Download 8/2/2011 Geometry Curriculum Mapping 1 Quarter Geometry

Geometry
Curriculum Mapping
8/2/2011
Quarter
1
1
Geometry Standards
Deconstructed Standards
I Can
Unit 1: Congruence, Proof, and
Constructions
Describe the undefined terms: point, line, and
distance along a line in a plane. (Knowledge)
I can describe the terms: point, line, and distance
along a line in a plane.
Unit 1: Congruence, Proof, and
Constructions
Identify and use properties of: Vertical angles,
Parallel lines with transversals, All angle
relationships, Corresponding angles, Alternate
interior angles, Perpendicular bisector, Equidistant
from endpoint (Knowledge)
I can identify and use properties of: vertical angles,
parallel lines with transversals, all angle
relationships, corresponding angles, alternate
interior angles, perpendicular bisector, equidistant
from endpoint.
1
Vocabulary
Point
Circle
Distance
G.CO.1 Know precise definitions of angle, circle, Define perpendicular lines, parallel lines, line
Ray
perpendicular line,
segments, and angles. (Knowledge)
I can define perpendicular lines, parallel lines, line
Plane
parallel line, and line segment, based on the
segments, and angles.
Line
undefined notions of point, line, distance along a Define circle and the distance around a circular arc.
Perpendicular Lines
line, and distance around a circular arc.
(Knowledge)
I can define circle and the distance around a circular Parallel Lines
arc.
*Skew Lines
Line Segment
Angle
*Circular Arc
* Theorem
* Alternate Exterior Angles
* Vertical Angles
G.CO.9 Prove theorems about lines and angles.
* Transversals
Theorems include:
* Corresponding Angles
vertical angles are congruent; when a
* Alternate Interior Angles
transversal crosses parallel lines,
Prove vertical angles are congruent. (Reasoning)
* Perpendicular Bisector
alternate interior angles are congruent and
I can prove vertical angles are congruent.
* Equidistant
corresponding angles are congruent; points on a Prove corresponding angles are congruent when
* Proof (2-column, paragraph, flow
perpendicular bisector of a line segment are
two parallel lines are cut by a transversal and
chart)
exactly those equidistant from the segment’s
converse. (Reasoning)
I can prove corresponding angles are congruent
* Linear Pair
endpoints.
when two parallel lines are cut by a transversal and * Consecutive Interior Angles
Prove alternate interior angles are congruent when converse.
two parallel lines are cut by a transversal and
converse. (Reasoning)
I can prove alternate interior angles are congruent
when two parallel lines are cut by a transversal and
Prove points are on a perpendicular bisector of a
converse.
line segment are exactly equidistant from the
segments endpoint. (Reasoning)
I can prove points are on a perpendicular bisector of
a line segment are exactly equidistant from the
segments endpoint.
*Bolded word
Use Marzano 6 Step Process
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 1: Congruence, Proof, and
Constructions
Deconstructed Standards
I Can
G.CO.9 (Standard continued)
From Appendix A: Encourage multiple ways of
writing proofs, such as in narrative paragraphs,
using flow diagrams, in two-column format, and
using diagrams without words. Students should be
encouraged to focus on the validity of the
underlying reasoning while exploring a variety of
formats for expressing that reasoning.
Unit 1: Congruence, Proof, and
Constructions
Explain the construction of geometric figures using I can explain the construction of geometric figures
a variety of tools and methods. (Knowledge)
using a variety of tools and methods.
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.
Apply the definitions, properties and theorems
about line segments, rays and angles to support
geometric constructions. (Reasoning)
Unit 1: Congruence, Proof, and
Constructions
Perform geometric constructions including: 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, using a variety
of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic
geometric software, etc.).
1
G.CO.12 (Standard continues)
1
*Bolded word
Use Marzano 6 Step Process
Apply properties and theorems about parallel and
perpendicular lines to support constructions.
(Reasoning)
I can apply the definitions, properties and theorems
about line segments, rays and angles to support
geometric constructions.
I can apply properties and theorems about parallel
and perpendicular lines to support constructions.
