Download CCRS Standard - Franklin County Schools

Geometry
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
1. Know precise
definitions of angle,
circle, perpendicular
line, parallel line, and
line segment based
on the undefined
notions of point, line,
distance along a line,
and distance around
a circular arc.
Congruence
Experiment with
transformations
in the plane.
Geometry
G-CO.1
Evidence of Student
Attainment
Students:
Given undefined
notions of point, line,
distance along a line,
and distance around a
circular arc,
Develop precise
definitions of angle,
circle, perpendicular
line, parallel line, and
line segment,
Identify examples
and non-examples of
angles, circles,
perpendicular lines,
parallel lines, and line
segments.
2. Represent
transformations in
the plane using, e.g.,
transparencies and
geometry software;
describe
transformations as
functions that take
points in the plane as
inputs and give other
points as outputs.
Compare
transformations that
preserve distance and
angle to those that
do not (e.g.,
translation versus
horizontal stretch).
Congruence
Experiment with
transformations
in the plane.
Geometry
G-CO.2
Franklin County Schools
Students:
Given a variety of
transformations
(translations, rotations,
reflections, and
dilations),
Represent the
transformations in the
plane using a variety of
methods (e.g.,
technology,
transparencies, semitransparent mirrors
(MIRAs), patty paper,
compass),
Describe
Teacher
Vocabulary
Knowledge
Skills
Students know: Students are able
to:
Undefined
notions of point,
line, distance
along a line, and
distance around
a circular arc,
Use known and
developed
definitions and
logical connections
to develop new
definitions.
Properties of
a mathematical
definition, i.e.
the smallest
amount of
information and
properties that
are enough to
determine the
concept. (Note:
may not include
all information
related to
concept).
Students know: Students are able
to:
Characteristi
cs of
transformations
(translations,
rotations,
reflections, and
dilations),
Accurately
perform dilations,
rotations,
reflections, and
translations on
objects in the
coordinate plane
Methods for with and without
technology,
representing
transformations,
Communicate
Characteristi the results of
cs of functions, performing
transformations on
objects and their
Conventions
Understanding
Resources
Students
understand that:
Geometric
definitions are
developed from a
few undefined
notions by a logical
sequence of
connections that
lead to a precise
definition,
A precise
definition should
allow for the
inclusion of all
examples of the
concept, and
require the
exclusion of all
non-examples.
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aligned to this
standard.
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Resources
Students
understand that:
Mapping one
point to another
Click below to
through a series of
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transformations can
resources
be recorded as a
aligned to this
function,
standard.
Some
transformations
(translations,
rotations, and
reflections)
preserve distance
and angle measure,
and the image is

