Geometry - Get Now DOC

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					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).
[G-CO2]

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]



 Core     What Does It Mean?                    Pre-Requisite Skills  What Does Mastery                 Future Math
 Standard                                                             Look Like?
 1        Know precise definitions        -Exposure to (recognize -Students will be                     Conics
          of angle, circle,               picture of or be able to able to define,                      Unit Circle
          perpendicular line, parallel    draw a picture of) an       draw, and                         Systems of equations
          line, and line segment,         angle, circle,              recognize picture of              (graphing )
          based on the undefined          perpendicular line,         selected definitions
          notions of point, line,         parallel line, line         (an angle, circle,
          distance along a line, and      segment, point, line,       perpendicular line,
          distance around a circular
                                          distance along a line,      parallel line, line
          arc.
                                          and distance around a segment, point,
                                          circular arc.               line, distance along
                                                                      a line, and distance
                                                                      around a circular
                                                                      arc).
 2            Represent transformations -Vocabulary                   -Students should be               Vectors
              using a variety of methods (transformation,             able to complete a                Graph transformations
              (ELMO, Geometer’s           coordinate, distance,       representation of a               Matrices
              Sketchpad); using the       angles, vertices)           transformation and
              coordinates of two figures, -Coordinate system (x- compare distance
              describe the change in x    axis, y-axis, origin, plot, and angles of both
              and y values; compare       point)                      figures
              distance and angles of      -Introduction to
              both figures                Geometer’s Sketchpad
 3            Given a rectangle,          -Vocabulary (rectangle, -Students should be                   Calculus rotations ( about
              parallelogram, trapezoid,   parallelogram,              able to describe                  the axes )
              or regular polygon,         trapezoid, regular          rotations and                     Pre-Cal/algebra graph
              describe the rotations and  polygon, rotations, and reflections on the                    transformations
              reflections that carry it   reflections)                coordinate plane.                 Matrices
              onto itself.                -Coordinate system (x-
                                          axis, y-axis, origin, plot,
                                              point)
                                              -Basic degree changes
                                              -Know how to describe
                                              changes in the x and y
                                              values
 4           Develop definitions of           -Vocabulary (angles,          -Students should be      Calculus rotations ( about
             rotations, reflections, and      circles, perpendicular        able to identify         the axes )
             translations in terms of         lines, parallel lines, and    examples of              Pre-Cal/algebra graph
             angles, circles,                 line segments)                rotations,               transformations
             perpendicular lines,             -Recognize examples           reflections, and         Matrices
             parallel lines, and line         of rotations,                 translations and
             segments.                        translations, and             figures of angles,
                                              reflections                   circles,
                                              -Recognize figures of         perpendicular lines,
                                              angles, circles,              parallel lines, and
                                              perpendicular lines,          line segments to
                                              parallel lines, and line      develop their own
                                              segments                      definitions.
 5           Given a geometric figure         -Recognize geometric          -Students should be      Calculus rotations ( about
             and a rotation, reflection,      figures.                      able to draw a           the axes )
             or translation, draw the         -Recognize rotation,          transformed figure       Pre-Cal/algebra graph
             transformed figure using,        reflection, and               using graph paper,       transformations
             e.g., graph paper, tracing       translation.                  tracing paper, or        Matrices
             paper, or geometry               -Coordinate system (x-        geometry software.
             software. Specify a              axis, y-axis, origin, plot,   -Students should be
             sequence of
                                              point)                        able to describe the
             transformations that will
                                                                            order of
             carry a given figure onto
             another.                                                       transformations
                                                                            that will carry a
                                                                            given figure onto
                                                                            another on the
                                                                            coordinate plane.


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 (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motions. [G-CO8]

 Core     What Does It Mean?                  Pre-Requisite Skills          What Does Mastery        Future Math
 Standard                                                                   Look Like?
