with Teaching Notes - Appalachian State University · Web viewConstruct perpendicular lines, including the perpendicular bisector of a line segment. Using a compass and straightedge,

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 Teaching Notes and Examples

G.1.1 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.

Level 1: Know that a point has position, no thickness or distance. A line is made of infinitely many points, and a line segment is a subset of the points on a line with endpoints. A ray is defined as having a point on one end and a continuing line on the other.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.

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?

G.1.2 Represent transformations in the plane using 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).

Level 2: Describe and compare geometric and algebraic transformations on a set of points as inputs to produce another set of points as outputs, to include translations and horizontal and vertical stretching.

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.

G.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Level 2: Describe the rotations and reflections of a rectangle, parallelogram, trapezoid, or regular polygon that maps each figure onto itself, beginning and ending with the same geometric shape.

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).

G.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Level 2: Students should understand and be able to explain that when a figure is reflected about a line, the segment that joins the preimage point to its corresponding image point is perpendicularly 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Level 2: Using Interactive Geometry Software or graph paper, perform the following transformations on the triangle ABC with coordinates A(4,5), B(8,7) and C(7,9). First, reflect the triangle over the line y=x. Then rotate the figure 180° about the origin. Finally, translate the figure up 4 units and to the left 2 units.Write the algebraic rule in the form (x,y)→(x’,y’) that represents this composite transformation.

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.

Geometry: CongruenceG.2 Understand congruence in terms of rigid motions.

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and ExamplesG.2.1 Use geometric descriptions of rigid motions to transform figures and to predict

Level 2: Use descriptions of rigid motions to move figures in a coordinate plane, and predict the effects rigid motion has on figures

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

Updated: October, 2013 Page 1

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.

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?

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.

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.

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)

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 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: CongruenceG.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 ExamplesG.3.1 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular

Level 1: Prove that any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the line.Pre-ASE, students have already experimented with these angle/line properties. The focus at this level is to prove these properties, not just to use and know them.

Level 1 Example:A carpenter is framing a wall and wants to make sure his the edges of his wall are parallel. He is using a cross-brace as show in the diagram below. What are several different ways he could verify that the edges are parallel? Can you write a formal argument to show that these sides are parallel? Pair up with another student who created a different argument


NC Adult Education Standards for ASE MA 4 Geometry bisector of a line segment are exactly those equidistant from the segment’s endpoints.

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 to180º; 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?

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?

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.

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: CongruenceG.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 ExamplesG.4.1 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing 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.

Copy a congruent segment Copy a congruent angle. Bisect a segment Bisect an angle Construct perpendicular lines, including the perpendicular bisector of a line segment. Using a compass and straightedge, construct a

perpendicular bisector of a segment. Prove/justify why this process provides the perpendicular bisector. Given a triangle, construct the circumcenter and justify/prove why the process gives the point that is equidistant from the vertices. Given a triangle, construct the incenter and justify/prove why the process gives the point that is equidistant from the sides of the triangle. Construct a line parallel to a given line through a point not on the line.***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.

G.4.2 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Level 2: Using a compass and straightedge or Interactive Geometry Software, construct an equilateral triangle so that each vertex of the equilateral triangle is on the circle. Construct an argument to show that your construction method will produce an equilateral triangle. If your method and/or argument are different than another students’, 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.

Level 2 Example:Donna is building a swing set. She wants to countersink the nuts by drilling a hole that will perfectly circumscribe the hexagonal nuts that she is using. Each side of the regular hexagonal nut is 8mm. Construct a diagram that shows what size drill she should use to countersink the nut.


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 Teaching Notes and Examples

G.5.1 Verify experimentally the properties of dilations given by a center and a scale factor.a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Level 2:a. Given a center of a dilation and a scale factor, verify using coordinates that when dilating a figure in a coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to it’s image when the dilation is preformed. However, a segment that passes through the center does not change. See Diagram 1.

