Download CCRS Standard - the Franklin County Schools Website

PRE CAL
CCRS Standard
1. (+) Find the
conjugate of a
complex number; use
conjugates to find
moduli and quotients
of complex numbers.
Mathematics CCRS Standards and Alabama COS
Standard ID
The Complex
Number System
Perform arithmetic
operations with
complex numbers.
Number/Quantity
N-CN.3
Evidence of
Student
Attainment
Students:
Given a complex
number,

Teacher Vocabulary



Find the
conjugate
and the
modulus.
Knowledge

Students know:
Given a quotient of
complex numbers,
Franklin County Schools
Multiply by 1
by
multiplying
the
numerator
and
denominator
by the
conjugate of
the
denominator
to write the
indicated
quotient as
a single
complex
number.
Understanding
Resources
Students
understand that:
Conjugate
Complex
number
Modulus/Moduli


Skills


The
definition
of the
conjugate
of a
complex
number.
A complex Students are able
to:
number
divided by
itself
 Find the
equals 1.
product of
two
The
complex
product of
numbers.
a complex
number
and its
conjugate
is a real
number
(the
square of
the
modulus).

The
modulus of
a complex
number
correspond
s to the
magnitude
of a vector
and,
therefore,
is useful in
the
geometric
representa Click below to
tion of
access all ALEX
complex resources aligned
numbers. to this standard.
Mathemati
cal
 ALEX
convention
Resources
is that
radical
expression
s are not
left in
denominat
ors to
facilitate
numerical
approxima
tions;
therefore,
since the i
is equal to
the square
root of -1,
convention
al form
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding

2. (+) Represent
complex numbers on
the complex plane in
rectangular and polar
form (including real
and imaginary
numbers), and explain
why the rectangular
and polar forms of a
given complex number
The Complex
Number System
Represent complex
numbers and their
operations on the
complex.
Number/Quantity
N-CN.4
Franklin County Schools
Students:
Given any complex
number,

Graphically
represent
the number
on the
complex
plane in
Students know:

In the
complex
plane the
horizontal
axis is the
real axis
(a) and
the vertical
Students are able
to:

Use
trigonomet
ry to find
the
measures
of angles
and
says that i
does not
appear in
the
denominat
or of a
fraction.
Different
forms of a
complex
number
quotient
(indicated
quotient,
single
complex
number)
may be
more
useful for
various
purposes.
Students
understand that:

Resources
Click below to
access all ALEX
resources aligned
A complex
to this standard.
number (a
+bi) can
 ALEX
be
Resources
graphed in
a
rectangula
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
represent the same
number.

rectangular
and polar
form.
Justify that
the
rectangular
(a+bi) and
polar forms
(z = r(cosθ
+isinθ)) are
equivalent
Teacher Vocabulary
Knowledge



3. (+) Represent
addition, subtraction,
multiplication, and
conjugation of
complex numbers
geometrically on the
complex plane; use
properties of this
representation for
computation.
Example: (-1 + √3 i)3
= 8 because (-1 + √3
i) has modulus 2 and
The Complex
Number System
Represent complex
numbers and their
operations on the
complex.
Number/Quantity
N-CN.5
Franklin County Schools
axis is the
imaginary
axis (b).
Trigonome
tric
techniques
for finding
measures
of angles
and
coordinate
s on the
unit circle.
The
characteris
tics of the
polar
coordinate
system.
Technique
s for
plotting
polar
coordinate
s.
Students know:

Complex
numbers
are
represente
d
geometrica
lly in the
complex
plane with
the real
part
Skills



coordinate
s on the
unit circle.
Use the
Pythagore
an
Theorem
to find the
lengths of
sides of a
right
triangle.
Convert
between
polar and
rectangula
r forms.
Plot polar
coordinate
s.
Students are able
to:

Add,
subtract,
and
multiply
the
component
parts of
complex
numbers
to find
Understanding


Resources
r
coordinate
system as
(a, b).
A complex
number
may be
represente
d in the
plane
using
equivalent
polar and
rectangula
r
coordinate
s.
Different
representa
tions of a
complex
number
may be
more
useful for
various
purposes.
Students
understand that:

Click below to
Different access all ALEX
representa resources aligned
to this standard.
tions of
mathemati
cal
 ALEX
concepts
Resources
(e.g.,
algebraic
and
geometric
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
argument 120°.


Franklin County Schools
measured
on the xaxis and
the
imaginary
is
represente
d on the yaxis.
Complex
numbers
can be
added or
subtracted
by
combining
the real
parts and
the
imaginary
parts or by
using
vector
procedures
geometrica
lly (end-toend,
parallelogr
am rule).
The
product of
complex
numbers
in polar
form may
be found
by
multiplying
the
magnitude
s and
adding the
Skills




sums,
differences
, and
products.
Identify
the
conjugate
of a
complex
number
and use
this as a
computatio
nal aid,
e.g., to
find a
quotient of
complex
numbers.
Represent
complex
numbers in
the
complex
plane.
To add
and
subtract
complex
numbers
geometrica
lly.
Multiply
complex
numbers in
polar form.
Understanding

representa
tion of
complex
numbers)
reveal
different
features of
the
concept
and each
may
facilitate
computatio
n and
sense
making in
different
settings.
Mathemati
cs is a
coherent
whole and
structure
within
mathemati
cs allows
for
procedures
from one
area to be
used in
another
(e.g.,
coordinate
geometry
and the
complex
plane,
vectors
and
complex
numbers,
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
arguments
.
4. (+) Calculate the
distance between
numbers in the
complex plane as the
modulus of the
difference, and the
midpoint of a segment
as the average of the
numbers at its
endpoints.
The Complex
Number System
Represent complex
numbers and their
operations on the
complex.
Number/Quantity
N-CN.6

5. Determine
numerically,
algebraically, and
graphically the limits
of functions at specific
values and at infinity.
(Alabama)
The Complex
Number System
Represent complex
numbers and their
operations on the
complex.
Number/Quantity
Franklin County Schools
or plotting
of a
conjugate
of a
complex
number in
transforma
tional
geometry).
Students
understand that:
Students know:

Students are able
Complex to:
numbers
can be
 Add and
represente
subtract
d
complex
geometrica
numbers.
lly.
 Find the
Complex
modulus of
numbers
a complex
can be
number.
added and

Represent
subtracted
complex
either
numbers
geometrica
geometrica
lly or
lly.
algebraical
ly.
Students know:

How to
graph the
families of
functions.
Resources
Students are able
to:

Identify
continuity
and
discontinui

Representi
ng
complex
numbers
on a
rectangula
r
coordinate
system
allows the
use of
techniques
developed
for real
numbers
to find
distances
and
midpoints.
Students
understand:

That
functions
have
various
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
N-CN.10

Apply limits
in problems
involving
convergence
and
divergence.
(Alabama)
6. (+) Recognize
vector quantities as
having both
magnitude and
direction. Represent
vector quantities by
directed line
segments, and use
appropriate symbols
for vectors and their
magnitudes (e.g., v,
|v|, ||v||, v).
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge


Vector and Matrix
Quantities
Perform operations on
vectors.
Number/Quantity
N-VM.1
Franklin County Schools
How to
factor,
simplify,
and
rationalize
functions.
How to
evaluate
functions
at a given
input
value.
Students know:


The
difference
between a
ray and a
directed
line
segment.
When
drawing a
vector on
the xyplane,
magnitude
is
represente
d by a
distance.
Skills

ty (if
discontinu
ous,
identify
the type).
Calculate
limits.
Students are able
to:



Locate
vectors on
the xyplane.
Use the
unit circle
to find
trigonomet
ric ratios
for values
in all four
quadrants.
Use the
Pythagore
an
Theorem
to find the
lengths of
sides of a
right
Understanding

types of
continuity
and
discontinui
ty.
That limits
can be
determine
d
numericall
y,
graphically
, and
algebraicall
y.
Resources
Resources
Students
understand:


Vectors
represent
magnitude
and
direction. Click below to
access all ALEX
When
resources aligned
vectors are
to this standard.
represente
d in
 ALEX
component
Resources
form,
ensuing
computatio
ns and
application
s can be
accomplish
ed.
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
triangle.
7. (+) Find the
components of a
vector by subtracting
the coordinates of an
initial point from the
coordinates of a
terminal point.
Vector and Matrix
Quantities
Perform operations on
vectors.
Number/Quantity
N-VM.2
Students
understand:

Students know:

8. (+) Solve problems
involving velocity and
other quantities that
can be represented by
vectors.
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.3
Franklin County Schools
If a vector
is
transposed
in the xyplane, it
retains its
magnitude
and
direction.
Students know:


The
interpretati
on of
forces as
vectors.
Vectors
can be
Students are able
to:

Transpose
a vector
from one
position to
another
position in
the xyplane.
Students are able
to:


Write and
represent
a given
force as a
vector.
Combine

Vectors
having the
same
magnitude
and
direction
are
equivalent Click below to
regardless access all ALEX
of where resources aligned
they are in to this standard.
the xyplane.
 ALEX
Resources
Vectors in
standard
position
have a
terminal
point that
is equal to
the
component
s of the
vector.
Students
understand:

Click below to
access all ALEX
resources aligned
The result to this standard.
of
combining
 ALEX
multiple
Resources
vector
forces
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
used to
represent
forces.
9. (+) Add and
subtract vectors.



(+) Add
vectors endto-end,
componentwise, and by
the
parallelogram
rule.
Understand
that the
magnitude of
a sum of two
vectors is
typically not
the sum of
the
magnitudes.
(+) Given
two vectors
in magnitude
and direction
form,
determine
the
magnitude
and direction
of their sum.
(+)
Understand
vector
subtraction v
– w as v +
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.4
Franklin County Schools
Understanding
vector
quantities.
Resources
creates a
net
magnitude
and
direction.
Students
understand:
Students know:


The
aspects of
end-toend,
componen
twise, and
the
parallelogr
am rule
involving
vectors.
The
additive
inverse of
a vector
has the
same
magnitude
but the
opposite
direction
(θ +
180⁰).
Students are able
to:




Draw and
find the
diagonal of
a
parallelogr
am.
Represent
vectors on
an xyplane.
Find the
component
s of a
vector
given the
direction
and
magnitude
.
Find the
additive
inverse of
a vector.



There are
multiple
ways to
find the
sum and
difference
of a pair of
vectors.
The
magnitude
Click below to
of the sum
access all ALEX
of two
resources aligned
vectors will
to this standard.
not be the
same as
 ALEX
the sum of
Resources
the
magnitude
s unless
the vectors
are in the
same
direction.
The vector
with the
larger
magnitude
will have
the
greatest
effect on
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
(–w), where
–w is the
additive
inverse of w,
with the
same
magnitude as
w and
pointing in
the opposite
direction.
Represent
vector
subtraction
graphically by
connecting
the tips in
the
appropriate
order, and
perform
vector
subtraction
componentwise.
the result.
10. (+) Multiply a
vector by a scalar.

Vector and Matrix
Quantities
Perform operations on
matrices and use
(+)
matrices in
Represent
applications.
scalar
multiplication Number/Quantity
graphically by N-VM.5
scaling
vectors and
possibly
reversing
their
direction;
perform
Franklin County Schools
Resources
Students are able
to:
Students know:


The
representa
tion of
vectors
graphically
on the xyplane.

Find the
component
s of a
vector.
Find the
magnitude
and
direction
of a
vector.
Students
understand:

Scalar
multiplicati
on results
in a
dilation of
the
original
vector
where a
scalar
greater
than 1
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard

Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
scalar
multiplication
componentwise, e.g., as
c(vx, vy) =
(cvx, cvy).
(+) Compute
the
magnitude of
a scalar
multiple cv
using ||cv||
= |c|v.
Compute the
direction of
cv knowing
that when
|c|v ≠ 0, the
direction of
cv is either
along v (for c
> 0) or
against v (for
c < 0).
11. (+) Use matrices
to represent and
manipulate data, e.g.,
to represent payoffs or
incidence relationships
in a network.
Understanding


Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.6
Franklin County Schools
Students know:

The
aspects of
a matrix
with
regard to
entries,
rows,
columns,
Students are able
to:

Translate
data into a
matrix.
would
increase
the
magnitude
and a
scalar from
0 to 1
would
decrease
the
magnitude
.
A negative
scalar
would
reverse
the
direction
(θ + 180⁰).
The
absolute
value of
the scalar
has the
resulting
affect on
the
magnitude
.
Students
understand:

Resources
That a
matrix is a
tool that
can help to
organize,
manipulate
, and
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
dimension
s,
elements,
and
subscript
notations.
12. (+) Multiply
matrices by scalars to
produce new matrices,
e.g., as when all of
the payoffs in a game
are doubled.
13. (+) Add, subtract,
and multiply matrices
of appropriate
dimensions.
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.7
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.8
Franklin County Schools
interpret
data.
Students are able
to:
Students know:



The
aspects of
a matrix
with
regard to
entries,
rows,
columns,
dimension
s,
elements,
and
subscript
notations.
Students know:
The
aspects of
a matrix
with
regard to
entries,
rows,
Resources
Write
percents
as
decimals.
Increase
or
decrease
an amount
by
multiplying
by a
percent
(i.e.,
increase of
10%
would
multiply by
1.1).
Students are able
to:

Strategicall
y choose
and apply
appropriat
e
representa
Students
understand:


Multiplying
a matrix
by a scalar
affects
every
Click below to
element in access all ALEX
the matrix resources aligned
equally.
to this standard.
Scalar
multiplicati
 ALEX
on is a tool
Resources
that allows
all
elements
of a matrix
to be
changed in
a simple
manner.
Students
understand:

Matrix
addition
and
subtraction
may be
performed
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
columns,
dimension
s,
elements,
and
subscript
notations.
tions of
matrices
on which
arithmetic
operations
can be
performed.
Understanding


14. (+) Understand
that, unlike
multiplication of
numbers, matrix
multiplication for
square matrices is not
a commutative
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
Franklin County Schools
Students know:

Commutati
ve and
associative
properties
of
Students are able
to:

Add,
subtract,
and
multiply
only if the
matrices
have the
same
dimensions
.
Matrix
multiplicati
on can be
performed
only when
the
number of
columns in
the first
matrix
equal the
number of
rows in the
second
matrix.
There are
many
contextual
situations
where
arithmetic
operations
on
matrices
allow us to
solve
problems.
Students
understand:

Resources
Click below to
access all ALEX
resources aligned
Properties to this standard.
that apply
to other
 ALEX
number
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
operation, but still
N-VM.9
satisfies the
associative and
distributive properties.
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge


15. (+) Understand
that the zero and
identity matrices play
a role in matrix
addition and
multiplication similar
to the role of 0 and 1
in the real numbers.
The determinant of a
square matrix is
nonzero if and only if
the matrix has a
multiplicative inverse.
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.10
Understanding
matrices.
Resources
systems do
not
necessarily
apply with
matrices.
Resources
Students
understand:
Students know:



Franklin County Schools
multiplicati
on.
Distributiv
e property.
The
aspects of
a square
matrix.
Skills
The
additive
Students are able
and
to:
multiplicati
ve identity
 Find the
properties
determina
for real
nt of a
numbers.
square
The
matrix.
aspects of
 Find the
the zero
multiplicati
and
ve inverse
identity
of a
matrices.
square
A matrix
matrix.
multiplied
 Add and
by its
multiply
multiplicati
matrices.
ve inverse
equals the
identity
matrix.



Identity
properties
that apply
to other
number
systems
apply to
matrices.
The
multiplicati
ve inverse
property
that
applies to
other
number
systems
applies to
matrices.
A matrix
with a
determina
nt equal to
zero does
not have a
multiplicati
ve inverse
analogous
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding

16. (+) Multiply a
vector (regarded as a
matrix with one
column) by a matrix of
suitable dimensions to
produce another
vector. Work with
matrices as
transformations of
vectors.
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.11
Students know:




Franklin County Schools
Conditions
under
which
matrix
multiplicati
on is
defined.
Technique
s for
adding
and
multiplying
matrices.
Technique
s for scalar
multiplicati
on.
Technique
s for
performing
translation
s,
rotations,
reflections,
Students are able
to:




Write a
vector in
matrix
notation.
Determine
if matrix
multiplicati
on is
defined for
a given
product of
a matrix
by a
vector.
Multiply a
vector by a
matrix.
Add,
multiply,
and
perform
scalar
multiplicati
Resources
to zero in
the real
number
system not
having a
multiplicati
ve inverse.
Division by
zero in the
real
number
system is
undefined.
Students
understand:

Mathemati
cal
representa
tions such
as vectors
and
matrices
may be
used in a
wide
variety of
settings to
model and
solve real
world
problems
and this
modeling
is
facilitated
by fluent
use of
techniques
for
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
and
dilations.
Understanding
on on
matrices.

17. (+) Work with 2 ×
2 matrices as a
transformations of the
plane, and interpret
the absolute value of
the determinant in
terms of area.
Vector and Matrix
Quantities
Perform operations on
matrices and use
matrices in
applications.
Number/Quantity
N-VM.12
Students know:



Franklin County Schools
Conditions
under
which
matrix
multiplicati
on is
defined.
Technique
s for
adding
and
multiplying
matrices.
Technique
Students are able
to:


Find the
area of a
parallelogr
am in a
coordinate
plane by
composing
or
decomposi
ng shapes.
Multiply a
vector by a
matrix.
Resources
working
with these
representa
tions.
Transform
ations of
multiple
vectors
may be
accomplish
ed by the
use of
matrix
multiplicati
on as a
means of
transformi
ng all
vectors at
the same
time in a
similar
manner.
Students
understand:

Multiplicati
Click below to
on of a
vector by a access all ALEX
resources aligned
2X2
matrix is a to this standard.
transforma
tion on the
 ALEX
coordinate
Resources
plane and
this
transforma
tion can be
represente
d in matrix
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge

18. (+) Represent a
system of linear
equations as a single
matrix equation in a
vector variable.
Reasoning with
Equations &
Inequalities
Solve systems of
equations.
Algebra
Franklin County Schools
s for
performing
translation
s and
rotations.
Procedures
for finding
the value
of the
determina
nt of a 2 X
2 matrix.
Students know:

The
structure
of a matrix
equation is
Skills

Understanding
Find the
value of
the
determina
nt of a 2 X
2 matrix.

Students are able
to:

Transform
a system
of linear
form or
algebraicall
y by
performing
the matrix
multiplicati
on.
When a
polygon is
transforme
d by 2 X 2
matrix
multiplicati
on in the
coordinate
plane, the
area of the
image
polygon
may be
found by
multiplying
the
absolute
value of
the
determina
nt of the
transforma
tion matrix
by the
area of the
original
polygon.
Students
understand that:

Resources
Multiple
representa
tions of
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
A-REI.8
19. (+) Find the
inverse of a matrix if it
exists and use it to
solve systems of linear
equations (using
technology for
matrices of dimension
3 × 3 or greater).
Reasoning with
Equations &
Inequalities
Solve systems of
equations.
Algebra
A-REI.9
the
product of
the
coefficient
matrix on
the left
and the
vector
variable on
the right is
equal to
the
constant
vector.
Students know:



Franklin County Schools
Skills
Procedures
for
converting
systems of
equations
to matrix
equations.
Conditions
that
determine
the inverse
of a matrix
exists.
Technique
s for
determinin
g the
Understanding
equations
to a matrix
equation.