 6        Use translation, rotation,          -translation                  Students will            Pre-Engineering
          reflection                          -rotation                     demonstrate that         Computer Science
                                              -reflection                   they can take a figure   and Animation
                                              -Congruence                   and apply a
                                                                            transformation.
 7           Show that a triangle that        -Vocabulary (rotation,   -Students should be            Pre-Engineering
             has been rotated,                reflection, translation, able to explain why a          Computer Science
             reflected, and translated is     angles, corresponding    triangle that has              and Animation
             still congruent to the           sides, corresponding     been rotated,
             original because they have       angles, congruency)      reflected, or
             corresponding pairs of                                    translated is still
             sides and angles.                                         congruent to the
                                                                       original.
 8           Use rotation, reflection,        -Vocabulary (rotation, -Students should be              Pre-Engineering
             and translation to prove         reflection, translation, able to prove a                Computer Science
             triangle congruence (ASA,        congruency, angle,       triangle that has              and Animation
             SAS, and SSS).                   side)                    been rotated,                  Trigonometry (
                                              -Coordinate system (x- reflected, or                    understanding law
                                              axis, y-axis, origin,    translated is                  of sine, cosine)
                                              plot, point)             congruent to the               Unit Circle
                                                                       original by ASA, SAS,
                                                                       or SSS.


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; 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; 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]

 Core     What Does It Mean?           Pre-Requisite Skills            What Does Mastery        Future math
 Standard                                                              Look Like?
 9        Recognize angle pairs and -Vocabulary                        Students will be         Architecture
          what is true when the lines Vertical angles                  able to develop and
          are parallel. Use theorems Transversal                       solve algebraic
          and definition of bisectors.  Parallel lines                 equations based on
                                        Congruent                      the relationships
                                        Alternate interior             that exist between
                                       angles                          the angles /
                                       Corresponding                   segments
                                       angles
                                        Perpendicular                  Students will be
                                        Bisector                       able to prove the
                                        Segment                        alternate exterior
                                       Equidistant                     angle theorem by
                                       End point                       using the
                                       Same-Side interior              corresponding
                                       angles                          angles postulate
                                       Alternate exterior              and vertical angles.
                                       angles
                                     Supplementary         Students should be
                                    -Developing            able to use the
                                    equations based        bisector definition
                                    on known               and bisector
                                    relationships          theorems in related
                                                           proofs.
10   Properties of triangles will   -Vocabulary            Students will be        Unit Circle
     be discovered through           Interior angles       able to use more        Trigonometry
     logical reasoning. This         Base angles           than one method to      Word problems
     includes the fact that the      Isosceles triangles   prove the triangle      Engineering
     sum of the interior angles      Midsegment            sum theorem;            Architecture
     of a triangle equal 180         Medians               Students will be        Art
     degrees, the base angles                              able to prove the
     of an isosceles triangle are                          base angles
     congruent, and a                                      theorem; prove the
     midsegment is parallel to                             midsegment
     the third side and half its                           theorem
     length.
11   Prove theorems about           -Vocabulary            Students will be        Word problems
     parallelograms                  Parallelogram         able to use a variety   Engineering
                                     Congruent             of methods to           Architecture
                                     Opposite angles       prove relationships     Art
                                     Diagonals             about
                                     Bisect                parallelograms
                                     Rectangles
                                     Opposite Sides
                                    -Triangle
                                    congruence
                                    theorems AAS, SAS
                                    etc
                                    - CPCTC
12   Use a compass and              -Vocabulary            Students will be        Engineering
     straight edge, paper             Circle Center        able to accurately      CAD
     folding, reflectors,             Arc                  do the following        Architecture
     software (Geometer’s             Angle                constructions:          Construction/design
     Sketchpad) to create             Perpendicular            - Copy a
     geometric constructions          Bisector                     segment
                                      Parallel                 - Copy an
                                      Segment                      angle
                                      Radius                   - Bisect a
                                     -Skills                       segment
                                     Use a compass             - Bisect an
                                    and straight edge              angle
                                                               - Perpendicul
                                                                   ar lines
                                                               - Parallel lines
                                                                   ( through a
                                                                   pt not on
                                                                   the line)
                                                               - Perpendicul
                                                                   ar bisector
                                                                  of a segment
13   Construct an equilateral      -Vocabulary           Students will be      Unit circle
     triangle, square, and          Equilateral          able to accurately    Calculus
     regular hexagon inscribed     triangle              construct an          Engineering
     in a circle                    Square               equilateral triangle,
                                   Regular               square, and regular
                                   Inscribed             hexagon inscribed
                                   Circle                in a circle.
