
Complete the
square in quadratic
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to this standard.

ALEX
Resources
Making
connections among
Click below to
equivalent
expressions reveals access all ALEX
the roles of important resources aligned
mathematical features to this standard.
of a problem.
 ALEX
Resources
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
square in a
quadratic
expression to
reveal the
maximum or
minimum value of
the function it
defines.
Evidence of Student
Attainment
Teacher
Vocabulary
square in a quadratic
expression to reveal the
maximum or minimum
value of the function it
defines,
Knowledge
Skills
Understanding
Resources
equivalent form of
expressions.
that same expression
to solve a given
problem.
Justify their selection
of a form for an
expression by explaining
which features of the
expression are revealed
by the particular form
and how these features
aid in resolving a
problem situation.
c. Determine a
quadratic equation
when given its
graph or roots.
(Alabama)
d. Use the
properties of
exponents to
transform
expressions for
exponential
functions.
Example: The
expression 1.15t
can be rewritten as
(1.151/12)12t ≈
1.01212t to reveal
the approximate
equivalent monthly
interest rate if the
annual rate is 15%.
10. Understand
that polynomials
form a system
analogous to the
integers; namely,
they are closed
under the
operations of
addition,
subtraction, and
Arithmetic
with
Polynomials
and Rational
Expressions
Perform
arithmetic
operations on
polynomials.
(Linear and
Franklin County Schools
Students:
Polynomials
Use the repeated
Closure
reasoning from prior
knowledge of properties
of arithmetic on integers
to progress consistently
to rules for arithmetic on
polynomials,
Students know:
Students are able to: Students understand
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that:
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Corresponding
Communicate the
resources aligned
rules of arithmetic of connection between There is an
to this standard.
integers, specifically the rules for
operational
what it means for
arithmetic on integers connection between
the integers to be
and the
the arithmetic on
 ALEX
closed under
corresponding rules integers and the
Resources
addition, subtraction, for arithmetic on
arithmetic on
and multiplication,
polynomials.
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
multiplication; add, quadratic.)
subtract, and
Algebra
multiply
A-APR.1A
polynomials.
Accurately perform
combinations of
operations on various
polynomials.
and not under
division,
11. Create
equations and
inequalities in one
variable, and use
them to solve
problems. Include
Creating
Equations*
Create equations
that describe
numbers or
relationships.
(Linear,
Students:
Given a contextual
situation that may
include linear, quadratic,
exponential, or rational
functional relationships in
one variable,
Student know:
Students are able to: Students understand
that:
Write equations
or inequalities in one
variable that
accurately model
contextual situations.
Algebra
A-CED.1A
Model the
relationship with
equations or inequalities
and solve the problem
presented in the
contextual situation for
the given variable.
When the
situation presented
in a contextual
problem is most
accurately modeled
by a linear,
quadratic,
exponential, or
rational functional
relationship.
Students know:
Students are able to: Students understand
that:
equations arising
from linear and
quadratic functions,
and simple rational
and exponential
functions.
quadratic, and
exponential
(integer inputs
only); for
Standard 13,
linear only.)
Resources
polynomials,
Accurately
Procedures for
perform combinations
performing addition, of operations on
subtraction, and
various polynomials.
multiplication on
polynomials.
Features of a
contextual problem
can be used to create
a mathematical model Click below to
for that problem.
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resources aligned
to this standard.

ALEX
Resources
(Please Note: This
standard must be taught
in conjunction with the
standard that follows).
12. Create
equations in two or
more variables to
represent
relationships
between quantities;
graph equations on
coordinate axes
with labels and
scales.
Creating
Equations*
Create equations
that describe
numbers or
relationships.
(Linear,
quadratic, and
exponential
(integer inputs
only); for
Franklin County Schools
Students:
Given a contextual
situation expressing a
relationship between
quantities with two or
more variables,
Model the
relationship with
equations and graph the
relationship on
When a
particular two
variable equation
accurately models
the situation
presented in a
contextual problem.
Click below to
Write equations in
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There are
two variables that
resources aligned
accurately model
relationships among to this standard.
contextual situations, features of a
contextual problem, a
 ALEX
Graph equations created mathematical
Resources
model for that
involving two
problem, and a graph
variables on
coordinate axes with of that relationship.
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Standard 13,
linear only.)
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
coordinate axes with
labels and scales.
Skills
Understanding
Resources
appropriate scales
and labels.
Algebra
A-CED.2A
(Please Note: This
standard must be taught
in conjunction with the
preceding standard).
13. Represent
constraints by
equations or
inequalities, and by
systems of
equations and/or
inequalities and
interpret solutions
as viable or
nonviable options
in a modeling
context.
Example:
Represent
inequalities
describing
nutritional and cost
constraints on
combinations of
different foods.
Creating
Equations*
Create equations
that describe
numbers or
relationships.
(Linear,
quadratic, and
exponential
(integer inputs
only); for
Standard 13,
linear only.)
Algebra
A-CED.3A
Students:
Given a contextual
situation involving
constraints,
Write equations or
inequalities or a system
of equations or
inequalities that model
the situation and justify
each part of the model in
terms of the context,
Solve the equation,
inequalities or systems
and interpret the solution
in the original context
including discarding
solutions to the
mathematical model that
cannot fit the real world
situation (e.g., distance
cannot be negative),
Solve a system by
graphing the system on
the same coordinate grid
and determine the
point(s) or region that
satisfies all members of
the system,
Franklin County Schools
Constraint
Students know:
Students are able to: Students understand
that:
When a
particular system of
two variable
equations or
inequalities
accurately models
the situation
presented in a
contextual problem,
Graph equations
and inequalities
involving two
variables on
coordinate axes,
Which points in
the solution of a
system of linear
inequalities need to
be tested to
maximize or
minimize the variable
of interest.
A symbolic
representation of
relevant features of a
real world problem
can provide for
resolution of the
Identify the
region that satisfies problem and
both inequalities in a interpretation of the
situation and solution,
system,
Identify the
point(s) that
maximizes or
minimizes the variable
of interest in a system
of inequalities,
Test a
mathematical model
using equations,
inequalities, or a
system against the
constraints in the
context and interpret
the solution in this
context.

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Representing
resources aligned
a physical
to this standard.
situation
with a
 ALEX
mathematica
Resources
l model
requires
consideratio
n of the
accuracy and
limitations of
the model.
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
Determine the
point(s) of the region
satisfying all members of
the system that
maximizes or minimizes
the variable of interest in
the case of a system of
inequalities.
14. 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.
Creating
Equations*
Create equations
that describe
numbers or
relationships.
(Linear,
quadratic, and
exponential
(integer inputs
only); for
Standard 13,
linear only.)
Algebra
A-CED.4A
15. Explain each
step in solving a
simple equation as
following from the
equality of
numbers asserted
at the previous
step, starting from
the assumption
that the original
equation has a
Reasoning
with Equations
& Inequalities
Understand
solving equations
as a process of
reasoning and
explain the
reasoning.
