Download RCHS Rev. 06/2011 Geometry A Unit One Congruence Length of

RCHS Rev. 06/2011
Geometry
Unit One
Length of
Unit
KY Common Core
Academic
Standards
4 Weeks
Cluster –
Experiment with
transformations
in the plane.
G.CO.1
Know precise definitions
of angle, circle,
perpendicular line,
parallel line, and line
segment based on the
undefined notions of a
point, line, distance
along a line and
distance around a
circular arc.
Rowan County Senior High School 2010-2011
Congruence
Vocabulary/Learning Targets
Vocabulary:
Angle, circle, perpendicular line, parallel line,
undefined terms, arc, ray, vertex, equidistant, intersect,
right angle, line segment, point, line distance,
transformations,
Functions, translation, horizontal stretch, rotation,
reflection, dilation, image, preimage, rigid motion,
Input, output, coordinates, angle measure
Unit Learning targets
Students will be able to:

G.CO.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)






A
Activities/Assessments/
Resources
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Power point presentations
Sage and Scribe
I will describe the undefined terms: point, line,
and distance alone a line in a plane.
I will define perpendicular lines, parallel lines,
line segments, and angles.
I will define circle and the distance around a
circular arc.
I will represent transformations in the plane
using, e.g., transparencies and geometry
software.
I will write functions to represent
transformations.
I will compare transformations that preserve
distance and angle to those that do not(e.g.,
translation versus horizontal stretch)
I will describe the different types of
transformations including translations,
reflections, rotations, and dilations.
Page 1
RCHS Rev. 06/2011
Cluster –
Experiment with
transformations
in the plane.
Geometry
Vocabulary:
Rectangle, parallelogram, trapezoid, isosceles trapezoid, regular
polygon, rotational symmetry, reflection symmetry, mapped onto
G.CO.3
Given a rectangle,
parallelogram,
trapezoid, or regular
polygon, describe the
rotations and reflections
that carry it onto itself.
Rotation, reflection, translation, perpendicular bisector, line
segment, preimage, image, parallel lines, angle, center of
rotation, map, figure, transformation, composition.
G.CO.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.
Rowan County Senior High School 2010-2011
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Power point presentations
Sage and Scribe
Unit Learning targets
Students will be able to:

G.CO.4
Develop definitions of
rotations, reflections
and translations in
terms of angles, circles,
perpendicular lines,
parallel lines and line
segments.
A

I will describe the rotations and/or reflections that carry it
onto itself when given a rectangle.
I can calculate the number of lines of reflection
symmetry and the degree of rotational symmetry of any
regular polygon.

I will develop definitions of rotations, reflections and
translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.

I can draw a specific transformation when given a
geometric figure and a rotation, reflection, or translation.
I can predict and verify the sequence of transformations
(a composition) that will map a figure onto another.

Page 2
RCHS Rev. 06/2011
Cluster –
Understanding
congruence in
terms of rigid
motion.
G.CO.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.
Geometry
Vocabulary:
Congruence, composition, rigid motions, map, reflection,
rotation, translation, transformation, angle measure, distance
Rigid motions, reflection, rotation, translation, distance, angle
measure, congruent, composition, map, figure, corresponding
sides, corresponding angles, triangle






Rowan County Senior High School 2010-2011
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Power point presentations
Sage and Scribe
Unit Learning targets
Students will be able to:

G.CO.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.
A
I can define rigid motions as reflections, rotations,
translations, and combinations of these, all of which
preserve distance and angle measure.
I can define congruent figures as figures that have the
same shape and size and state that a composition of
rigid motions will map one congruent figure onto the
other.
I can predict the composition of transformations that will
map a figure onto a congruent figure.
I can determine if two figures are congruent by
determining if rigid motions will turn one figure into the
other.
I can identify corresponding sides and corresponding
angles of congruent triangles.
I can explain that in a pair of congruent triangles,
corresponding sides are congruent(distance is
preserved) and corresponding angles are
congruent(angle measure is preserved).
I can demonstrate that when distance is
preserved(corresponding sides are congruent) and
angle measure is preserved(corresponding angles are
congruent) the triangles must also be congruent.
Page 3
RCHS Rev. 06/2011
ClusterUnderstand
congruence in
terms of rigid
motion.
Cluster –
Prove geometric
theorems
Geometry
Vocabulary:
G.CO.8
Explain how the criteria
for triangle congruence
(ASA, SAS, and SSS)
follow from the definition
of congruence in terms
of rigid motions.
G.CO.9
Prove theorems about
lines and angles.
Theorems include:
vertical angles are
congruent; when a
transversal crosses
parallel lines, alternate
interioir 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.
Rigid motion, reflection, rotation, translation, congruent,
composition, map, SAS, ASA, SSS
Theorem, linear pair, vertical angles, alternate interior angles,
alternate exterior, same-side intrior angles, corresponding
angles, perpendicular bisector, supplementary angles,
complimentary angles, equidistant, congruent, adjacent,
consecutive/non-consecutive, reflection, Law of Syllogism
Unit Learning targets
Students will be able to:







Rowan County Senior High School 2010-2011
A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Power point presentations
Sage and Scribe
I can define rigid motions as reflections, rotations,
translations, and combinations of these, all of which
preserve distance and angle measure.
I can list the sufficient conditions to prove triangles are
congruent.
I can map a triangle with one of the sufficient conditions
( e.g., SSS) onto the original triangle and show that
corresponding sides and corresponding angles are
congruent.
I can identify and use the properties of congruence and
equality (reflexive, symmetric, transitive) in my proofs.
I can order statements based on the Law of Syllogism
when constructing my proof.
I can correctly interpret geometric diagrams by
identifying what can and cannot be assumed.
I can use theorems, postulates, or definitions to prove
theorems about lines and angles, including:
a) Vertical angles are congruent;
b) When a transversal crosses parallel lines
alternate interior angles are congruent and
corresponding angles are congruent, and sameside interior angles are supplementary;
c) Points on a perpendicular bisector of a line
segment are exactly those equidistant from the
segment’s endpoints.
Page 4
RCHS Rev. 06/2011
Cluster –
Prove geometric
theorems
G.CO.10
Prove theorems about
triangles. Theorems
include: measures of
interior angles of a
triangle sum to 180̊ ;
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.CO.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.
Rowan County Senior High School 2010-2011
Geometry
Vocabulary:
Midpoint, midsegment, isosceles triangle, median, centroid,
coordinate proof, adjacent, consecutive/ non-consecutive, Law
of Syllogism
Quadrilateral, parallelogram, rectangle, diagonals, distance
formula, midpoint formula, slope, bisector, congruence
properties
Unit Learning targets
Students will be able to:
 I can order statements based on the Law of Syllogism
when constructing my proof.
 I Can correctly interpret geometric diagrams (what can
and cannot be assumed).
 I Can use theorems, postulates, or definitions to prove
theorems about triangles, including
a) Measures of interior angles of a triangle sum to
180̊ ;
b) Base angles of isosceles triangles are
congruent;
c) The segment joining midpoints of two sides of a
triangle is parallel to the third side and half the
length;
d) The medians of a triangle meet at a point.

A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Power point presentations
Sage and Scribe
I can use theorems, postulates, or definitions to prove
theorems about parallelograms, including:
a) Prove opposide sides of a parallelogram are
congruent;
b) Prove opposite angles of a parallelogram are
congruent;
c) Prove the diagonals of a parallelogram bisect
each other;
d) Prove that rectangles are parallelograms with
congruent diagonals.
Page 5
RCHS Rev. 06/2011
Cluster –
Make geometric
constructions
G.CO.12
Make formal geometric
constructions with a
variety of tools and
methods (compass and
straightedge, string,
reflective devices, paper
folding, dynamic
geometric software,
etc.). Copying a
segment; copying an
angle; bisecting a
segment; bisecting an
angle; constructing
perpendicular lines,
including the
perpendicular bisector
of a line segment; and
constructing a line
parallel to a given line
through a point not on
the line.
G.CO.13
Construct an equilateral
triangle, a square, and a
regular hexagon
inscribed in a circle.
Geometry
Vocabulary:
Segment, angle, perpendicular lines, perpendicular bisector,
parallel lines, bisect, formal construction, informal construction,
compass, straightedge
Construction, equilateral triangle, square, regular hexagon,
inscribe, circle.
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Power point presentations
Sage and Scribe
Unit Learning targets
Students will be able to:








Rowan County Senior High School 2010-2011
A
I can identify the tools used in formal constructions.
I can use tools and methods to precisely copy a
segment, copy an angle, bisect a segment, bisect an
angle, construct perpendicular lines and bisectors, and
construct a line parallel to a given line through a point
not on the line.
I can informally perform the constructions listed above
using string, reflective devices, paper folding, and/or
dynamic geometric software.
I can define inscribed polygons (the vertices of the figure
must be points on the circle).
I can construct an equilateral triangle inscribed in a
circle.
I can construct a square inscribed in a circle.
I can construct a hexagon inscribed in a circle.
I can explain the steps to constructing an equilateral
triangle, a square, and a regular hexagon inscribed in a
circle.
Page 6
RCHS Rev. 06/2011
Geometry
Unit Two
Length of
Unit
4 weeks
Domain
Similarity,
Right
triangles, and
Trigonometry
A
Similarity, Right Triangles, and Trigonometry
KY Common Core
Academic
Standards
G.SRT.1a
Verify experimentally
the properties of
dilations given by a
center by a center and a
scale factor.
b. 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.
Rowan County Senior High School 2010-2011
Vocabulary
Unit Learning Targets
Vocabulary:
Image
Center
pre-image
similar figures
scale factor
Unit Learning targets
Students will be able to:

I will identify a dilation, stating its scale factor and
center.

I will verify experimentally that a dilated image is similar
to its pre-image by showing congruent corresponding
angles and proportional sides.

I will verify experimentally that a dilation takes a line not
passing through the center of the dilation to a parallel
line by showing the lines are parallel.

I will verify experimentally that dilation leaves a line
passing through the center of the dilation unchanged by
showing that it is the same line.
Activities/Assessments/
Resources
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Page 7
RCHS Rev. 06/2011
Geometry
Unit Two
Length of
Unit
Cont.
Domain
Similarity,
Right
triangles, and
Trigonometry
A
Similarity, Right Triangles, and Trigonometry
KY Common Core
Academic
Standards
G.SRT.1b
Verify experimentally
the properties of
dilations given by a
center and a scale
factor.
b. The dilation of a line
segment is longer or
shorter in the ratio given
by the scale factor.
G.SRT.2
Given two figures, use
the definition of
similarity in terms of
similarity
transformations to
decide if they are
similar; explain using
similarity
transformations the
meaning of similarity for
triangles as the equality
of all corresponding
pairs of angles and the
proportionality of all
corresponding pairs of
sides.
Rowan County Senior High School 2010-2011
Vocabulary
Unit Learning Targets
Vocabulary:
Image
Center
pre-image
similar figures
scale factor
Unit Learning targets
Students will be able to:




I will explain that the scale factor represents how many
times longer or shorter a dilated line segment is than it’s
pre-image.
I will verify experimentally that the dilation of a line
segment is longer or shorter in the ratio given by the
scale factor.
Activities/Assessments/
Resources
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
I will use similarity transformations to explain that
triangles are similar if all pairs of corresponding angles
are congruent and all corresponding pairs of sides are
proportional.
I will, when given two figures, decide if they are similar
by using the definition of similarity in terms of similarity
transformations.
Page 8
RCHS Rev. 06/2011
Domain
Similarity,
Right
triangles, and
Trigonometry
G.SRT.3
Use the properties of
similarity
transformations to
establish the AA
criterion for two
triangles to be similar.
Geometry
A
Vocabulary
Properties of similarity transformations
Pythagorean Theorem
postulates
AA similarity
theorems
Unit Learning targets
Students will be able to:
G.SRT.4
Prove theorems about
triangles. Theorems
include: a line parallel to
one side of a triangle
divides the other two
proportionally, and
conversely; the
Pythagorean Theorem
proved using triangle
similarity.


I will recall the properties of similarity transformations.
I will establish the AA criterion for similarity of triangles
by extending the properties of similarity transformations
to the general case of any two similar triangles.