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
transformations as
functions that take
points in the plane as
inputs and give other
points as outputs,
explain why this
satisfies the definition
of a function, and
adapt function notation
to that of a mapping
[e.g. F(x,y) → F(x+a,
y+b)],
Teacher
Vocabulary
Knowledge
of functions with corresponding
mapping
coordinates in the
notation.
coordinate plane,
including when the
transformation
preserves distance
and angle,
Use the
language and
notation of
functions as
mappings to
describe
transformations.
Compare
transformations that
preserve distance and
angle to those that do
not.
3. Given a rectangle,
parallelogram,
trapezoid, or regular
polygon, describe the
rotations and
reflections that carry
it onto itself.
Congruence
Experiment with
transformations
in the plane.
Geometry
G-CO.3
Students:
Given a collection of
figures that include
rectangles,
parallelograms,
trapezoids, or regular
polygons,
Identify which
figures that have
rotations or reflections
that carry the figure
onto itself,
Perform and
communicate rotations
and reflections that
map the object to
itself,
Distinguish these
transformations from
those which do not
Franklin County Schools
Skills
Students know: Students are able
to:
Characteristi
cs of
transformations
(translations,
rotations,
reflections, and
dilations),
Accurately
perform dilations,
rotations,
reflections, and
translations on
objects in the
coordinate plane
Characteristi with and without
cs of rectangles, technology,
parallelograms,
trapezoids, and Communicate
regular
the results of
polygons.
performing
transformations on
objects and their
corresponding
coordinates in the
coordinate plane.
Understanding
Resources
then congruent to
the pre-image,
while dilations
preserve angle but
not distance, and
the pre-image is
similar to the
image,
Distortions,
such as only a
horizontal stretch,
preserve neither.
Students
understand that:
Mapping one
point to another
through a series of
transformations can
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be recorded as a
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function,
resources
aligned to this
Since rotations
standard.
and reflections
preserve distance
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and angle measure,
Resources
the image is then
congruent.
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
carry the object back to
itself,
Describe the
relationship of these
findings to symmetry.
4. Develop definitions
of rotations,
reflections, and
translations in terms
of angles, circles,
perpendicular lines,
parallel lines, and line
segments.
Congruence
Experiment with
transformations
in the plane.
Geometry
G-CO.4
Students:
Students know: Students are able
to:
Use geometric
terminology (angles,
circles, perpendicular
lines, parallel lines, and
line segments) to
describe the series of
steps necessary to
produce a rotation,
reflection, or
translation,
Characteristi
cs of
transformations
(translations,
rotations,
reflections, and
dilations),
Use these
descriptions to
communicate precise
definitions of rotation,
reflection, and
translation.
5. Given a geometric
figure and a rotation,
reflection, or
translation, draw the
transformed figure
using, e.g., graph
paper, tracing paper,
or geometry
software. Specify a
sequence of
transformations that
Congruence
Experiment with
transformations
in the plane.
Geometry
G-CO.5
Franklin County Schools
Students:
Given a geometric
figure,
Produce the image
of the figure under a
rotation, reflection, or
translation using graph
paper, tracing paper, or
Students
understand that:
Accurately
Geometric
perform rotations, definitions are
reflections, and
developed from a
translations on
few undefined
objects with and
notions by a logical Click below to
without technology, sequence of
access all ALEX
connections that
resources
aligned to this
Properties of Communicate lead to a precise
definition,
standard.
a mathematical the results of
definition, i.e.,
performing
the smallest
transformations on A precise
 ALEX
amount of
objects,
definition should
Resources
information and
allow for the
properties that Use known and inclusion of all
are enough to
examples of the
developed
determine the
concept and require
definitions and
concept. (Note: logical connections the exclusion of all
may not include to develop new
non-examples.
all information
definitions.
related to
concept).
Students know: Students are able
to:
Students
understand that:
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Characteristi
resources
Accurately
The same
cs of
aligned to this
transformations perform rotations, transformation may
standard.
(translations,
reflections, and
be produced using
rotations,
translations on
a variety of tools,
 ALEX
reflections, and objects using graph but the geometric
Resources
dilations),
paper, tracing
sequence of steps
paper, or geometry that describe the
transformation is
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
will carry a given
figure onto another.
Evidence of Student
Attainment
Teacher
Vocabulary
geometry software,
Techniques
for producing
images under
transformations
using graph
paper, tracing
paper, or
geometry
software.
Describe and justify
the sequence of
transformations that
will carry a given figure
onto another.
6. Use geometric
descriptions of rigid
motions to transform
figures and to predict
the effect of a given
rigid motion on a
given figure; given
two figures, use the
definition of
congruence in terms
of rigid motions to
decide if they are
congruent.
Congruence
Understand
congruence in
terms of rigid
motions. (Build
on rigid motions
as a familiar
starting point for
development of
concept of
geometric
proof.)
Geometry
G-CO.6
Students:
Given geometric
descriptions of rigid
motions,
Predict the effect of
the rigid motion on a
given figure,
Produce the image
of a figure under the
transformation,
Compare and
contrast the predictions
to the actual
transformation.
Given two figures,
Determine if a
sequence of rotations,
reflections, and
translations will carry
the first to the second,
and if so justify their
congruence by the
definition of
congruence in terms of
rigid motions.
Franklin County Schools
Knowledge
Rigid motion
Skills
Understanding
software,
consistent,
Communicate
the results of
performing
transformations on
objects.
Any distance
preserving
transformation is a
combination of
rotations,
reflections, and
translations.
Students know: Students are able
to:
Students
understand that:
Characteristi
cs of
translations,
rotations, and
reflections
including the
definition of
congruence,
Use geometric
descriptions of rigid
motions to
accurately perform
these
transformations on
objects,
Any distance
preserving
transformation is a
combination of
rotations,
reflections, and
translations,
Communicate
the results of
performing
transformations on
objects.
If a series of
translations,
rotations, and
reflections can be
described that
transforms one
object exactly to a
second object, the
objects are
congruent.
Techniques
for producing
images under
transformations
using graph
paper, tracing
paper, compass,
or geometry
software,
Geometric
terminology
(e.g., angles,
circles,
perpendicular
lines, parallel
lines, and line
segments) which
describes the
series of steps
necessary to
Resources
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
produce a
rotation,
reflection, or
translation.
7. Use the definition
of congruence in
terms of rigid motions
to show that two
triangles are
congruent if and only
if corresponding pairs
of sides and
corresponding pairs
of angles are
congruent.
If and only if
Students:
Given a triangle and its
image under a
sequence of rigid
motions (translations,
on rigid motions reflections, and
as a familiar
translations),
Congruence
Understand
congruence in
terms of rigid
motions. (Build
starting point for
development of
concept of
geometric
proof.)
Geometry
G-CO.7
8. Explain how the
criteria for triangle
congruence, angleside-angle (ASA),
side-angle-side (SAS),
and side-side-side
(SSS), follow from
the definition of
congruence in terms
of rigid motions.
Congruence
Understand
congruence in
terms of rigid
motions. (Build
Verify that
corresponding sides
and corresponding
angles are congruent.
Use geometric
descriptions of rigid
motions to
accurately perform
these
transformations on
objects,
Communicate
the results of
performing
transformations on
objects.
Students
understand that:
If a series of
translations,
rotations, and
reflections can be
described that
transforms one
object exactly to a
second object, the
objects are
congruent.
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometric
terminology
which describes
the series of
steps necessary
to produce a
rotation,
reflection, or
translation.
Triangle
congruence
Use rigid motions
ASA
and the basic
properties of rigid
on rigid motions motions (that they
SAS
as a familiar
preserve distance and
starting point for angle), which are
development of assumed without proof SSS
concept of
to establish that the
geometric
usual triangle
proof.)
congruence criteria
Franklin County Schools
Characteristi
cs of
translations,
rotations, and
reflections
including the
definition of
congruence,
Techniques
for producing
images under
transformations,
Given two triangles
that have the same
side lengths and angle
measures, - Find a
sequence of rigid
motions that will map
one onto the other.
Students:
Students know: Students are able
to:
Students know: Students are able
to:
Basic
properties of
rigid motions
(that they
preserve
distance and
angle),
Use logical
reasoning to
connect geometric
ideas to justify
other results,
Perform rigid
Methods for motions of
Students
understand that:
It is beneficial
to have minimal
sets of
requirements to
justify geometric
results (e.g., use
ASA, SAS, or SSS
instead of all sides
and all angles for
Click below to
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resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Geometry
G-CO.8
Evidence of Student
Attainment
make sense and can
then be used to prove
other theorems.
Teacher
Vocabulary
Knowledge
presenting
logical reasoning
using assumed
understandings
to justify
subsequent
results.
Skills
geometric figures,
Determine
whether two plane
figures are
congruent by
showing whether
they coincide when
superimposed by
means of a
sequence of rigid
motions
(translation,
reflection, or
rotation),
Identify two
triangles as
congruent if the
lengths of
corresponding sides
are equal (SSS
criterion), if the
lengths of two pairs
of corresponding
sides and the
measures of the
corresponding
angles between
them are equal
(SAS criterion), or if
two pairs of
corresponding
angles are
congruent and the
lengths of the
corresponding sides
between them are
equal (ASA
criterion),
Apply the SSS,
Franklin County Schools
Understanding
congruence).
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
SAS, and ASA
criteria to verify
whether or not two
triangles are
congruent.
9. Prove theorems
about lines and
angles. Theorems
Congruence
Students:
Prove geometric
theorems.
Make, explain, and
include vertical angles (Focus on
justify (or refute)
are congruent; when validity of
conjectures about
a transversal crosses underlying
geometric relationships
parallel lines,
reasoning while with and without
alternate interior
using variety of technology,
angles are congruent
and corresponding
angles are congruent;
points on a
perpendicular
bisector of a line
segment are exactly
those equidistant
from the segment’s
endpoints.
ways of writing
proofs.)
Geometry
G-CO.9
Prove
Students know: Students are able
to:
Students
understand that:
Transversal
Requirement
Communicate Proof is
s for a
logical reasoning in necessary to
Alternate interior mathematical
proof,
a systematic way to establish that a
angles
present a
conjecture about a
mathematical proof relationship in
Techniques
Corresponding
mathematics is
for presenting a of geometric
angles
theorems,
always true, and
Explain the
proof of
may also provide
requirements of a
geometric
insight into the
Generate a
mathematical proof,
theorems.
mathematics being
conjecture about
Click below to
addressed.
geometric
access all ALEX
Present a complete
relationships that
resources
mathematical proof of
calls for proof.
aligned to this
geometry theorems
standard.
including the following:
vertical angles are
congruent; when a
 ALEX
transversal crosses
Resources
parallel lines, alternate
interior angles are
congruent and
corresponding angles
are congruent; points
on a perpendicular
bisector of a line
segment are exactly
those equidistant from
the segment’s
endpoints,
Critique proposed
proofs made by others.
10. Prove theorems
about triangles.
Congruence
Students:
Prove geometric
Franklin County Schools
Students know: Students are able
to:
Students
understand that:
Click below to
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Geometry
CCRS Standard
Theorems include
measures of interior
angles of a triangle
sum to 180°, base
angles of isosceles
triangles are
congruent, the
segment joining
midpoints of two
sides of a triangle is
parallel to the third
side and half the
length, the medians
of a triangle meet at
a point.
Mathematics CCRS Standards and Alabama COS
Standard ID
theorems.
(Focus on
validity of
underlying
reasoning while
using variety of
ways of writing
proofs.)
Geometry
G-CO.10
Evidence of Student
Attainment
Make, explain, and
justify (or refute)
conjectures about
geometric relationships
with and without
technology,
Explain the
requirements of a
mathematical proof,
Present a complete
mathematical proof of
geometry theorems
about triangles,
including the following:
measures of interior
angles of a triangle
sum to 180°; base
angles of isosceles
triangles are
congruent; the
segment joining
midpoints of two sides
of a triangle is parallel
to the third side and
half the length; the
medians of a triangle
meet at a point,
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
resources
aligned to this
Communicate Proof is
standard.
logical reasoning in necessary to
a systematic way to establish that a
present a
conjecture about a
 ALEX
mathematical proof relationship in
Resources
of
geometric
mathematics
is
Techniques
always true and
for presenting a theorems,
may also provide
proof of
insight into the
Generate a
geometric
mathematics being
theorems.
conjecture about
addressed.
geometric
Requirement
s for a
mathematical
proof,
relationships that
calls for proof.
Critique proposed
proofs made by others.
11. Prove theorems
Congruence
Students:
about parallelograms. Prove geometric
Theorems include
theorems.
Make, explain, and
opposite sides are
(Focus on
justify (or refute)
congruent, opposite validity of
conjectures about
angles are congruent, underlying
geometric relationships
the diagonals of a
reasoning while with and without
parallelogram bisect
using variety of
Franklin County Schools
Students know: Students are able
to:
Students
understand that:
Requirement
Communicate Proof is
s for a
mathematical
logical reasoning in necessary to
proof,
a systematic way to establish that a
present a
conjecture about a
mathematical
proof
relationship in
Techniques
Click below to
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resources
aligned to this
standard.