Students are able
to:


Perform
row
operations
on a
matrix to
find the
inverse of
the matrix.
Efficiently
and
accurately
find the
inverse of
a matrix
using
technology
Resources
linear
systems
are
needed to
facilitate
work with
more
complex
systems.
Matrix
representa
tion allows
for
solutions
with
systems
containing
any
number of
variables.
Resources
Students
understand that:


The
solutions
to a matrix
equation
are the
solutions
to the
system of
equations
that
produced
the matrix
equation.
Solving a
matrix
equation is
analogous
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge


Skills
inverse of
a matrix
(including
using
technology
).
Process for
using the
inverse of
a matrix to
find the
solution to
a matrix
equation
(2 x 2).
Matrix
multiplicati
on is not
commutati
ve,
specifically
Understanding
.

Resources
to solving
a linear
equation.
Technolog
y is a
useful tool
that
facilitates
investigati
on and,
once the
initial
process is
understoo
d, helps
find
solutions
of more
complex
problems.
A-1AX =
A-1B not
A-1AX =
BA-1.
20. Create graphs of
conic sections,
including parabolas,
hyperbolas, ellipses,
circles, and
degenerate conics,
from second-degree
equations. (Alabama)
Example: Graph x2 6x + y2 - 12y + 41 =
0 or y2 - 4x + 2y + 5
= 0.

Conic Sections
Understand the
graphs and equations
of conic sections.
(Alabama)
Functions
F-IF.7D
Formulate
equations of
Franklin County Schools
Students:
Given a symbolic
representation of a
function (including
linear, quadratic,
square root, cube
root, piecewisedefined functions,
polynomial,
exponential,
logarithmic,
trigonometric, and
(+) rational),
Produce an
Students know:




Vertex
form of a
parabola.
Standard
form of a
circle.
Vertex and
axis of
symmetry
of a
parabola.
Completin
Students are able
to:


Create
graphs of
parabolas
from
second
degree
equations.
Create
graphs of
hyperbolas
from
Students
understand:


Click below to
The
access all ALEX
determinin
resources aligned
g
to this standard.
characteris
tics of
 ALEX
conic
Resources
section
graphs.
The
parameter
s that
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
conic sections
from their
determining
characteristic
s. (Alabama)
Example:
Write the
equation of
an ellipse
with center
(5, -3), a
horizontal
major axis of
length 10,
and a minor
axis of length
4.
Answer: (x 5)/
(y + 3)/
25 +
4
= 1.
Evidence of
Student
Attainment
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:


Franklin County Schools
quadratic or
linear
(intercepts,
maxima, and
minima),
square root,
cube root,
and
piecewisedefined
functions,
including
step
functions
and absolute
value
functions
(descriptive
features
such as the
values that
are in the
range of the
Teacher Vocabulary
Knowledge

g the
square.
Factoring
a
quadratic
function.
Skills




second
degree
equations.
Create
graphs of
ellipses
from
second
degree
equations.
Create
graphs of
circles
from
second
degree
equations.
Create
graphs of
degenerat
e conics
from
second
degree
equations.
Formulate
equations
of conic
sections
given a set
of
determinin
g
characteris
tics.
Understanding
determine
the graph.
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment




21. (+) Graph rational
functions, identifying
zeros and asymptotes
when suitable
factorizations are
available, and showing
end behavior.
Interpreting
Functions
Analyze functions
using different
representations.
(Logarithmic and
trigonometric
functions.)
Franklin County Schools
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
function and
those that
are not),
polynomial
(zeros when
suitable
factorization
s are
available,
end
behavior),
(+) rational
(zeros and
asymptotes
when
suitable
factorization
s are
available,
end
behavior),
exponential
and
logarithmic
(intercepts
and end
behavior),
trigonometri
c functions
(period,
midline, and
amplitude).
Students:
Given a symbolic
representation of a
function (including
linear, quadratic,
square root, cube
root, piecewisedefined functions,
Students know:


Technique
s for
graphing,
Key
features of
Students are able
to:

Identify
the type of
function
from the
Students
understand that:

Key
features
are
different
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Functions
F-IF.7E
Evidence of
Student
Attainment
polynomial,
exponential,
logarithmic,
trigonometric, and
(+) rational),
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:


Franklin County Schools
quadratic or
linear
(intercepts,
maxima, and
minima),
square root,
cube root,
and
piecewisedefined
functions,
including
step
functions
and absolute
value
Teacher Vocabulary
Knowledge
Skills
graphs of
functions.


symbolic
representa
tion,
Manipulate
expression
s to reveal
important
features
for
identificati
on in the
function,
Accurately
graph any
relationshi
p.
Understanding

depending
on the
function,
Identifying
key
features of
functions
aid in
graphing
and
interpretin
g the
function.
Resources
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment




Franklin County Schools
functions
(descriptive
features
such as the
values that
are in the
range of the
function and
those that
are not),
polynomial
(zeros when
suitable
factorization
s are
available,
end
behavior),
(+) rational
(zeros and
asymptotes
when
suitable
factorization
s are
available,
end
behavior),
exponential
and
logarithmic
(intercepts
and end
behavior),
trigonometri
c functions
(period,
midline, and
amplitude).
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
PRE CAL
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
22. (+) Compose
functions.
Example: If T(y) is the
temperature in the
atmosphere as a
function of height, and
h(t) is the height of a
weather balloon as a
function of time, then
T(h(t)) is the
temperature at the
location of the
weather balloon as a
function of time.
Building Functions
Build a function that
models a relationship
between two
quantities.
Functions
F-BF.1C
Evidence of
Student
Attainment
Students:
Given a contextual
situation containing
two quantities,
Express a
relationship between
the quantities
through an explicit
expression using
function notation,
recursive process, or
steps for calculation,
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 to
combine the original
functions to model
the relationship of the
given quantities, (+)
standards not
covered.
Teacher Vocabulary
Explicit expression
Recursive process
Knowledge
Students know:

Decaying
exponential



Franklin County Schools
Technique
s for
expressing
functional
relationshi
ps (explicit
expression
,a
recursive
process, or
steps for
calculation
) between
two
quantities.
Technique
s to
combine
functions
using
arithmetic
operations
.
Technique
s to
compose
functions
using
algebraic
operations
.
Notation
for
function
compositio
n.
Skills
Understanding
Resources
Students are able
to:




Accurately
develop a
model that
shows the
functional Students
relationshi understand that:
p between
two
 Relationshi
quantities.
ps can be
Accurately
modeled
create a
by several
new
methods
function
(e.g.,
through
Click below to
explicit
arithmetic
expression access all ALEX
operations
resources aligned
or
of other
recursive to this standard.
functions.
process).
Accurately
 ALEX
 Arithmetic
create a
Resources
combinatio
new
ns and/or
function
compositio
through
n of
compositio
functions
n of other
may be
functions.
used to
When
improve
functions
the fit of a
are
model.
combined
to create a
new
function,
present an
argument
to show
how the
function
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
models the
relationshi
p between
the
quantities.
23. Determine the
inverse of a function
and a relation.
(Alabama)
Building Functions
Build new functions
from existing
functions.
Functions
F-BF.5A
Students:
Given a contextual
situation containing
two quantities,
Express a
relationship between
the quantities
through an explicit
expression using
function notation,
recursive process, or
steps for calculation,
Explicit expression
Recursive process
Decaying
exponential
Students know:

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 to
combine the original
functions to model
the relationship of the
given quantities, (+)
standards not
covered.
24. (+) Verify by
composition that one
Building Functions
Build new functions
Franklin County Schools
Students:
Given an invertible
Students
understand:
Inverse
The
Students are able
domain
and range to:
of a given
relation or
 Find the
function.
inverse of
Algebraic
relation/fu
properties.
nction.
Symmetry
about the
line y = x.
Students know:
Students are able
to:


The
graphical
and
algebraic
relationshi
p between
a function
and its
inverse.
The
process of
finding the
inverse of
a function.
Students
understand that:
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
Click below to
access all ALEX
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
function is the inverse from existing
of another.
functions.
Functions
F-BF.4B
Evidence of
Student
Attainment
function in algebraic
form,
Use algebraic
properties to find the
inverse of the given
function, (+)
standards not
covered.
Teacher Vocabulary
Knowledge





Franklin County Schools
Algebraic
properties
of equality
(Table 4).
Technique
s for
composing
functions.
The
compositio
n of a
function
and its
inverse is
the
identity
function.
When
(x,y) is a
point on
an
invertible
function,
(y,x) is a
point on
the
inverse.
In order
for a
function to
have an
inverse
function,
the
original
function
must have
a one-toone
correspon
Skills



Accurately
perform
algebraic
properties
to find the
inverse.
Accurately
identify
restrictions
on a noninvertible
function
that allow
it to be
invertible.
Accurately
find the
compositio
n of two
functions.
Understanding


Resources
resources aligned
to this standard.
The
inverse of
 ALEX
a function
interchang
Resources
es the
input and
output
values
from the
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.
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
dence.
25. (+) Read values of
an inverse function
from a graph or a
table, given that the
function has an
inverse.
Building Functions
Build new functions
from existing
functions.
Functions
F-BF.4C
Students:
Given an invertible
function in algebraic
form,
Use algebraic
properties to find the
inverse of the given
function, (+)
standards not
covered.
Inverse
Students know:





Franklin County Schools
Algebraic
properties
of equality
Students
(Table 4).
understand that:
Technique
Students are able
s for
 The
to:
composing
inverse of
functions.
a function
 Accurately
The
interchang
perform
compositio
es the
algebraic
n of a
input and
properties
function
output
to find the
and its
values
inverse.
inverse is
from the
 Accurately
the
original
identify
identity
function.
restrictions
function.

The
on a nonWhen
inverse of
invertible
(x,y) is a
a function
function
point on
must also
that allow
an
be a
it to be
invertible
function to
invertible.
function,
exist and
 Accurately
(y,x) is a
the
find the
point on
domain
compositio
the
may need
n of two
inverse.
to be
functions.
In order
restricted
for a
to make
function to
this occur.
have an
inverse
function,
the
original
function
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
must have
a one-toone
correspon
dence.
26. (+) Produce an
invertible function
from a non-invertible
function by restricting
the domain.
Building Functions
Build new functions
from existing
functions.
Functions
F-BF.4D
Students:
Given an invertible
function in algebraic
form,
Use algebraic
properties to find the
inverse of the given
function, (+)
standards not
covered.
Inverse
Students know:





Franklin County Schools
Algebraic
Students
properties
understand that:
of equality
Students are able
(Table 4).
 The
to:
Technique
inverse of
s for
a function
 Accurately
composing
interchang
perform
functions.
es the
algebraic
The
input and
properties
compositio
output
to
find
the
n of a
values
inverse.
function
from the
 Accurately
and its
original
identify
inverse is
function.
restrictions
the

The
on a nonidentity
inverse of
invertible
function.
a function
function
When
must also
that
allow
(x,y) is a
be a
it to be
point on
function to
invertible.
an
exist and
 Accurately
invertible
the
find the
function,
domain
compositio
(y,x) is a
may need
n of two
point on
to be
functions.
the
restricted
inverse.
to make
In order
this occur.
for a
function to
have an
inverse
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
function,
the
original
function
must have
a one-toone
correspon
dence.
27. (+) Understand
the inverse
relationship between
exponents and
logarithms, and use
this relationship to
solve problems
involving logarithms
and exponents.
Building Functions
Build new functions
from existing
functions.
Functions
F-BF.5
Students Know:




Franklin County Schools
Definition
of
Logarithm.
If bx = y,
then logb y
=x
Restriction
s on
domain
and range
of both
functions
(b > 0, b
not equal
to 1, y >
0).
Technique
s for
creating
graphs of
exponentia
l and
logarithmic
functions.
Situations
that can
be
modeled
Students are able
to:



Re-write
the
equivalent
inverse
function
from an
exponentia
l function
or
logarithm
function in
various
forms
(graphs,
tables, or
equations)
.
Model
contextual
situations
using
logarithmic
and
exponentia
l functions.
Solve
exponentia
l and
Students
understand that:


Equivalent
forms of a
function,
specifically
logarithmic
and
exponentia
l, may be
Click below to
useful at
access all ALEX
different
resources aligned
times to
to this standard.
solve
problems.
 ALEX
Although
Resources
the
algebraic
representa
tions look
different,
the
logarithmic
form and
exponentia
l form of
the same
relationshi
p are
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
by
exponentia
l and
logarithmic
functions.
28. Compare effects of
parameter changes on
graphs of
transcendental
functions. (Alabama)
Example: Explain the
relationship of the
graph y = ex-2 to the
graph y = ex.
Building Functions
Build new functions
from existing
functions.
Functions
F-BF.5B
logarithmic
equations
by
isolating
the
variable.
Resources
equivalent.
Students are able
to:

Students know:



Franklin County Schools
Understanding
Transform
ations
(shift,
stretch,
compress,
reflect).
Exponentia
l functions.
Logarithmi
c
functions.