(Master linear;
learn as general
Franklin County Schools
Students:
Students know:
Students are able to: Students understand
that:
Rearrange formulas
which arise in contextual
situations to isolate
variables that are of
interest for particular
problems. For example, if
the electric company
charges for power by the
formula COST = 0.03
KWH + 15, a consumer
may wish to determine
how many kilowatt hours
they may use to keep the
cost under particular
amounts, by considering
KWH< (COST - 15)/0.03
which would yield to
keep the monthly cost
under $75, they need to
use less than 2000 KWH.
Properties of
equality and
inequality (Tables 4
and 5).
Accurately
rearrange equations
or inequalities to
produce equivalent
forms for use in
resolving situations of
interest.
Students:
The structure of
mathematics allows
for the procedures
used in working with
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equations to also be
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valid when
resources aligned
rearranging formulas,
to this standard.
The isolated
variable in a formula
is not always the
unknown and
rearranging the
formula allows for
sense-making in
problem solving.
Inverse
operations
Justify each step in
the solution of equations Equality
by communicating their
understandings of
inverse operations as
well as other operation
properties (commutative,
associative, identity,
distributive) and
Students are able to: Students understand
that:
Accurately
Rules for
producing equivalent rearrange equations The structure of
equations (Table 4), to produce equivalent mathematics present
forms for use in
in the properties of
resolving situations of the operations can be
Properties of
interest,
used to maintain
addition and
equality while
multiplication (Table
Communicate
rearranging
3 ).
equations, as well as
reasoning behind

ALEX
Resources
Students know:
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
solution. Construct principle.)
a viable argument Algebra
to justify a solution A-REI.1
method.
properties of equality.
16. Solve linear
equations and
inequalities in one
variable, including
equations with
coefficients
represented by
letters.
Students:
Students know:
Students are able to: Students understand
that:
Solve linear
equations and
inequalities in one
variable including being
able to isolate variables
when coefficients are
letters.
Properties of
equality and
inequality (Tables 4
and 5).
Accurately use
the properties of
equality and
inequality (Tables 4
and 5) to produce
equivalent
expressions which
lead to solutions.
Reasoning
with Equations
& Inequalities
Solve equations
and inequalities
in one variable.
(Linear
inequalities;
literal that are
linear in the
variables being
solved for;
quadratics with
real solutions.)
each step conducted to justify steps in the
in producing each
finding of solutions of
equivalent form.
equations.
square to
transform any
quadratic
equation in x
into an
equation of the
form (x – p)2
= q that has
the same
solutions.
inequalities;
literal that are
linear in the
variables being
solved for;
quadratics with
real solutions.)
Algebra
A-REI.4
Franklin County Schools
The structure of
mathematics allows
for the procedures
used in working with
equations and
inequalities to also be
valid when
coefficients are
letters,
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
The solutions
arrived at though
algebraic
manipulations should
make the original
equation true.
Algebra
A-REI.3
17. Solve quadratic Reasoning
equations in one
with Equations
variable.
& Inequalities
Solve equations
and inequalities
a. Use the
in one variable.
method of
completing the (Linear
Resources
Students:
Solve quadratic
equations where both
sides of the equation
have evident square
roots by inspection,
Completing the
square
Quadratic
equations
Quadratic
formula
Transform quadratic
equations to a form
Complex
where the square root of solutions
each side of the equation
may be taken, including
Inspection
completing the square,
Use the method of
completing the square on
Students know:
Students are able to: Students understand
that:
Any real number
has two square
roots, that is, if a is
the square root of a
real number then so
is -a,
Accurately use
properties of equality
(Table 4) and other
algebraic
manipulations
including taking
square roots of both
sides of an equation,
The method for
completing the
square,
Notational
methods for
expressing complex
Solutions to a
quadratic equation
must make the
original equation true
and this should be
verified,
When the
Accurately
quadratic equation is
complete the square derived from a
on a quadratic
contextual situation,
polynomial as a
proposed solutions to
strategy for finding
the quadratic
solutions to quadratic equation should be
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
ALGI
CCRS Standard
b.
Mathematics CCRS Standards and Alabama COS
Standard ID
Derive the
quadratic
formula from
this form.
Solve
quadratic
equations by
inspection
(e.g., for x2 =
49), 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.
18. Prove that,
given a system of
two equations in
two variables,
replacing one
equation by the
sum of that
equation and a
multiple of the
other produces a
system with the
Evidence of Student
Attainment
Teacher
Vocabulary
the equation in standard
form (ax2+bx+c=0) to
derive the quadratic
formula,
Identify quadratic
equations which may be
solved efficiently by
factoring, and then use
factoring to solve the
equation,
Knowledge
equations,
A quadratic
equation in standard
form (ax2+bx+c=0)
has real roots when
b2-4ac is greater
than or equal to zero
and complex roots
when b2-4ac is less
than zero.
Factor quadratic
polynomials as a
Different
strategy for finding
solutions to quadratic procedures for solving
equations,
quadratic equations
are necessary under
Rewrite solutions different conditions,
Explain when the
roots are real or complex
for a given quadratic
equation, and when
complex write them as a
± bi for real numbers a
and b,
and linearquadratic.)
Algebra
A-REI.5
Franklin County Schools
Use the properties of
equality (Table 4) (where
A, B, C, D, E, and F are
real numbers) to show
Resources
verified within the
context given, as well
as mathematically,
to quadratic
If ab=0, then at
equations in useful
forms including a ± bi least one of a or b
and simplified radical must be zero (a=0 or
expressions,
b=0) and this is then
used to produce the
two solutions to the
Make strategic
choices about which quadratic equation,
procedures
Whether the roots
(inspection,
completing the
of a quadratic
square, factoring, and equation are real or
quadratic formula) to complex is
use to reach a
determined by the
solution to a
coefficients of the
quadratic equation. quadratic equation in
standard form
(ax2+bx+c=0).
Demonstrate that a
proposed solution to a
quadratic equation is
truly a solution by
making the original true.
Students:
Given a system of two
equations in the form
Ax + By = C
Dx + Ey = F
Understanding
numbers,
Use the quadratic
formula to solve
quadratic equations,
Reasoning
with Equations
& Inequalities
Solve systems of
equations.
(Linear-linear
Skills
System of two
equations in two
variables
Students know:
Students are able to: Students understand
that:
Click below to
access all ALEX
Appropriate use Accurately
resources aligned
When the
of properties of
perform the
to this standard.
addition and
operations of
properties of
multiplication (Table multiplication and
operations and
3) and equality
addition, and
equality are applied
a. ALEX
(Table 4).
techniques for
to systems of
Resources
manipulating
equations, the
equations.
resulting equations
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
same solutions.
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
that the same solution is
produced by replacing
one equation by the sum
of that equation and a
multiple of the other,
Understanding
Resources
have the same
solution as the
original.
For example, the system
2x + 3y = 18
4x + 2y = 20
is equivalent to 0x - 4y =
-16 (multiply first
equation by -2 add to the
second equation).
19. Solve systems
of linear equations
exactly and
approximately
(e.g., with graphs),
focusing on pairs of
linear equations in
two variables.
Reasoning
with Equations
& Inequalities
Solve systems of
equations.
(Linear-linear
and linearquadratic.)