From Appendix A: Build on prior student
experience with simple constructions. Emphasize
the ability to formalize and explain how these
I can emphasize the ability to formalize and explain
constructions result in the desired objects. Some of how these constructions result in the desired
these constructions are closely related to previous objects.
standards and can be introduced in conjunction
with them.
2
Vocabulary
Ray
* Geometric Construction
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 1: Congruence, Proof, and
Constructions
G.CO.13 Construct an equilateral triangle, a
square, and a regular
hexagon inscribed in a circle.
Deconstructed Standards
Note: Underpinning performance, reasoning, and
knowledge targets, if applicable, are addressed in
G.CO.12
I Can
3
Vocabulary
I can emphasize the ability to formalize and explain Equilateral Triangle
how these constructions result in the desired
* Inscribed Polygon
objects.
From Appendix A: Build on prior student
experience with simple constructions. Emphasize
the ability to formalize and explain how these
constructions result in the desired objects.
(Knowledge)
1
Some of these constructions are closely related to
previous standards and can be introduced in
conjunction with them.
Construct an equilateral triangle, a square and a
regular hexagon inscribed in a circle. (Product)
1
Unit 4: Connecting Algebra and Geometry
Through Coordinates
Recognize that slopes of parallel lines are equal.
(Knowledge)
G.GPE.5 Prove the slope criteria for parallel and
perpendicular lines and uses them to solve
geometric problems (e.g., find the equation of a
line parallel or perpendicular to a given line that
passes through a given point).
Recognize that slopes of perpendicular lines are
opposite reciprocals (i.e, the slopes of
perpendicular lines have a product of -1)
(Knowledge)
Find the equation of a line parallel to a given line
that passes through a given point.
(Knowledge)
I can recognize that slopes of parallel lines are equal. Opposite Reciprocals
Parallel Lines
Perpendicular Lines
I can recognize that slopes of perpendicular lines are Slope
opposite reciprocals .
Equations of Lines
Slope-Intercept Form
Point-Slope Form
Standard Form
I can find the equation of a line parallel to a given
line that passes through a given point.
From Appendix A: Relate work on parallel lines
in G.GPE.5 to work on A.REI.5 in High School
I can find the equation of a line perpendicular to a
Algebra 1 involving systems of equations having Find the equation of a line perpendicular to a given given line that passes through a given point.
no solution or infinitely many solutions.
line that passes through a given point.
(Knowledge)
I can prove the slope criteria for parallel and
perpendicular lines and use them to solve geometric
Prove the slope criteria for parallel and
problems.
perpendicular lines and use them to solve
geometric problems. (Reasoning)
*Bolded word
Use Marzano 6 Step Process
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Deconstructed Standards
Unit 4: Connecting Algebra and Geometry
Through Coordinates
Recall the definition of ratio. (Knowledge)
G.GPE.6 Find the point on a directed line
segment between two given points that
partitions the segment in a given ratio.
Recall previous understandings of coordinate
geometry. (Knowledge)
I Can
I can find the point on a directed line segment
between two given points that partitions the
segment in a given ratio.
4
Vocabulary
* Partition
Ratio
* Coordinate Geometry
I can recall the definition of a ratio.
Given a line segment (including those with positive I can recall previous understandings of coordinate
and negative slopes) and a ratio, find the point on geometry.
the segment that partitions the segment into the
given ratio. (Reasoning)
I can find the point on the segment that partitions
the segment into a given ratio.
1
Unit 1: Congruence, Proof, and
Constructions
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.
1,2
*Bolded word
Use Marzano 6 Step Process
Classify types of quadrilaterals. (Knowledge)
Explain theorems for parallelograms and relate to
figure. (Knowledge)
Use the principle that corresponding parts of
congruent triangles are congruent to solve
problems. (Reasoning)
Use properties of special quadrilaterals in a proof.
(Reasoning)
From Appendix A: Encourage multiple ways of
writing proofs, such as in narrative paragraphs,
using flow diagrams, in two-column format, and
using diagrams without words. Students should be
encouraged to focus on the validity of the
underlying reasoning while exploring a variety of
formats for expressing that reasoning.
I can classify types of quadrilaterals.
I can explain theorems for parallelograms and relate
to figure.
I can use the principle that corresponding parts of
congruent triangles are congruent to solve
problems.
