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NC Adult Education Standards for ASE MA 4 Geometry
Geometry: Congruence
G.1 Experiment with transformations in the plane.
Objectives
What Learner Should Know, Understand, and Be Able to Do
G.1.1 Know precise definitions of angle,
Level 1: Know that a point has position, no thickness or distance. A
circle, perpendicular line, parallel line, and
line is made of infinitely many points, and a line segment is a subset
line segment, based on the undefined
of the points on a line with endpoints. A ray is defined as having a
notions of point, line, distance along a line,
point on one end and a continuing line on the other.
and distance around a circular arc.
An angle is determined by the intersection of two rays.
A circle is the set of infinitely many points that are the same distance
from the center forming a circular are, measuring 360 degrees.
Perpendicular lines are lines in the interest at a point to form right
angles.
Parallel lines that lie in the same plane and are lines in which every
point is equidistant from the corresponding point on the other line.
G.1.2 Represent transformations in the
Level 2: Describe and compare geometric and algebraic
plane using transparencies and geometry
transformations on a set of points as inputs to produce another set
software; describe transformations as
of points as outputs, to include translations and horizontal and
functions that take points in the plane as
vertical stretching.
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).
G.1.3 Given a rectangle, parallelogram,
Level 2: Describe the rotations and reflections of a rectangle,
trapezoid, or regular polygon, describe the
parallelogram, trapezoid, or regular polygon that maps each figure
rotations and reflections that carry it onto
onto itself, beginning and ending with the same geometric shape.
itself.
G.1.4 Develop definitions of rotations,
Level 2: Students should understand and be able to explain that
reflections, and translations in terms of
when a figure is reflected about a line, the segment that joins the
angles, circles, perpendicular lines, parallel
preimage point to its corresponding image point is perpendicularly
lines, and line segments.
bisected by the line of reflection. When figures are rotated, the
points travel in a circular path over some specified angle of rotation.
When figures are translated, the segments of the preimage are
parallel to the corresponding segments of the image.
G.1.5 Given a geometric figure and a
Level 2: Using Interactive Geometry Software or graph paper,
rotation, reflection, or translation, draw the perform the following transformations on the triangle ABC with
transformed figure using, e.g., graph paper, coordinates A(4,5), B(8,7) and C(7,9). First, reflect the triangle over
tracing paper, or geometry software.
the line y=x. Then rotate the figure 180° about the origin. Finally,
Specify a sequence of transformations that
translate the figure up 4 units and to the left 2 units.
will carry a given figure onto another.
Write the algebraic rule in the form (x,y)→(x’,y’) that represents this
composite transformation.
Geometry: Congruence
G.2 Understand congruence in terms of rigid motions.
Objectives
What Learner Should Know, Understand, and Be Able to Do
G.2.1 Use geometric descriptions of rigid
Level 2: Use descriptions of rigid motions to move figures in a
motions to transform figures and to predict coordinate plane, and predict the effects rigid motion has on figures
Updated: October, 2013
Teaching Notes and Examples
Definitions are used to begin building blocks for proof. Infuse these
definitions into proofs and other problems. Pay attention to
Mathematical practice 3 “Construct viable arguments and critique the
reasoning of others: Understand and use stated assumptions, definitions
and previously established results in constructing arguments.” Also
mathematical practice number six says, “Attend to precision:
Communicate precisely to others and use clear definitions in discussion
with others and in their own reasoning.”
Level 1 Example: How would you determine whether two lines are parallel
or perpendicular?
Level 2 Example:
Using Interactive Geometry Software perform the following dilations
(x,y)→(4x,4y) (x,y)→(x,4y) (x,y)→(4x, y) on the triangle defined by the
points (1,1) (6,3) (2,13).
Compare and contrast the following from each dilation: angle measure ,
side length, and perimeter.
Level 2 Example: Given combinations of rotations and reflections,
illustrate each combination with a diagram. Where a combination is not
possible, give examples to illustrate why that will not work (coordinate
arguments).
The expectation for Geometry is to build on student experience with rigid
motions from earlier grades. Point out the basis of rigid motions in
geometric concepts, e.g., translations move points a specified distance
along a line parallel to a specified line; rotations move objects along a
circular arc with a specified center through a specified angle.
Teaching Notes and Examples
Level 2 Example: Consider parallelogram ABCD with coordinates A(2,-2),
B(4,4), C(12,4) and D(10,-2). Perform the following transformations. Make
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NC Adult Education Standards for ASE MA 4 Geometry
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.
in the coordinate plane.
Use this fact knowing rigid transformations preserve size and shape
or distance and angle measure, to connect the idea of congruency
and to develop the definition of congruent.
G.2.2 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.
G.2.3 Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from
the definition of congruence in terms of
rigid motions.
Level 2: Use the definition of congruence, based on rigid motion, to
show two triangles are congruent if and only if their corresponding
sides and corresponding angles are congruent.
predictions about how the lengths, perimeter, area and angle measures
will change under each transformation.
a. A reflection over the x-axis.
b. A rotation of 270° about the origin.
c. A dilation of scale factor 3 about the origin.
d. A translation to the right 5 and down 3.
Verify your predictions. Compare and contrast which transformations
preserved the size and/or shape with those that did not preserve size
and/or shape. Generalize, how could you determine if a transformation
would maintain congruency from the preimage to the image?
Level 2 Example: Using Interactive Geometry Software or graph paper
graph the following triangle M(-1,1), N(-4,2) and P(-3,5) and perform the
following transformations. Verify that the preimage and the image are
congruent. Justify your answer.
a. (x,y)→(x+2, y-6)
b. (x,y)→(-x,y)
c. (x,y)→(-y,x)
Level 2: Use the definition of congruence, based on rigid motion, to
develop and explain the triangle congruence criteria; ASA, SSS, SAS,
AAS, and HL.
Students should connect that these triangle congruence criteria are
special cases of the similarity criteria in GSRT.3.
ASA and AAS are modified versions of the AA criteria for similarity.
Students should note that the “S” in ASA and AAS has to be present
to include the scale factor of one, which is necessary to show that it
is a rigid transformation. Students should also investigate why SSA
and AAA are not useful for determining whether triangles are
congruent.
Students can make sense of this problem by drawing diagrams of
important features and relationships and using concrete objects or
pictures to help conceptualize and solve this problem.
Level 2 Example: Andy and Javier are designing triangular gardens for
their yards. Andy and Javier want to determine if their gardens that they
build will be congruent by looking at the measures of the boards they will
use for the boarders, and the angles measures of the vertices. Andy and
Javier use the following combinations to build their gardens.
Will these combinations create gardens that enclose the same area? If so,
how do you know?

Each garden has length measurements of 12ft, 32ft and 28ft.

Both of the gardens have angle measure of 110°, 25° and 45°.

One side of the garden is 20ft another side is 30ft and the angle
between those two boards is 40°.

One side of the garden is 20ft and the angles on each side of that
board are 60° and 80°.