Analyze
and
describe
the
transforma
tion of the
graphs,
equations, Students
and tables understand:
of values
Click below to
of
 The
access all ALEX
transcende
transforma resources aligned
ntal
tion of
to this standard.
functions
transcende
based on
ntal
 ALEX
parameter
functions
Resources
changes.
based on
Describe
parameter
the
changes.
differences
in the
domain,
range,
intercepts,
asymptote
s, end
behavior,
and
increasing
and
decreasing
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
intervals
as a result
of
parament
changes in
transcende
ntal
functions.
29. Determine the
amplitude, period,
phase shift, domain,
and range of
trigonometric
functions and their
inverses. (Alabama)
Trigonometric
Functions
Recognize attributes
of trigonometric
functions and solve
problems involving
trigonometry.
(Alabama)
Functions
F-TF.10
Students are able
to:

Students know:


How to
graph
trigonomet
ric
functions
including
amplitude,
midline,
and
period.
How to
graph the
inverses of
sine,
cosine,
and
tangent
functions.


Franklin County Schools
Find
amplitude,
period,
phase
shift,
domain,
and range
of the sine
and cosine
functions.
Find
period,
phase
shift,
domain
and range
of the
tangent,
cotangent,
secant,
and
cosecant
functions.
Find the
domain
and range
of the
trigonomet
ric inverse
Students
understand:



Amplitude,
period,
phase
shift,
domain,
and range
of the sine
and cosine
functions.
Period,
phase
shift,
domain
and range
of the
tangent,
cotangent,
secant,
and
cosecant
functions.
The
interval
defined by
the
domain
and range
of the
trigonomet
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
functions.
30. Use the sum,
difference, and halfangle identities to find
the exact value of a
trigonometric function.
(Alabama)
Trigonometric
Functions
Recognize attributes
of trigonometric
functions and solve
problems involving
trigonometry.
(Alabama)
Functions
F-TF.11
a.
b.
Trigonometric
Functions
Recognize attributes
of trigonometric
functions and solve
problems involving
Solve application- trigonometry.
(Alabama)
based problems
Functions
involving
F-TF.12
parametric
equations.
(Alabama)
Solve applied
problems that
include sequences
Franklin County Schools
ric inverse
functions.
Students know:




31. Utilize parametric
equations by graphing
and by converting to
rectangular form.
(Alabama)
Resources
How to
simplify
Students are able Students
radicals.
to:
How to
understand:
rationalize
radicals.
 Evaluate
 The sum
trigonomet
How to
identity.
ric
use the
 The
functions
unit circle
difference
using sum,
to
identity.
difference,
determine

The halfand halftrigonomet
angle
angle
ric values
identity.
identities.
at specific
angles.
Special
right
triangles.
Students know:


How to
model
motion
using
quadratic
functions.
Algebraic
manipulati
on of
equations.
Students are able
to:



Develop
parametric
equations
for a given
situation.
Graph the
parametric
equations.
Answer
questions
in the
context of
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
Students
understand:


Parametric
equations
can be
used to
model and
evaluate
trajectory
of
projectiles.
Parametric
equations
can be
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
with recurrence
relations.
(Alabama)
Understanding
the
parametric
problem.

32. (+) Use special
triangles to determine
geometrically the
values of sine, cosine,
tangent for π/3, π/4,
and π/6, and use the
unit circle to express
the values of sine,
cosines, and tangent
for x, π + x, and 2π –
x in terms of their
values for x, where x
is any real number.
Trigonometric
Functions
Extend the domain of
trigonometric
functions using the
unit circle.
Functions
F-TF.3
Students are able
to:

Students know:


The
relationshi
p between
the
lengths of
the sides
of a 4545-90 and
30-60-90
triangle.
The basic
trig ratios.


Franklin County Schools
Find the
value of
sine,
cosine,
and
tangent of
π/3, π/4
and π/6
using
special
triangles.
Locate an
angle in
standard
position in
the unit
circle.
Convert
between
degrees
and
radians.
Resources
used to
model and
evaluate
range of
projectiles.
Which
situtations
paramatric
modeling
is
appropriat
e.
Students
understand that:

For an
angle in
standard
position,
the point
where the
terminal
ray
intersects
the unit
circle has
an xcoordinate
which is
the value
of the
cosine of
the angle
and a ycoordinate
which is
the value
of the sine
of the
angle.
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
PRE CAL
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Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding

33. (+) Use the unit
circle to explain
symmetry (odd and
even) and periodicity
of trigonometric
functions.
Trigonometric
Functions
Extend the domain of
trigonometric
functions using the
unit circle.
Functions
F-TF.4
Trigonometric
Functions
Franklin County Schools
Patterns
that can
be
identified
on the unit
circle allow
for
application
of right
triangle
trigonomet
ry to
angles of
all sizes.
Students
understand that:
Students Know:


34. (+) Understand
that restricting a
Resources
The
characteris
tics of the
unit circle.
The
characteris
tics of
even and
odd
functions.
Students Know:
Students are able
to:


Transpose
coordinate
s from the
unit circle
to a graph
on the xyplane.
Use the
unit circle
to verify
even and
odd trig
identities.
Students are able
to:


Basic trig
functions
can be
classified
as either
even or
odd.
The
repetitive
nature of
the unit
circle
creates
periodic
functions
which is
displayed
in their
graphs.
Students
understand that:
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
Resources
Click below to
access all ALEX
PRE CAL
CCRS Standard
trigonometric function
to a domain on which
it is always increasing
or always decreasing
allows its inverse to be
constructed.
Mathematics CCRS Standards and Alabama COS
Standard ID
Model periodic
phenomena with
trigonometric
functions.
Functions
F-TF.6
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Characteri
stics of an
always
increasing
or always
decreasing
function.
When a
function
has an
inverse
that is also
a function.