ALEX
Geometry
CCRS Standard
each other, and
conversely,
rectangles are
parallelograms with
congruent diagonals.
Mathematics CCRS Standards and Alabama COS
Standard ID
ways of writing
proofs.)
Geometry
G-CO.11
Evidence of Student
Attainment
technology,
Teacher
Vocabulary
Knowledge
Skills
Understanding
for presenting a of geometric
proof of
theorems,
geometric
theorems.
Generate a
conjecture about
geometric
relationships that
calls for proof.
mathematics is
always true and
may also provide
insight into the
mathematics being
addressed.
Students:
Students know: Students are able
to:
Students
understand that:
Make and justify
formal geometric
constructions with a
variety of tools and
methods (e.g.,
compass and
straightedge, string,
reflective devices,
paper folding, dynamic
geometric software,
etc.) including the
following: Copying a
segment; copying an
angle; bisecting a
segment; bisecting an
angle; constructing
Methods for
accurately using
tools to perform
geometric
constructions,
including
compass and
straightedge,
string, reflective
devices, paper
folding, and
dynamic
geometric
software,
Explain the
requirements of a
mathematical proof,
Present a complete
mathematical proof of
geometry theorems
about parallelograms,
including the following:
opposite sides are
congruent, opposite
angles are congruent,
the diagonals of a
parallelogram bisect
each other, and
conversely, rectangles
are parallelograms with
congruent diagonals,
Resources
Resources
Critique proposed
proofs made by others.
12. Make formal
geometric
constructions with a
variety of tools and
methods such as
compass and
straightedge, string,
reflective devices,
paper folding, and
dynamic geometric
software, etc.
Congruence
Make geometric
constructions.
(Formalize and
explain
processes.)
Geometry
G-CO.12
Constructions include
copying a segment;
copying an angle;
bisecting a segment;
bisecting an angle;
constructing
perpendicular lines,
Franklin County Schools
Choose and use Limiting oneself
appropriate
to a specific tool or
construction tools set of tools to
strategically to
perform a
perform geometric geometric
constructions,
construction
illuminates
important
Use logical
mathematical
reasoning and
properties of and features of the
object being
relationships
between geometric constructed,
figures to justify
geometric
Methods for constructions.
Different tools
for geometric
Click below to
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resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
including the
perpendicular
bisector of a line
segment; and
constructing a line
parallel to a given line
through a point not
on the line.
Evidence of Student
Attainment
Teacher
Vocabulary
perpendicular lines,
including the
perpendicular bisector
of a line segment; and
constructing a line
parallel to a given line
through a point not on
the line,
Knowledge
Skills
justifying a
geometric
construction
using geometric
properties.
Compare and
contrast different
methods for doing the
same construction, and
identify geometric
properties that justify
steps in the
constructions.
13. Construct an
equilateral triangle, a
square, and a regular
hexagon inscribed in
a circle.
Congruence
Make geometric
constructions.
(Formalize and
explain
processes.)
Geometry
G-CO.13
Students:
Similarity,
Right
Construct
Use tools (e.g.,
Inscribed
compass, straight
edge, geometry
software) and
geometric relationships
to construct regular
polygons inscribed in
circles,
Franklin County Schools
Students:
Given a center of
Resources
constructions may
offer different
levels of precision
in the construction
and the purpose of
the construction
should help
determine the tool
of choice,
Properties of
geometric figures
can and should be
used to verify the
correctness of
geometric
constructions
regardless of the
construction tool or
method used.
Explain and justify
the sequence of steps
taken to complete the
construction.
14. Verify
experimentally the
Understanding
Dilations
Students know: Students are able
to:
Students
understand that:
Properties of
Choose and use Properties of
regular
polygons,
appropriate
geometric figures
construction tools can and should be
used to verify the
Characteristi strategically to
perform
geometric
correctness of
cs of inscribed
constructions,
geometric
figures,
constructions
regardless of the
Communicate
Methods for
construction tool or
accurately using with logical
method used.
reasoning
the
tools to perform
series
of
steps
geometric
necessary for
constructions.
constructing an
inscribed figure and
the justification for
each step.
Students know: Students are able
to:
Students
understand that:
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resources
aligned to this
standard.

ALEX
Resources
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Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
properties of dilations Triangles, &
given by a center and Trigonometry
a scale factor.
Understand
similarity in
a. A dilation takes a terms of
line not passing similarity
transformations.
through the
Geometry
center of the
G-SRT.1
dilation to a
b.
parallel line and
leaves a line
passing through
the center
unchanged.
The dilation of a
line segment is
longer or shorter
in the ratio given
by the scale
factor.
Evidence of Student
Attainment
Teacher
Vocabulary
dilation, a scale factor, Center
and a polygonal image,
Scale factor
Create a new
image by extending a
line segment from the
center of dilation
through each vertex of
the original figure by
the scale factor to find
each new vertex,
Present a
convincing argument
that line segments
created by the dilation
are parallel to their
pre-images unless they
pass through the
center of dilation, in
which case they remain
on the same line,
Find the ratio of
the length of the line
segment from the
center of dilation to
each vertex in the new
image and the
corresponding segment
in the original image
and compare that ratio
to the scale factor,
Knowledge
Skills
Understanding
Resources
resources
A dilation uses aligned to this
standard.
a center and line
segments through
vertex points to
 ALEX
create an image
Resources
which is similar to
the original image
Accurately find but in a ratio
the length of line specified by the
scale factor,
Dilations
segments and
may be
ratios of line
The ratio of the
performed on
segments,
polygons by
line segment
drawing lines
Communicate formed from the
through the
center of dilation to
with logical
center of dilation reasoning a
a vertex in the new
and each vertex conjecture of
image and the
of the polygon generalization from corresponding
then marking off experimental
vertex in the
a line segment results.
original image is
changed from
equal to the scale
the original by
factor.
the scale factor.
Methods for
finding the
length of line
segments (both
in a coordinate
plane and
through
measurement),
Accurately
create a new image
from a center of
dilation, a scale
factor, and an
image,
Conjecture a
generalization of these
results for all dilations.
15. Given two figures,
use the definition of
similarity in terms of
similarity
Similarity,
Right
Triangles, &
Trigonometry
Franklin County Schools
Students:
Given two figures,
Determine if they
Similarity
transformation
Students know: Students are able
to:
Students
understand that:
Properties of
Apply rigid
rigid motions
A figure that
Click below to
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resources
aligned to this
Geometry
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
transformations to
decide if they are
similar; explain using
similarity
transformations the
meaning of similarity
for triangles as the
equality of all
corresponding pairs
of angles and the
proportionality of all
corresponding pairs
of sides.
Understand
similarity in
terms of
similarity
transformations.
Geometry
G-SRT.2
Evidence of Student
Attainment
Teacher
Vocabulary
are similar by
demonstrating whether
one figure can be
obtained from the
other through a dilation
and a combination of
translations,
reflections, and
rotations.
Knowledge
Skills
Understanding
Resources
and dilations,
motion and dilation may be obtained
standard.
to a figure,
from another
through a dilation
Definition of
 ALEX
and a combination
Explain and
similarity in
Resources
terms of
justify whether or of translations,
similarity
not one figure can reflections, and
transformations, be obtained from rotations is similar
another through a to the original,
Techniques combination of rigid
motion and dilation. When a figure
for producing
is similar to another
images under a
the measures of all
dilation and rigid
corresponding
motions.
angles are equal,
and all of the
corresponding sides
are in the same
proportion.
Given a triangle,
Produce a similar
triangle through a
dilation and a
combination of
translations, rotations,
and reflections,
Demonstrate that a
dilation and a
combination of
translations,
reflections, and
rotations maintain the
measure of each angle
in the triangles and all
corresponding pairs of
sides of the triangles
are proportional.
16. Use the
properties of
similarity
transformations to
establish the angleangle (AA) criterion
for two triangles to
be similar.
Similarity,
Right
Triangles, &
Trigonometry
Understand
similarity in
terms of
similarity
transformations.
Geometry
Franklin County Schools
Students:
Given two triangles,
Explain why if the
measures of two angles
from one triangle are
equal to the measures
of two angles from
another triangle, then
measures of the third
AA criterion
Students know: Students are able
to:
The sum of
the measures of
the angles of a
triangle is 180
degrees,
Explain and
justify why the
third pair of
corresponding
angles of two
Properties of triangles must be
equal if each of the
rigid motions
Students
understand that:
It is beneficial
to have minimal
sets of
requirements to
justify geometric
results (i.e., use AA
instead of all sides
Click below to
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resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
G-SRT.3
Evidence of Student
Attainment
Teacher
Vocabulary
angles must be equal
to each other,
Knowledge
and dilations.
Use this established
fact and the properties
of a similarity
transformation to
demonstrate that the
Angle-Angle (AA)
criterion for similar
triangles is sufficient.
17. Prove theorems
about triangles.
Similarity,
Right
Theorems include a
Triangles, &
line parallel to one
Trigonometry
side of a triangle
Prove theorems
divides the other two involving
proportionally, and
similarity.
conversely; the
Geometry
Pythagorean
G-SRT.4
Theorem proved
using triangle
similarity.
Students:
Given a triangle and a
line parallel to one of
the sides,
Prove the other
two sides are divided
proportionally by using
AA, similarity
properties, previously
proven theorems and
properties of equality
(Table 4).
Given a triangle with
two of the sides divided
proportionally,
Prove the line
dividing the sides is
parallel to the third
side of the triangle.
Franklin County Schools
Skills
Students know: Students are able
to:
Properties of
similar triangles
and methods of
showing that
triangles are
similar,
Resources
other two
proportional and all
corresponding pairs angles congruent
are equal,
for similarity),
Justify through
the use of rigid
motion and dilation
why corresponding
sides of triangles
are in the same
proportion if the
measures of two
pairs of
corresponding
angles are equal.
Theorem
Understanding
If the measures
of two angles of
one triangle are
equal to the
measures of two
angles of another
triangle, then the
triangles are similar
and the similarity of
the triangles can be
justified through
similarity
transformations.
Students
understand that:
Apply
Triangle
properties of similar similarity may be
triangles to justify used to justify
relationships of the theorems involving
sides of a triangle, the connection
Click below to
between the
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proportion of sides
Properties of Explain and
resources
and whether or not
equality (Table justify that a line
aligned to this
4),
passing through the the line dividing the standard.
triangle divides the sides is parallel to
the other side of
sides
Previously
 ALEX
the triangle,
proven theorems proportionally, if
Resources
and
only
if,
the
line
including those
is parallel to a side Through the
concerning
of the triangle,
use of similar
parallel lines.
triangles, a right
triangle may be
Justify the
divided into two
Pythagorean
Theorem through right triangles
the use of similar which are similar to
the original right
triangles.
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
18. Use congruence
and similarity criteria
for triangles to solve
problems and to
prove relationships in
geometric figures.
Similarity,
Right
Triangles, &
Trigonometry
Prove theorems
involving
similarity.
Geometry
G-SRT.5
Use similar
triangles and
properties of
equality
(Table 4) to
prove the
Pythagorean
Theorem.
Students:
Given a contextual
situation involving
triangles,
Determine
solutions to problems
by applying congruence
and similarity criteria
for triangles to assist in
solving the problem,
Justify solutions
and critique the
solutions of others.
Given a geometric
figure,
Establish and
justify relationships in
the figure through the
use of congruence and
similarity criteria for
triangles.
Franklin County Schools
Resources
triangle; therefore
the corresponding
sides must be
proportional and
may be used to
prove the
Pythagorean
Theorem,
Given a right triangle,