Two sides measure 16ft and 18ft and the non-included angle of the
garden measures 30°.
The instructional expectation for Geometry is to understand that rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions
(that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can
then be used to prove other theorems.
Geometry: Congruence
G.3 Prove geometric theorems: Using logic and deductive reasoning, algebraic and geometric properties, definitions, and proven theorems to draw conclusions.
Encourage multiple ways of writing proofs, to include paragraph, flow charts, and two-column format. These proof standards should be woven throughout the course. Students should be making
arguments about content throughout their geometry experience. The focus is not the particular content items that they are proving. However, the focus is on the idea that students are proving
geometric properties. Pay close attention to the mathematical practices especially number three, “Construct viable argument and critique the reasoning of others.”
Objectives
What Learner Should Know, Understand, and Be Able to Do
Teaching Notes and Examples
G.3.1 Prove theorems about lines and
Level 1: Prove that any point equidistant from the endpoints of a
Level 1 Example:
angles. Theorems include: vertical angles
segment lies on the perpendicular bisector of the line.
A carpenter is framing a wall and wants to make sure his the edges of his
are congruent; when a transversal crosses
Pre-ASE, students have already experimented with these angle/line
wall are parallel. He is using a cross-brace as show in the diagram below.
parallel lines, alternate interior angles are
properties. The focus at this level is to prove these properties, not
What are several different ways he could verify that the edges are
congruent and corresponding angles are
just to use and know them.
parallel? Can you write a formal argument to show that these sides are
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Geometry
congruent; points on a perpendicular
bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
parallel? Pair up with another student who created a different argument
than yours, and critique their reasoning. Did you need to modify the
diagram in anyway to help your argument?
G.3.2 Prove theorems about triangles.
Theorems include: measures of interior
angles of a triangle sum to
180º; 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.
Level 2: Using any method you choose, construct the medians of a
triangle. Each median is divided up by the centroid.
Investigate the relationships of the distances of these segments. Can
you create a deductive argument to justify why these relationships
are true? Can you prove why the medians all meet at one point for all
triangles? Extension: using coordinate geometry, how can you
calculate the coordinate of the centroid? Can you provide an
algebraic argument for why this works for any triangle?
G.3.3 Prove theorems about
parallelograms. Theorems include: opposite
sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram
bisect each other, and conversely,
rectangles are parallelograms with
congruent diagonals.
Jerry is laying out the foundation for a rectangular foundation for an
outdoor tool shed. He needs ensure that it is indeed fulfills the
definition of a rectangle. The only tools he brought with him are pegs
(for nailing in the ground to mark the corners), string and a tape
measure. Create a plan for Jerry to follow so that he can be sure his
foundation is rectangular. Justify why your plan works. Discuss your
method with another student to make sure your plan is error proof.
Level 2 Example: Using Interactive Geometry Software or tracing paper,
investigate the relationships of sides and angles when you connect the
midpoints of the sides of a triangle. Using coordinates can you justify why
the segment that connects the midpoints of two of the sides is parallel to
the opposite side. If you have not done so already, can you generalize
your argument and show that it works for all cases? Using coordinates
justify that the segment that connects the midpoints of two of the sides is
half the length of the opposite side. If you have not done so already, can
you generalize your argument and show that it works for all cases?
When teaching, connect this standard with G.2.3 and use triangle
congruency criteria to determine all of the properties of parallelograms
and special parallelograms.
Geometry: Congruence
G.4 Make geometric constructions: Create formal geometric constructions using a compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software.
Objectives
What Learner Should Know, Understand, and Be Able to Do
Teaching Notes and Examples
G.4.1 Make formal geometric constructions 
Copy a congruent segment
with a variety of tools and methods

Copy a congruent angle.
(compass and straightedge, string,

Bisect a segment
reflective devices, paper folding, dynamic

Bisect an angle
geometric software, etc.). Copying a

Construct perpendicular lines, including the perpendicular bisector of a line segment. Using a compass and straightedge, construct a
segment; copying an angle; bisecting a
perpendicular bisector of a segment. Prove/justify why this process provides the perpendicular bisector.
segment; bisecting an angle; constructing

Given a triangle, construct the circumcenter and justify/prove why the process gives the point that is equidistant from the vertices.
perpendicular lines, including the

Given a triangle, construct the incenter and justify/prove why the process gives the point that is equidistant from the sides of the triangle.
perpendicular bisector of a line segment;