Algebra
A-REI.6
Students:
Given systems of linear
equations,
Choose an
appropriate method for
solving (e.g.,
substitution, elimination,
and graphing),
Solve and justify
solutions,
Provide reasonable
approximations when
appropriate on a graph.
20. Solve a simple
system consisting
of a linear equation
and a quadratic
equation in two
variables
algebraically and
graphically.
Reasoning
with Equations
& Inequalities
Solve systems of
equations.
(Linear-linear
and linearquadratic.)
Franklin County Schools
Students:
Given a system of a
linear equation and a
quadratic equation,
Solve the system
algebraically by
Students know:
Students are able to: Students understand
that:
Appropriate use
of properties of
addition and
multiplication (Table
3) and equality
(Table 4),
Accurately
perform the
operations of
multiplication and
addition, and
techniques for
manipulating
equations,
Techniques for
producing and
interpreting graphs
of linear equations,
Graph linear
equations precisely,
The solution of a
linear system is the
set of all ordered
Click below to
pairs that satisfy both access all ALEX
equations,
resources aligned
to this standard.
Solving by
graphing often leads
 ALEX
to approximate
Resources
solutions,
The conditions
under which a
system of linear
equations has 0, 1,
or infinite solutions.
Use estimation to A system of linear
find approximate
equations will have 0,
solutions on a graph. 1, or infinite
solutions.
Students know:
Students are able to: Students understand Click below to
that:
access all ALEX
resources aligned
Accurately use
to this standard.
properties of equality Solutions of a
Appropriate use
of properties of
equality (Table 4),
Techniques to
solve quadratic
(Table 4) to solve a system of equations is
system of a linear and the set of all ordered
a quadratic equation, pairs that make both
equations true

ALEX
Resources
ALGI
CCRS Standard
Example: Find the
points of
intersection
between the line y
= –3x and the
circle x2 + y2 = 3.
Mathematics CCRS Standards and Alabama COS
Standard ID
Algebra
A-REI.7
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
Graph linear and simultaneously,
quadratic equations
precisely and
The conditions
A system
under which a linear interpret the results. consisting of a linear
equation and a
equation and a
quadratic equation
quadratic equation
have 0, 1, or 2
will have 0,1, or 2
solutions,
solutions.
substitution,
equations,
Graph the linear
equation and the
quadratic equation on
the same Cartesian
plane, and identify the
intersection point(s),
Make sense of the
existence of 0, 1, or 2
solutions to the system
by explaining the
relationship of the
solutions to the graph,
Techniques for
producing and
interpreting graphs
of linear and
quadratic equations.
Verify that the
proposed solutions
satisfy both equations.
21. Understand
that the graph of
an equation in two
variables is the set
of all its solutions
plotted in the
coordinate plane,
often forming a
curve (which could
be a line).
Reasoning
with Equations
& Inequalities
Represent and
solve equations
and inequalities
graphically.
(Linear and
exponential;
learn as general
principle.)
Algebra
A-REI.10
22. 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
Reasoning
with Equations
& Inequalities
Represent and
solve equations
and inequalities
graphically.
(Linear and
Franklin County Schools
Curve (which
Students:
Given an equation in two could be a line)
variables,
Verify that any
ordered pair that makes
the equation true is a
point on the graph,
Students are able to: Students understand
that:
Appropriate
methods to find
ordered pairs that
satisfy an equation,
Accurately find
ordered pairs that
satisfy the equation,
Techniques to
graph the collection
of ordered pairs to
form a curve.
Show that there are
an infinite number of
ordered pairs that satisfy
the equation.
Students: Given two
functions (linear,
polynomial, rational,
absolute value,
exponential, and
logarithmic) that
intersect (e.g., y= 3x and
y= 2x), - Graph each
Students know:
Functions
Students know:
Successive
approximations
Defining
characteristics of
linear, polynomial,
rational, absolute
value, exponential,
and logarithmic
Linear functions
Click below to
An equation in
access all ALEX
two variables has an resources aligned
infinite number of
to this standard.
Accurately graph solutions (ordered
the ordered pairs and pairs that make the
 ALEX
equation true), and
form a curve.
Resources
those solutions can
be represented by the
graph of a curve.
Students are able to: Students understand
Click below to
that:
access all ALEX
Determine a
resources aligned
solution or solutions When two
of a system of two
functions are equal, to this standard.
functions,
the x coordinate(s) of
the intersection of
those functions is the

ALEX
ALGI
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
Evidence of Student
Attainment
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.★
exponential;
learn as general
principle.)
function and identify the
intersection point(s), Explain solutions for f(x)
= g(x) as the xcoordinate of the points
of intersection of the
graphs, and explain
solution paths (e.g., the
values that make 3x = 2x
true, are the x-coordinate
intersection points of
y=3x and y=2x, - Use
technology, tables, and
successive
approximations to
produce the graphs, as
well as to determine the
approximation of
solutions.
Polynomial
functions
23. Graph the
solutions to a linear
inequality in two
variables as a halfplane (excluding
the boundary in the
case of a strict
inequality), and
graph the solution
set to a system of
linear inequalities
in two variables as
the intersection of
the corresponding
half-planes.
Reasoning
with Equations
& Inequalities
Represent and
solve equations
and inequalities
graphically.
(Linear and
Students:
Given a linear inequality
in two variables or a
system of linear
inequalities,
Half-planes
Students know:
Students are able to: Students understand
that:
System of linear
inequalities
When to include
and exclude the
boundary of linear
inequalities,
Accurately graph
a linear inequality and Solutions to a
identify values that
linear inequality result Click below to
make the inequality in the graph of a half- access all ALEX
resources aligned
true (solutions),
plane,
to this standard.
Algebra
A-REI.11A
exponential;
learn as general
principle.)
Teacher
Vocabulary
Rational
functions
Absolute value
functions
Exponential
functions
Logarithmic
functions
Graph solutions and
solution sets using the
appropriate notation
(dotted or solid line).
Knowledge
graphs,
Skills
Accurately use
technology to
produce graphs and
tables for linear,
polynomial, rational,
absolute value,
exponential, and
logarithmic functions,
Methods to use
technology, tables,
and successive
approximations to
produce graphs and
tables for linear,
polynomial, rational,
Accurately use
absolute value,
exponential, and
technology to
logarithmic functions. approximate solutions
on graphs.
Understanding
value that produces
the same output (yvalue) for both
functions,
Resources
Technology is
useful to quickly and
accurately determine
solutions and produce
graphs of functions.
Techniques to
Solutions to a
graph the boundaries Find the
of linear inequalities, intersection of
system of linear
multiple linear
inequalities are the
Methods to find inequalities to solve a intersection of the
solutions of each
solution regions of a system.
inequality in the
linear inequality and
system.
systems of linear
Algebra
A-REI.12
Resources

ALEX
Resources
inequalities.