                                   Hexagon
14   Verify experimentally the     -Vocabulary           Students will be       Engineering
     properties of dilations        Scale factor         able to draw and       CAD
     given by a center and a        Dilation             describe dilations (   Architecture
     scale factor                   Center of dilation   properties) using      Graphing
                                    Parallel Line        various tools (        Transformations
                                    Ratio                software/compass
                                                         & straight edge)




15   Determine if two figures      -Vocabulary           Students will be       Daily life
     are similar, explain why       Similarity           able to determine      (proportion
     two triangles are similar.     Similarity           and explain why        concept)
                                   transformation        two triangles are      All other Math
                                    Similar              similar using          classes
                                    Proportional         transformations
                                    Corresponding        and corresponding
                                   angles                parts.
                                    Corresponding
                                   sides
                                    Using cross
                                   products to solve
                                   proportions
16   Use the properties of         -Vocabulary           Students will be       CAD/ Engineering
     similarity transformations     Similar              able to discover,      Design
     to establish the AA            Transformation       recognize, and use     College Algebra
     criterion for two triangles   -sum of angles of a   the AA criterion for
     to be similar                 triangle              triangle similarity.
17   Prove theorems about          SAS, AAS, SAS, SSS,   Students will be       Proof concepts used
     triangles                     HL                    able to prove          in all future math
                                   Vocabulary            theorems about         courses/life
                                   -Hypotenuse           triangles including    Unit Circle
                                   -Angle                but not limited to     Calculus
                                   -Side                 the Triangle angle     Trigonometry
                                   -Congruent            bisector theorem       Physics
                                  -proportion           and the
                                  -similarity           Pythagorean
                                  -Pythagorean          theorem using
                                  Theorem               triangle similarity.
18   Using the fact that two      -proving triangles    Students will be        Calculus
     triangles are congruent or are congruent           able to produce a       Proof concepts used
     similar to prove other facts -Using CPCTC          proof in which          in all future math
     about various geometric      -Recognizing          several steps are       courses
     figures.                     properties of other   required including
                                  geometric figures     using congruent
                                                        and / or similar
                                                        triangles.
19   Trigonometric ratio          -right triangle       Students will be        Algebra II
     definitions can be derived   -sine,cosine,         able to                 Precalculus
     from the similarity of the   tangent               demonstrate the         Calculus
     side ratios in right         -opposite,            connection              Probability/Stats
     triangles as properties of   adjacent,             between the             Alg. Connections
     the angles.                  hypotenuse            trigonometric ratios    Trigonometry
                                  -ratio                and the side ratios
                                  -similarity           in right triangles as
                                                        a property of the
                                                        acute angles.
20   Explain and use the          -sine                 Students will           Unit Circle
     relationship between the     -cosine               demonstrate that        Trigonometry
     sine and cosine of           -complementary        the sine of one of      Calculus
     complementary angles         -opposite,            the two acute           Algebra II
                                  adjacent,             angles is the cosine    Alg. Connections
                                  hypotenuse            of the other and
                                                        vice versa.