Construct a line parallel to a given line through a point not on the line.
and constructing a line parallel to a given
***It makes sense to combine this standard with G.3.1 and G.3.2 and have students make arguments about why these constructions work.
line through a point not on the line.
G.4.2 Construct an equilateral triangle, a
Level 2: Using a compass and straightedge or Interactive Geometry
Level 2 Example:
square, and a regular hexagon inscribed in a Software, construct an equilateral triangle so that each vertex of the
Donna is building a swing set. She wants to countersink the nuts by
circle.
equilateral triangle is on the circle. Construct an argument to show
drilling a hole that will perfectly circumscribe the hexagonal nuts that she
that your construction method will produce an equilateral triangle. If
is using. Each side of the regular hexagonal nut is 8mm. Construct a
your method and/or argument are different than another students’,
diagram that shows what size drill she should use to countersink the nut.
understand his/her argument and decide whether it makes sense and
look for any flaws in their reasoning. Repeat this process for a square
inscribed in a circle and a regular hexagon inscribed in a circle.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Geometry
Similarity, Right Triangles, and Trigonometry
G.5 Understand similarity in terms of similarity transformations.
Objectives
What Learner Should Know, Understand, and Be Able to Do
G.5.1 Verify experimentally the properties
Level 2:
Diagram 1
of dilations given by a center and a scale
a. Given a center of a dilation and a scale factor, verify using
factor.
coordinates that when dilating a figure in a coordinate plane, a
a. A dilation takes a line not passing
segment of the pre-image that does not pass through the center of
through the center of the dilation to a
the dilation, is parallel to it’s image when the dilation is preformed.
parallel line, and leaves a line passing
However, a segment that passes through the center does not change.
through the center unchanged.
See Diagram 1.
b. The dilation of a line segment is longer or
Rule for the transformation is (x,y)→(⅓x,⅓y).
shorter in the ratio given by the scale
Comment on the slopes and lengths of the segments that make up
factor.
the triangles. What patterns do you notice? How could you be
confident that this pattern would be true for all cases? See Diagram 2
Diagram 2
Rule for the transformation is (x,y)→(.x,.y).
Comment on the slopes and lengths of the segments that make up
the triangles.
1
Teaching Notes and Examples
y
x
2
4
6
−1
−2
−3
y
b. Given a center and a scale factor, verify using coordinates, that
when performing dilations of the preimage, the segment that
becomes the image, is longer or shorter based on the ratio given by
the scale factor.
What patterns do you notice? How could you be confident that this
pattern would be true for all cases?
Students should have already had experience with dilations and
other transformations in GLE 8 and proportional reasoning in GLE 6-7
G.5.2 Given two figures, use the definition
of similarity in terms of similarity
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.
G.5.3 Use the properties of similarity
transformations to establish the AA
criterion for two triangles to be similar.
Use the idea of geometric transformations to develop the definition
of similarity.
Given two figures determine whether they are similar and explain
their similarity based on the congruency of corresponding angles and
the proportionality of corresponding sides.
x
2
4
6
8
−2
−4
Instructional note: The ideas of congruency and similarity are related. It is
important for students to connect that congruency is a special case of
similarity with a scale factor of one. Therefore these similarity rules can
be expanded to work for congruency in triangles. AA similarity is the
foundation for ASA and AAS congruency theorems.
Knowing from the definition of a dilation, angle measures are preserved
and sides change by a multiplication of scale factor k.
Use the properties of similarity transformations to develop the
criteria for proving similar triangles by AA, SSS, and SAS.
Connect this standard with standard G-SRT.4
Level 3: Given that VMNP is a dilation of VABC with scale factor k, use
properties of dilations to show that the AA criterion is sufficient to prove
similarity.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Geometry
Similarity, Right Triangles, and Trigonometry
G.6 Define trigonometric ratios and solve problems involving right triangles.
Objectives
What Learner Should Know, Understand, and Be Able to Do
Teaching Notes and Examples
G.6.1 Understand that by similarity, side
Level 2: Using corresponding angles of similar right triangles, show
There are three points on a line that goes through the origin, (5,12)
ratios in right triangles are properties of the that the relationships of the side ratios are the same, which leads to
y
(10,24) (15,36). Sketch this graph. How do the ratios of
compare? Why
angles in the triangle, leading to definitions
the definition of trigonometric ratios for acute angles.
x
of trigonometric ratios for acute angles.
Students should use their knowledge of dilations and similarity to
does this make sense? Call the distance from the origin to each point
justify why these triangles are congruent. It is not expected at this
stage that they will know the Triangle Similarity Theorems.
y
x
(“r”). Find the r for each point. Find the ratios of
and
.
r
G.6.2 Explain and use the relationship
between the sine and cosine of
complementary angles.
G.6.3 Use trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems.
Level 2
Apply both trigonometric ratios and Pythagorean Theorem to solve
application problems involving right triangles.
r
Level 2 Example: An onlooker stands at the top of a cliff 119 meters above
the water’s surface. With a clinometer, she spots two ships due west.
The angle of depression to each of the sailboats is 11 and 16. Calculate
the distance between the two sailboats. What is the distance from the
onlooker’s eyes to each of the sailboats? What is the difference of those
distances? Explain why or why not this difference is not the same as the
distance between the two sailboats.
Similarity, Right Triangles, and Trigonometry
G.7 Apply trigonometry to general triangles.
Objectives
G.7.1 (+) Derive the formula A=½ ab sin (C)
for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to
the opposite side.
G.7.2 (+) Understand and apply the Law of
Sines and the Law of Cosines to find
unknown measurements in right and nonright triangles (e.g., surveying problems,
resultant forces).
What Learner Should Know, Understand, and Be Able to Do
Level 2: For a triangle that is not a right triangle, draw an auxiliary
line from a vertex, perpendicular to the opposite side and derive the
formula, A=½ ab sin (C), for the area of a triangle, using the fact that
the height of the triangle is, h=a sin(C). The focus is on deriving the
formula, not using it.
Level 2: A surveyor standing at point C is measuring the length of a
property boundary between two points located at A and B. Explain
what measurements he is able to collect using his transit. Create a
plan for this surveyor to find the length of the boundary between A
and B. How does the surveyor use the law of sines and/or cosines in
this problem? Will your process develop a reliable answer? Justify.
Teaching Notes and Examples
Expressing Geometric Properties with Equations
G.8 Use Coordinates to Prove Simple Geometric Theorems Algebraically
Objectives
What Learner Should Know, Understand, and Be Able to Do
Teaching Notes and Examples
G.8.1 Use coordinates to prove simple
Level 1: Use the concepts of slope and distance to prove that a figure
Level 1 Example: The coordinates are for a quadrilateral, (3, 0), (1, 3), (-2,
geometric theorems algebraically. For
in the coordinate system is a special geometric shape.
1), and (0,-2). Determine the type of quadrilateral made by connecting
example, prove or disprove that a figure
these four points? Identify the properties used to determine your
defined by four given points in the
classification. You must give confirming information about the polygon. .
coordinate plane is a rectangle; prove or
Level 1 Example: If Quadrilateral ABCD is a rectangle, where A(1, 2), B(6,
disprove that the point (1, √3) lies on the
0), C(10,10) and D(?, ?) is unknown.
circle centered at the origin and containing

Find the coordinates of the fourth vertex.
the point (0, 2).

Verify that ABCD is a rectangle providing evidence related to the
sides and angles.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Geometry
G.8.2 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).
G.8.3 Find the point on a directed line
segment between two given points that
partitions the segment in a given ratio.
G.8.4 Use coordinates to compute
perimeters of polygons and areas of
triangles and rectangles, e.g., using the
distance formula.
Level 1
Use the formula for the slope of a line to determine whether two
lines are parallel or perpendicular. Two lines are parallel if they have
the same slope and two lines are perpendicular if their slopes are
opposite reciprocals of each other. In other words the product of the
slopes of lines that are perpendicular is (-1). Find the equations of
lines that are parallel or perpendicular given certain criteria.
Given two points on a line, find the point that divides the segment
into an equal number of parts. If finding the mid-point, it is always
halfway between the two endpoints. The x-coordinate of the midpoint will be the mean of the x-coordinates of the endpoints and the
y-coordinate will be the mean of the y-coordinates of the endpoints.
At this level, focus on finding the midpoint of a segment.
Students should find the perimeter of polygons and the area of
triangles and rectangles using coordinates on the coordinate plane.
Level 1 Example: Suppose a line k in a coordinate plane has slope c .