Understanding
The same
theorem may be
proven in many
different ways (i.e.,
the Pythagorean
Theorem).
Congruence and
similarity criteria for
triangles
Students know: Students are able
to:
Criteria for
congruent (SAS,
ASA, AAS, SSS)
and similar (AA)
triangles and
transformation
criteria,
Students
understand that:
Accurately
Congruence
solve a contextual and similarity
problem by
criteria for triangles
applying the criteria may be used to find
of congruent and solutions of
similar triangles,
contextual
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problems,
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Techniques Provide
resources
to apply criteria justification for the Relationships in aligned to this
of congruent
solution process,
geometric figures standard.
and similar
may be proven
triangles for
through the use of
Analyze the
 ALEX
solving a
solutions of others congruent and
Resources
contextual
similar triangles.
and explain why
problem.
their solutions are
valid or invalid,
Justify
relationships in
geometric figures
through the use of
congruent and
similar triangles.
Geometry
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
19. Understand that
by similarity, side
ratios in right
triangles are
properties of the
angles in the triangle
leading to definitions
of trigonometric
ratios for acute
angles.
Similarity,
Right
Triangles, &
Trigonometry
Define
trigonometric
ratios and solve
problems
involving right
triangles.
Geometry
G-SRT.6
Evidence of Student
Attainment
Students:
Given a collection of
right triangles,
Construct similar
right triangles of
various sizes for each
right triangle given,
Teacher
Vocabulary
Knowledge
Skills
Understanding
Side ratios
Students know: Students are able
to:
Trigonometric
ratios
Techniques
Accurately find The ratios of
to construct
similar triangles, the side ratios of
the sides of right
triangles,
triangles are
dependent on the
Properties of
size of the angles
similar triangles. Explain and
justify relationships of the triangle.
Compare the ratios
of the sides of the
original triangles to the
ratios of the sides of
the similar triangles,
Resources
Students
understand that:
between the side
ratios of a right
triangle and the
angles of a right
triangle.
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resources
aligned to this
standard.
Communicate
observations made
about changes (or no
change) to such ratios
as the length of the
side opposite an angle
to the hypotenuse, or
the side opposite the
angle to the side
adjacent, as the size of
the angle changes or in
the case of similar
triangles, remains the
same,

ALEX
Resources
Summarize these
observations by
defining the six
trigonometric ratios.
20. Explain and use
the relationship
between the sine and
cosine of
complementary
angles.
Similarity,
Right
Triangles, &
Trigonometry
Define
trigonometric
ratios and solve
Franklin County Schools
Students:
Given a right triangle,
Sine
Cosine
Explain why the
two smallest angles
Complementary
must be complements, angles
Students know: Students are able
to:
Students
understand that:
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resources
Methods for
The sine of an aligned to this
finding sine and Accurately
standard.
cosine ratios in a solve a contextual angle is equal to
right triangle
problem by using the cosine of the
(e.g., use of
the sine and cosine complement of the
 ALEX
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
problems
involving right
triangles.
Geometry
G-SRT.7
Evidence of Student
Attainment
Compare the side
ratios of
opposite/hypotenuse
and
adjacent/hypotenuse
for each of these
angles and discuss
conclusions.
Given a contextual
situation involving right
triangles,
Teacher
Vocabulary
Skills
Knowledge
triangle
properties:
similarity;
Pythagorean
Theorem;
isosceles and
equilateral
characteristics
for 45-45-90 and
30-60-90
triangles and
technology for
others).
ratios,
Understanding
Resources
angle,
Resources
Justify solutions Switching
and discuss other between using a
possible solutions given angle or its
through the use of complement and
complementary
between sine or
angles and the sine cosine ratios may
or cosine ratios.
be used when
solving contextual
problems.
Compare solutions
to the situation using
the sine of the given
angle and the cosine of
its complement.
21. Use trigonometric
ratios and the
Pythagorean
Theorem to solve
right triangles in
applied problems.★
Similarity,
Right
Triangles, &
Trigonometry
Define
trigonometric
ratios and solve
problems
involving right
triangles.
Geometry
G-SRT.8
Students:
Given a contextual
situation involving right
triangles,
Create a drawing to
model the situation,
Find the missing
sides and angles using
trigonometric ratios
and the Pythagorean
Theorem,
Use the above
information to interpret
results in the context of
the situation.
Franklin County Schools
Students know: Students are able
to:
Methods of
using the
trigonometric
ratios to solve
for sides or
angles in a right
triangle,
The
Pythagorean
Theorem and its
use in solving
for unknown
parts of a right
triangle.
Create an
accurate diagram
to model a
contextual situation
involving right
triangles and use it
to solve the right
triangles,
Students
understand that:
Unknown parts
of right triangles
may be found
through the use of
trigonometric
ratios, Pythagorean
Theorem, or a
combination of
both,
Identify the
trigonometric ratio Right triangles
useful to solve for a may be used to
particular unknown model and solve
part of a right
contextual
triangle and use
situations.
that ratio to
accurately solve for
the unknown part.
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resources
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
ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
use the
Pythagoren
Theorem to find
unknown sides of a
right triangle
explain the
solution in terms of
the given
contextual situation
22. (+) Derive the
formula A = 1/2 ab
sin(C) for the area of
a triangle by drawing
an auxiliary line from
a vertex
perpendicular to the
opposite side.
Similarity,
Right
Triangles, &
Trigonometry
Apply
trigonometry to
general
triangles.
Geometry
G-SRT.9
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standard.
23. (+) Prove the
Laws of Sines and
Cosines and use them
to solve problems.
Similarity,
Right
Triangles, &
Trigonometry
Apply
trigonometry to
general
triangles.
Geometry
G-SRT.10
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resources
aligned to this
standard.
24. (+) Understand
and apply the Law of
Sines and the Law of
Cosines to find
unknown
measurements in
right and non-right
triangles (e.g.,
surveying problems,
resultant forces).
Similarity,
Right
Triangles, &
Trigonometry
Apply
trigonometry to
general
triangles.
Geometry
G-SRT.11
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resources
aligned to this
standard.
25. Prove that all
Circles
Franklin County Schools



Students:
Students know: Students are able
Students
ALEX
Resources
ALEX
Resources
ALEX
Resources
Click below to
Geometry
CCRS Standard
circles are similar.
Mathematics CCRS Standards and Alabama COS
Standard ID
Understand and
apply theorems
about circles.
Geometry
G-C.1
Evidence of Student
Attainment
Teacher
Vocabulary
Given a collection of
circles,
Skills
Knowledge
to:
Techniques
to create
dilations,
Show that in each
case there exists a
transformation that
consists of a dilation
and a combination of
rigid motions that will
take one of the circles
to any of the others,
Similar
figures have the
same shape,
A figure
transformed by
a dilation and
any combination
of rigid motions
will be similar to
the image.
Verify that the ratio
of the circumference of
the circles created
through dilations is
equal to the ratio of
the radii of the circles,
which is the same as
the scale factor of the
dilation,
Understanding
Resources
understand that:
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resources
Accurately
Any circle can aligned to this
create circles by
be created through standard.
making dilations of a dilation of
a given circle,
another circle and a
 ALEX
combination of rigid
Resources
Communicate motions,
logical reasoning in
a systematic way to Any geometric
present a
figure that is fully
mathematical proof defined by a single
of geometric
parameter will be
theorems.
similar to all other
figures in that class
(squares,
equilateral
triangles, circles,
parabola, etc.)
Explain with logical
reasoning how a circle
is fully defined by a
single parameter "r" so
the only nontranslational changes
that can be made is
alteration of "r", which
changes the size and
not the shape and
therefore the circles
are similar.
26. Identify and
describe relationships
among inscribed
angles, radii, and
chords. Include the
relationship between
central, inscribed,
Circles
Understand and
apply theorems
about circles.
Geometry
G-C.2
Franklin County Schools
Students:
Given circles with two
points on the circle,
Central angles
Students know: Students are able
to:
Students
understand that:
Inscribed angles
Definitions
and
characteristics of
central,
inscribed, and
Relationships
that exist among
inscribed angles,
radii, and chords
Compare the
Circumscribed
measures of the angles angles
(with and without
Explain and
justify possible
relationships
among central,
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resources
aligned to this
standard.