21   *Use the trigonometric       -sine                 Students will be        Unit Circle
     ratios and the               -cosine               able to use the         Trigonometry
     Pythagorean theorem to       -tangent              trigonometric ratios    Calculus
     solve right triangles in     -acute angles         to solve real world     Algebra II
     applied problems.            -right triangle       problems such as        Alg. Connections
                                                        measuring ‘tall’
                                                        objects, finding a
                                                        distance, etc.
22   +Derive the formula          -vocabulary           Students will be        Algebra II
     A=1/2abSin(C) by drawing     -sine                 able to draw an         Precalculus
     an auxiliary line from a     -auxiliary line       auxiliary line in a     Calculus
     vertex perpendicular to      -vertex               triangle in such a      Trigonmetry
     the opposite side            -perpendicular        way that the            Physics
                                  -opposite side        formula
                                  -derive               A=1/2abSin(C) is
                                                        derived.
23   +Prove the laws of Sines     -sine                 Students will be        Algebra II
     and Cosines and use them     -cosine               able to prove the       Precalculus
     to solve problems            -acute angle          Law of Sines and        Calculus
                                                        Cosines.                Trigonometry
                                                        Additionally, they      Physics
                                                        will be able to use
                                                         them to solve
                                                         problems.
24   +Understand and apply          -Law of Sines        Students will be        Algebra II
     the Law of Sines and           -Law of Cosines      able to use these       Precalculus
     Cosines to find unknown        -Acute angles        Laws to solve           Calculus
     measurements in any                                 problems such as        Trigonometry
     triangle.                                           surveying problems,     Physics
                                                         resultant forces,
                                                         etc. The student
                                                         should be able to
                                                         use any kind of
                                                         triangle.
25   Prove that all circles are     -circle              Students will be        Conics
     similar                        -radius              able to                 Trigonometry
                                    -diameter            demonstrate or          PreCalculus
                                    -circumference       explain why all
                                    -pi                  circles are similar.
26   Identify and describe          -radius              Students will be        Engineering
     relationships among            -chord               able to describe and    Computer
     inscribed angles, radii, and   -circle              use the relationship    Animations
     chords                         -inscribed           between inscribed,      Vectors
                                    -circumscribed       central, and            Unit Circle
                                    -central angle       circumscribed           Calculus
                                    -right angle         angles, show that       Trigonometry
                                    -perpendicular       angles inscribed in a   Physics
                                    -tangent             semicircle are right
                                    -intersection        angles (inscribed on
                                                         a diameter), radius
                                                         is tangent to a
                                                         tangent at the
                                                         intersection point.
27   Construct the inscribed        -inscribed           Students will be        Engineering
     and circumscribed circles      -circumscribed       able to use either      CAD
     of a triangle and prove        -quadrilateral       compass and             Architecture
     properties of angles for a     -opposite,           straight edge or
     quadrilateral inscribed in a   consecutive angles   software to prove
     circle.                                             properties of
                                                         inscribed and
                                                         circumscribed
                                                         triangles.
28   +Construct a tangent line      -tangent             Students will use a     Calculus
     from a point outside a         -circle              compass and             PreCalculus
     given circle to the circle                          straight edge or
                                                         software to
                                                         construct the
                                                         tangent to a circle
                                                         from a point
                                                         outside the circle.
29   Derive using similarity the    -similarity          Students will be        College Algebra
     fact that the length of the    -sector ( Area )     able to show that       Unit Circle
     arc intercepted by an          -arc ( length )      the length of the arc
     angle is proportional to       -circle               is proportional to
     the radius and define the      -proportion           the radius and the
     radian measure of the          -radius               radian measure is
     angle as the constant of       -intercepted arc      the constant of
     proportionality; derive the    -radius               proportionality.
     formula for the area of a      -radian measure       Students should be
     sector.                        -constant             able to derive the
                                                          formula for the area
                                                          of a sector.