d
What is the slope of a line parallel to k? Why must this be the case?
What is the slope of a line perpendicular to k? Why does this seem
reasonable?
Level 1 Example: Two points A(0, -4) , B(2, -1) determines a line, AB.
What is the equation of the line AB? What is the equation of the line
perpendicular to AB passing through the point (2,-1)?
Level 1 Example: There is a situation in which two lines are perpendicular
but the product of their slopes is not (-1). Explain the situation in which
this happens.
Level 1 Example: If you are given the midpoint of a segment and one
endpoint. Find the other endpoint.
a. midpoint: (6, 2) endpoint: (1, 3) b. midpoint: (-1, -2) endpoint: (3.5, -7)
Level 2 Example: If Jennifer and Jane are best friends. They placed a map
of their town on a coordinate grid and found the point at which each of
their house lies. If Jennifer’s house lies at (9, 7) and Jane’s house is at (15,
9) and they wanted to meet in the middle, what are the coordinates of
the place they should meet?
Level 1 Example: John was visiting three cities that lie on a coordinate grid
at (-4, 5), (4, 5), and (-3, -4). If he visited all the cities and ended up where
he started, what is the distance in miles he traveled?
Expressing Geometric Properties with Equations
G.9 Translate between the geometric description and the equation for a conic section.
Objectives
What Learner Should Know, Understand, and Be Able to Do
Teaching Notes and Examples
G.9.1 Derive the equation of a circle of
Level 2
Given an equation of a circle, complete the square to find the center and
given center and radius using the
Use the Pythagorean Theorem to derive the equation of a circle,
radius of a circle.
Pythagorean Theorem; complete the
given the center and the radius.
Given a coordinate and a distance from that coordinate develop a rule
square to find the center and radius of a
that shows the locus of points that is that given distance from the given
circle given by an equation.
point (based on the Pythagorean theorem).
Geometric Measure and Dimension
G.10 Explain Volume Formulas and Use Them to Solve Problems
Objectives
What Learner Should Know, Understand, and Be Able to Do
G.10.1 Give an informal argument for the
Understanding the formula for the circumference of a circle, you can
formulas for the circumference of a circle,
either begin with the measure of the diameter or the measure of the
area of a circle, volume of a cylinder,
radius. Take those measurements and measure around the outside
pyramid, and cone. Use dissection
of a circle. The diameter will go around a little over 3 times, which
arguments, Cavalieri’s principle, and
indicates 𝐶 = 𝜋𝑑. The radius will go around half of the circle a little
informal limit arguments.
over 3 times, therefore 𝐶 = 2𝜋𝑟. This can either be done using pipe
cleaners or string and a measuring tool.
Understanding the volume of a cylinder is based on the area of a
Updated: October, 2013
Teaching Notes and Examples
Understanding the formula for the circumference of a circle can be taught
using the diameter of the circle or the radius of the circle. Measure either
the radius or diameter with a string or pipe cleaner. As you measure the
distance around the circle using the measure of the diameter, you will
find that it’s a little over three, which is pi. Therefore the circumference
can be written as 𝐶 = 𝜋𝑑. When measuring the circle using the radius,
you get a little over 6, which is 2𝜋. Therefore, the circumference of the
circle can also be expressed using 𝐶 = 2𝜋𝑟.
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NC Adult Education Standards for ASE MA 4 Geometry
G.10.2 Use volume formulas for cylinders,
pyramids, cones, and spheres to solve
problems.
circle, realizing that the volume is the area of the circle over and over
again until you’ve reached the given height, which is a simplified
version of Cavalieri’s principle. In Cavalieri’s principle, the crosssections of the cylinder are circles of equal area, which stack to a
specific height. Therefore the formula for the volume of a cylinder is
𝑉 = 𝐵ℎ. Informal limit arguments are not the intent at this level.
Understanding the formula for the area of a circle can be shown using
dissection arguments. First dissect portions of the circle like pieces of a
pie. Arrange the pieces into a curvy parallelogram as indicated below.
Formulas for pyramids, cones, and spheres will be given.
Level 1 Example: Given the formula 𝑉 = 𝐵𝐻, for the volume of a cone,
3
where B is the area of the base and H is the height of the. If a cone is
inside a cylinder with a diameter of 12in. and a height of 16 in., find the
volume of the cone.
Geometric Measure and Dimension
G.11 Visualize relationships between two-dimensional and three-dimensional objects.
Objectives
What Learner Should Know, Understand, and Be Able to Do
G.11.1 Identify the shapes of twoLevel 2: Given a three- dimensional object, identify the shape made
dimensional cross-sections of threewhen the object is cut into cross-sections.
dimensional objects, and identify threeLevel 2: When rotating a two- dimensional figure, such as a square,
dimensional objects generated by rotations know the three-dimensional figure that is generated, such as a
of two-dimensional objects.
cylinder. Understand that a cross section of a solid is an intersection
of a plane (two-dimensional) and a solid (three-dimensional).
Modeling with Geometry
G.12 Apply geometric concepts in modeling situations.
Objectives
What Learner Should Know, Understand, and Be Able to Do
G.12.1 Use geometric shapes, their
Level 2 Use geometric shapes, their measures, and their properties to
measures, and their properties to describe
describe objects (e.g., modeling a soda can or the paper towel roll as
objects (e.g., modeling a tree trunk or a
a cylinder).
human torso as a cylinder).
G.12.2 Apply concepts of density based on
area and volume in modeling situations
(e.g., persons per square mile, BTUs per
cubic foot).
Level 2
Use the concept of density when referring to situations involving
area and volume models, such as persons per square mile.
G.12.3 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).
Level 2: Solve design problems by designing an object or structure
that satisfies certain constraints, such as minimizing cost or working
with a grid system based on ratios (i.e., The enlargement of a picture
using a grid and ratios and proportions)
Updated: October, 2013
1
Teaching Notes and Examples
Teaching Notes and Examples
Level 2 Example: Consider a rectangular swimming pool 30 feet long and
20 feet wide. The shallow end is 3½ feet deep and extends for 5 feet.
Then for 15 feet (horizontally) there is a constant slope downwards to the
10 foot-deep end.

Sketch the pool and indicate all measures on the sketch.

How much water is needed to fill the pool to the top? To a level 6
inches below the top?

One gallon of pool paint covers approximately 75 sq. feet of surface.
How many gallons of paint are needed to paint the inside walls of the
pool? If the pool paint comes in 5-gallon cans, how many cans are
needed?

How much material is needed to make a rectangular pool cover that
extends 2 feet beyond the pool on all sides?