I will prove theorems involving similarity about triangles.
(Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and
conversely; the Pythagorean Theorem proved using
triangle similarity.)
I will recall postulates, theorems, and definitions to
prove theorems about triangles.

G.SRT.5
Use congruence and
similarity criteria for
triangles to solve
problems and to prove
relationships in
geometric figures.
Rowan County Senior High School 2010-2011



Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
I will recall congruence and similarity criteria for
triangles.
I will use congruency and similarity theorems for
triangles to solve problems.
I will use congruency and similarity theorems for
triangles to prove relationships in geometric figures.
Page 9
RCHS Rev. 06/2011
Geometry
Domain
Vocabulary
Similarity,
Right
triangles, and
Trigonometry
Trigonometric ratios
sine
Complementary angles
G.SRT.6
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.SRT.8
Use trigonometric ratios
and the Pythagorean
Theorem to solve right
triangles in applied
problems.







Rowan County Senior High School 2010-2011
tangent
Unit Learning targets
Students will be able to:

G.SRT.7
Explain and use the
relationship between
the sine and cosine of
complementary angles.
cosine
A
I will name the sides of a right triangle as related to an
acute angle.
I will recognize that if two right triangles have a pair of
acute, congruent angles that the triangles are similar.
I will compare common ratios for similar right triangles
and develop a relationship between the ratio and the
acute angle leading to the trigonometry ratios.
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
I will use the relationship between the sine and cosine of
complementary angles.
I will explain how the sine and cosine of complementary
angles are related to each other.
I will recognize which methods could be used to solve
right triangles in applied problems.
I will solve for an unknown angle or side of a right
triangle using sine, cosine, and tangent.
I will apply right triangle trigonometric ratios and the
Pythagorean Theorem to solve right triangles in applied
problems.
Page 10
RCHS Rev. 06/2011
Geometry
Domain
Vocabulary
Modeling with
geometry
Measures of geometric shapes, properties of geometric shapes
density volume
typographical grid
design problems
G.MG.1
Use geometric shapes,
their measures, and
their properties to
describe objects (e.g.,
modeling a tree trunk or
a human torso as a
cylinder)
G.MG.2
Apply concepts of
density based on area
and volume in modeling
situations (e.g., persons
per square mile, BTUs
per cubic foot).*
*Modeling Standard)
G.MG.3
Apply geometric
methods to solve design
problems (e.g.,
designing an object or
structure to satisfy
physical constraints or
minimize cost; working
with typographic grid
systems based on
ratios). * (*Modeling
Standard)
Rowan County Senior High School 2010-2011
Unit Learning targets
Students will be able to:






I will use measures and properties of geometric shapes
to describe real world objects.
I will, when given a real world object, classify the object
as a known geometric shape – use this to solve
problems in context.
A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Focus on situations well modeled
by trigonometric ratios for acute
angles.
I will define density.
I will apply concepts of density based on area and
volume to model real-life situations (e.g., persons per
square mile, BTUs per cubic foot).
I will describe a typographical grid system.
I will apply geometric methods to solve design problems.
Design an object or structure to
satisfy physical constraints or
minimize cost; working with
typographic grid systems based on
ratios.
Page 11
RCHS Rev. 06/2011
Geometry
A
Vocabulary
Domain
Similarity,
right triangles,
and
trigonometry
Right triangle trigonometry
G.SRT.9 (+)
Derive the formula A =
½ ab sin(c) for the area
of a triangle by drawing
an auxiliary line from a
vertex perpendicular to
the opposite side.
G.SRT.10 (+)
Prove the Laws of Sines
and Cosines and use
them to solve problems
G.SRT.11 (+)
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).
Rowan County Senior High School 2010-2011
Law of Sines Law of Cosines
Unit Learning targets
Students will be able to:


I will recall right triangle trigonometry to solve
mathematical problems.
I will derive the formula A = ½ sin(c) for the area of a
triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.



I will prove the Law of Sines.
I will prove the Law of Cosines.
I will recognize when the Law of Sines or Law of
Cosines can be applied to a problem and solve
problems in context using them.

I will 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).