30   Derive the equation of a       -circle equation      Students will be        Algebra II
     circle given a center and      -completing the       able to derive the      PreCalculus
     radius using the               square                equation of a circle    Conics
     Pythagorean theorem.           -Pythagorean          with given radius
     Complete the square to         theorem               and center using
     find the center and radius     -radius               the Pythagorean
     of a circle given by an        -circle center        theorem. Students
     equation.                      -diameter             will be able to
                                                          complete the
                                                          square to find the
                                                          center and radius of
                                                          a circle.
31   Use coordinates to prove       -solving equations    Students should be      Algebra II
     simple geometric               -coordinate proofs    able to use algebra     Precalculus
     theorems algebraically.        -slope                and algebraic
                                    -parallel &           formulas to prove
                                    perpendicular lines   or disprove
                                    -Pythagorean          properties of circles
                                    Theorem               and quadrilaterals.
                                    -distance formula (
                                    between points)
                                    -quadrilateral
                                    definitions
                                    -quadrilateral
                                    properties
                                    -graphing circles
32   Prove the slope criteria for   -slope                Students will be        All future math
     parallel and perpendicular     -parallel             able to determine       courses
     lines and use them to          -perpendicular        and use the
     solve geometric problems.      -graphing lines       relationship
                                    -equations of lines   between the slopes
                                                          of parallel and
                                                          perpendicular lines.
33   Find the point on a            -line segment         Students will be        Algebra II
     directed line segment          -point                able to locate a
     between two given points       -ratio                point on a segment
     that partitions the            -proportion           using a ratios and
     segment in a given ratio.                            proportions
34   Use coordinates to             -distance formula     Students will be        All future math
     compute perimeters of          -perimeter            able to calculate       courses
     polygons and areas of          -polygon              perimeters of
     triangles and rectangles.      -area                 polygons and areas
                                   -triangle            of rectangles and
                                   -rectangle           triangles by using
                                                        the distance
                                                        formula for
                                                        example.
35   Determine areas and           -perimeter           Students will be      All future math
     perimeters of regular         -area                able to calculate the courses
     polygons including            -inscribed           perimeter and areas
     inscribed and                 -circumscribed       of regular polygons.
     circumscribed polygons,       -polygon
     given the coordinates of      -vertex
     vertices or other
     characteristics
36   Give an informal argument     -circle              Students will be         Algebra II
     for the formulas for the      -circumference       able to use              Physics
     circumference of a circle,    -area                dissection               Calculus
     area of a circle, volume of   -volume              arguments,               PreCalculus
     a cylinder, pyramid, and      -cylinder            cavalieri’s principle,
     cone.                         -pyramid             and informal limit
                                   -cone                arguments to derive
                                                        or defend the
                                                        formulas for
                                                        circumference,
                                                        area, and volume.
37   Use volume formulas for       -volume              Students will be         Calculus
     cylinders, pyramids, cones,   -cylinder            able to use the          PreCalculus
     and spheres to solve          -cone                volume formulas for
     problems                      -sphere              cylinders, cones,
                                                        and spheres to
                                                        solve problems.
38   Determine the relationship    -similarity          Students will be         Calculus
     between surface areas of      -area                able to use the fact     Construction
     similar figures and           -volume              that two figures are     Algebra II
     volumes of similar figures    -ratio               similar to calculate     Word Problems
                                   -proportion          volume or area.          Architecture
39   Identify the shapes of two-   -circle              Students will be         Computer
     dimensional cross-sections    -ellipse             able to determine        Animation
     of three-dimensional          -parabola            the names of cross       Computer Science
     objects, and identify 3-D     -hyperbola           sections of 3-D          CAD
     objects generated by          -square              objects.                 Airforce/military
     rotations of two-             -rectangle                                    tests
     dimensional objects.          -parallelogram                                Conics
                                   -triangle
                                   -various and
                                   sundry 3-D objects
40   Use geometric shapes,         -solid geometry      Students will be         CAD
     their measures, and their     shapes               able to use              Computer Science
     properties to describe        -volume              composite figures        Art
     objects                       -area                to model and             Architecture
                                   -measuring           estimate area and
                                                        volume of various
                                                      3-D objects.