24. Understand
that a function
from one set
(called the domain)
to another set
Interpreting
Functions
Understand the
concept of a
function and use
Franklin County Schools
Students:
Given input/output
relations between two
variables in graphical
form, verbal description,
Domain
Students know:
Range
Distinguishing
characteristics of
Students are able to: Students understand
Click below to
that:
access all ALEX
Accurately graph
resources aligned
functions when given Functions are
to this standard.
relationships between
ALGI
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
(called the range)
assigns to each
element of the
domain exactly one
element of the
range. If f is a
function and x is an
element of its
domain, then f(x)
denotes the output
of f corresponding
to the input x. The
graph of f is the
graph of the
equation y = f(x).
function
notation. (Learn
as general
principle; focus
on linear and
exponential and
on arithmetic
and geometric
sequences.)
Functions
F-IF.1
Evidence of Student
Attainment
set of ordered pairs, or
algebraic model,
Distinguish between
those that are functions
and non-functions.
Given a functional
relationship,
Teacher
Vocabulary
Knowledge
functions,
Skills
function notation,
Conventions of
function notation,
Accurately
determine domain
and range values
from function
In graphing
functions the ordered notation.
pairs are (x,f(x)) and
the graph is y = f(x).
Determine that
exactly one element of
the range (output) is
assigned to each element
of the domain (input) by
the function,
Understanding
two variables that
have a unique
characteristic: that for
each input there
exists exactly one
output,
Resources
a.
ALEX
Resources
Function notation
is useful to see the
relationship between
two variables when
the unique output for
each input relation is
satisfied.
Represent the
function with a graph
and with functional
notation.
25. Use function
notation, evaluate
functions for inputs
in their domains,
and interpret
statements that
use function
notation in terms of
a context.
Interpreting
Functions
Understand the
concept of a
function and use
function
notation. (Learn
as general
principle; focus
on linear and
exponential and
on arithmetic
and geometric
sequences.)
Functions
F-IF.2
Franklin County Schools
Students:
Given a contextual
situation that may be
represented as a
function,
Use function notation
to model the situation,
Evaluate the function
to produce outputs when
given a value in the
domain,
Explain in the original
context the meaning of
the output when related
to the input.
Function notation Students know:
Distinguishing
characteristics of a
function,
Conventions of
function notation.
Students are able to: Students understand
that:
Accurately use
function notation to
model physical
situations,
Accurately
evaluate function
equations given
values in the domain.
Function notation
is useful to see the
relationship between
two variables when
the unique output for
each input relation is
satisfied.
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
ALGI
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
26. Recognize that
sequences are
functions,
sometimes defined
recursively, whose
domain is a subset
of the integers.
Example: The
Fibonacci sequence
is defined
recursively by f(0)
= f(1) = 1, f(n+1)
= f(n) + f(n-1) for
n ≥ 1.
Interpreting
Functions
Understand the
concept of a
function and use
function
notation. (Learn
27. 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.
Interpreting
Functions
Interpret
functions that
arise in
applications in
terms of the
context. (Linear,
as general
principle; focus
on linear and
exponential and
on arithmetic
and geometric
sequences.)
Evidence of Student
Attainment
Students:
Given a sequence,
Generate and justify
a function that relates
the number of the term
to the value of the term
in the sequence.
Teacher
Vocabulary
Understanding
Students know:
Students are able to: Students understand
that:
Recursively
Distinguishing
characteristics of a
function,
Relate the
number of the term
to the value of the
term in a sequence
and express the
relation in functional
notation.
Domain
Distinguishing
characteristics of
function notation,
Resources
Each term in the Click below to
domain of a sequence access all ALEX
defined as a function resources aligned
is unique and
to this standard.
consecutive.

Distinguishing
characteristics of
generating
sequences.
Students:
Given a function that
models a relationship
between two quantities,
Produce the graph
and table of the function
and show the key
exponential, and features (intercepts;
quadratic.)
intervals where the
Functions
function is increasing,
F-IF.4A
decreasing, positive, or
negative; relative
maximums and
Key features
minimums; symmetries;
include intercepts;
end behavior; and
intervals where the
periodicity) that are
function is
appropriate for the
increasing,
function.
Franklin County Schools
Skills
Sequence
Functions
F-IF.3
decreasing,
positive, or
negative; relative
maximums and
minimums;
symmetries; end
behavior; and
Knowledge
Given key features from
verbal description of a
relationship,
ALEX
Resources
Function
Students know:
Students are able to: Students understand
that:
Key features
Key features of
function graphs (i.e.,
intercepts; intervals
where the function is
increasing,
decreasing, positive,
or negative; relative
maximums and
minimums;
symmetries; end
behavior; and
periodicity),
Accurately graph
The relationship
any relationship,
between two
variables determines
Interpret key
features of a graph. the key features that
need to be used when
interpreting and
Click below to
producing the graph. access all ALEX
Methods of
modeling
relationships with a
graph or table.
resources aligned
to this standard.

ALEX
Resources
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
periodicity.★
Sketch a graph with
the given key features.
28. Relate the
domain of a
function to its
graph and, where
applicable, to the
quantitative
relationship it
describes.
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.★
Interpreting
Functions
Interpret
functions that
arise in
applications in
terms of the
context. (Linear,
Students:
Given a contextual
situation that is
functional,
29. 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.★
Interpreting
Functions
Interpret
functions that
arise in
applications in
terms of the
context. (Linear,
Students:
Given an interval on a
graph or table,
Function
exponential, and
quadratic.)
Functions
F-IF.6A
Given a graph of
contextual situation,
Estimate the rate of
change between intervals
that are appropriate for
the summary of the
context.
Students are able to: Students understand
that:
Techniques for
graphing functions,
Interpret the
domain from the
context,
Students know:
Students are able to: Students understand
that:
Different contexts
produce different
domains and graphs, Click below to
Techniques for
access all ALEX
Produce a graph
determining the
Function notation resources aligned
domain of a function of a function based
to this standard.
from its context.
on the context given. in itself may produce
graph points which
 ALEX
should not be in the
Resources
graph as the domain
is limited by the
context.
Model the situation
with a graph and
construct the graph
exponential, and based on the parameters
quadratic.)
given in the domain of
Functions
the context.
F-IF.5A
Calculate the average
rate of change within the
interval.
Students know:
Average rate of
change
Techniques for
graphing,
Calculate rate of
The average
change over an
interval on a table or provides information Click below to
graph,
on the overall
Techniques for
access all ALEX
changes within an
finding a rate of
resources aligned
Estimate a rate of interval, not the
change over an
to this standard.
details within the
interval on a table or change over an
graph,
interval on a graph. interval (an average
 ALEX
of the endpoints of an
Resources
interval does not tell
Techniques for
you the significant
estimating a rate of
features within the
change over an
interval).
interval on a graph.
Click below to
Franklin County Schools
ALGI
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
Evidence of Student
Attainment
30. Graph functions
expressed
symbolically and
show key features
of the graph, by
hand in simple
cases and using
technology for
more complicated
cases.★
Interpreting
Functions
Analyze
functions using
different
representations.
(Linear,
Students:
Given a symbolic
representation of a
function (including linear,
quadratic, square root,
cube root, piecewisedefined functions,
polynomial, exponential,
logarithmic,
trigonometric, and (+)
rational),
a.
b.
c.
exponential,
quadratic,
absolute value,
step, piecewisedefined.)
Graph linear
and quadratic Functions
functions and F-IF.7A
show
intercepts,
maxima, and
minima.