I can use properties of special quadrilaterals in a
proof.
I can write proofs in multiple ways, such as in
narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words.
I can focus on the validity of the underlying
reasoning while exploring a variety of formats for
expressing that reasoning.
* Proof (2-column, paragraph, flow chart)
* CPCTC (Corresponding Parts of
Congruent Triangles are Congruent)
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Deconstructed Standards
I Can
Unit 1: Congruence, Proof, and
Constructions
Identify the hypothesis and conclusion of a
theorem. (Knowledge)
I can identify the hypothesis and conclusion of a
theorem.
G.CO.10 Prove theorems about triangles.
Theorems include: measures of interior angles
of a triangle sum to 180°; 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.
Design an argument to prove theorems about
triangles. (Reasoning)
I can design an argument to prove theorems about
triangles.
Analyze components of the theorem. (Reasoning)
I can analyze components of the theorem.
Prove theorems about triangles. (Reasoning)
I can prove theorems about triangles.
From Appendix A: Encourage multiple ways of
writing proofs, such as in narrative paragraphs,
using flow diagrams, in two-column format, and
using diagrams without words.
.
5
Vocabulary
Midpoint
Median of a Triangle
* Exterior Angle Theorem
Triangle Sum Theorem
Isosceles Triangle Theorem (Base Angle
Theorem)
* Points of Concurrency
Centroid
In center
Circumcenter
Orthocenter
I can write proofs in multiple ways, such as in
narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words.
Students should be encouraged to focus on the
validity of the underlying reasoning while exploring
a variety of formats for expressing that reasoning. I can focus on the validity of the underlying
Implementations of G.CO.10 may be extended to reasoning while exploring a variety of formats for
include concurrence of perpendicular bisectors and expressing that reasoning.
angle bisectors as preparation for G.C.3 in Unit 5.
Unit 2: Similarity, Proof, and Trigonometry
2
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.
*Bolded word
Use Marzano 6 Step Process
Recall postulates, theorems, and definitions to
prove theorems about triangles. (Knowledge)
I can recall postulates, theorems, and definitions to * Postulate
prove theorems about right triangles.
* Axiom
Triangle Similarity
Prove theorems involving similarity about triangles. I can prove theorems involving similarity about
(Reasoning)
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.)
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 2: Similarity, Proof, and Trigonometry
G.SRT.5 Use congruence and similarity criteria
for triangles to solve problems and to prove
relationships in geometric figures.
2
Deconstructed Standards
2
I can recall congruence and similarity criteria for
triangles.
Use congruency and similarity theorems for
triangles to solve problems. (Reasoning)
I can use congruency and similarity theorems for
triangles to solve problems.
Use congruency and similarity theorems for
triangles to prove relationships in geometric
figures. (Reasoning)
I can use congruency and similarity theorems for
triangles to prove relationships in geometric figures.
Recall previous understandings of coordinate
I can recall previous understandings of coordinate
geometry (including, but not limited to: distance,
geometry.
G.GPE.4 Use coordinates to prove simple
midpoint and slope formula, equation of a line,
geometric theorems algebraically. For example, definitions of parallel and perpendicular lines, etc.)
prove or disprove that a figure defined by four (Knowledge)
given points in the coordinate plane is a
rectangle; prove or disprove that the point (1,
Use coordinates to prove simple geometric
√3) lies on the circle centered at the origin and theorems algebraically. (Reasoning)
I can use coordinates to prove simple geometric
containing the point (0, 2).
theorems algebraically.
For example, prove or disprove that a figure
defined by four given points in the coordinate plane
is a rectangle; prove or disprove that the point (1,
√3) lies on the circle centered at the origin and
containing the point (0, 2).
From Appendix A: Include simple proofs involving
circles.
*Bolded word
Use Marzano 6 Step Process
Vocabulary
I Can
Recall congruence and similarity criteria for
triangles. (Knowledge)
Unit 5: Circles With and Without Coordinates
6
Congruence
Coordinate Geometry
Proof
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Deconstructed Standards
Informally use rigid motions to take angles to
I can informally use rigid motions to take angles to
angles and segments to segments (from 8th grade). angles and segments to segments (from 8th grade).