Rule for the transformation is (x,y)→(⅓x,⅓y).Comment on the slopes and lengths of the segments that make up the triangles. What patterns do you notice? How could you be confident that this pattern would be true for all cases? See 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.

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

Diagram 1

2 4 6

−3

−2

−1

1

x

y

Diagram 2

2 4 6 8

−4

−2

x

y

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.

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.

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.

G.5.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

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.


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 ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Level 2: Using corresponding angles of similar right triangles, show that the relationships of the side ratios are the same, which leads to the definition of trigonometric ratios for acute angles.Students should use their knowledge of dilations and similarity to justify why these triangles are congruent. It is not expected at this stage that they will know the Triangle Similarity Theorems.

There are three points on a line that goes through the origin, (5,12)

(10,24) (15,36). Sketch this graph. How do the ratios of compare? Why does this make sense? Call the distance from the origin to each point

(“r”). Find the r for each point. Find the ratios of and .G.6.2 Explain and use the relationship between the sine and cosine of complementary angles.

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.

G.6.3 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Level 2Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right triangles.

Similarity, Right Triangles, and TrigonometryG.7 Apply trigonometry to general triangles.

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and ExamplesG.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.

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.

G.7.2 (+) 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).

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.

Expressing Geometric Properties with EquationsG.8 Use Coordinates to Prove Simple Geometric Theorems Algebraically

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and ExamplesG.8.1 Use coordinates to prove simple geometric theorems algebraically. For example, prove or 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).

Level 1: Use the concepts of slope and distance to prove that a figure in the coordinate system is a special geometric shape.

Level 1 Example: The coordinates are for a quadrilateral, (3, 0), (1, 3), (-2, 1), and (0,-2). Determine the type of quadrilateral made by connecting these four points? Identify the properties used to determine your classification. You must give confirming information about the polygon. .Level 1 Example: If Quadrilateral ABCD is a rectangle, where A(1, 2), B(6, 0), C(10,10) and D(?, ?) is unknown. Find the coordinates of the fourth vertex. Verify that ABCD is a rectangle providing evidence related to the

sides and angles.


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).

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.

Level 1 Example: Suppose a line k in a coordinate plane has slope cd .

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.

G.8.3 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

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 mid-point 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.

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?

G.8.4 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Students should find the perimeter of polygons and the area of triangles and rectangles using coordinates on the coordinate plane.

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 EquationsG.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 ExamplesG.9.1 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.

Level 2Use the Pythagorean Theorem to derive the equation of a circle, given the center and the radius.

Given an equation of a circle, complete the square to find the center and radius of a circle.

Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is that given distance from the given point (based on the Pythagorean theorem).

Geometric Measure and DimensionG.10 Explain Volume Formulas and Use Them to Solve Problems

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and ExamplesG.10.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Understanding the formula for the circumference of a circle, you can either begin with the measure of the diameter or the measure of the radius. Take those measurements and measure around the outside of a circle. The diameter will go around a little over 3 times, which indicates C=πd. The radius will go around half of the circle a little over 3 times, therefore C=2πr . This can either be done using pipe cleaners or string and a measuring tool.

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


NC Adult Education Standards for ASE MA 4 Geometry Understanding the volume of a cylinder is based on the area of a 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 cross-sections 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.

circle can also be expressed using C=2πr .

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.

G.10.2 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Formulas for pyramids, cones, and spheres will be given.Level 1 Example: Given the formula V=1

3BH , for the volume of a

cone, 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 DimensionG.11 Visualize relationships between two-dimensional and three-dimensional objects.

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and ExamplesG.11.1 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Level 2: Given a three- dimensional object, identify the shape made when the object is cut into cross-sections.Level 2: When rotating a two- dimensional figure, such as a square, know the three-dimensional figure that is generated, such as a cylinder. Understand that a cross section of a solid is an intersection of a plane (two-dimensional) and a solid (three-dimensional).

Modeling with GeometryG.12 Apply geometric concepts in modeling situations.