ALEX
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
and circumscribed
angles; inscribed
angles on a diameter
are right angles; the
radius of a circle is
perpendicular to the
tangent where the
radius intersects the
circle.
Evidence of Student
Attainment
technology) formed by
creating radii to the
given points, creating
chords from a third
point on the circle to
the given points, and
creating tangents from
a third point outside
the circle to the given
points, and conjecture
about possible
relationships among
the angles,
Use logical
reasoning to justify (or
deny) the conjectures
(in particular justify
that an inscribed angle
is one half the central
angle cutting off the
same arc, and the
circumscribed angle
cutting off that arc is
supplementary to the
central angle relating
all three).
Given circles with
chords from a point on
the circle to the
endpoints of a
diameter,
Find the measure
of the angles (with and
without technology),
conjecture about and
explain possible
relationships,
Franklin County Schools
Teacher
Vocabulary
Knowledge
circumscribed
angles in a
circle,
Techniques
to find measures
of angles
including using
technology
(dynamic
geometry
software).
Skills
inscribed, and
circumscribed
angles sharing
intersection points
on the circle,
Accurately find
measures of angles
(including using
technology
(dynamic geometry
software)) formed
from inscribed
angles, radii,
chords, central
angles,
circumscribed
angles, and
tangents
Understanding
may be used to find
the measures of
other angles when
appropriate
conditions are
given,
Identifying and
justifying
relationships exist
in geometric
figures.
Resources
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
Use logical
reasoning to justify (or
deny) the conjectures
(in particular justify
that an inscribed angle
on a diameter is a right
angle).
Given a circle with a
tangent and radius
intersecting at a point
on the circle,
Find the measure
of the angle at the
intersection point (with
and without
technology), conjecture
about and explain
possible relationships,
Use logical
reasoning to justify (or
deny) the conjectures
(in particular justify
that the radius of a
circle is perpendicular
to the tangent where
the radius intersects
the circle.
27. Construct the
inscribed and
circumscribed circles
of a triangle, and
prove properties of
angles for a
quadrilateral inscribed
in a circle.
Circles
Understand and
apply theorems
about circles.
Geometry
G-C.3
Franklin County Schools
Students:
Given a triangle,
Use tools (e.g.,
compass, straight
edge, geometry
software) to construct
inscribed and
circumscribed circles,
Construct
Students know: Students are able
to:
Inscribed and
Techniques
circumscribed circles to find
of a triangle
circumcenter
and incenter of
a triangle, and
Quadrilateral
to use this in
inscribed in a circle
inscribing or
circumscribing
Students
understand that:
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resources
Use appropriate Every triangle aligned to this
tools to accurately has a point which is standard.
construct inscribed equidistant from
and circumscribed each vertex of the
 ALEX
circles of a triangle, triangle and a point
Resources
which is equidistant
from each side of
Explain and
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Explain and justify
the sequence of steps
taken to complete the
construction.
Knowledge
Skills
Understanding
Resources
the triangle,
justify the steps
the triangle,
that are used when
Properties of creating the
Opposite angles
inscribed angles. construction,
of a quadrilateral
inscribed in a circle
Apply the
are supplementary.
properties of
inscribed angles to
reach a conclusion.
Given a quadrilateral
inscribed in a circle,
Conjecture possible
relationships among
the angles through the
use of inscribed angles
use logical reasoning to
justify (or deny) the
conjectures (in
particular, justify that
diagonally opposite
angles of a
quadrilateral are
supplementary).
28. (+) Construct a
tangent line from a
point outside a given
circle to the circle.
Circles
Understand and
apply theorems
about circles.
Geometry
G-C.4
Click below to
access all ALEX
resources
aligned to this
standard.

29. Derive, using
similarity, the fact
that the length of the
arc intercepted by an
angle is proportional
to the radius, and
define the radian
measure of the angle
as the constant of
proportionality;
derive the formula for
Circles
Find arc lengths
and areas of
sectors of
circles. (Radian
introduced only
as unit of
measure.)
Geometry
G-C.5
Franklin County Schools
Students:
Given an arc
intercepted by an
angle,
Similarity
Students know: Students are able
to:
Constant of
proportionality
Techniques
to use dilations
(including using
dynamic
geometry
software) to
create circles
with arcs
Use dilations to
Sector
create arcs intercepted
by the same central
angle with radii of
various sizes (including
Reason from
progressive
examples using
dynamic geometry
software to form
conjectures about
relationships
Students
understand that:
Radians
measure the ratio
of the arc length to
the radius for an
intercepted arc,
The ratio of the
ALEX
Resources
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
the area of a sector.
Evidence of Student
Attainment
using dynamic
geometry software),
and use the ratios of
the arc lengths and
radii to make
conjectures regarding
possible relationship
between the arc length
and the radius,
Justify the
conjecture for the
formula for any arc
length (i.e., since 2πr is
the circumference of
the whole circle, a
piece of the circle is
reduced by the ratio of
the arc angle to a full
angle (360)),
Find the ratio of
the arc length to the
radius of each
intercepted arc and use
the ratio to name the
angle calling this the
radian measure of the
angle by extending the
definition of one radian
as the angle which
intercepts an arc of the
same length as the
radius,
Develop the
formula for the area of
a sector by interpreting
a circle as a complete
revolution and a sector
as a fractional part of a
revolution.
Franklin County Schools
Teacher
Vocabulary
Knowledge
intercepted by
same central
angles,
Skills
Understanding
Techniques
to find arc
length,
among arc length, area of a sector to
central angles, and the area of a circle
the radius,
is proportional to
the ratio of the
central angle to a
Use logical
reasoning to justify complete
revolution.
(or deny) these
Formulas for
area and
circumference of
a circle.
conjectures and
critique the
reasoning
presented by
others,
Interpret a
sector as a portion
of a circle, and use
the ratio of the
portion to the
whole circle to
create a formula for
the area of a
sector.
Resources
Geometry
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
30. Derive the
equation of a circle of
given center and
radius using the
Pythagorean
Theorem; complete
the square to find the
center and radius of a
circle given by an
equation.
Expressing
Geometric
Properties
with
Equations
Translate
between the
geometric
description and
the equation for
a conic section.
Geometry
G-GPE.1
Evidence of Student
Attainment
Students:
Given the center (h,k)
and radius (r) of a
circle,
Explain and justify
that every point on the
circle is a combination
of a horizontal and
vertical shift from the
center with a length
equal to the radius,
Create a right
triangle from the center
of a circle to a general
point on the circle, and
show that the legs of
the right triangle are
the absolute values of
x-h and y-k, and the
hypotenuse is r, then
apply Pythagorean
theorem to show that
r2 = (x - h)2 + (y - k)2.
Teacher
Vocabulary
Knowledge
Skills
Students know: Students are able
to:
Key features
Create a right
of a circle,
triangle in a circle
using the horizontal
The
and vertical shifts
Pythagorean
from the center as
Theorem,
the legs and the
radius of the circle
The
as the hypotenuse,
technique of
completing the
Convert an
square.
equation of a circle
from general form
to standard form
using the method
of completing the
square.
Given an equation of a
circle in general form,
Understanding
Resources
Students
understand that:
Circles
represent a fixed
distance in all
directions in a
plane from a given
point, and a right
triangle may be
created to show the
relationship of the
horizontal and
vertical shift to the
Click below to
distance,
access all ALEX
resources
Rewriting
aligned to this
algebraic
standard.
expressions or
equations in
 ALEX
equivalent forms
Resources
often reveals
significant features
of the expression,
(i.e., circles written
in standard form
are useful for
recognizing the
center and radius
of a circle).
Complete the
square to rewrite the
equation in the form r2
= (x - h)2 + (y - k)2
and determine the
center and radius.
31. Use coordinates
to prove simple
geometric theorems
algebraically.
Example: Prove or
Expressing
Geometric
Properties
with
Equations
Franklin County Schools
Simple geometric Students know: Students are able
Students:
Given coordinates and theorems
to:
geometric theorems
Relationship
and statements defined
s (e.g. distance, Accurately
on a coordinate
slope of line)
determine what
Students
understand that:
Modeling
geometric figures
Click below to
access all ALEX
resources
aligned to this
standard.
Geometry
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
disprove that a figure
defined by four given
points in the
coordinate plane is a
rectangle; prove or
disprove that the
point (1, √3) lies on
the circle centered at
the origin and
containing the point
(0, 2).
Use coordinates
to prove simple
geometric
theorems
algebraically.
(Include
distance
formula; relate
to Pythagorean
Theorem.)
Evidence of Student
Attainment
system,
Use the coordinate
system and logical
reasoning to justify (or
deny) the statement or
theorem, and to
critique arguments
presented by others.
Geometry
G-GPE.4
Teacher
Vocabulary
Knowledge
Skills
Understanding
between sets of information is
or relationships on
points,
needed to prove or a coordinate graph
disprove a
assists in
statement
or
determining truth
Properties of
theorem,
of a statement or
geometric
theorem,
shapes,
Resources