How many 6-inch square ceramic tiles are needed to tile the top 18
inches of the inside faces of the pool? If the lowest line of tiles is to
be in a contrasting color, how many of each tile are needed?
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NC Adult Education Standards for ASE MA 4 Statistics and Probability
Interpreting Categorical and Quantitative Data
SP.1 Summarize, represent, and interpret data on a single count or measurement variable.
What Learner Should Know, Understand, and
Objectives
Teaching Notes and Examples
Be Able to Do
SP.1.1 Represent data with
Construct appropriate graphical displays (dot
Level 1 Example: Make a dot plot of the number of siblings that members of your class have.
plots on the real number line
plots, histogram, and box plot) to describe sets
(Level I)
(dot plots, histograms, and
of data values.
Level 1 Example: Create a frequency distribution table and histogram for the following set of data:
box plots).
Age (in months) of First Steps
13 9 12 11 10 8.5 14 9 12.5 10 13.5 9.5 6 7.5 15 9 8 11.5 10 12 10.5 11 13 12.5
Level 1 Example: Construct a box plot of the number of buttons each of your classmates has on their
clothing today.
SP.1.2 Use statistics
Understand which measure of center and which Level 1 Example: You are planning to take on a part time job as a waiter at a local restaurant. During your
appropriate to the shape of
measure of spread is most appropriate to
interview, the boss told you that their best waitress, Jenni, made an average of $70 a night in tips last week.
the data distribution to
describe a given data set. The mean and
However, when you asked Jenni about this, she said that she made an average of only $50 per night last
compare center (median,
standard deviation are most commonly used to
week. She provides you with a copy of her nightly tip amounts from last week (see below). Calculate the
mean) and spread
describe sets of data. However, if the
mean and the median tip amount.
(interquartile range, standard
distribution is extremely skewed and/or has
a. Which value is Jenni’s boss using to describe the average tip? Why do you think he chose this
deviation) of two or more
outliers, it is best to use the median and the
value?
different data sets.
interquartile range to describe the distribution
b. Which value is Jenni using? Why do you think she chose this value?
since these measures are not sensitive to
c. Which value best describes the typical amount of tips per night? Explain why.
outliers.
Sunday $50, Monday $45, Wednesday $48, Friday $125, Saturday $85
Select the appropriate measures to describe and
compare the center and spread of two or more
Level 1 Example: A teacher wanted to evaluate the background knowledge in three broad topics of the
data sets in context.
students entering her course. In the first class she gave a multiple choice pre-test with ten questions on each
topic and recorded the number of correct answers for each of the fifteen students in her class. Use
appropriate statistics to construct an argument for which of the three topics needs the most emphasis in her
course.
Topic A: 7, 6, 8, 5, 5, 5, 3, 5, 7, 6, 9, 2, 6, 7, 8
Topic B: 1, 9, 6, 8, 7, 8, 9, 6, 5, 6, 9, 9, 7, 4, 9
Topic C: 1, 2, 10, 5, 1, 2, 4, 9, 10, 0, 10, 7, 3, 3, 10
SP.1.3 Interpret differences in Understand and be able to use the context of
Level 1 Examples:
shape, center, and spread in
the data to explain why its distribution takes on
Create a data set based on test scores that illustrates the following
the context of the data sets,
a particular shape (e.g. are there real-life limits
a. A skewed left distribution. b. A skewed right distribution c. A symmetrical distribution
accounting for possible effects to the values of the data that force skewness?
Comment on what the distribution tells you about the difficulty level of the test.
of extreme data points
are there outliers?)
On last week’s math test, Mrs. Smith’s class had an average of 83 points with a standard deviation of 8
(outliers).
points. Mr. Tucker’s class had an average of 78 points with a standard deviation of 4 points. Which class was
Understand that the higher the value of a
more consistent with their test scores? How do you know?
measure of variability, the more spread out the
data set is.
The heights of Washington High School’s basketball players are: 5 ft 9in, 5 ft 4in, 5 ft 7 in, 5ft 6 in, 5 ft. 5 in,
5 ft. 3 in, and 5 ft 7 in. A student transfers to Washington High and joins the basketball team. Her height is 6
Explain the effect of any outliers on the shape,
ft. 10in.
center, and spread of the data sets.
a. What affect does her height have on the team’s height distribution and stats (center and spread)?
b. How many players are taller than the new mean team height?
c. Which measure of center most accurately describes the team’s average height? Explain.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Statistics and Probability
SP.1.4 Use the mean and
standard deviation of a data
set to fit it to a normal
distribution and to estimate
population percentages.
Recognize that there are data
sets for which such a
procedure is not appropriate.
Use calculators, spreadsheets,
and tables to estimate areas
under the normal curve.
a. Understand the characteristics of the
Standard Normal Distribution (μ = 0, σ = 1),
including symmetry, the Empirical Rule (6895-99.7 rule) and the fact that the mean =
median = mode.
b. Find the number of standard deviations a
value is from the mean by calculating its zscore.
c. Use the Empirical Rule to estimate
population percentages for a set of data that
is approximately normally distributed.
d. Understand that population percentages
correspond to areas under the normal
distribution curve between given values.
e. Use a table of z-scores, spreadsheets and
calculators to find areas under the curve to
estimate population percentages. Interpret
these percentages in context.
f. Understand that the use of the normal
distribution to estimate population
percentages is only appropriate for moundshaped, symmetrical distributions.
Level 3 Examples:
a. Describe the characteristics of the Standard Normal Distribution.
b. IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15. How
many standard deviations below the mean is an IQ score of 75?
c. IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15.
What percent of the population has IQ scores above 115? What percent have IQ scores between 85 and
130?
d/e The heights of adult males are approximately normally distributed with μ = 70 in. and σ = 3 in. What
percentage of adult males are between 5 and 6 ft tall? What percentage of adult males are 7 ft tall or
taller?
f. Which of the following sets of data are approximately normally distributed?