Graph square
root, cube
root, and
piecewisedefined
functions,
including step
functions and
absolute value
functions.
Graph
exponential
and
logarithmic
functions,
showing
intercepts and
end behavior,
and
trigonometric
functions,
showing
period,
midline, and
Franklin County Schools
Produce an accurate
graph (by hand in simple
cases and by technology
in more complicated
cases) and justify that
the graph is an alternate
representation of the
symbolic function,
Identify key features
of the graph and connect
these graphical features
to the symbolic function,
specifically for special
functions:
a.
b.
quadratic or
linear
(intercepts,
maxima, and
minima),
square root,
cube root, and
piecewisedefined
functions,
including step
functions and
absolute value
Teacher
Vocabulary
Knowledge
Students know:
Techniques for
graphing,
Key features of
graphs of functions.
Skills
Understanding
tudents are able to:
Resources
Students understand access all ALEX
that:
resources aligned
to this standard.
Identify the type
of function from the Key features are
symbolic
different depending
 ALEX
representation,
on the function,
Resources
Manipulate
Identifying key
expressions to reveal features of functions
important features for aid in graphing and
identification in the
interpreting the
function,
function.
Accurately graph
any relationship.
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
amplitude.
c.
d.
e.
f.
31. Write a
function defined by
an expression in
different but
equivalent forms to
reveal and explain
different properties
of the function.
a.
Use the
process of
Interpreting
Functions
Analyze
functions using
different
representations.
(Linear,
exponential,
quadratic,
absolute value,
step, piecewisedefined.)
Franklin County Schools
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
functions
(descriptive
features such as
the values that
are in the range
of the function
and those that
are not),
polynomial
(zeros when
suitable
factorizations
are available,
end behavior),
(+) rational
(zeros and
asymptotes
when suitable
factorizations
are available,
end behavior),
exponential and
logarithmic
(intercepts and
end behavior),
trigonometric
functions
(period, midline,
and amplitude).
Students:
Given a contextual
situation containing a
function defined by an
expression,
Zeros
Students know:
Students are able to: Students understand
that:
Extreme values
Techniques to
factor and complete
the square,
Accurately
manipulate (e.g.,
factoring, completing
the square)
expressions using
appropriate
technique(s) to reveal
key properties of a
function.
Symmetry
Use algebraic
Exponential
properties to rewrite the
expression in a form that growth or decay
makes key features of
the function easier to
Properties of
exponential
expressions,
Algebraic
An expression
may be written in
various equivalent
forms,
Some forms of
the expression are
more beneficial for
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
ALGI
CCRS Standard
b.
Mathematics CCRS Standards and Alabama COS
Standard ID
factoring and Functions
completing the F-IF.8A
square in a
quadratic
function to
show zeros,
extreme
values, and
symmetry of
the graph, and
interpret these
in terms of a
context.
Use the
properties of
exponents to
interpret
expressions for
exponential
functions.
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.
32. Compare
properties of two
functions each
represented in a
different way
(algebraically,
Interpreting
Functions
Analyze
functions using
different
representations.
Franklin County Schools
Evidence of Student
Attainment
find,
Teacher
Vocabulary
Knowledge
properties of equality
(Table 4).
Manipulate a
quadratic function by
factoring and completing
the square to show
zeros, extreme values,
and symmetry of the
graph,
Skills
Understanding
Resources
revealing key
properties of the
function.
Explain and justify
the meaning of zeros,
extreme values, and
symmetry of the graph in
terms of the contextual
situation,
Apply exponential
properties to expressions
and explain and justify
the meaning in a
contextual situation.
Students:
Given two functions
represented differently
(algebraically,
graphically, numerically
in tables, or by verbal
Students know:
Techniques to
find key features of
functions when
Students are able to: Students understand
that:
Click below to
access all ALEX
Accurately
determine which key Functions can be resources aligned
to this standard.
features are most
written in different
ALGI
CCRS Standard
graphically,
numerically in
tables, or by verbal
descriptions).
Example: Given a
graph of one
quadratic function
and an algebraic
expression for
another, say which
has the larger
maximum.
Mathematics CCRS Standards and Alabama COS
Standard ID
(Linear,
exponential,
quadratic,
absolute value,
step, piecewisedefined.)
Functions
F-IF.9A
Evidence of Student
Attainment
Teacher
Vocabulary
descriptions),
Knowledge
Skills
Understanding
presented in
different ways,
appropriate for
but equivalent ways
comparing functions, (algebraically,
graphically,
numerically in tables,
Techniques to
Manipulate
convert a function to functions algebraically or by verbal
descriptions),
a different form
to reveal key
(algebraically,
functions,
Different
graphically,
numerically in tables, Convert a
representations of
or by verbal
function to a different functions may aid in
descriptions).
comparing key
form (algebraically,
features of the
graphically,
numerically in tables, functions.
Use key features to
compare the functions,
Explain and justify
the similarities and
differences of the
functions.
Resources

ALEX
Resources
or by verbal
descriptions) for the
purpose of comparing
it to another function.
33. Write a
function that
describes a
relationship
between two
quantities.★
a.
b.
Determine an
explicit
expression, a
recursive
process, or
steps for
calculation
from a
context.
Combine
standard
function types
using
arithmetic
operations.
Example: Build
a function that
Building
Functions
Build a function
that models a
relationship
between two
quantities. (For
Students:
Given a contextual
situation containing two
quantities,
Explicit
expression
Recursive
process
Express a
relationship between the Decaying
standards 33
quantities through an
exponential
and 34, linear,
explicit expression using
exponential, and function notation,
quadratic.)
recursive process, or
Functions
steps for calculation,
F-BF.1
Franklin County Schools
Explain and justify
how the expression or
process models the
relationship between the
given quantities,
Create a new
function by using
standard function types
and arithmetic operations
Students know:
Students are able to: Students understand
that:
Techniques for
expressing functional
relationships (explicit
expression, a
recursive process, or
steps for calculation)
between two
quantities,
Accurately
develop a model that
shows the functional
relationship between
two quantities,
Techniques to
combine functions
using arithmetic
operations.
Relationships can
be modeled by
several methods
(e.g., explicit
expression or
Click below to
Accurately create recursive process),
access all ALEX
a new function
resources aligned
Arithmetic
through arithmetic
to this standard.
operations of other
combinations of
functions,
functions may be
 ALEX
used to improve the
Resources
fit of a model.
Present an
argument to show
how the function
models the
relationship between
the quantities.
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
models the
temperature of
a cooling body
by adding a
constant
function to a
decaying
exponential,
and relate
these functions
to the model.
34. Write
arithmetic and
geometric
sequences both
recursively and
with an explicit
formula, use them
to model situations,
and translate
between the two
forms.★
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
to combine the original
functions to model the
relationship of the given
quantities, (+) standards
not covered.
Building
Functions
Build a function
that models a
relationship
between two
quantities. (For
standards 33
and 34, linear,
exponential, and
quadratic.)
Functions
F-BF.2
35. 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
Evidence of Student
Attainment
Building
Functions
Build new
functions from
existing
functions.