ALEX
Resources
Accurately find
Coordinate the needed
information and
graphing rules
and techniques, explain and justify
conclusions,
Geometric
theorems may be
proven or disproven
by examining the
properties of the
Techniques
geometric shapes
Communicate
for presenting a
logical reasoning in in the theorem
proof of
a systematic way to through the use of
geometric
appropriate
present a
theorems.
mathematical proof algebraic
techniques.
of geometric
theorems.
32. Prove the slope
criteria for parallel
and perpendicular
lines, and use them
to solve geometric
problems (e.g., find
the equation of a line
parallel or
perpendicular to a
given line that passes
through a given
point).
Expressing
Geometric
Properties
with
Equations
Use coordinates
to prove simple
geometric
theorems
algebraically.
(Include
distance
formula; relate
to Pythagorean
Theorem.)
Geometry
G-GPE.5
Slope criteria for Students know:
parallel and
perpendicular lines
Techniques
Create lines parallel
to find the slope
to the given line and
of a line,
compare the slopes of
parallel lines by
Key features
examining the rise/run
needed to solve
ratio of each line,
geometric
Students:
Given a line,
Create lines
perpendicular to the
given line by rotating
the line 90 degrees and
compare the slopes by
examining the rise/run
ratio of each line,
Use understandings
of similar triangles and
logical reasoning to
prove that parallel lines
Franklin County Schools
Students are able
to:
Explain and
justify conclusions
reached regarding
the slopes of
parallel and
perpendicular lines,
Students
understand that:
Relationships
exist between the
slope of a line and
any line parallel or Click below to
perpendicular to
access all ALEX
that line,
resources
problems,
aligned to this
Apply slope
Slope criteria
standard.
criteria
for
parallel
for
parallel
and
Techniques
for presenting a and perpendicular perpendicular lines
 ALEX
lines to accurately may be useful in
proof of
Resources
find the solutions of solving geometric
geometric
geometric problems problems.
theorems.
and justify the
solutions,
Communicate
logical reasoning in
a systematic way to
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
have equal slopes and
the slopes of
perpendicular lines are
negative reciprocals.
Skills
Understanding
Resources
present a
mathematical proof
of geometric
theorems.
Given a geometric
problem involving
parallel or
perpendicular lines,
Apply the
appropriate slope
criteria to solve the
problem and justify the
solution including
finding equations of
lines parallel or
perpendicular to a
given line.
33. Find the point on
a directed line
segment between
two given points that
partitions the
segment in a given
ratio.
Expressing
Geometric
Properties
with
Equations
Use coordinates
to prove simple
geometric
theorems
algebraically.
(Include
distance
formula; relate
to Pythagorean
Theorem.)
Geometry
G-GPE.6
Franklin County Schools
Directed line
Students:
Given two points and a segment
ratio that partitions the
segment between the Partitions
points,
Construct a circle
using one of the given
points as the center
and the distance
between the points as
the radius,
Construct a dilation
of the circle using the
given ratio as the scale
factor and find the
intersection between
the dilation and the
equation of the line
passing through the
Students know: Students are able
to:
Techniques
for finding the
distance
between two
points and the
equation of a
line passing
through two
points,
Accurately find
the distance
between two points
and the equation of
a line passing
through two points,
Students
understand that:
A radius of a
circle may be used
to show the
distance between Click below to
access all ALEX
two points,
resources
aligned to this
A dilation of a
standard.
Accurately find circle may be used
the equation of a to partition a line
dilation of a circle, segment by making c. ALEX
Resources
it the radius, in a
given ratio.
Find the
Forms for
writing the
equation of a
circle dependent intersection
on the
point(s) of a line
information
and a circle.
given to find the
equation of the
dilation of a
Geometry
CCRS Standard
34. Use coordinates
to compute
perimeters of
polygons and areas of
triangles and
rectangles, e.g.,
using the distance
formula.★
Mathematics CCRS Standards and Alabama COS
Standard ID
Expressing
Geometric
Properties
with
Equations
Use coordinates
to prove simple
geometric
theorems
algebraically.
(Include
distance
formula; relate
to Pythagorean
Theorem.)
Geometry
G-GPE.7
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
given points,
circle,
Justify and explain
the reasons for each
step in the process of
finding a point that
partitions a segment in
a given ratio.
Techniques
to find the
intersection
between a line
and a circle.
Students:
Given a contextual
situation that requires
the perimeter and/or
area of a polygon as
part of its solution,
Students know: Students are able
to:
Find the solution to
the situation through
the use of coordinates
and the distance
formula as appropriate,
through modeling the
situation in a Cartesian
coordinate system and
explain and justify the
solution.
Understanding
Resources
Students
understand that:
The distance
formula and its Create
applications,
geometric figures
on a coordinate
Techniques system from a
contextual
for coordinate
situation,
graphing.
Contextual
situations may be
modeled in a
Cartesian
coordinate system, Click below to
access all ALEX
resources
Coordinate
aligned to this
Accurately find modeling is
standard.
the perimeter of
frequently useful to
polygons and the visualize a situation
 ALEX
area of triangles
and to aid in
Resources
and rectangles
solving contextual
from the
problems.
coordinates of the
shapes,
Explain and
justify solutions in
the original context
of the situation.
35. Determine areas
and perimeters of
regular polygons,
including inscribed or
circumscribed
polygons, given the
coordinates of
vertices or other
characteristics.
(Alabama)
Geometric
Measurement
& Dimension
Use coordinates
to prove simple
geometric
theorems
algebraically.
(Alabama)
Geometry
Franklin County Schools
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
G-GMD.5
36. Give an informal
argument for the
formulas for the
circumference of a
circle; area of a
circle; and volume of
a cylinder, pyramid,
and cone. Use
Geometric
Measurement
& Dimension
Explain volume
formulas and
use them to
solve problems.
Geometry
dissection arguments, G-GMD.1
Cavalieri’s principle,
and informal limit
arguments.
Students:
Given a circle,
Dissection
arguments
Use repeated
reasoning from multiple
examples of the ratio
of circle circumference
to the diameter, to
informally conjecture
that the circumference
of any circle is a little
more than three times
the diameter,
Cavalieri’s
Principle
Divide the circle
into an equal number
of sectors, and
rearrange the sectors
to form a shape that is
approaching a
parallelogram,
Make conjectures
about the area and
perimeter of the new
shape as the number of
sectors becomes larger,
and relate those
conjectures to the
original circle.
Given a cylinder,
Explain how a
cylinder could be
divided into an infinite
number of circles, and
the area of those
circles multiplied by the
Franklin County Schools
Cylinder
Pyramid
Cone
Students know: Students are able
to:
Students
understand that:
Techniques
Geometric
to find the area Accurately
and perimeter of decompose circles, shapes may be
parallelograms, cylinders, pyramids, decomposed into
and cones into
other shapes which
may be useful in
Techniques other geometric
creating formulas,
to find the area shapes,
of circles or
polygons.
Explain and
justify how the
formulas for
circumference of a
circle, area of a
circle, and volume
of a cylinder,
pyramid, and cone
may be created
from the use of
other geometric
shapes.
Geometric
shapes may be
divided into an
infinite number of
smaller geometric
shapes, and the
combination of
those shapes
maintain the area
and volume of the
original shape.
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
height is the volume of
the cylinder, and use
Cavalieri’s Principle to
demonstrate that if two
solids have the same
height and the same
cross-sectional area at
every level, then they
have the same volume.
Given a pyramid or
cone,
Explain that the
shapes could be
divided into crosssections, and the area
of the cross-sections is
decreasing as the
cross-sections become
further away from the
base, and the area of
an infinite number of
cross-sections is the
volume of a pyramid or
cone.
37. Use volume
formulas for
cylinders, pyramids,
cones, and spheres to
solve problems.★
Geometric
Measurement
& Dimension
Explain volume
formulas and
use them to
solve problems.
Geometry
G-GMD.3
Students:
Given a contextual
situation that requires
finding the volume of a
cylinder, pyramid,
cone, or sphere as part
of its solution,
Use an appropriate
shape or 2-D drawing
to model the situation,
Solve using the
Franklin County Schools
Students know: Students are able
to:
Volume
formulas for
cylinders,
pyramids, cones,
and spheres,
Students
understand that:
Click below to
A contextual
access all ALEX
situation involving resources
cylinders, pyramids, aligned to this
cones, and spheres standard.
may be modeled by
shapes or 2-D
Techniques
 ALEX
drawings, and the
for modeling 3-D
Resources
Use the model model may provide
objects with
shapes or 2-D
or drawing to find insight into the
drawings, and
values needed for solution of the
for using these use in the volume
Accurately
model a contextual
situation with a
cylinder, pyramid,
cone, sphere, or a
2-D drawing,
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
appropriate formula,
Justify and explain
the solution and
solution path in the
context of the given
situation.
38. Determine the
relationship between
surface areas of
similar figures and
volumes of similar
figures. (Alabama) ★
39. Identify the
shapes of twodimensional crosssections of threedimensional objects,
and identify threedimensional objects
generated by
rotations of twodimensional objects.
Teacher
Vocabulary
Knowledge
models to
identify specific
values for use in
volume
formulas.
Skills
Understanding
formula,
problem,
Accurately find
a solution to the
given situation, and
explain the solution
in the context of
the situation.
Formulas are
useful for efficiency
when many
problems of the
same type need to
be solved.
Geometric
Measurement
& Dimension
Explain volume
formulas and
use them to
solve problems.
Geometry
G-GMD.6
Geometric
Measurement
& Dimension
Visualize
relationships
between twodimensional and
threedimensional
objects.
Geometry
G-GMD.4
Click below to
access all ALEX
resources
aligned to this
standard.

