# 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
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
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
-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
-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
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
-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
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
angle as the constant of       -intercepted arc      the constant of
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
of a circle given by an        -circle center        theorem. Students
equation.                      -diameter             will be able to
complete the
square to find the
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
-distance formula (
between points)
definitions
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,
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
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
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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