Weights of adult females

Incomes of NFL football players

Diameters of trees in Umstead Park

Shoe sizes of sophomore males at your school

Female ages at marriage
Interpreting Categorical and Quantitative Data
SP.2 Summarize, represent, and interpret data on two categorical and quantitative variables.
What Learner Should Know, Understand, and Be Able to
Objectives
Teaching Notes and Examples
Do
SP.2.1 Summarize categorical
Create a two-way frequency table from a set of data on two
Level I Example: Make a two-way frequency table for the following set of data. Use the
data for two categories in
categorical variables. Calculate joint, marginal, and
following age groups:
two-way frequency tables.
conditional relative frequencies and interpret in context.
3-5, 6-8, 9-11, 12-14, 15-17
Interpret relative frequencies
Joint relative frequencies are compound probabilities of
.
Youth Soccer League
in the context of the data
using AND to combine one possible outcome of each
Gender Age
Gender Age
Gender Age
Gender Age
Gender Age
(including joint, marginal, and
categorical variable (P(A and B)). Marginal relative
M
4
F
7
M
17
M
5
F
10
M
7
M
7
M
16
M
9
M
6
conditional relative
frequencies are the probabilities for the outcomes of one
F
8
F
15
F
14
F
13
F
4
frequencies). Recognize
of the two categorical variables in a two-way table,
F
6
M
13
M
14
M
15
M
5
possible associations and
without considering the other variable. Conditional
M
4
M
12
F
12
M
17
M
9
trends in the data.
relative frequencies are the probabilities of one particular
F
10
M
15
F
8
M
12
M
10
outcome of a categorical variable occurring, given that
F
11
F
16
M
13
F
13
F
15
one particular outcome of the other categorical variable
Use
the
frequency
table
to
answer
the
following
questions:
has already occurred.
a. What is the relative frequency of players who are male and 9-11 years old? (joint relative
Recognize associations and trends in data from a two-way
frequency)
table.
b. What is percentage of female players that are 15-17 years old? (conditional relative
frequency)
c. What percentage of league members are male? (marginal relative frequency)
Updated: October, 2013
Page 9
NC Adult Education Standards for ASE MA 4 Statistics and Probability
SP.2.2 Represent data on two
quantitative variables on a
scatter plot, and describe how
the variables are related.
a. Fit a function to the data;
use functions fitted to data
to solve problems in the
context of the data. Use
given functions or choose a
function suggested by the
context. Emphasize linear
and exponential models.
b. Informally assess the fit of
a
c. function by plotting and
analyzing residuals.
d. Fit a linear function for a
scatter plot that suggests a
linear association.
Create a scatter plot from two quantitative variables.
Describe the form, strength, and direction of the
relationship between the two variables in context.
a. Determine which type of function best models a set of
data. Fit this type of function to the data and interpret
constants and coefficients in the context of the data (e.g.
slope and y-intercept of linear models, base/growth or
decay rate and y-intercept of exponential models). Use
the fitted function to make predictions and solve
problems in the context of the data.
b. Calculate the residuals for the data points fitted to a
function. A residual is the difference between the actual
y-value and the predicted y-value (� − �), which is a
measure of the error in prediction. (Note: � is the symbol
for the predicted y-value for a given x-value.) A residual is
represented on the graph of the data by the vertical
distance between a data point and the graph of the
function.
c. Create and analyze a residual plot. A residual plot is a
graph of the x-values vs. their corresponding residuals.
(Note that some computer software programs plot � vs.
residual instead of x vs. residual. However, the
interpretation of the residual plot remains the same.) If
the residual plot shows a balance between positive and
negative residuals and a lack of a pattern, this indicates
that the model is a good fit. For more accurate
predictions, the size of the residuals should be small
relative to the data.
d. For data sets that appear to be linear, use algebraic
methods and technology to fit a linear function to the
data. To develop the concept of LSRL, begin by finding the
centroid (�, �) and selecting another point to fit a line
through the center of the data. (Note: When describing a
set of one-variable data, the mean is the most common
predictor of a value in that data set. Therefore, the
centroid is a logical choice for a point on the line of best
fit because it uses the average of the x-values and the
average of the y-values.) Find the sum of the squared
errors of this line and compare to lines fitted to the same
set of data (but a different second point) by others. The
Least Squares Regression Line is a line that goes through
the centroid and minimizes the sum of these squared
errors.
Level 1 Examples:
In a keyboarding lab, the instructor gives an assignment and records the number of errors along
with the amount of practice each student has completed. The data for each student is
Student Practice time (hours) Number of errors
Amy
6
2
Bob
3
3
Edna
4
4
George
6
0
Jean
3
3
Nancy
2
6
Steve
5
1
Zoey
6
3
Describe, in context, the form, strength, and direction of a scatterplot of the above data.
What type of function models the data found in the scatterplot above? Find the function that
best describes the data. What is the meaning of the slope and y-intercept in the context of the
problem? Use the model to predict Connie’s earnings for selling 100 services.
Calculate the residuals from the plot above. What do they represent? Are the points with
negative residuals located above or below the regression line?
What is the sum of the squared residuals of the linear model that represents the situation
described
above? Can you find a different line that gives a smaller sum? Explain.
Below is the data for the 1919 season and World Series batting averages for nine White Sox
players.
a.
b.
c.
Season
Batting
Average
.319
.279
.275
.290
.351
.302
.256
.282
.296
World Series
Batting
Average
.226
.250
.192
.233
.375
.056
.080
.304
.324
Create a scatter plot for the data provided. Is there a linear association? Explain.
What is the Least Squares Regression Line that models this data?
How do you know this equation is the line of best fit to model the data?
Interpreting Categorical and Quantitative Data
SP.3 Interpret linear models.
Updated: October, 2013
Page 10
NC Adult Education Standards for ASE MA 4 Statistics and Probability
Objectives
SP.3.1 Interpret the slope
(rate of change) and the
intercept (constant term) of a
linear model in the context of
the data.
SP.3.2 Compute (using
technology) and interpret the
correlation coefficient of a
linear fit.
SP.3.3 Distinguish between
correlation and causation.
What Learner Should Know, Understand, and
Be Able to Do
Understand that the key feature of a linear
function is a constant rate of change. Interpret
in the context of the data, i.e. as x increases (or
decreases) by one unit, y increases (or
decreases) by a fixed amount. Interpret the yintercept in the context of the data, i.e. an initial
value or a one-time fixed amount.
Understand that the correlation coefficient, r, is
a measure of the strength and direction of a
linear relationship between two quantities in a
set of data. The magnitude (absolute value) of r
indicates how closely the data points fit a linear
pattern. If r = 1, the points all fall on a line. The
closer � is to 1, the stronger the correlation.
The closer � is to zero, the weaker the
correlation. The sign of r indicates the direction
of the relationship – positive or negative.
Understand that just because two quantities
have a strong correlation, we cannot assume
that the explanatory (independent) variable
causes a change in the response (dependent)
variable. The best method for establishing
causation is to conduct an experiment that
carefully controls for the effects of lurking
variables. If this is not feasible or ethical,
causation can be established by a body of
evidence collected over time (e.g. smoking
causes cancer).
Teaching Notes and Examples
Level 1 Example: The equation � = 40 + 2� represents a pay plan offered to employees who collect credit
card applications. What do the numbers in the rule tell you about the relationship between daily pay and
the number of credit card applications collected?
Level 1 Example: A medical researcher records the weights and blood pressures of 8 subjects as shown
below.
Subject:
1
2
3
4
5
6
7
8
Weight:
132 209 142
184 159
180
161 173
Systolic blood pressure:
78
96
79
88
81
93
82
90
a. Using technology, make a scatterplot for the two rankings.
b. Predict the value of the correlation coefficient (r). Use the scatterplot to help explain your answer.
c. Find the Least Squares Regression Line that models this set of data.
d. Using technology, determine the correlation coefficient and interpret what it means in this context.
Level 1 Examples:
When you have an association between two variables, how can you determine if the association is a result
of a cause-and-effect relationship?
There is a strong positive association between the number of firefighters at a fire and the amount of
damage. John said “This means that firefighters must be the cause of the damage at a fire.” Is John correct in
his reasoning? Explain why or why not.
Making Inferences and Justifying Conclusions
SP.4 Understand and evaluate random processes underlying statistical experiments.
What Learner Should Know, Understand, and
Objectives
Teaching Notes and Examples
Be Able to Do
SP.4.1 Understand statistics as a. Explain the difference between a
Level 3 Examples:
a process for making
population parameter and a sample
a. What is the difference between a population parameter and a sample statistic?
inferences about population
statistic.
b. Describe the process of statistical inference.
parameters based on a
b. Understand that random samples tend to
c. For statistical inference, why is it important that a sample be representative of the population it is
random sample from that
be representative of the population they
drawn from?
population.
are drawn from and therefore we can draw d. What kind of sampling method would you use for each of the following situations? Explain why in each
conclusions about the population based on
case.
the sample. If conclusions cannot be drawn

To determine which gubernatorial candidate voters are most likely to choose in the next election.
from the random sample, discuss why and

To determine the quality of potato chips being produced at a factory.
propose a better way to select a random

To determine the average size of bass fish in a lake.
sample.