Students:
Given 3-D objects,
Conjecture about
the characteristics of
geometric shapes
formed if a crosssection of a 3-D shape
is taken,
Take 2-D crosssections at different
angles of cut,
Explain the shape
formed by taking 2-D
cross-sections,
Compare and
contrast the figures
formed when the angle
Franklin County Schools
Resources
ALEX
Resources
Two-dimensional Students know: Students are able
cross-sections
to:
Students
understand that:
Characteristi
Two-dimensional cs of 2-D and 3- Conjecture
objects
D geometric
about the
objects,
characteristics of
geometric shapes
Threedimensional objects Techniques formed from taking
a cross-section of a
for finding a
3-D shape, or
cross-section
of
Rotations
rotating a 2-D
a 3-D object,
shape about a line,
3-D objects can
be created from 2D plane figures
Click below to
through
access all ALEX
transformations
such as rotations, resources
aligned to this
Cross-sections standard.
Techniques
for rotating a 2- Accurately
D object about a determine the
geometric shapes
line.
formed from taking
a cross-section of a
3-D shape, or
rotating a 2-D
shape about a line.
of 3-D objects can
be formed in a
variety of ways,
depending on the
angle of the cut
with the base of
the object.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
of the cut changes.
Given 2-D objects,
Conjecture about
the characteristics of
geometric shapes
formed from rotating a
2-D shape about a line,
Rotate the object
about given lines,
Explain the 3-D
objects formed if the 2D object is rotated
about a line.
40. Use geometric
shapes, their
measures, and their
properties to describe
objects (e.g.,
modeling a tree trunk
or a human torso as
a cylinder).★
Modeling with
Geometry
Apply geometric
concepts in
modeling
situations.
Geometry
G-MG.1
Students:
Given a real-world
object,
Students know: Students are able
to:
Techniques
to find measures Model a realof geometric
world object
shapes,
through the use of
a geometric shape,
Select an
appropriate geometric
shape to model the
object,
Properties of
Justify the
geometric
shapes.
model by
connecting its
measures and
properties to the
object.
Provide a
description of the
object through the
measures and
properties of the
geometric shape which
is modeling the object,
Franklin County Schools
Geometric
shapes may be
used to model realClick below to
world objects,
access all ALEX
resources
Attributes of
aligned to this
geometric figures standard.
help us identify the
figures and find
 ALEX
their measures,
Resources
therefore matching
these figures to
real world objects
allows the
application of
geometric
techniques to real
world problems.
Explain and justify
the model which was
selected.
41. Apply concepts of Modeling with Students:
Students
understand that:
Density
Students know: Students are able
Students
Click below to
Geometry
Mathematics CCRS Standards and Alabama COS
CCRS Standard
Standard ID
density based on area
and volume in
modeling situations
(e.g., persons per
square mile, British
Thermal Units (BTUs)
per cubic foot).★
Geometry
Apply geometric
concepts in
modeling
situations.
Geometry
G-MG.2
Evidence of Student
Attainment
Teacher
Vocabulary
Given a contextual
situation involving
density,
Skills
Knowledge
to:
Understanding
understand that:
Geometric
concepts of area Accurately
and volume,
model a situation
involving density,
Situations
involving density
may be modeled
through a
Properties of
Justify how the representation of a
rates,
concentration per
model is an
unit of area or unit
accurate
Modeling
representation of of volume.
techniques.
the given situation.
Model the situation
by creating an average
per unit of area or unit
of volume,
Generate questions
raised by the model
and defend answers
they produce to the
generated questions
(e.g., should population
density be given per
square mile or per
acre? What insights
might one yield over
the other?),
Resources
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Explain and justify
the model in terms of
the original context.
42. Apply geometric
methods to solve
design problems
(e.g., designing an
object or structure to
satisfy physical
constraints or
minimize cost;
working with
typographic grid
systems based on
ratios).★
Modeling with
Geometry
Apply geometric
concepts in
modeling
situations.
Geometry
G-MG.3
Students:
Given a contextual
situation involving
design problems,
Geometric
methods
Students know: Students are able
to:
Students
understand that:
Properties of
Design problems geometric
Accurately
Design
shapes,
model and solve a problems may be Click below to
access all ALEX
Create a geometric
design problem,
modeled with
resources
method to model the
geometric
methods,
Characteristi
aligned to this
situation and solve the
Justify how
cs of a
standard.
problem,
Geometric
mathematical
their model is an
model.
accurate
models may have
 ALEX
Explain and justify
representation of physical
Resources
the model which was
the given situation. constraints,
created to solve the
problem.
Models
represent the
mathematical core
of a situation
Franklin County Schools
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
Understanding
Resources
without extraneous
information, for the
benefit in a
problem solving
situation.
43. Understand the
conditional probability
of A given B as P(A
and B)/P(B), and
interpret
independence of A
and B as saying that
the conditional
probability of A given
B is the same as the
probability of A, and
the conditional
probability of B given
A is the same as the
probability of B.
Conditional
Probability &
the Rules of
Probability
Understand
independence
and conditional
probability and
use them to
interpret data.
(Link to data
Conditional
Students:
Given scenarios
probability
involving two events A
and B both when A and Independence
B are independent and
when A and B are
dependent,
Determine the
probability of each
individual event, then
from simulations limit the sample space
or experiments.) to those outcomes
Statistics &
where B has occurred
Probability
and calculate the
S-CP.3
probability of A,
compare the P(A) and
the P(A given B), and
explain the equality or
difference in the
original context of the
problem,
Students know: Students are able
to:
Methods to
find probability
of simple and
compound
events,
Techniques
to find
conditional
probability.
Justify that P(A
given B) =
P(A∩B)/P(B).
44. Construct and
interpret two-way
frequency tables of
data when two
categories are
associated with each
Conditional
Probability &
the Rules of
Probability
Understand
independence
Franklin County Schools
Two way
Students:
Given a situation in
frequency tables
which it is meaningful
to collect categorical
Sample space
data for two categories
Students
understand that:
Accurately
determine the
probability of
simple and
compound events,
The
independence of
two events is
determined by the
effect that one
event has on the
outcome of another
Accurately
event,
Click below to
determine the
access all ALEX
conditional
The occurrence resources
probability P(A
given B)from a
of one event may aligned to this
standard.
sample space or
or may not
from the knowledge influence the
of P(A∩B) and the likelihood that
 ALEX
P(B).
another event
Resources
occurs (e.g.,
successive flips of a
coin - the first toss
exerts no influence
on whether a head
occurs on the
second, drawing an
ace from a deck
changes the
probability that the
next card drawn is
an ace).
Students know: Students are able
to:
Students
understand that:
Techniques
to construct
two-way
frequency
Two-way
frequency tables
show conditional
Accurately
construct a twoway frequency
Click below to
access all ALEX
resources
aligned to this
standard.
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
object being
classified. Use the
two-way table as a
sample space to
decide if events are
independent and to
approximate
conditional
probabilities.
Example: Collect data
from a random
sample of students in
your school on their
favorite subject
among math, science,
and English. Estimate
the probability that a
randomly selected
student from your
school will favor
science given that the
student is in tenth
grade. Do the same
for other subjects and
compare the results.
and conditional
probability and
use them to
interpret data.
(Link to data
Statistics &
Probability
S-CP.4
probabilities of simple
events and conditional
events from the table,
explain whether the
events are independent
based on the context
and the probability
calculations.
45. Recognize and
explain the concepts
of conditional
probability and
independence in
everyday language
and everyday
situations.
Example: Compare
the chance of having
lung cancer if you are
a smoker with the
chance of being a
smoker if you have
lung cancer.
Conditional
Probability &
the Rules of
Probability
Understand
independence
and conditional
probability and
use them to
interpret data.
(Link to data
Conditional
Students:
Given a contextual
probability
situation and scenarios
involving two events,
Independence
Collect data and
create two-way
frequency tables,
Independent
tables,
Conditional
probabilities
Techniques
to find simple
and conditional
probability in
two-way
frequency
tables.
from simulations
or experiments.) Determine
Explain the
meaning of
independence from a
formula perspective
P(A∩B) = P(A) x P(B)
and from the intuitive
from simulations notion that A occurring
or experiments.) has no impact on
Statistics &
whether B occurs or
Probability
not,
S-CP.5
Compare these two
Franklin County Schools
Skills
Knowledge
probability and can
be used to test for
Accurately find independence.
simple and
conditional
probability from a
two-way frequency
table.
Resources
table,
Students know: Students are able
to:
Possible
relationships and
differences
between the
simple
probability of an
event and the
probability of an
event under a
condition.
Understanding