(Knowledge)
G.CO.8 Explain how the criteria for triangle
I can formally use dynamic geometry software or
congruence (ASA, SAS, and SSS) follow from the Formally use dynamic geometry software or
straightedge and compass to take angles to angles
definition of congruence in terms of rigid
straightedge and compass to take angles to angles and segments to segments.
motions.
and segments to segments. (Knowledge)
2,3
Explain how the criteria for triangle congruence
(ASA, SAS, SSS) follows from the definition of
congruence in terms of rigid motions (i.e. if two
angles and the included side of one triangle are
transformed by the same rigid motion(s) then the
triangle image will be congruent to the original
triangle). (Reasoning)
Unit 4: Connecting Algebra and Geometry
Through Coordinates
2,3
I can explain how the criteria for triangle
congruence (ASA, SAS, SSS) follows from the
definition of congruence in terms of rigid motions.
*Bolded word
Use Marzano 6 Step Process
Rigid Motion
ASA
SAS
AAS
SSS
HL
HA
LL
LA
I can reason from the basic properties of rigid
motions .
Recall previous understandings of coordinate
I can recall previous understandings of coordinate
geometry (including, but not limited to: distance,
geometry.
midpoint and slope formula, equation of a line,
G.GPE.4 Use coordinates to prove simple
definitions of parallel and perpendicular lines, etc.)
geometric theorems algebraically. For example, (Knowledge)
prove or disprove that a figure defined by four
given points in the coordinate plane is a
Use coordinates to prove simple geometric
rectangle; prove or disprove that the point (1,
theorems algebraically.
I can use coordinates to prove simple geometric
√3) lies on the circle centered at the origin and For example, prove or disprove that a figure
theorems algebraically.
containing the point (0, 2).
defined by four given points in the coordinate plane
is a rectangle; prove or disprove that the point (1,
√3) lies on the circle centered at the origin and
containing the point (0, 2). (Reasoning)
e.g., derive the equation of a line through 2 points
using similar right triangles. (Reasoning)
Vocabulary
I Can
Unit 1: Congruence, Proof, and
Constructions
From Appendix A: Rigid motions are at the
foundation of the definition of congruence.
Students reason from the basic properties of
rigid motions (that they preserve distance and
angle), which are assumed without proof. Rigid
motions and their assumed properties can be
used to establish the usual triangle congruence
criteria, which can then be used to prove other
theorems.
7
Distance Formula
Midpoint Formula
Slope Formula
Parallel Line
Perpendicular Line
* Theorem
Coordinate Geometry
* Proof
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Technology
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 5: Circles With and Without Coordinates
Deconstructed Standards
Define inscribed and circumscribed circles of a
triangle. (Knowledge)
G.C.3 Construct the inscribed and circumscribed
circles of a triangle,
Recall midpoint and bisector definitions.
and prove properties of angles for a
(Knowledge)
quadrilateral inscribed in a circle.
Define a point of concurrency. (Knowledge)
Prove properties of angles for a quadrilateral
inscribed in a circle. (Reasoning)
2,3
Construct inscribed circles of a triangle.
(Performance)
Construct circumscribed circles of a triangle.
(Performance)
Unit 1: Congruence, Proof, and
Constructions
3
Vocabulary
I can define inscribed and circumscribed circles of a
triangle.
I can recall midpoint and bisector definitions.
I can define a point of concurrency.
I can prove properties of angles for a quadrilateral
inscribed in a circle.
I can construct inscribed circles of a triangle.
I can construct circumscribed circles of a triangle.
I can describe the different types of transformations * Transformation
including translations, reflections, rotations and
* Translation
dilations.
* Rotation
G.CO.2 Represent transformations in the plane
* Reflection
using, e.g., transparencies and geometry
Describe transformations as functions that take
* Dilation
software; describe transformations as functions points in the coordinate plane as inputs and give
I can describe transformations as functions that use * Rigid Motion
that take points in the plane as inputs and give other points as outputs (Knowledge)
points in the coordinate plane as inputs and produce * Non-Rigid Motion
other points as outputs. Compare
points as outputs.