I will determine from given measurements in right and
non-right triangles whether it is appropriate to use the
Law of Sines or Cosines
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
With respect to the general case of
Laws of Sines and Cosines, the
definition of sine and cosine must
be extended to obtuse angles.
Page 12
RCHS Rev. 06/2011
Geometry
Unit Three
Length of
Unit
Domain
Geometric
Measurement
and
Dimension
Vocabulary/Learning Targets
Vocabulary:
formulas cylinders pyramids cones spheres
two-dimensional three-dimensional cross-sections
rotations
geometric shapes and their properties
G.GMD.3
Use volume formulas
for cylinders, pyramids,
cones, and spheres to
solve problems.
*(*Modeling Standard)
G.GMD.4
Identify the shapes of
two-dimensional crosssections of threedimensional objects,
and identify threedimensional objects
generated by rotations
of two-dimensional
objects.
Domain –
Modeling with
Geometry
Geometric Measurement and Dimension
KY Common Core
Academic
Standards
4 weeks
G.MG.1
Use geometric shapes,
their measures, and
their properties to
describe objects (e.g.,
modeling a tree trunk or
a human torso as a
cylinder). *(* Modeling
standard)
Rowan County Senior High School 2010-2011
A
Unit Learning Targets
Students will be able to







Utilize the appropriate formula for volume depending on
the figure.
Use volume formulas for cylinders, pyramids, cones,
and spheres to solve contextual problems.
Relate the shapes of two-dimensional cross-sections to
their three-dimensional objects
Discover three-dimensional objects generated by
rotations of two-dimensional objects.
Use strategies to help visualize relationships between
two-dimensional and three-dimensional objects.
Activities/Assessments/
Resources
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Informal arguments for area and
volume formulas can make use of
the way in which area and volume
scale under similarity
transformations: when one figure in
the plane results from another by
applying a similarity transformation
with scale factor K, its area is K2
times the area of the first. Similarly,
volumes of solid figures scale by K3
under a similarity transformations
with scale factor K.
Given a real world object, classify the object as a known
geometric shape; use this to solve problems in context.
Use measures and properties of geometric shapes to
describe real world objects.
Page 13
RCHS Rev. 06/2011
Geometry
Unit Four
Length of
Unit
4 weeks
Cluster – Use
coordinates to
prove simple
geometric
theorems
algebraically.
A
Expressing Geometric Properties with Equations
KY Common Core
Academic
Standards
Vocabulary/Learning Targets
Vocabulary:
Coordinate geometry distance, midpoint, slope, equation of a
line, parallel, perpendicular, proof
G.GPE.4
Use coordinates to
prove simple geometric
theorems algebraically.
For example, prove or
disprove that a figure
defined by four given
points in the coordinate
plane is a rectangle;
prove or disprove that
the point (1,
) lies on
the circle centered at
the origin and
containing the point
(0,2).
Rowan County Senior High School 2010-2011
Unit Learning Targets
Students will be able to


Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a
figure defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the point
(1,
) lies on the circle centered at the origin and
containing the point (0,2).
Recall previous understandings of coordinate geometry
(including but not limited to: distance, midpoint and
slope formula, equations of a line, definitions of parallel
and perpendicular lines, etc.)
Activities/Assessments/
Resources
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Page 14
RCHS Rev. 06/2011
Geometry
A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Vocabulary:
Ratio , coordinate geometry, positive slope, negative slope
perimeter area
distance formula parabola focus
directrix
Unit Learning Targets
Students will be able to



Recall the definition of ratio.
Recall previous understandings of coordinate geometry.
Given a line segment (including those with positive and
negative slopes) and a ratio, find the point on the
segment that partitions the segment into the given ratio.

Use the coordinates of the vertices of a polygon to find
the necessary dimensions for finding the perimeter.
Use the coordinates of the vertices of a triangle to find
the necessary dimensions (base, height) for finding the
area (i.e., the distance between vertices by counting,
distance formula, Pythagorean Theorem, etc.)
Use the coordinates of the vertices of a rectangle to find
the necessary dimensions (base, height) for finding the
area (i.e., the distance between vertices by counting,
distance formula).
Formulate a model of figures in contextual problems to
compute area and/or perimeter.



