Download GEOMETRY--2013

GEOMETRY
The Geometry course builds on Algebra I concepts and increases students’ knowledge of shapes and their
properties through geometry-based applications, many of which are observable in aspects of everyday life. This
knowledge helps develop visual and spatial sense and strong reasoning skills. The Geometry course requires
students to make conjectures and to use reasoning to validate or negate these conjectures. The use of proofs and
constructions is a valuable tool that enhances reasoning skills and enables students to better understand more
complex mathematical concepts. Technology should be used to enhance students’ mathematical experience, not
replace their reasoning abilities. Because of its importance, this Euclidean geometry course is required of all
students receiving an Alabama High School Diploma.
School systems may offer Geometry and Geometry A and Geometry B. Content standards 1, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 26, 30, 31, 32, 33, and 34 must be taught in the Geometry A course. Content standards 2,
12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 35, 36, 37, 38, 39, 40, 41, 42, and 43 must be taught in the
Geometry B course.
Students will:
GEOMETRY
Congruence
Experiment with transformations in the plane.
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. [G-CO1]
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs. Compare
transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2]
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry
it onto itself. [G-CO3]
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments. [G-CO4]
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. [G-CO5]
 ACTIVITY: Use the TI-83/84 graphing calculators to graph functions and then practice
editing the lists of ordered pairs to graph images which are transformations of these
functions. A sheet of instructions for a simple activity involving transformations can be
found below. All of the keystrokes for the calculator are given. The transformations can be
extended to include composition of more than one transformation, etc.
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Transformations on the TI-83/84 Graphing Calculators
By
Linda Bridges
1. Make a list of ordered pairs that will be used to graph a particular object.
2.
Enter those ordered pairs into L1 and L2 on the calculator. Go to STAT and
then ENTER. If there are numbers already in L1 and L2, clear them. To
clear the lists, highlight upward to the name of the list (L1 or L2) and then
hit CLEAR and ENTER. Once lists are cleared, enter the X-coordinates into
L1 and the y-coordinates of the ordered pairs into L2.
3. To graph the ordered pairs, make sure all other graphs are turned off. To do this, hit the
“y=” key on the first row and hit “clear” until all graphs have been cleared. Also, the stat
plots must be turned off. To do this, hit “2nd” and “y=” and then scroll to number 4 “Plotsoff”
and hit ENTER twice until it says “done”. Now you have all graphs cleared out and are ready
to graph the ordered pairs. Hit “2nd” and “y=” again and then ENTER. Scroll over to “ON”
and then press ENTER until “on” is highlighted. Under the “type”, choose the 2nd type of
graph (scatterplot with connected dots) and then press enter until it is the only graph type
highlighted. Under “xlist” type L1 and under “ylist”, type L2. To type L1, you must first hit
the “2nd” key and then the “1” key. To type L2, you must first hit the “2nd” key and then the
“2” key. Choose a “mark” to be used to graph the points. You may choose the square, the
cross, or the dot. VERY IMPORTANT: Before attempting to look at the graph, hit the
ZOOM key on the top row and then scroll down to number 9, “ZoomStat”, and press enter.
You should now see the ordered pairs you placed in the lists graphed and the dots connected
with line segments to form the picture.
4. To observe transformations, you must enter more data into lists 3 and 4. KEEP LISTS 1
and 2 AS THEY ARE. DO NOT DELETE OR CHANGE THESE TWO LISTS.
TRANSLATIONS: Hit the “Stat” key and ENTER and then go to L3 and L4. If you need to
clear these lists, highlight the list name at the top and then hit “clear” and “enter”. Then
suppose you want to move everything to the right 2 places and down 4 places. Then you need
to have “X+2” and “Y-4”. You do not have to change each individual ordered pair. GO to L3
and arrow upward to the list name and then press ENTER. You should now see the cursor at
the bottom of your screen and it is waiting for you to give it a formula. Since you want to add
2 to your x values and these are in L1, type in “L1 +2” and then press ENTER.
REMEMBER: To type L1, you must hit the “2nd” key and then the “1” key. After you have
typed “L1+2” and pressed ENTER, you should immediately see L3 filled with the L1 values
plus 2. Then go to L4 and scroll upward and highlight the list name and push ENTER. You
now want to tell the calculator to take everything in the y list which is L2 and subtract 4.
Thus, after you have highlighted L4 and hit ENTER, you should now see the cursor at the
bottom. Type “L2-4” and hit ENTER and then list 4 should be filled will all your y values
minus 4. To see the graph of these, go to “2nd” and “y=” and then scroll down to Plot2. DO
NOT TURN PLOT ONE OFF___KEEP IT ON. Scroll to Plot 2 and press ENTER. Follow
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the same instructions as before to turn it on and then choose the 2nd option under the type.