* Scale Factor
transformations that preserve distance and
Represent transformations in the plane using, e.g.,
* Composition of Transformations
angle to those that do not (e.g., translation
transparencies and geometry software.
* Vectors
versus horizontal stretch).
(Reasoning)
I can represent transformations in the plane using
technology.
Write functions to represent transformations.
(Reasoning)
I can write functions to represent transformations.
*Bolded word
Use Marzano 6 Step Process
Describe the different types of transformations
including translations, reflections, rotations and
dilations. (Knowledge)
I Can
8
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 1: Congruence, Proof, and
Constructions
Deconstructed Standards
Compare transformations that preserve distance
and angle to those that do not (e.g., translation
versus horizontal stretch)
I Can
9
Vocabulary
I can compare transformations that preserve
distance and angle to those that do not.
G.CO.2 Standard (continued)
From Appendix A: Build on student experience with I can build on my experiences with rigid motions
rigid motions from earlier grades. Point out the
from earlier grades and point out the basis of rigid
basis of rigid motions in geometric concepts, e.g,
motions in geometric concepts.
translations move points a specific distance along a
line parallel to a specified line; rotations move
objects along a circular arc with a specified center
through a specified angle.
3
Unit 1: Congruence, Proof, and
Constructions
Given a rectangle, parallelogram, trapezoid, or
regular polygon, describe the rotations and/or
reflections that carry it onto itself. (Knowledge)
G.CO.3 Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the
From Appendix A: Build on student experience with
rotations and reflections that carry it onto itself. rigid motions from earlier grades. Point out the
basis of rigid motions in geometric concepts, e.g,
translations move points a specific distance along a
line parallel to a specified line; rotations move
objects along a circular arc with a specified center
through a specified angle.
3
*Bolded word
Use Marzano 6 Step Process
I can describe the rotations and reflections that
carries a polygon, e.g. a rectangle, parallelogram,
trapezoid, or regular shape, onto itself.
I can point out the basis of rigid motions in
geometric concepts, e.g, translations move points a
specific distance along a line parallel to a specified
line; rotations move objects along a circular arc with
a specified center through a specified angle.
Regular Polygon
Rectangle
Trapezoid
Parallelogram
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 1: Congruence, Proof, and
Constructions
G.CO.4 Develop definitions of rotations,
reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and
line segments.
3
Unit 1: Congruence, Proof, and
Constructions
3
Deconstructed Standards
Recall definitions of angles, circles, perpendicular
and parallel lines and line segments. (Knowledge)
Vocabulary
* Transformation
* Translation
* Rotation
Develop definitions of rotations, reflections and
I can develop definitions of rotations, reflections and * Reflection
translations in terms of angles, circles,
translations in terms of angles, circles, perpendicular * Dilation
perpendicular lines, parallel lines and line
lines, parallel lines and line segments.
* Rigid Motion
segments. (Reasoning)
* Non-Rigid Motion
I can point out the basis of rigid motions in
* Scale Factor
From Appendix A: Build on student experience with geometric concepts, e.g., translations move points a * Composition of Transformations
rigid motions from earlier grades. Point out the
specific distance along a line parallel to a specified * Vectors
basis of rigid motions in geometric concepts, e.g., line; rotations move objects along a circular arc with
translations move points a specific distance along a a specified center through a specified angle.
line parallel to a specified line; rotations move
objects along a circular arc with a specified center
through a specified angle.
Given a geometric figure and a rotation, reflection
or translation, draw the transformed figure using,
e.g. graph paper, tracing paper or geometry
G.CO.5 Given a geometric figure and a rotation, software. (Knowledge)
reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or Draw a transformed figure and specify the
geometry software. Specify a sequence of
sequence of transformations that were used to
transformations that will carry a given figure
carry the given figure onto the other. (Reasoning)
onto another.
From Appendix A: Build on student experience with
rigid motions from earlier grades. Point out the
basis of rigid motions in geometric concepts, e.g.,
translations move points a specific distance along a
line parallel to a specified line; rotations move
objects along a circular arc with a specified center
through a specified angle.
*Bolded word
Use Marzano 6 Step Process
I Can
10
I can recall definitions of angles, circles,
perpendicular and parallel lines and line segments.
I can draw the transformed figure in multiple ways.