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and ExamplesG.12.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Level 2 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a soda can or the paper towel roll as a cylinder).

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?

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 2Use 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)


NC Adult Education Standards for ASE MA 4 Geometry


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilityInterpreting Categorical and Quantitative Data

SP.1 Summarize, represent, and interpret data on a single count or measurement variable.

Objectives What Learner Should Know, Understand, and Be Able to Do Teaching Notes and Examples

SP.1.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

Construct appropriate graphical displays (dot plots, histogram, and box plot) to describe sets of data values.

Level 1 Example: Make a dot plot of the number of siblings that members of your class have.(Level I)Level 1 Example: Create a frequency distribution table and histogram for the following set of data:Age (in months) of First Steps13 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 12.5Level 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 appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Understand which measure of center and which measure of spread is most appropriate to describe a given data set. The mean and standard deviation are most commonly used to describe sets of data. However, if the distribution is extremely skewed and/or has outliers, it is best to use the median and the interquartile range to describe the distribution since these measures are not sensitive to outliers.Select the appropriate measures to describe and compare the center and spread of two or more data sets in context.

Level 1 Example: You are planning to take on a part time job as a waiter at a local restaurant. During your interview, the boss told you that their best waitress, Jenni, made an average of $70 a night in tips last week. However, when you asked Jenni about this, she said that she made an average of only $50 per night last week. She provides you with a copy of her nightly tip amounts from last week (see below). Calculate the mean and the median tip amount.

a. Which value is Jenni’s boss using to describe the average tip? Why do you think he chose this value?

b. Which value is Jenni using? Why do you think she chose this value?c. Which value best describes the typical amount of tips per night? Explain why.

Sunday $50, Monday $45, Wednesday $48, Friday $125, Saturday $85

Level 1 Example: A teacher wanted to evaluate the background knowledge in three broad topics of the 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, 9Topic C: 1, 2, 10, 5, 1, 2, 4, 9, 10, 0, 10, 7, 3, 3, 10

SP.1.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Understand and be able to use the context of the data to explain why its distribution takes on a particular shape (e.g. are there real-life limits to the values of the data that force skewness? are there outliers?)

Understand that the higher the value of a measure of variability, the more spread out the data set is.

Explain the effect of any outliers on the shape, center, and spread of the data sets.

Level 1 Examples:Create a data set based on test scores that illustrates the following a. A skewed left distribution. b. A skewed right distribution c. A symmetrical distributionComment on what the distribution tells you about the difficulty level of the test.

On last week’s math test, Mrs. Smith’s class had an average of 83 points with a standard deviation of 8 points. Mr. Tucker’s class had an average of 78 points with a standard deviation of 4 points. Which class was more consistent with their test scores? How do you know?

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 ft. 10in.

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.


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilitySP.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 (68-95-99.7 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 z-score.

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 mound-shaped, 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. Whatpercentage 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 DataSP.2 Summarize, represent, and interpret data on two categorical and quantitative variables.


SP.2.1 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, andconditional relative frequencies). Recognize possible associations and trends in the data.

Create a two-way frequency table from a set of data on two categorical variables. Calculate joint, marginal, and conditional relative frequencies and interpret in context. Joint relative frequencies are compound probabilities of using AND to combine one possible outcome of each categorical variable (P(A and B)). Marginal relative frequencies are the probabilities for the outcomes of one of the two categorical variables in a two-way table, without considering the other variable. Conditional relative frequencies are the probabilities of one particular outcome of a categorical variable occurring, given that one particular outcome of the other categorical variable has already occurred.Recognize associations and trends in data from a two-way table.