Then you must type in L3 (“2nd” and “3”) for the x-list and then L4 (“2nd” and “4”) for the ylist. IMPORTANT: Choose a “mark” different from the one that you chose for the original
figure so that you will be able to distinguish the new figure from the original figure when both
are graphed together on the same screen. Then go to ZOOM and choose number 9 and then
hit ENTER. You can see both graphs and how the second graph is shifted 2 to the right and
down 4. You can do other translations in a similar manner.
For instance, “L1-3” and “L2+5” placed as formulas into L3 and L4 would move the original
graph 3 to the left and up 5.
5. REFLECTIONS about Y-axis. To see reflections, leave L1 and L2 as they
are. DO NOT CHANGE THESE LISTS. To reflect about the y-axis, you
must have (-X,Y) as the ordered pairs. Thus go to L3 and L4 and clear those
out. Then highlight L3 name at the top and press ENTER. Type in negative
sign and then L1 and press ENTER. Remember there is a difference
between the negative sign next to the enter key and the subtraction operation
symbol. You must type in “- L1”, since you want all your x-values to be the
opposite sign.
GO to L4 and highlight its name at the top and press ENTER and then type
in “L2” and press ENTER since you want the y values to stay the same. Then
go to “2nd” and “y=” and then go immediately to ZOOM and then #9 and
press enter. (You have already told it to graph all four lists in two different statplots.
You do not have to tell it to do that again). However, you must tell the calculator to
ZOOMSTAT each time before you view another graph.
6. REFLECTIONS about the X-axis. To see reflections, leave L1 and L2 as they are. DO
NOT CHANGE THESE LISTS. To reflect about the x-axis, you must have (X,-Y) as the
ordered pairs. Thus go to L3 and L4 and clear those out. Then highlight L3 name at the top
and press ENTER. Type in “L1” and press ENTER. GO to L4 and highlight its name at the
top and press ENTER and then type in “negative sign and L2” and press ENTER since you
want the y values to be the opposite of the original y values. Then go to “2nd” and “y=” and
then go immediately to ZOOM and then #9 and press ENTER. (You have already told it to
graph all four lists in two different statplots. You do not have to tell it to do that again).
However, you must tell the calculator to ZOOMSTAT each time before you view another
graph.
7. ROTATION of 180 degrees about the origin. If you clear L3 and L4 and then set L3 equal
to the negative of L1 and set L4 equal to the negative of L2, you should be able to see a
rotation of 180 degrees.
8. ROTATION OF 90 Degrees Counterclockwise about the origin. If you clear L3 and L4
and then set L3 equal to the negative of L2 and set L4 equal to L1, you should be able to see a
rotation of 90 degrees counterclockwise.
9. DILATIONS: To see a dilation (stretching or shrinking) of the figure, try multiplying both
L1 and L2 by a constant---either less than 1 or greater than 1---to see how the size of the
object changes. This is a good exercise for talking about scale factors and similar figures.
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OTE: I started out with a picture of a house, using these original ordered pairs.
L1
L2
3
1
3
6
6
9
7
7
7
10
8
10
8
6
8
1
3
1
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 A quick overview of various transformations can be found at:
http://www.regentsprep.org/Regents/math/algtrig/ATP9/funclesson1.htm
Summarizes and illustrates:
Reflections over the x-axis, -f(x); reflections over the y-axis, f(-x); translations, f(x+a) and
f(x) + a; horizontal and vertical stretches or compressions, f(ax) or af(x);
 Additional information can be found at:
http://www.regentsprep.org/Regents/math/algtrig/ATP9/reviewTranformations.htm
Summarizes line reflections (in the y-axis, in the x-axis, in y = x, in y= -x); point reflections
in the origin; rotations of 90 degrees, 180 degrees, and 270 degrees; dilations of scale factor
k; and translations. The website also summarizes which transformations are isometries.
 A composition of transformations occurs when two or more transformations are combined
to form a new transformation. Watch the order. Perform the transformation indicated on
the right first, then do the transformation indicated by the one on the left.
For instance, ry-axis o rx-axis = rorigin
This implies that a reflection in the x-axis followed by a reflection in the y-axis is the same as a
single transformation of a reflection in the origin.
Likewise, rx-axis o T(2,3) indicates a translation of (x+2, y+3) followed by a reflection in the
x-axis.
A glide reflection is an example of a composition of transformations. A glide reflection is the
composition of a glide (translation) and a reflection across a line parallel to the direction of
the translation. A glide reflection is an isometry, since translations and reflections are both
isometries. (The composition of isometries is an isometry).