Trigonometric
Functions
Model periodic
phenomena with
trigonometric
functions.
Functions
F-TF.7
Franklin County Schools
Resources
resources aligned


35. (+) Use inverse
functions to solve
trigonometric
equations that arise in
modeling contexts;
evaluate the solutions
using technology, and
interpret them in
Understanding
Students Know:

Periodic
situations
are best
modeled
by trig
Identify
intervals
where the
function is
increasing
or
decreasing
.
Identify
the
domain of
a function
that will
produce an
inverse.
Produce
different
models of
functions
and their
resulting
inverse.
Students are able
to:


Solve a
trig
equation.
Interpret


Due to the to this standard.
periodic
 ALEX
nature of
trig
Resources
functions,
domains
have to be
restricted
in order to
produce a
one-to-one
relationshi
p.
While
there are
many
different
intervals
that are
always
increasing
or always
decreasing
, there are
convention
al choices
for the
restricted
domain of
the trig
functions.
Students
understand that:

Click below to
access all ALEX
resources aligned
There are to this standard.
periodic
phenomen
 ALEX
a in real
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
terms of the
context.★
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge


36. (+) Prove the
addition and
subtraction formulas
for sine, cosine, and
tangent and use them
to solve problems.
Trigonometric
Functions
Prove and apply
trigonometric
identities.
Functions
F-TF.9
Franklin County Schools
functions.
Solutions
that are
mathemati
cally
possible
may not
be
physically
possible in
the
context of
the
problem.
Determine
when
technology
is
appropriat
e to
finding a
solution.
Students know:

Finding
the ratio of
a the sum
or
difference
Skills



the
meaning
of the
solution.
Interpret a
domain in
a
contextual
situation.
Use
technology
to solve a
trig
equation.
Translate a
computed
solution to
an
equivalent
solution
that fits in
the
physical
domain
(i.e., If -30
degrees is
a solution,
then
another
solution
can be 330
degrees).
Students are able
to:

Simplify
expression
s with trig
ratios.
Understanding

life that
can be
modeled
by trig
functions.
From the
many
solutions
that result
from a
periodic
function,
only some
may be
logical
given the
context.
Students
understand that:

Resources
Knowing
the exact
value of
the trig
Resources
Click below to
access all ALEX
resources aligned
to this standard.

ALEX
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge


of two
angles is
more
complex
than just
adding or
subtractin
g their
ratios.
Procedures
for proving
formulas.
Know the
exact
values of
the trig
ratios of
the special
angles
using the
unit circle.
Skills


When
given a
trig ratio
for an
angle and
the
quadrant,
determine
any other
needed
trig ratio
using a
reference
angle.
Convert
radians to
degrees
and
degrees to
radians.
Understanding


Franklin County Schools
ratios of
the special
angles,
along with
the
addition
and
subtraction
formulas,
allow for
determinat
ion of the
exact
values for
the trig
ratios of
other
angles.
Creating
equivalent
trig
expression
s often
involves
the use of
trig
identities
and other
trig ratios
where the
resulting
expression
doesn't
have a
similar
appearanc
e to the
original
expression
.
Proof is
necessary
Resources
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
to
establish
that a
conjecture
about a
relationshi
p in
mathemati
cs is
always
true and
may
provide
insight into
the
mathemati
cs being
addressed.
37. Derive the
equation of a parabola
given a focus and
directrix.
Expressing
Geometric
Properties with
Equations
Translate between
the geometric
description and the
equation for a conic
section.
Geometry
G-GPE.2
Students:
Given the focus and
directrix of a
parabola,
Use the distance
formula to write an
equation for all points
which are equidistant
from a given focus
and directory,
Use logical
repeated reasoning to
generalize to a
general point and
general directrix,
developing the
formula for the
equation of a
Franklin County Schools
Parabola
Focus
Students know:

Directrix

The
distance
formula
for points
in the
coordinate
plane and
methods
of finding
the
distance
between a
point and
a line,
Properties
of equality
(Table 4).
Students are able
to:

Create and
justify a
formula
through
algebraic
manipulati
on after
setting the
distances
between
two points
and
between a
point and
a line
equal to
each
Students
understand that:

The points
that satisfy
the
Click below to
equation access all ALEX
of a
resources aligned
parabola to this standard.
represent
all of the
 ALEX
points that
Resources
are
equidistant
from a
given point
and a
given line.
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
parabola,
Understanding
Resources
other.
Compare and
contrast parabola
equations for
horizontal and
vertical directrices.
38. (+) Derive the
equations of ellipses
and hyperbolas given
the foci, using the fact
that the sum or
difference of distances
from the foci is
constant.
Expressing
Geometric
Properties with
Equations
Translate between
the geometric
description and the
equation for a conic
section.
Geometry
G-GPE.3
Students are able
to:

Students know:



Franklin County Schools
The
characteris
tics of an
ellipse and
a
hyperbola.
Distance
formula
for finding
the
distance
between
two points
on the
coordinate
plane.
Pythagore
an
Theorem.

Create and
justify a
formula
through
algebraic
manipulati
on after
finding the
sum of the
distances
from the
foci to a
point on
the ellipse
and setting
the sum
equal to a
constant.
Create and
justify a
formula
through
algebraic
manipulati
on after
finding the
difference
of the
distances
from the
foci to a
Students
understand:


The points
that satisfy
the
equation
of the
ellipse are
all the
points
where the
sum of the
Click below to
distances
access all ALEX
from the
resources aligned
foci is
to this standard.
constant.
The points
 ALEX
that satisfy
Resources
the
equation
of the
hyperbola
are all the
points
where the
absolute
value of
the
difference
of the
distances
from the
foci is
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
point on
the
hyperbola
and setting
the
absolute
value of
the
difference
equal to a
constant.
39. (+) Give an
informal argument
using Cavalieri’s
principle for the
formulas for the
volume of a sphere
and other solid
figures.
Geometric
Measurement &
Dimension
Explain volume
formulas and use
them to solve
problems.
Geometry
G-GMD.2
always a
constant.
Students
understand that:

Students know:



Franklin County Schools
Resources
Attributes
of solid
Students are able
figures.
to:
Volume
formulas
 Make an
for a cone
informal
and a
argument.
cylinder

Perform
and other
geometric
solid
transforma
figures.
tions.
The
aspects of
Cavalieri's
principle.

There are
relationshi
ps
between
the sizes
of various
solid
figures
(i.e. cone
and
sphere,
pyramid
and prism,
etc.).
Cavalieri’s
principle
applies for
all types of
solids. As
long as
they have
the same
height and
the same
cross
sectional
Click below to
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
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Resources
PRE CAL
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Standard ID
Evidence of
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Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
area, they
will have
the same
volume.
40. (+) Define a
random variable for a
quantity of interest by
assigning a numerical
value to each event in
a sample space; graph
the corresponding
probability distribution
using the same
graphical displays as
for data distributions.
Using Probability
to Make Decisions
Calculate expected
values and use them
to solve problems.
Statistics & Probability
S-MD.1
Students
understand that:

Students know:


The sum
of all the
probabilitie
s in the
sample
space will
equal 1.
The
difference
between
experimen
tal and
theoretical
probabilitie
s.
Students are able
to:


Determine
the
probabilitie
s of any
event in
the sample
space.
Create a
data
distribution
for a
sample
space and
show it
graphically
.