ALEX
Resources
Students
understand that:
Communicate The occurrence
the concepts of
of one event may
conditional
or may not
probability and
influence the
independence using likelihood that
everyday language another event
by discussing the occurs (e.g.,
impact of the
successive flips of a
occurrence of one coin - first toss
event on the
exerts no influence
likelihood of the
on whether a head
other occurring.
occurs on the
second, drawing an
ace from a deck
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
interpretations within
the context of the
scenario.
Understanding
Resources
changes the
probability that the
next card drawn is
an ace),
Events are
independent if the
occurrence of one
does not affect the
probability of the
other occurring.
46. Find the
conditional probability
of A given B as the
fraction of B’s
outcomes that also
belong to A, and
interpret the answer
in terms of the
model.
Conditional
Probability &
the Rules of
Probability
Use the rules of
probability to
compute
probabilities of
compound
events in a
uniform
probability
model.
Statistics &
Probability
S-CP.6
Students:
Given a contextual
situation consisting of
two events,
Determine the
probability of each
individual event, then
limit the sample space
to those outcomes
where B has occurred
and calculate the
probability of A,
compare the P(A) and
the P(A given B), and
explain the equality or
difference in the
original context of the
problem,
Determine the
probability of each
individual event, then
limit the sample space
to those outcomes
where B has occurred
and calculate the P(A
and B), compare the
ratio of P(A and B) and
P(B) to P(A given B),
Franklin County Schools
Conditional
probability
Students know: Students are able
to:
Possible
relationships and
differences
between the
simple
probability of an
event and the
probability of an
event under a
condition.
Accurately
determine the
probability of
simple and
compound events,
Accurately
determine the
conditional
probability P(A
given B) from a
sample space or
from the knowledge
of P(A∩B) and the
P(B).
Students
understand that:
Conditional
probability is the
probability of an
event occurring
given that another
event has occurred.
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Skills
Knowledge
Understanding
Resources
and explain the
equality or difference in
the original context of
the problem.
47. Apply the
Addition Rule, P(A or
B) = P(A) + P(B) –
P(A and B), and
interpret the answer
in terms of the
model.
Conditional
Probability &
the Rules of
Probability
Use the rules of
probability to
compute
probabilities of
compound
events in a
uniform
probability
model.
Statistics &
Probability
S-CP.7
Students:
Given a contextual
situation consisting of
two events,
Addition Rule
Determine the
simple probability of
each event,
Students know: Students are able
to:
Students
understand that:
Techniques
for finding
probabilities of
simple and
compound
events.
Formulas are
useful to generalize
regularities, but
must be justified,
Accurately
determine the
probability of
simple and
compound events.
The Addition
Rule may be used
for finding
compound
probability.
Determine the P(A
or B) and P(A and B),
Interpret the
Addition Rule by
counting outcomes in
the four events A, B, A
and B, A or B and
showing the
relationship to P(A or
B) = P(A) + P(B) - P(A
and B),
Click below to
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resources
aligned to this
standard.

ALEX
Resources
Interpret the
Addition Rule in the
case that the P(A and
B) = 0.
48. (+) Apply the
general Multiplication
Rule in a uniform
probability model,
P(A and B) =
P(A)P(B|A) =
P(B)P(A|B), and
interpret the answer
in terms of the
model.
Conditional
Probability &
the Rules of
Probability
Use the rules of
probability to
compute
probabilities of
compound
events in a
uniform
Franklin County Schools
Students:
Given a contextual
situation consisting of
two events A and B,
Use the definition of
conditional probability
P(B|A) = P(A and
B)/P(A) to determine
the probability of the
Uniform
probability
model
General
Multiplication
Rule
Probability
Students are able
Students know: to:
Techniques
for finding
probabilities
of simple
and
conditional
events.


Determine
the
probability
of a single
event.
Determine
the
Students
understand that:
The general
Multiplication
Rule for
probability is a
manipulation
of the formula
for conditional
Click below to
access all ALEX
resources
aligned to this
standard.

ALEX
Resources
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
probability
model.
Statistics &
Probability
S-CP.8
49. (+) Use
permutations and
combinations to
compute probabilities
of compound events
and solve problems.
Conditional
Probability &
the Rules of
Probability
Use the rules of
probability to
compute
probabilities of
compound
events in a
uniform
probability
model.
Statistics &
Probability
S-CP.9
Evidence of Student
Attainment
Teacher
Vocabulary
compound event (A
and B) when the
P(A|B) and the P(A)
are known or may be
determined. Interpret
the probability as it
relates to the context.
Simple events
Skills
probability
of a
conditional
event.
Conditional
events
Students are able
Students know: to:
Students:
Given a contextual
situation,
Choose the
appropriate
counting technique
(permutation or
combination),
Find the number of
ways an event(s)
can occur,
Use these counts
to determine
probabilities of the
event, including
compound events.
Franklin County Schools
Knowledge
Permutations
Combinations
Compound
events
Probability
Possible
outcomes
Order is the
determining
factor in
whether a event
requires a
permutation or a
combination to
count the
number of
possible
outcomes of the
event.
Techniques for
finding
probabilities of
simple and
compound
events.
Techniques for
finding the
number of
permutations or
combinations of
an event.
Evaluate
factorial
expressions.
Apply the
multiplication
and addition
rules to
determine
probabilities.
Interpret and
apply the
different
notations for
combinations
and
permutations.
Perform
procedures to
evaluate
expressions
involving the
number of
combinations
and
Understanding
Resources
probability.
The formula
P(A and B) =
P(A)P(B|A) will
always apply
regardless of
whether the
events are
independent or
dependent.
Students
understand that:
There are
contextual
situations that
can be
interpreted
through the
use of
combinations Click below to
and
access all ALEX
permutations. resources
aligned to this
The contextual standard.
situation
determines
 ALEX
whether
Resources
combinations
or
permutations
must be
utilized.
Mathematics is
a coherent
whole.
Structure
within
mathematics
Geometry
CCRS Standard
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Knowledge
Skills
permutations
of n things
taken r at a
time.
50. (+) Use
probabilities to make
fair decisions (e.g.,
drawing by lots, using
a random number
generator).
Using
Probability to
Make
Decisions
Use probability
to evaluate
outcomes of
decisions.
(Introductory;
apply counting
rules.)
Statistics &
Probability
S-MD.6
Resources
allows for
procedures or
models from
one concept to
be applied
elsewhere
(e.g., Pascal's
triangle as it
applies to the
number of
combinations).
Students
understand that:
Students:
Given a contextual
situation in which a
decision needs to be
made,
Use a random
probability selection
model to produce
unbiased decisions.
Franklin County Schools
Understanding
Fair decisions
Probability
Fair decisions
Random
Students are able
to:
Students know:
The
characteristics of
a random
sample.
Randomly
select a sample
from a
population
(using
technology
when
appropriate).
Multiple factors
may ultimately
determine the
decision one
makes other
than the
probability of
events, such as Click below to
access all ALEX
ethical
resources
constraints,
social policy, or aligned to this
standard.
feelings of
others.
 ALEX
 Probabiliti Resources
es can be
used to
explain
why a
decision
was
considered
to be fair
or
objective.
Geometry
CCRS Standard
51. (+) Analyze
decisions and
strategies using
probability concepts
(e.g., product testing,
medical testing,
pulling a hockey
goalie at the end of a
game).
Mathematics CCRS Standards and Alabama COS
Standard ID
Evidence of Student
Attainment
Teacher
Vocabulary
Using
Probability to
Make
Decisions
Use probability
to evaluate
outcomes of
decisions.
(Introductory;
apply counting
rules.)
Statistics &
Probability
S-MD.7
Understanding
Resources
Students
understand that:

Students know:

Students:
Given a contextual
situation in which a
decision needs to be
made,

Franklin County Schools
Skills
Knowledge
Use
probability
concepts to
analyze,
justify, and
make
objective
decisions.

Probability
Techniq
ues for
finding
probabi
lities of
simple,
compou
nd, and
conditio
nal
events
and
from
probabi
lity
distribu
tions.
Students are able
to:



Choose
the
appropriat
e
probability
concept
for the
given
situation.
Use and
apply the
selected
probability
rule.
Communic
ate the
reasoning
behind
decisions.

Objective
decision
making
can be
mathemati
cally
based,
often
using
analysis
involving Click below to
probability access all ALEX
concepts. resources
Multiple
aligned to this
factors
standard.
may
ultimately
 ALEX
determine
Resources
the
decision
one makes
other than
the
probability
of events,
such as
ethical
constraints
, social
policy, or
feelings of
others.