 An isometry is a transformation in which the preimage and the image are congruent.
Direct Isometry - orientation is preserved - the order of the lettering in
the figure and the image are the same, either both clockwise or both counterclockwise.
Opposite Isometry - orientation is not preserved - the order of the lettering is reversed,
either clockwise becomes counterclockwise or counterclockwise becomes clockwise.
 The following website gives a concise review of all transformations and the isometry status
of each—whether the transformation is an opposite isometry, direct isometry, or not an
isometry.
http://www.regentsprep.org/regents/math/geometry/GT5/properties.htm
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 ACTIVITY: Using masking or colored tape, divide the classroom floor into a coordinate
plane, with the tape indicating where the X and Y axes are located. (Be sure to identify the
orientation of these axes, by labeling the positive x-axis and the positive y-axis.) The
students’ desks will then be divided up into the four quadrants. Students will be asked to
identify who is their: reflection over the y-axis, reflection over the x-axis, point reflection in
the origin, translation under a given rule, etc. Then a student will be asked to tell how they
could be transformed to another student’s position, etc.
 ACTIVITY: Matching Exercise. Students will be given a card with a set of points whose
coordinates are given. Then students will be given another card with a transformation
activity on it—for instance, “dilation with a scale factor of 3” or “ reflection in the line y = x,
translate (x-3, y+4), etc. Students will then identify their new points after the
transformation has been performed.
 ACTIVITY: Use Geometer’s SketchPad software to experiment with transformations.
Steps for using the software to investigate transformations can be found at
http://mathbits.com/MathBits/GSP/ScratchTransformations.htm
Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for
development of concept of geometric proof.)
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion
on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are
congruent. [G-CO6]
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent. [G-CO7]
8. Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-sideside (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8]
 Congruence between two geometric objects can be defined as a rigid motion of the plane
that maps one of the objects onto the other.
 ACTIVITY: Students will use straws, scissors, and rulers to cut 2 inch, 3 inch, and 4 inch
segments. Then they will form a triangle by stringing these together with twine. Are all
triangles formed by different groups the same? Identify congruent angles, congruent sides,
etc. Use this knowledge of congruent figures to find missing side lengths and missing angle
measures of figures known to be congruent. Then students will construct triangles with
two and three inch sides and a 60 degree included angle. Are all the triangles formed just
alike? Then they will do a similar exploration with ASA and AAS congruence conjectures
to see if these conditions always produce congruent triangles. Then they will explore the
AAA and the SSA options to see if these alway produce congruent triangles. (Of course the
AAA and the SSA situations do not always produce congruent triangles. )
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Prove geometric theorems. (Focus on validity of underlying reasoning while using variety of ways of writing
proofs.)
9. 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; and points
on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. [G-CO9]
10. 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, and the medians of a triangle meet at a point. [G-CO10]
11. 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. [G-CO11]
Make geometric constructions. (Formalize and explain processes.)
12. Make formal geometric constructions with a variety of tools and methods such as compass and straightedge,
string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [G-CO12]
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13]
Please note: If you do not know where the center of the circle is, you may draw two chords, segment AB and
segment BC. Then construct the perpendicular bisector of each chord. They should meet at the center.
 To inscribe a regular hexagon in a circle, first locate the center of the circle. Make a point
anywhere on the circle. This will be the first vertex of the hexagon. Set the compass on this
point and set the width of the compass to be the distance between the first vertex and the
center of the circle. Now, with the tip of the compass on the first vertex, mark an arc on the
circle with the compass equal to the distance between the center of the circle and the first
vertex. This will be the second vertex of the hexagon. Continue doing this until all 6 vertices
have been located. Connect the points to form the 6 sides of the hexagon. (Since each side of
a regular hexagon is equal to the distance from the center to any vertex, this construction
simply sets the compass width to that radius, and then steps that length off around the circle
to create the six vertices of the hexagon.)
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 To inscribe an equilateral triangle in a circle, start with an inscribed hexagon (constructed
with the above instructions). Then draw the line segments indicated below to form an
equilateral triangle.
 To inscribe a square in a circle, draw a diameter of the circle. Then construct another
diameter perpendicular to the first one. Then connect vertices as shown in the diagram to
make a square.
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Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
14. Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT1]
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. [G-SRT1a]
 A dilation is a transformation that produces an image that is the same shape as the original, but is a
different size. (A figure and its dilation are similar). The center of dilation is a fixed point in the
plane about which all points are expanded or contracted. It is the only point that does not change in
a dilation.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. [G-SRT1b]
15. 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-SRT2]
16. Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two
triangles to be similar. [G-SRT3]
Prove theorems involving similarity.
17. Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides
the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle
similarity. [G-SRT4]
 To prove the Pythagorean Theorem using similarity, start with right triangle ABC with right angle
at C. Draw an altitude from point C to the hypotenuse. Then it can be shown that Triangle ABC is
similar to Triangle CBH and Triangle ABC is similar to Triangle ACH.
Since Triangle ABC is similar to Triangle CBH, it follows that AB/CB = BC/BH and c/a =a/e and
a2=ec.
Since Triangle ABC is similar to Triangle ACH, it follows that AB/AC= AC/AH and c/b=b/d and
b2=cd.
Hence, a2+b2=ec+cd=c(e+d)=c(c)=c2.
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18. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures. [G-SRT5]
Define trigonometric ratios and solve problems involving right triangles.
19. Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle leading to definitions of trigonometric ratios for acute angles. [G-SRT6]
20. Explain and use the relationship between the sine and cosine of complementary angles. [G-SRT7]
21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.* [G-SRT8]
Apply trigonometry to general triangles.
22. (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [G-SRT10]
23. (+) 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).
[G-SRT11]
Circles
Understand and apply theorems about circles.
24. Prove that all circles are similar. [G-C1]
25. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between
central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a
circle is perpendicular to the tangent where the radius intersects the circle. [G-C2]
26. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle. [G-C3]
27. (+) Construct a tangent line from a point outside a given circle to the circle. [G-C4]
Find arc lengths and areas of sectors of circles. (Radian introduced only as unit of measure.)
28. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius,
and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a
sector. [G-C5]
Expressing Geometric Properties With Equations
Translate between the geometric description and the equation for a conic section.
29. 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. [G-GPE1]
Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to
Pythagorean Theorem.)
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30. Use coordinates to prove simple geometric theorems algebraically. [G-GPE4]
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).
31. 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-GPE5]
32. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
[G-GPE6]
 Label the unknown point as (x,y). Use the fact that all three points must be collinear and the
slope between any 2 of the points is the same to write an equation relating x and y. Find the
distance between the left endpoint and (x,y) and then the distance between the right
endpoint and (x,y). Set the ratio of these two distances equal to the desired ratio. Solve for
x and y to find the coordinates of the desired point.
 Consider the point P(xP , yP) that lies on the line segment AB joining points A(xA ,
yB) and B(xA , yB):
Point P divides AB in the ratio k1 : k2 , and the coordinates of point P are given by
these formulas:
The blue formula is for three dimensional space, where you have 'z' coordinates as
well as 'x' and 'y'.
Source for above formula and information:
http://www.teacherschoice.com.au/Maths_Library/Analytical%20Geometry/AnalGeom_3.htm
33. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.* [G-GPE7]
Use coordinates to prove simple geometric theorems algebraically. (ALABAMA)
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34. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the
coordinates of vertices or other characteristics.
Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems.
35. Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a
cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. [GGMD1]
 Using a limit process, one can find the formula for the area of a circle by looking at the
areas of regular polygons inscribed in the circle and determining what happens as the
number of sides of the regular polygons continues to increase.

Cavalieri’s Principle:
2-dimensional case: Suppose two regions in a plane are included between two parallel lines in
that plane. If every line parallel to these two lines intersects both regions in line segments of equal
length, then the two regions have equal areas.
3-dimensional case: Suppose two regions in three-space (solids) are included between two
parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections
of equal area, then the two regions have equal volumes.
To see an explanation of how to use Cavalieri’s principle to find the volume of a sphere, look
at this site:
http://en.wikipedia.org/wiki/Cavalieri's_principle
For a general explanation of Cavalieri’s principle, see:
http://www.shmoop.com/common-core-standards/ccss-hs-g-gmd-2.html
For demos: http://learni.st/learnings/29642-a-demonstration-of-cavalieri-s-principle
Demo of same volumes and Cavalieri’s principle from Wikipedia
Applet: http://www.cut-the-knot.org/Curriculum/Calculus/Cavalieri.shtml
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36. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [G-GMD3]
37. Determine the relationship between surface areas of similar figures and volumes of similar figures. )
Visualize relationships between two-dimensional and three-dimensional objects.
38. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. [G-GMD4]
 Consider using computer software to generate 3-D solids of revolutions by rotating twodimensional objects, so that students can mentally visualize these cross sections more
easily.
Modeling With Geometry
Apply geometric concepts in modeling situations.
39. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).* [G-MG1]
40. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,
British Thermal Units (BTUs) per cubic foot).* [G-MG2]
41. 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).* [G-MG3]
STATISTICS AND PROBABILITY
Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules.)
42. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]
43. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game). [S-MD7]
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