To determine the average number of televisions per household in the US.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Statistics and Probability
c.
SP.4.2 Decide if a specified
model is consistent with
results from a given data
generating process, e.g., using
simulation. For example, a
model says a spinning coin
falls heads up with probability
0.5. Would a result of 5 tails in
a row cause you to question
the model?
Demonstrate understanding of the
different kinds of sampling methods
(simple random sample, systematic random
sample, stratified random sample, cluster
or multistage sample, convenience
sample).
Use data-generating processes such as
simulations to evaluate the validity of a
statistical model.
Level 3 Example:
Jack rolls a 6 sided die 15 times and gets the following results:
4, 6, 1, 3, 6, 6, 2, 5, 6, 5, 4, 1, 6, 3, 2
Based on these results, is Jack rolling a fair die? Justify your answer using a simulation.
Making Inferences and Justifying Conclusions
SP.5 Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
Objectives
What Learner Should Know, Understand, and Be Able to Do
Teaching Notes and Examples
SP.5.1 Recognize the purposes a. Students should understand that sample surveys are used to
Level 3 Example:
of and differences among
collect data from human subjects to describe the population of
For each of the following situations, decide whether an experiment or an
sample surveys, experiments,
interest. Experiments and observational studies are used to
observational study is more
and observational studies;
establish a cause and effect relationship.
appropriate to determine if there is a cause and effect relationship:
explain how randomization
b. In an experiment, a treatment is imposed on the experimental
a. Cell phone use and brain tumors
relates to each.
units. In an observational study, the treatment is not imposed but
b. Use of a fertilizer and growth of plants
the relationship between the variables of interest is observed
(e.g. smoking and birth defects).
c. Students should understand that in sample surveys
randomization occurs when the sample is selected. For surveys,
randomization ensures that the sample is representative of the
population it is drawn from. With experiments, randomization
occurs when experimental units are assigned to treatments
(randomized comparative experiment) in order to ensure that the
treatment groups are equivalent. In an observational study, there
is no random assignment of treatments. (For example, when
looking at the relationship between smoking and cancer, we do
not “assign” people to be smokers; they choose to be one or the
other. We then observe the rate of cancer for smokers and for
non –smokers and compare.)
d. Given a situation, decide whether an experiment or observational
study is more appropriate to establish a cause and effect
relationship.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Statistics and Probability
SP.5.2 Use data from a sample
survey to estimate a
population mean or
proportion; develop a margin
of error through the use of
simulation models for random
sampling.
SP.5.3 Use data from a
randomized experiment to
compare two treatments; use
simulations to decide if
differences between
parameters are significant.
Given data from a sample survey, calculate the sample mean or
sample proportion and use it to estimate the population value.
Use simulation to collect multiple samples. Calculate the sample mean
or proportion for each and use this information to determine a
reasonable margin of error for the population estimate.
Determine whether one treatment is more effective than another
treatment in a randomized experiment.
Use simulations to generate data simulating application of two
treatments. Use results to evaluate significance of differences.
Level 3 Examples:
You conduct a survey at your school to determine who is the students’ favorite
instructor. You randomly survey 20 students and receive the responses below:
Gabe Sarah Gabe Lola Andy Andy Gabe Lola Andy Sarah
Gabe Lola Andy Gabe Gabe Sarah Andy Gabe Lola Sarah
How likely is it that Gabe is the favorite of most students at this school? Create an
argument justifying your conclusion. For the situation above, use technology to
generate random numbers to simulate multiple samples of 20 votes. Calculate the
proportion of votes for Gabe in each sample and use this information to determine a
reasonable margin of error for the proportion of students whose favorite instructor is
Gabe.
Level 3 Examples: Sal purchased two types of plant fertilizer and conducted an
experiment to see which fertilizer would be best to use in his greenhouse. He planted
20 seedlings and used Fertilizer A on ten of them and Fertilizer B on the other ten. He
measured the height of each plant after two weeks. Use the data below to determine
which fertilizer Sal should use. Write a letter to Sal describing your recommendation.
Be sure to explain fully how you arrived at your conclusion.
Plant Height (cm)
Fertilizer A 23.4 30.1 28.5 26.3 32.0 29.6 26.8 25.2 27.5 30.8
Fertilizer B 19.8 25.7 29.0 23.2 27.8 31.1 26.5 24.7 21.3 25.6
SP.5.4 Evaluate reports based
on data.
Evaluate reports based on data on multiple aspects (e.g. experimental
design, controlling for lurking variables, representativeness of
samples, choice of summary statistics, etc.)
Use the list above to generate simulated treatment results by randomly selecting ten
plant heights from the twenty plant heights listed. (Mix them all together.) Calculate
the average plant height for each “treatment” of ten plants. Find the difference
between consecutive pairs of “treatment” averages and compare. Does your
simulated data provide evidence that the difference in average plant heights using
Fertilizer A and Fertilizer B is significant? Explain.
Level 3 Example:
Find a statistical report based on an experiment and evaluate the experimental
design.
Conditional Probability and the Rules of Probability
SP.6 Understand independence and conditional probability and use them to interpret data.
What Learner Should Know, Understand, and
Objectives
Teaching Notes and Examples
Be Able to Do
SP.6.1 Describe events as
a. Define the sample space for a given
Level 2 Examples:
subsets of a sample space (the
situation.
a. What is the sample space for rolling a die?
set of outcomes) using
b. Describe an event in terms of categories or
b. What is the sample space for randomly selecting one letter from the word MATHEMATICS?
characteristics (or categories)
characteristics of the outcomes in the
c. Describe different subsets of outcomes for rolling a die using a single category or characteristic.
of the outcomes, or as unions,
sample space.
d. Describe the following subset of outcomes for choosing one card from a standard deck of cards as the
intersections, or complements c. Describe an event as the union,
intersection of two events: {queen of hearts, queen of diamonds}.
of other events (“or,” “and,”
intersection, or complement of other
“not”).
events.
Updated: October, 2013
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NC Adult Education Standards for ASE MA 4 Statistics and Probability
SP.6.2 Understand that two
events A and B are
independent if the probability
of A and B occurring together
is the product of their
probabilities, and use this
characterization to determine
if they are independent.
SP.6.3 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.
SP.6.4 Construct and interpret
two-way frequency tables of
data when two categories are
associated with each object
being classified. Use the twoway table as a sample space
to decide if events are
independent and to
approximate conditional
probabilities. For 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.
Updated: October, 2013
Understand that two events A and B are
independent if and only if P(A and B) = P(A) x
P(B).
Level 2 Examples:
For the situation of drawing a card from a standard deck of cards, consider the two events of “draw a
diamond” and “draw an ace.” Determine if these two events are independent.
Determine whether two events are independent
using the Multiplication Rule (stated above).