(Linear,
Students:
Given a contextual
situation that is
sequential (arithmetic
and geometric),
Create both recursive
and explicit models for
the sequence,
Recursively
Explicit formula
Students:
Given a function in
algebraic form,
Even and odd
functions
Students are able to: Students understand
that:
Techniques to
translate between
recursive and explicit
formulas for
sequences
(geometric and
arithmetic).
Use the
Arithmetic and
properties of
operations and
geometric sequences
equality (Tables 3 and can be expressed
4) and knowledge of with a recursive
recursive functions to model or explicit
justify that an explicit formula, and each
formula that models a form may have
sequence is
benefits to aid in
equivalent to a
understanding or
recursive model.
interpreting the
situation.
Students know:
Students are able to: Students understand
that:
Explain and justify
the relationship between
the recursive and explicit
forms that model the
situation.
Graph the function,
f(x), conjecture how the
graph of f(x) + k, k f(x),
exponential,
f(kx), and f(x + k) for
quadratic, and
specific values of k(both
absolute value; positive and negative)
for standard 36a, will change from f(x),
linear only.)
and test the conjectures,
Functions
Franklin County Schools
Arithmetic and
Students know:
geometric sequences
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
Accurately graph
Click below to
Graphs of
functions,
access all ALEX
functions may be
resources aligned
Check conjectures shifted, stretched, or to this standard.
reflected by adding or
Methods of using about how a
technology to graph parameter change in multiplying the input
 ALEX
or output of a
functions,
a function changes
Resources
the graph and critique function by a constant
value,
the reasoning of
Techniques to
others
about
such
identify even and
Even and odd
odd functions both
Graphing
techniques of
functions,
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
the effects on the F-BF.3A
graph using
technology. Include
recognizing even
and odd functions
from their graphs
and algebraic
expressions for
them.
Evidence of Student
Attainment
Teacher
Vocabulary
Describe how the
graphs of the functions
were affected (e.g.,
horizontal and vertical
shifts, horizontal and
vertical stretches, or
reflections),
Knowledge
algebraically and
from a graph.
Skills
shifts,
Identify shifts,
stretches, or
reflections between
graphs,
Understanding
Resources
functions may be
identified from a
graph or algebraic
form of a function.
Determine when
a function is even or
odd.
Use technology to
explain possible effects
on the graph from adding
or multiplying the input
or output of a function by
a constant value,
Recognize if a
function is even or odd.
Given the graph of a
function and the graph of
a translation, stretch, or
reflection of that
function,
Determine the value
which was used to shift,
stretch, or reflect the
graph,
Recognize if a
function is even or odd.
36. Find inverse
functions.
a.
Building
Functions
Build new
functions from
Solve an
equation of the existing
form f(x) = c functions.
(Linear,
for a simple
function f that exponential,
Franklin County Schools
Students:
Given an invertible
function in algebraic
form,
Use algebraic
properties to find the
inverse of the given
Inverse
Students know:
Students are able to: Students understand Click below to
that:
access all ALEX
resources aligned
Algebraic
Accurately
The inverse of a to this standard.
properties of equality perform algebraic
(Table 4).
properties to find the function interchanges
inverse.
the input and output
 ALEX
values from the
Resources
ALGI
CCRS Standard
has an inverse
and write an
expression for
the inverse.
Example: f(x)
=2 x3 or f(x) =
(x+1)/(x–1)
for x ≠ 1.
Mathematics CCRS Standards and Alabama COS
Standard ID
b.
c.
Teacher
Vocabulary
Knowledge
Skills
quadratic, and
function, (+) standards
absolute value; not covered.
for standard 36a,
linear only.)
Linear,
Quadratic, &
Exponential
Models*
Construct and
compare linear,
quadratic, and
exponential
models and solve
Prove that
problems.
linear
functions grow Functions
F-LE.1
by equal
differences
over equal
intervals, and
that
exponential
functions grow
by equal
factors over
equal intervals.
Recognize
situations in
which one
quantity
changes at a
constant rate
per unit
interval
relative to
another.
Recognize
Franklin County Schools
Students:
Given a linear or
exponential function,
Create a sequence
from the functions and
examine the results to
demonstrate that linear
functions grow by equal
differences, and
exponential functions
grow by equal factors
over equal intervals,
Use slope-intercept
form of a linear function
and the general definition
of exponential functions
to justify through
algebraic rearrangements
that linear functions grow
by equal differences, and
exponential functions
grow by equal factors
over equal intervals.
Given a contextual
situation modeled by
functions,
Understanding
Resources
original function,
The inverse of a
function must also be
a function to exist
and the domain may
need to be restricted
to make this occur.
Functions
F-BF.4
37. Distinguish
between situations
that can be
modeled with linear
functions and with
exponential
functions.
a.
Evidence of Student
Attainment
Linear functions
Students know:
Students are able to: Students understand
that:
Exponential
functions
Key components
of linear and
exponential
functions,
Accurately
determine
relationships of data
from a contextual
situation to determine
if the situation is one
in which one quantity
changes at a constant
rate per unit interval
relative to another
(linear),
Properties of
operations and
equality (Tables 3
and 4).
Linear functions
have a constant value
added per unit
interval, and
exponential functions
have a constant value
multiplied per unit
interval,
Click below to
Distinguishing key access all ALEX
features of and
resources aligned
Accurately
categorizing functions to this standard.
determine
facilitates
relationships of data mathematical
 ALEX
from a contextual
modeling and aids in
Resources
situation to determine problem resolution.
if the situation is one
in which one quantity
grows or decays by a
constant percent rate
per unit interval
relative to another
(exponential).
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
situations in
which a
quantity grows
or decays by a
constant
percent rate
per unit
interval
relative to
another.
38. 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).
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Understanding
Resources
Determine if the
change in the output per
unit interval is a constant
being added or multiplied
to a previous output, and
appropriately label the
function as linear,
exponential, or neither.
Linear,
Quadratic, &
Exponential
Models*
Construct and
compare linear,
quadratic, and
exponential
models and solve
problems.
Functions
F-LE.2
Students:
Given a contextual
situation shown by a
graph, a description of a
relationship, or two
input-output pairs,
Arithmetic and
Students know:
geometric sequences
Create a linear or
exponential function that
models the situation,
Create arithmetic and
geometric sequences
from the given situation,
That linear
functions grow by
equal differences
over equal intervals,
and that exponential
functions grow by
equal factors over
equal intervals,
Properties of
arithmetic and
geometric
sequences.
Justify the equality of
the sequences and the
functions mathematically
and in terms of the
original sequence.
39. Observe using
graphs and tables
that a quantity
increasing
exponentially
eventually exceeds
a quantity
Skills
Linear,
Quadratic, &
Exponential
Models*
Construct and
compare linear,
quadratic, and
Franklin County Schools
Increasing
Students:
Given a quantity
exponentially
increasing exponentially
and a quantity increasing
as a polynomial function
(e.g., linearly,
quadratically),
Students know:
Students are able to: Students understand
that:
Accurately
Linear and
recognize
relationships within
exponential functions
data and use that
may be used to
relationship to create model data that is
a linear or
presented as a graph, Click below to
access all ALEX
exponential function a description of a
to model the data of relationship, or two resources aligned
to this standard.
a contextual situation. input-output pairs
(include reading these
from a table),
 ALEX
Resources
Linear functions
have a constant value
added per unit
interval, and
exponential functions
have a constant value
multiplied per unit
interval.