Level I 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 Gender Age Gender Age Gender Age Gender AgeM 4 F 7 M 17 M 5 F 10M 7 M 7 M 16 M 9 M 6F 8 F 15 F 14 F 13 F 4F 6 M 13 M 14 M 15 M 5M 4 M 12 F 12 M 17 M 9F 10 M 15 F 8 M 12 M 10F 11 F 16 M 13 F 13 F 15

Use the frequency table to answer the following questions:a. What is the relative frequency of players who are male and 9-11 years old? (joint relative

frequency)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)


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilitySP.2.2 Represent data on twoquantitative 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 (𝑦 − 𝑦), which is a measure of the error in prediction. (Note: 𝑦 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 𝑦 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 (𝑥, 𝑦) 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 describedabove? 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.

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

a. Create a scatter plot for the data provided. Is there a linear association? Explain.b. What is the Least Squares Regression Line that models this data?c. How do you know this equation is the line of best fit to model the data?


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilityInterpreting Categorical and Quantitative Data

SP.3 Interpret linear models.


SP.3.1 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

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 y-intercept in the context of the data, i.e. an initial value or a one-time fixed amount.

Level 1 Example: The equation 𝐶 = 40 + 2𝑛 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?

SP.3.2 Compute (using technology) and interpret the correlation coefficient of a linear fit.

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 𝑟 is to 1, the stronger the correlation. The closer 𝑟 is to zero, the weaker the correlation. The sign of r indicates the direction of the relationship – positive or negative.

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 8Weight: 132 209 142 184 159 180 161 173Systolic 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.

SP.3.3 Distinguish between correlation and causation.

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).

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 ConclusionsSP.4 Understand and evaluate random processes underlying statistical experiments.


SP.4.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

a. Explain the difference between a population parameter and a sample statistic.

b. Understand that random samples tend to be representative of the population they are drawn from and therefore we can draw conclusions about the population based on the sample. If conclusions cannot be drawn from the random sample, discuss why and

Level 3 Examples:a. What is the difference between a population parameter and a sample statistic?b. Describe the process of statistical inference.c. For statistical inference, why is it important that a sample be representative of the population it is

drawn from?d. What kind of sampling method would you use for each of the following situations? Explain why in each

case. To determine which gubernatorial candidate voters are most likely to choose in the next election. To determine the quality of potato chips being produced at a factory.


NC Adult Education Standards for ASE MA 4 Statistics and Probabilitypropose a better way to select a random sample.

c. Demonstrate understanding of the different kinds of sampling methods (simple random sample, systematic random sample, stratified random sample, cluster or multistage sample, convenience sample).

To determine the average size of bass fish in a lake. To determine the average number of televisions per household in the US.

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?

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 ConclusionsSP.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 ExamplesSP.5.1 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

a. Students should understand that sample surveys are used to collect data from human subjects to describe the population of interest. Experiments and observational studies are used to establish a cause and effect relationship.

b. In an experiment, a treatment is imposed on the experimental units. In an observational study, the treatment is not imposed but 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.

Level 3 Example: For each of the following situations, decide whether an experiment or an observational study is moreappropriate to determine if there is a cause and effect relationship:

a. Cell phone use and brain tumorsb. Use of a fertilizer and growth of plants


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilitySP.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.

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.

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 SarahGabe 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.

SP.5.3 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

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: 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 30.8Fertilizer B 19.8 25.7 29.0 23.2 27.8 31.1 26.5 24.7 21.3 25.6

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.

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.)

Level 3 Example:Find a statistical report based on an experiment and evaluate the experimental design.

Conditional Probability and the Rules of ProbabilitySP.6 Understand independence and conditional probability and use them to interpret data.


SP.6.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

a. Define the sample space for a given situation.

b. Describe an event in terms of categories or characteristics of the outcomes in the sample space.

c. Describe an event as the union, intersection, or complement of other events.

Level 2 Examples:a. What is the sample space for rolling a die?b. What is the sample space for randomly selecting one letter from the word MATHEMATICS?c. Describe different subsets of outcomes for rolling a die using a single category or characteristic.d. Describe the following subset of outcomes for choosing one card from a standard deck of cards as the

intersection of two events: {queen of hearts, queen of diamonds}.


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilitySP.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.