Rowan County Senior High School 2010-2011
Page 15
RCHS Rev. 06/2011
Cluster – Use
coordinates to
prove simple
geometric
theorems
algebraically
Geometry
Vocabulary:
Parallel lines slope perpendicular lines
reciprocals
Equation ratio distance Pythagorean Theorem polygon
area
G.GPE.5
Prove the slope criteria
for parallel and
perpendicular lines and
use them to solve
geometric problems
(e.g., find the equations
of a line parallel or
perpendicular to a given
line that passes through
a given point).
G.GPE.6
Find the point on a
directed line segment
between two given
points that partitions the
segment in a given
ratio.
G.GPE.7
Use coordinates to
compute perimeters of
polygons and area of
triangles and
rectangles, e.g., using
the distance formula.
*(*Modeling Standard)
Rowan County Senior High School 2010-2011
Unit Learning Targets
Students will be able to
 Recognize that slopes of parallel lines are equal.
 Recognize that slopes of perpendicular lines are
opposite reciprocals(i.e. the slopes of perpendicular
lines have a product of -1)
 Find the equation of a line parallel to a given line that
passes through a given point.
 Find the equation of a line perpendicular to a given line
that passes through a given point. Prove the slope
criteria for parallel and perpendicular lines and use them
to solve geometric problems.


Recall the definition of ratio.
Given a line segment (including those with positive and
negative slopes) and a ratio, find the point on the
segment that partitions the segment into the given ratio.

Use the coordinates of the vertices of a polygon to find
the necessary dimentsions for finding the perimeter (ie.
The distance between vertices).
Use the coordinates of the vertices of a triangle to find
the necessary dimensions (base, height) for finding the
area ( i.e. the distance between vertices by counting,
distance formula, Pythagorean Theorem, etc.).
Formulate a model of figures in contextual problems to
compute area and/or perimeter.


A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Be sure to relate this to systems of
equations having no solutions or
infinitely many solutions
Distance formula and its connection
with the Pythagorean theorem.
Page 16
RCHS Rev. 06/2011
Cluster –
Translate
between the
geometric
description
and the
equation for
a conic
section.
Geometry
Vocabulary:
Parabola focus directrix
Unit Learning Targets
Students will be able to:
G.GPE.2
Derive the equation of a
parabola given a focus
and directrix.
Rowan County Senior High School 2010-2011


Define a parabola including the relationship of the focus
and the equation of the directrix to the parabolic shape.
Derive the equation of parabola given the focus and
directrix.
A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
The directrix should be parallel to a
coordinate axis.
Page 17
RCHS Rev. 06/2011
Geometry
Unit Five
Length of
Unit
A
Circles
KY Common Core
Academic
Standards
Vocabulary/Learning Targets
Activities/Assessments/
Resources
4 weeks
Cluster –
Understand
and apply
theorems
about circles.
G.C.1
Prove that all circles are
similar.
Vocabulary:
Circle circumference diameter dilation inscribed angles
Radii chords central angles circumscribed angles
Diameter tangent right angles point of tangency
Unit Learning Targets
Students will be able to

G.C.2
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.
Rowan County Senior High School 2010-2011







Compare the ratio of the circumference of a circle to the
diameter of the circle.
Discuss, develop and justify this ratio for several circles.
Determine that this ratio is constant for all circles.
Recognize when figures are similar. When one figure is
a dilation of the other.
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Identify inscribed angles, radii, chords, central angles,
circumscribed angles, diameter, tangent.
Recognize that inscribed angles on a diameter are right
angles.
Recognize that radius of a circle is perpendicular to the
radius at the point of tangency.
Examine the relationship between central, inscribed and
circumscribed angles by applying theorems about their
measures.
Page 18
RCHS Rev. 06/2011
Geometry
Vocabulary:
Tangent radius perpendicular bisector midpoint
Arc
Inscribed circles
circumscribed circles
Midpoint
bisector point of concurrency
Cluster –
Understand
and apply
theorems
about circles.
Unit Learning Targets
Students will be able to
G.C.3
Construct the inscribed
and circumscribed
circles of a triangle, and
prove properties of
angles for a
quadrilateral inscribed
in a circle.
G.C.4 (+)
Construct a tangent line
from a point outside a
given circle to the circle.
A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions




Define inscribed and circumscribed circles of a triangle.
Recall midpoint and bisector definitions.
Define a point of concurrency.
Prove properties of angles for a quadrilateral inscribed
in a circle.
Construct inscribed circles of a
triangle.
Construct circumscribed circles of a
triangle.