Students are able to: Students understand
Click below to
that:
access all ALEX
Techniques to
Accurately create
resources aligned
graph and create
graphs and tables for Exponential
to this standard.
tables for
exponential and
functions grow at a
exponential and
polynomial functions, faster rate than
 ALEX
polynomial functions.
polynomial functions
ALGI
CCRS Standard
increasing linearly,
quadratically, or
(more generally) as
a polynomial
function.
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
Use the graphs
after some point in
and tables to present their domain.
a convincing
argument that the
exponential function
eventually exceeds
the polynomial
function.
exponential
models and solve Construct graphs and
problems.
tables that demonstrate
Functions
the exponential function
F-LE.3
will exceed the
polynomial function at
some point,
Resources
Present a convincing
argument that this must
be true for all polynomial
functions.
40. Interpret the
Linear,
parameters in a
Quadratic, &
linear or
Exponential
exponential
Models*
function in terms of Interpret
a context.
expressions for
functions in
terms of the
situation they
model. (Linear
and exponential
of form f(x) = bx
+ k.)
Interpreting
Categorical &
Quantitative
Data
Summarize,
represent, and
interpret data on
a single count or
measurement
variable.
Statistics &
Probability
S-ID.1
Students:
Given numerical data in
any form (e.g., all real
numbers),
Franklin County Schools
Parameters
Create a function
that models the situation,
Define and justify the
parameters (all constants
used to define the
function) in terms of the
original context.
Functions
F-LE.5
41. Represent data
with plots on the
real number line
(dot plots,
histograms, and
box plots).
Students:
Given a contextual
situation that may be
modeled by a linear or
exponential function,
Organize and display
the data using plots on a
real number line,
including dot plots,
histograms, and box
plots.
Students know:
Students are able to: Students understand
that:
Key components
of linear and
exponential
functions.
Communicate the
meaning of defining
values (parameters
and variables) in
functions used to
model contextual
situations in terms of
the original context.
Dot plots
Students know:
Histograms
Techniques for
Choose from
constructing dot
among data display
plots, histograms,
(dot plots,
and box plots from a histograms, box
set of data.
plots) to convey
significant features of
data,
Box plots
Sense making in
mathematics requires
that meaning is
attached to every
value in a
mathematical
expression.
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
Students are able to: Students understand
that:
Accurately
construct dot plots,
histograms, and box
Click below to
Sets of data can access all ALEX
be organized and
resources aligned
displayed in a variety to this standard.
of ways each of which
provides unique
 ALEX
perspectives of the
Resources
data set,
Data displays help
in conceptualizing
ALGI
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
plots.
42. Use statistics
appropriate to the
shape of the data
distribution to
compare center
(median, mean)
and spread
(interquartile
range, standard
deviation) of two or
more different data
sets.
43. Interpret
differences in
shape, center, and
spread in the
context of the data
sets, accounting for
possible effects of
extreme data
points (outliers).
Understanding
ideas and in solving
problems.
Interpreting
Categorical &
Quantitative
Data
Summarize,
represent, and
interpret data on
a single count or
measurement
variable.
Statistics &
Probability
S-ID.2
Students:
Given two or more
different data sets,
Center
Students know:
Students are able to: Students understand
that:
Median
Compare the center
(median, mean) and the
spread (interquartile
range, standard
deviation) of the data
sets to describe
differences and
similarities of the data
sets.
Mean
Techniques to
calculate the center
and spread of data
sets,
Accurately find
the center (median
and mean) and
spread (interquartile
range and standard
deviation) of data
sets,
Interpreting
Categorical &
Quantitative
Data
Summarize,
represent, and
interpret data on
a single count or
measurement
variable.
Statistics &
Probability
S-ID.3
Outliers
Students:
Given multiple data sets,
Students know:
Center
Recognize and
explain the differences in Shape
shape, center, and
spread, including effects
Spread
of outliers.
Techniques to
calculate the center
and spread of data
sets,
Spread
Interquartile
range
Standard
deviation
Methods to
compare data sets
based on measures
of center (median,
mean) and spread
(interquartile range
and standard
deviation) of the
data sets.
Franklin County Schools
Categorical data
Students:
Given categorical data for
Multiple data sets
can be compared by
making observations
about the center and
spread of the data,
The center and
spread of multiple
data sets are used to
justify comparisons of
the data.
Students know:
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Resources
Students are able to: Students understand
that:
Accurately
identify differences in Differences in the
shape, center, and
shape, center, and
spread when
spread of data sets
comparing two or
can result from
more
data
sets,
various causes,
Methods to
including outliers and
compare attributes
clustering.
(e.g. shape, median, Accurately
mean, interquartile identify outliers,
range, and standard
deviation) of the
Explain, with
data sets,
justification, why
Methods to
identify outliers.
44. Summarize
Interpreting
categorical data for Categorical &
Present viable
arguments and
critique arguments of
others from the
comparison of the
center and spread of
multiple data sets.
Resources
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Resources
there are differences
in the shape, center,
and spread of data
sets.
Students are able to: Students understand Click below to
that:
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ALGI
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
two categories in
two-way frequency
tables. Interpret
relative frequencies
in the context of
the data (including
joint, marginal, and
conditional relative
frequencies).
Recognize possible
associations and
trends in the data.
Quantitative
Data
Summarize,
represent, and
interpret data on
two categorical
and quantitative
variables. (Linear
focus, discuss
general
principle.)
Statistics &
Probability
S-ID.5
Evidence of Student
Attainment
two categories,
Create two-way
frequency tables,
Find relative
frequencies using ratios,
Recognize and justify
possible relationships and
patterns in the data by
examining the joint,
marginal, and conditional
relative frequencies.
Teacher
Vocabulary
Two-way
frequency Tables
Relative
frequency
Joint frequency
Marginal
frequency
Conditional
relative frequency
Knowledge
Skills
Characteristics of Accurately
a two-way frequency construct frequency
table,
tables,
Methods for
Accurately
converting frequency construct relative
tables to relative
frequency tables,
frequency tables,
That the sum of
the frequencies in a
row or a column
gives the marginal
frequency,
Techniques for
finding conditional
relative frequency,
Accurately find
the joint, marginal,
and conditional
relative frequencies,
Recognize and
explain possible
associations and
trends in the data.
Techniques for
finding the joint
frequency in tables.
45. Represent data
on two quantitative
variables on a
scatter plot, and
describe how the
variables are
related.
a.
Fit a function
to the data;
use functions
fitted to data
to solve
problems in
the context of
the data. Use
Interpreting
Categorical &
Quantitative
Data
Summarize,
represent, and
interpret data on
two categorical
and quantitative
variables. (Linear
focus, discuss
general
principle.)