Create and prove two events are independent from drawing a card from a standard deck.
a.
Level 3 Example:
Understand that the conditional probability
of event A given event B has already
happened is given by the formula:
P(A|B) =
b.
c.
a.
b.
c.
P(A and B)
P(B)
Understand that two events A and B are
independent if and only if P(A|B) =
P(A) and P(B|A) = P(B)
In other words, the fact that one of the
events happened does not change the
probability of the other event happening.
Prove that two events A and B are
independent by showing that 𝑃(𝐴|𝐵) =
𝑃(𝐴) 𝑎𝑛𝑑 𝑃(𝐵|𝐴) = 𝑃(𝐵)
Create a two-way frequency table from a
set of data on two categorical variables.
Determine if two categorical variables are
independent by analyzing a two-way table
of data collected on the two variables.
Calculate conditional probabilities based on
two categorical variables and interpret in
context.
For the situation of drawing a card from a standard deck of cards, consider the two events of “draw a spade”
and “draw a king.” Prove that these two events are independent.
Create and prove two events are dependent from drawing a card from a standard deck.
Level 1 Example:
Make a two-way frequency table for the following set of data. Use the following age groups:
3-5, 6-8, 9-11, 12-14, 15-17
.
Youth Soccer League
Gender Age
M
4
M
7
F
8
F
6
M
4
F
10
F
11
Gender Age
F
7
M
7
F
15
M
13
M
12
M
15
F
16
Gender Age
M
17
M
16
F
14
M
14
F
12
F
8
M
13
Gender Age
M
5
M
9
F
13
M
15
M
17
M
12
F
13
Gender Age
F
10
M
6
F
4
M
5
M
9
M
10
F
15
Level 3 Example:
Use the frequency table to answer the following questions:
a. Given that a league member is female, how likely is she to be 9-11 years old?
b. What is the probability that a league member is aged 9-11?
c. Given that a league member is aged 9-11, what is the probability that a member of this league is
a female?
d. What is the probability that a league member is female?
e. Are the events “9-11 years old” and “female” independent? Justify your answer.
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NC Adult Education Standards for ASE MA 4 Statistics and Probability
SP.6.5 Recognize and explain
the concepts of conditional
probability and independence
in everyday language and
everyday situations.
Given an everyday situation describing two
events, use the context to construct an
argument as to whether the events are
independent or dependent. For 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.)
Level 2 Examples:
Felix is a good chess player and a good math student. Do you think that the events “being good at playing
chess” and “being a good math student” are independent or dependent? Justify your answer.
Juanita flipped a coin 10 times and got the following results: T, H, T, T, H, H, H, H, H, H. Her math partner
Harold thinks that the next flip is going to result in tails because there have been so many heads in a row. Do
you agree? Explain why or why not.
Conditional Probability and the Rules of Probability
SP.7 Use the rules of probability to compute probabilities of compound events in a uniform probability model.
What Learner Should Know, Understand, and
Objectives
Teaching Notes and Examples
Be Able to Do
SP.1 Find the conditional
Understand that when finding the conditional
Level 2 Example: Peter has a bag of marbles. In the bag are 4 white marbles, 2 blue marbles, and 6 green
probability of A given B as the probability of A given B, the sample space is
marbles. Peter randomly draws one marble, sets it aside, and then randomly draws another marble. What is
fraction of B’s outcomes that
reduced to the possible outcomes for event B.
the probability of Peter drawing out two green marbles?
also belong to A, and interpret Therefore, the probability of event A happening
the answer in terms of the
is the fraction of event B’s
model.
outcomes that also belong to A.
SP.7.2 Apply the Addition
Understand that two events A and B are
Level 2 Examples:
Rule, P(A or B) = P(A) + P(B) –
mutually exclusive if and only if P(A and B) = 0.
Given the situation of rolling a six-sided die, determine whether the following pairs of events are disjoint:
P(A and B), and interpret the
In other words, mutually exclusive events
a. rolling an odd number; rolling a five
answer in terms of the model. cannot occur at the same time.
b. rolling a six; rolling a prime number
c. rolling an even number; rolling a three
Determine whether two events are disjoint
d. rolling a number less than 4; rolling a two
(mutually exclusive).
Given the situation of drawing a card from a standard deck of cards, calculate the probability of the
Given events A and B, calculate P(A or B) using
following:
the Addition Rule.
a. drawing a red card or a king
b. drawing a ten or a spade
c. drawing a four or a queen
d. drawing a black jack or a club
e. drawing a red queen or a spade
SP.7.3 Apply the general
Calculate probabilities using the General
Multiplication Rule in a
Multiplication Rule. Interpret in context.
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.
SP.7.4 Use permutations and
Identify situations as appropriate for use of a
combinations to compute
permutation or combination to calculate
probabilities of compound
probabilities. Use permutations and
events and solve problems.
combinations in conjunction with other
probability methods to calculate probabilities of
compound events and solve problems.
Updated: October, 2013
Page 15
NC Adult Education Standards for ASE MA 4 Statistics and Probability
Using Probability to Make Decisions
SP.8 Calculate expected values and use them to solve problems.
Objectives
SP.8.1 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.
SP.8.2 Calculate the expected value of a random variable; interpret it as
the mean of the probability distribution.
SP.8.3 Develop a probability distribution for a random variable defined
for a sample space in which theoretical probabilities can be calculated;
find the expected value. For 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.
SP.8.4 Develop a probability distribution for a random variable defined
for a sample space in which probabilities are assigned empirically; find
the expected value. For 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.
What Learner Should Know, Understand, and Be Able
to Do
Understand what a random variable is and the
properties of a random variable.
Given a probability situation (theoretical or empirical),
be able to define a random variable, assign probabilities
to it’s sample space, create a table and graph of the
distribution of the random variable.
Calculate and interpret in context the expected value of
a random variable.
Develop a theoretical probability distribution and find
the expected value.
Teaching Notes and Examples
Develop an empirical probability distribution and find the
expected value.
Using Probability to Make Decisions
SP.9 Use probability to evaluate outcomes of decisions.
Objectives
SP.9.1 Weigh the possible outcomes of a decision by assigning
probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the
expected winnings from a state lottery ticket or a game at a fast-food
restaurant.
b. Evaluate and compare strategies on the basis of expected values. For
example, compare a high-deductible versus a low-deductible automobile
insurance policy using various, but reasonable, chances of having a minor
or a major accident.
SP.9.2 Use probabilities to make fair decisions (e.g., drawing by lots,
using a random number generator).
SP.9.3 Analyze decisions and strategies using probability concepts (e.g.,
product testing, medical testing, pulling a hockey goalie at the end of a
game).
What Learner Should Know, Understand, and Be Able
to Do
Set up a probability distribution for a random variable
representing payoff values in a game of chance.
Teaching Notes and Examples
Make decisions based on expected values. Use expected
values to compare long- term benefits of several
situations.
Explain in context decisions made based on expected
values.
Source: NC Public School Common Core Standards for Mathematics and Unpacking the Common Core Standards for Mathematics
Updated: October, 2013
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