Understand that two events A and B are independent if and only if P(A and B) = P(A) x P(B).

Determine whether two events are independent using the Multiplication Rule (stated above).

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.

Create and prove two events are independent from drawing a card from a standard deck.

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.

a. Understand that the conditional probability of event A given event B has already happened is given by the formula:

P (A|B )= P(A∧B)P (B)

b. Understand that two events A and B are independent if and only if P (A|B )=P (A )∧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.

c. Prove that two events A and B are independent by showing that P (A|B )=P (A )∧P (B|A )=P(B)

Level 3 Example:

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.

SP.6.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way 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 giventhat the student is in tenth grade. Do the same for other subjects and compare the results.

a. Create a two-way frequency table from a set of data on two categorical variables.

b. Determine if two categorical variables are independent by analyzing a two-way table of data collected on the two variables.

c. Calculate conditional probabilities based on two categorical variables and interpret in context.

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 Gender Age Gender Age Gender Age Gender AgeM 4 F 7 M 17 M 5 F 10M 7 M 7 M 16 M 9 M 6F 8 F 15 F 14 F 13 F 4F 6 M 13 M 14 M 15 M 5M 4 M 12 F 12 M 17 M 9F 10 M 15 F 8 M 12 M 10F 11 F 16 M 13 F 13 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.


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilitySP.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 playingchess” 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 partnerHarold 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 ProbabilitySP.7 Use the rules of probability to compute probabilities of compound events in a uniform probability model.


SP.1 Find the conditional probability of A given B as thefraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

Understand that when finding the conditional probability of A given B, the sample space is reduced to the possible outcomes for event B. Therefore, the probability of event A happening is the fraction of event B’s outcomes that also belong to A.

Level 2 Example: Peter has a bag of marbles. In the bag are 4 white marbles, 2 blue marbles, and 6 green marbles. Peter randomly draws one marble, sets it aside, and then randomly draws another marble. What is the probability of Peter drawing out two green marbles?

SP.7.2 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.

Understand that two events A and B are mutually exclusive if and only if P(A and B) = 0. In other words, mutually exclusive events cannot occur at the same time.

Determine whether two events are disjoint (mutually exclusive).

Given events A and B, calculate P(A or B) using the Addition Rule.

Level 2 Examples:Given the situation of rolling a six-sided die, determine whether the following pairs of events are disjoint:

a. rolling an odd number; rolling a fiveb. rolling a six; rolling a prime numberc. rolling an even number; rolling a threed. rolling a number less than 4; rolling a two

Given the situation of drawing a card from a standard deck of cards, calculate the probability of the following:

a. drawing a red card or a kingb. drawing a ten or a spadec. drawing a four or a queend. drawing a black jack or a clube. drawing a red queen or a spade

SP.7.3 Apply the generalMultiplication 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.

Calculate probabilities using the General Multiplication Rule. Interpret in context.

SP.7.4 Use permutations and combinations to compute probabilities of compound events and solve problems.

Identify situations as appropriate for use of a permutation or combination to calculate probabilities. Use permutations and combinations in conjunction with other probability methods to calculate probabilities ofcompound events and solve problems.


NC Adult Education Standards for ASE MA 4 Statistics and ProbabilityUsing Probability to Make Decisions

SP.8 Calculate expected values and use them to solve problems.


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.

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.

SP.8.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Calculate and interpret in context the expected value of a random variable.

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.

Develop a theoretical probability distribution and find the expected value.

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.

Develop an empirical probability distribution and find the expected value.

Using Probability to Make DecisionsSP.9 Use probability to evaluate outcomes of decisions.


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.

Set up a probability distribution for a random variable representing payoff values in a game of chance.

SP.9.2 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Make decisions based on expected values. Use expected values to compare long- term benefits of several situations.

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).

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


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with Teaching Notes - Appalachian State University · Web viewConstruct perpendicular lines, including the perpendicular bisector of a line segment. Using a compass and straightedge,