Statistics &
Probability
S-ID.6
given functions
Franklin County Schools
Students:
Given a data set of two
quantitative variables,
Create a scatter plot
(with and without
technology),
Create a function
which best fits the data
(linear, quadratic, and
exponential models),
Compare the graphs
of the scatter plot and
function to see the fit to
Quantitative
variables
Scatter plot
Residuals
Understanding
Two-way
frequency tables may
be used to represent
categorical data,
Resources
resources aligned
to this standard.

ALEX
Resources
Relative
frequency tables
show the ratios of the
categorical data in
terms of joint,
marginal, and
conditional relative
frequencies,
Two-way
frequency or relative
frequency tables may
be used to aid in
recognizing
associations and
trends in the data.
Students know:
Students are able to: Students understand
that:
Techniques for
creating a scatter
plot,
Accurately create
a scatter plot of data, Functions are
used to create
Click below to
Correctly choose equations
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representative of
a function to fit the
resources aligned
ordered pairs of data,
scatter plot,
to this standard.
Techniques for
fitting various
functions (linear,
quadratic,
Make reasonable
exponential) to data, assessments on the
fit of the function to
Methods for
the data by
using residuals to
examining residuals,
judge the closeness
of the fit of the
Accurately fit a
Residuals may be
c.
examined to analyze
how well a function
fits the data,
When a linear
association is
ALEX Resources
ALGI
CCRS Standard
b.
c.
Mathematics CCRS Standards and Alabama COS
Standard ID
or choose a
function
suggested by
the context.
Emphasize
linear,
quadratic, and
exponential
models.
Evidence of Student
Attainment
the original data,
Teacher
Vocabulary
Knowledge
Skills
Understanding
function to the
original data.
linear function to data suggested, a linear
when there is
function can be fit to
evidence of a linear the scatter plot to aid
association.
in modeling the
relationship.
Students know:
Students are able to: Students understand
that:
Fit a linear function
to the data if the scatter
plot indicates a linear
association.
Resources
Informally
assess the fit
of a function
by plotting and
analyzing
residuals.
Fit a linear
function for a
scatter plot
that suggests
a linear
association.
46. Interpret the
slope (rate of
change) and the
intercept (constant
term) of a linear
model in the
context of the data.
Interpreting
Categorical &
Quantitative
Data
Interpret linear
models.
Statistics &
Probability
S-ID.7
Students:
Given a contextual
situation that yields a
data set of ordered pairs
that suggests a linear
relationship,
Fit a linear function
to the data,
Determine the slope
and intercept of that
function,
Interpret the slope
and intercept of the
linear function in the
context of the data.
Franklin County Schools
Techniques for
creating a scatter
plot,
Accurately create
a scatter plot of data Linear functions
are used to model
data that have a
Accurately fit
relationship that
Techniques for
linear functions to
closely resembles a
fitting a linear
scatter plots,
linear relationship,
function to a scatter
plot,
Correctly find the
slope and intercept of The slope and
intercept of a linear
Methods to find linear functions,
function may be
the slope and
interpreted as the
intercept of a linear Justify and
rate of change and
function.
explain the relevant
the zero point
connections slope and (starting point).
intercept of the linear
function to the data.
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Resources
ALGI
CCRS Standard
47. Compute (using
technology) and
interpret the
correlation
coefficient of a
linear fit.
Mathematics CCRS Standards and Alabama COS
Standard ID
Interpreting
Categorical &
Quantitative
Data
Interpret linear
models.
Statistics &
Probability
S-ID.8
Evidence of Student
Attainment
Teacher
Vocabulary
Correlation
Students:
Given a contextual
coefficient
situation that yields a
data set of ordered pairs Linear fit
that suggests a linear
relationship,
Find the correlation
coefficient of a linear fit
using technology,
Knowledge
49. Describe events
as subsets of a
sample space (the
set of outcomes),
using
characteristics (or
categories) of the
outcomes, or as
unions,
intersections, or
complements of
other events (“or,”
“and,” “not”).
Interpreting
Categorical &
Quantitative
Data
Interpret linear
models.
Statistics &
Probability
S-ID.9
Students:
Given situations where
two variables are
correlated,
Conditional
Probability &
the Rules of
Probability
Understand
independence
and conditional
probability and
use them to
interpret data.
(Link to data
Subsets
Students:
Given scenarios involving
chance,
Sample space
Franklin County Schools
Causation
Explain why the
correlation does not
mean that changes in
one variable cause the
changes in the other
variable.
Determine the
Unions
sample space and a
variety of simple and
Intersections
compound events that
may be defined from the
Complements
sample space,
from simulations
or experiments.) Use the language of
Statistics &
Correlation
union, intersection, and
complement
Understanding
Resources
Students know:
Students are able to: Students understand
that:
The relationship
among the shape of
the scatter plot, the
value of the
correlation
coefficient, and the
strength of the linear
relationship in the
data.
Use technology
accurately to find the Correlation
Click below to
correlation coefficient, coefficients are used access all ALEX
to measure the
resources aligned
strength of the linear to this standard.
Justify and
relationship in a set of
communicate
conclusions about the data,
 ALEX
relationship of the
Resources
data based upon the Technology aids
correlation coefficient. in finding the
correlation coefficient
of a linear fit.
Communicate the
relationship of the
correlation coefficient to
the data.
48. Distinguish
between
correlation and
causation.
Skills
Students know:
Students are able to: Students understand
that:
Click below to
Interpretations of Justify that two
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correlation and
variables that show a Correlation may resources aligned
causation.
strong correlation do be an indication of
to this standard.
not necessarily show causation, but may
that one is the cause also be an indication
 ALEX
of the other.
of influence of other
Resources
variables or
coincidence.
Students know:
Students are able to: Students understand
that:
Methods for
describing events
from a sample space
using set language
(subset, union,
intersection,
complement).
Interpret the
given information in
the problem,
Accurately
determine the
probability of the
scenario.
Set language can Click below to
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be useful to define
events in a probability resources aligned
to this standard.
situation and to
symbolize
relationships of
 ALEX
events.
Resources
ALGI
CCRS Standard
50. Understand
that two events A
and B are
independent if the
probability of A and
B occurring
together is the
product of their
probabilities, and
use this
characterization to
determine if they
are independent.
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Probability
S-CP.1
appropriately to define
events.
Conditional
Probability &
the Rules of
Probability
Understand
independence
and conditional
probability and
use them to
interpret data.
(Link to data
Independent
Students:
Given scenarios involving
two events,
Explain the meaning
of independence from a
formula perspective
P(A∩B) = P(A) x P(B)
and from the intuitive
notion that A occurring
has no impact on
from simulations whether B occurs or not,
or experiments.)
Statistics &
Compare these two
Probability
interpretations within the
S-CP.2
context of the scenario.
Knowledge
Students know:
Skills
Resources
Students are able to: Students understand
that:
Methods to find Interpret the
probability of simple given information in
and compound
the problem,
events.
Accurately
determine the
probability of simple
and compound
events,
Accurately
calculate the product
of the probabilities of
two events.

Understanding
Events are
Click below to
independent if one
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occurring does not
affect the probability resources aligned
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of the other
occurring, and that
this may be
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demonstrated
Resources
mathematically by
showing the truth of
P(A∩B) = P(A) x P(B).
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Resources