41   Apply concepts of density     -density           Students will be          Physics
     based on area and volume      -ratios            able to model             Chemistry
     in modeling situations (      -proportions       density using             Precalculus
     persons per square miles,     -area              geometric concepts.       Calculus
     BTUs per cubic feet)          -volume
42   Apply geometric methods       ( possible )       Students will be          Physics
     to solve design problems      -area              able to use               Architecture
     (e.g., designing an object    -perimeter         geometric concepts        Engineering
     or structure to satisfy       -minimum           to design solutions       CAD
     physical constraints or       -maximum           to real life              Design/planning
     minimize cost; working        -distance          problems. This            Real life budget
     with typographic grid         -measurements      would be a good           management
     systems based on ratios).*                       opportunity to use
                                                      graphing
                                                      calculators,
                                                      spreadsheets, or
                                                      GSP
43   Understand the                -Probability       Students will be          Algebra II
     conditional probability of    -independent       able to calculate the     PreCalculus
     A given B as P(A and          variable           probability of            Probability/Stats
     B)/P(B), and interpret        -dependent         independent of
     independence of A and B       variable           events.
     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.

44   Construct and interpret       -frequency table   Students will be          Statistics
     two-way frequency tables      -tally marks       able to collect, tally,   Economics/Govern
     of data when two              -probability       and interpret             ment/Social Studies
     categories are associated     -independent       statistical data.
     with each object being        variable
     classified. Use the two-      -dependent
     way table as a sample         variable
     space to decide if events     -random
     are independent and to
     approximate conditional
     probabilities.
45   Recognize and explain the     -probability       Students should be        Statistics
     concepts of conditional       -independent       able to compare           Business
     probability and               variable           and analyze               Probability
     independence in everyday      -dependent         statistical data to       Economics
     language and everyday         variable           determine if events
     situations.                                      are dependent or
                                                      independent.
46   Find the conditional          -conditional       Students should be        Statistics
     probability of A given B as   probability        able to find the          Business
     the fraction of B's             -model/modeling     probability of         Probability
     outcomes that also belong       -fractions          dependent events.      Economics
     to A, and interpret the         -outcome
     answer in terms of the
     model.
47   Apply the Addition Rule,        -probability        Students will be       Statistics
     P(A or B) = P(A) + P(B) - P(A   -independent        able to model and      Business
     and B), and interpret the       events              also show              Probability
     answer in terms of the          -dependent events   mathematically the     Economics
     model.                          -fractions          probability of
                                                         events A or B.
48   (+) Apply the general           -probability        Students will be       Statistics
     Multiplication Rule in a        -independent        able to model and      Business
     uniform probability model,      events              also show              Probability
     P(A and B) = P(A)P(B|A) =       -dependent events   mathematically the     Economics
     P(B)P(A|B), and interpret       -fractions          probability of
     the answer in terms of the                          events A and B.
     model.
49   (+) Use permutations and        -combination        Students should be     Computer Science (
     combinations to compute         -permutation        able to calculate a    Gaming industry)
     probabilities of compound       -probability        combination and a      Statistics
     events and solve                                    permutation and        Probability
     problems.                                           solve related          Business
                                                         problems
50   (+) Use probabilities to        -probability        Students should be     Government
     make fair decisions (e.g.,      -random             able to use            Statistics
     drawing by lots, using a                            probability to make    Economics
     random number                                       a fair decision in a   Probability
     generator).                                         given random
                                                         situation.
51   (+) Analyze decisions and    -probability           Students will be       Government
     strategies using probability                        able to use            Statistics
     concepts (e.g., product                             probability to         Economics/
     testing, medical testing,                           determine              Marketing
     pulling a hockey goalie at                          appropriate            Probability
     the end of a game).                                 strategies.            Sports and/or
                                                                                Medicine

				
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