Synthesize theorems that apply to circles and tangents,
such as: tangent drawn from a common external point
are congruent, a radius is perpendicular.
Construct the perpendicular isector
of the line segment between the
center C to the outside point P.
Construt arcs on circle C from the
midpoint Q, have length CQ.
G.C.5
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
constraint of
proportionality; derive
the formula for the area
of a sector.
Rowan County Senior High School 2010-2011


Determine the constant of proportionality(scale factor)
Explain that 1̊ = /180 radians.

Justify the radii of any two circles (r1 and r2) and the arc
lengths (s1 and s2) determined by congruent central
angles are proportional, such that r1/s1 = r2/s2 .

Verify that the constant of a proportion is the same as
the radian measure, Ѳ, of the given central angle.
Conclude s = r Ѳ.
Construct the tangent line.
Emphasize the similarity of all
circles. Note that by similarity of
sectors with the same central angle,
arc lengths are proportional to the
radius. Use this as a basis for
introducing radian as a unit of
measure. It is not intended that it
be applied to the development of
circular trigonometry in this course.
Page 19
RCHS Rev. 06/2011
Geometry
A
Vocabulary:
Cluster –
Translate
between the
geometric
description
and the
equation for a
conic section.
G.GPE.1
Derive the equation of a
circle of given center
and radius using the
Pythagorean Theorem;
complete the square to
find the center and
radius of a circle given
an equation.
G.GPE.4
Use coordinates to
prove simple geometric
theorems algebraically.
For example, prove or
disprove that a figure
defined by four given
points in the coordinate
plane is a rectangle;
prove or disprove that
the point
(1, ) lies
on the circle centered at
the origin and
containing the point
(0,2)
Rowan County Senior High School 2010-2011
Equation of a circle
complete the square
Sector radian distance midpoint slope
perpendicular lines
parallel lines
Unit Learning Targets
Students will be able to





Define a circle.
Complete the square of a quadratic equation.
Derive equation of a circle using the Pythagorean
theorem – given coordinates of the center and length of
the radius.
Determine the center and radius by completing the
square.
Use coordinates to prove simple geometric theorems
algebraically. For example prove that a figure is a
rectangle given 4 coordinates or prove that a point lies
on a circle given the coordinates for its center.
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Emphasize the similarity of all
circles. Note that by similarity of
sectors with the same central angle,
arc lengths are proportional to the
radius. Use this as a basis for
introducing radians as a unit of
measure. It is not intended that it
be applied to the development of
circular trigonometry in this course.
Include simple proofs that involve
circles.
Page 20
RCHS Rev. 06/2011
Cluster –
apply
geometric
concepts in
modeling
situations
Geometry
G.MG.1
Use geometric shapes,
their measures, and
their properties to
describe objects (e.g.,
modeling a tree trunk or
a human torso as a
cylinder). * (*Modeling
Standard)
Vocabulary:
Geometric shape
Unit Learning Targets
Students will be able to


Unit Three
Length of
Unit
KY Common Core
Academic
Standards
Rowan County Senior High School 2010-2011
Use measures and properties of geometric shapes to
describe real world objects.
Given a real world object, classify the object as a known
geometric shape – use this to solve problems in context.
A
Word wall
Think-Pair Share
Anticipation Guides
Admit/exit slips
Jigsaw
Double entry organizer
Note-taking Guide
Win-GEOM
Geogebra
Smartboard interactions
Focus on situations in which the
analysis of circles is required.
Geometric Measurement and Dimension
Vocabulary/Learning Targets
Activities/Assessments/
Resources
Page 21
RCHS Rev. 06/2011
Rowan County Senior High School 2010-2011
Geometry
A
Page 22