Franklin County Schools
Probabilitie
s can be
calculated
using a
theoretical
model,
experiment
, or
simulation,
as well as
using the
Click below to
results
access all ALEX
from data
resources aligned
collection.
to this standard.
When the
number of
 ALEX
trials
Resources
approache
s infinity,
the
experiment
al
probability
will
approach
the
theoretical
probability
(Law of
Large
Numbers).
When
symmetry
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
Resources
or a trend
exist in a
probability
distribution
, it may
aid in
interpretati
on.
41. (+) Calculate the
expected value of a
random variable;
interpret it as the
mean of the
probability
distribution.
Using Probability
to Make Decisions
Calculate expected
values and use them
to solve problems.
Statistics & Probability
S-MD.2
Students know:


42. (+) Develop a
probability distribution
for a random variable
defined for a sample
Using Probability
to Make Decisions
Calculate expected
values and use them
Franklin County Schools
The sum
of all the
probabilitie
s in the
sample
space will
equal 1.
In a
probability
distributio
n, the
probabilitie
s of values
of the
random
variable
are
weighted
by their
relative
frequency
of
occurrence
.
Students know:

Distinguish
Students are able
to:


Find the
probability Students
distribution understand that:
for a
sample
 The mean
space.
Click below to
of the
Find the
probability access all ALEX
expected
distribution resources aligned
value of a
to this standard.
is a
probability
weighted
distribution
 ALEX
average in
as the sum
which each
Resources
of each
outcome is
value of
weighted
the
by its
random
probability.
variable
multiplied
by its
associated
probability.
Students are able
to:
Students
understand that:
Click below to
access all ALEX
resources aligned
to this standard.
PRE CAL
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
space in which
theoretical
probabilities can be
calculated; find the
expected value.
Example: Find the
theoretical probability
distribution for the
number of correct
answers obtained by
guessing on all five
questions of a
multiple-choice test
where each question
has four choices, and
find the expected
grade under various
grading schemes.
to solve problems.
Statistics & Probability
S-MD.3
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge


ing
characteris
tics of
empirical
and
theoretical
probabilitie
s.
Theoretical
probabilitie
s can be
mathemati
cally
calculated
to obtain
an exact
value, not
an
approxima
tion.
The
expected
value of a
probability
distributio
n is the
mean of
that
probability
distributio
n.
Skills




Franklin County Schools
Determine
probabilitie
s from
contexts
that can
be
modeled
by
probability
distribution
s which
may be
uniform,
binomial,
etc.
Describe a
probability
distribution
via a
function,
graph, or
table for a
random
variable
defined on
a sample
space
where the
theoretical
values are
obtainable.
Find the
expected
value of a
probability
distribution
.
Identify
which
values of a
Understanding



The rules
for finding
the
probability
of a
compound
event may
be
extended
to
determine
the
probability
of being
above or
below a
certain
criterion in
a
probability
distribution
.
The
expected
value is
the
anticipated
average
outcome if
the event
is repeated
a large
number of
times.
Probabilitie
s obtained
by a
simulation
or
experiment
Resources

ALEX
Resources
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
Skills
Understanding
probability
distribution
fit into
certain
intervals
(e.g., If a
grade of
70% is
needed to
pass, what
is the
probability
of passing
when
guessing
on a
true/false
test?).
43. (+) Develop a
probability distribution
for a random variable
defined for a sample
space in which
probabilities are
assigned empirically;
find the expected
value.
Example: Find a
current data
distribution on the
number of TV sets per
household in the
United States, and
calculate the expected
number of sets per
household. How many
TV sets would you
expect to find in 100
randomly selected
households?
Using Probability
to Make Decisions
Calculate expected
values and use them
to solve problems.
Statistics & Probability
S-MD.4
Franklin County Schools
Students know:



Distinguish
ing
characteris
tics of
empirical
and
theoretical
probabilitie
s.
Empirical
probabilitie
s are
approxima
tions to
the
theoretical
probability
.
The
expected
Students are able
to:



Determine
empirical
probabilitie
s.
Determine
the
expected
value of a
probability
distribution
.
Extrapolat
e the
expected
value of a
random
variable to
a different
sized
Resources
are not
necessarily
equal to
the
theoretical
probability.
Students
understand that:


Probabilitie
s obtained
by a
simulation
Click below to
or
access all ALEX
experiment
resources aligned
are not
to this standard.
necessarily
equal to
 ALEX
the
Resources
theoretical
probability.
Values
obtained
using the
expected
value and
proportion
s are
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
value of a
probability
distributio
n is the
mean of
that
probability
distributio
n.
44. (+) Weigh the
possible outcomes of a
decision by assigning
probabilities to payoff
values.
a.
b.
Using Probability
to Make Decisions
Use probability to
evaluate outcomes of
decisions.
Statistics & Probability
Find the expected S-MD.5
payoff for a game
of chance.
Examples: Find
the expected
winnings from a
state lottery ticket
or a game at a
fast-food
restaurant.
Evaluate and
compare
strategies on the
basis of expected
values.
Example:
Compare a highdeductible versus
a low-deductible
automobile
insurance policy
using various, but
reasonable,
chances of having
a minor or a
Franklin County Schools
Skills
population.
Students know:



Understanding
Interpretat Students are able
to:
ion of
probabilitie
s and odds
 Determine
associated
the
with a
probability
game of
of an
chance.
event and
A situation
expected
in which a
value of a
decision
random
has to be
variable.
made has
 Find or
an
research
expected
the
payoff that
probability
is based
and
on the
expected
probability
value of an
of each
event.
outcome
 Communic
and the
ate the
value of
reasoning
the
behind
associated
decisions.
outcome.
Multiple
factors
may
Resources
merely
estimates
of actual
values.
Students
understand that:


Objective
decisions
can be
based on
mathemati
cal
informatio
n obtained
by using Click below to
probabilitie access all ALEX
s and
resources aligned
expected to this standard.
values.
Multiple
 ALEX
factors
Resources
may
ultimately
determine
the
decision
one makes
other than
the
probability
of events,
such as
ethical
constraints
PRE CAL
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
major accident.
Evidence of
Student
Attainment
Teacher Vocabulary
Knowledge
ultimately
determine
the given
probability
of an
event.
Skills
Understanding
Resources
, social
policy, or
feelings of
others.
Click below to
access all ALEX
resources aligned
to this standard.

Franklin County Schools
ALEX
Resources