* Transformation
* Translation
* Rotation
* Reflection
* Dilation
I can draw a transformed figure and specify the
* Rigid Motion
sequence of transformations that were used to carry * Non-Rigid Motion
the given figure onto the other.
* Scale Factor
* Composition of Transformations
I can point out the basis of rigid motions in
* Vectors
geometric concepts.
Resources
Technology
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Deconstructed Standards
I Can
Unit 1: Congruence, Proof, and
Constructions
Use geometric descriptions of rigid motions to
transform figures.
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.
Predict the effect of a given rigid motion on a given I can predict the effect of a given rigid motion on a
figure.
given figure.
3
Unit 1: Congruence, Proof, and
Constructions
G.CO.6 Standard (continued)
3
*Bolded word
Use Marzano 6 Step Process
11
Vocabulary
I can use geometric descriptions of rigid motions to * Congruence
transform figures.
* Composition of Transformations
Define congruence in terms of rigid motions (i.e.
two figures are congruent if there exists a rigid
motion, or composition of rigid motions, that can
take one figure to the second).
I can define congruence in terms of rigid motions
Decide if two figures are congruent in terms of rigid
motions (it is not necessary to find the precise
transformation(s) that took one figure to a second,
only to understand that such a transformation or
composition exists).
I can decide if two figures are congruent in terms of
rigid motions (it is not necessary to find the precise
transformation(s) that took one figure to a second,
only to understand that such a transformation or
composition exists).
From Appendix A: Rigid motions are at the
foundation of the definition of congruence.
Students reason from the basic properties of rigid
motions (that they preserve distance and angle),
which are assumed without proof. Rigid motions
are their assumed properties can be used to
establish the usual triangle congruence criteria,
which can then be used to prove other theorems.
I can reason from the basic properties of rigid
motions, which can then be used to prove other
theorems.
Resources
Technology
Resources
Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Deconstructed Standards
I Can
12
Vocabulary
Unit 1: Congruence, Proof, and
Constructions
Identify corresponding angles and sides of two
triangles. (Knowledge)
I can identify corresponding angles and sides of two * Corresponding Parts
triangles.
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.
Identify corresponding pairs of angles and sides of
congruent triangles after rigid motions.
(Knowledge)
I can identify corresponding pairs of angles and sides
of congruent triangles after rigid motions.
Unit 1: Congruence, Proof, and
Constructions
From Appendix A: Rigid motions are at the
foundation of the definition of congruence.
Students reason from the basic properties of rigid
motions (that they preserve distance and angle),
which are assumed without proof. Rigid motions
and their assumed properties can be used to
establish the usual triangle congruence criteria,
which can then be used to prove other theorems.
3
G.CO.7 (Standard continue)
3
*Bolded word
Use Marzano 6 Step Process
I can use the definition of congruence in terms of
Use the definition of congruence in terms of rigid rigid motions to show that two triangles are
motions to show that two triangles are congruent if congruent if corresponding pairs of sides and
corresponding pairs of sides and corresponding
corresponding pairs of angles are congruent.
pairs of angles are congruent. (Reasoning)
I can use the definition of congruence in terms of
Use the definition of congruence in terms of rigid rigid motions to show that if the corresponding pairs
motions to show that if the corresponding pairs of of sides and corresponding pairs of angles of two
sides and corresponding pairs of angles of two
triangles are congruent then the two triangles are
triangles are congruent then the two triangles are congruent.
congruent. (Reasoning)
I can justify congruency of two triangles using
transformations.
I can reason from the basic properties of rigid
motions (that they preserve distance and angle),
which are assumed without proof. Rigid motions and
their assumed properties can be used to establish
the usual triangle congruence criteria, which can
then be used to prove other theorems.
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Geometry
Curriculum Mapping
8/2/2011
Quarter
3
Geometry Standards
Deconstructed Standards
Unit 2: Similarity, Proof, and Trigonometry
Define image, pre-image, scale factor, center, and
similar figures as they relate to transformations.
(Knowledge)
Unit 2: Similarity, Proof, and Trigonometry
Explain that the scale factor represents how many
times longer or shorter a dilated line segment is
than its pre-image. (Knowledge)
I Can
I can define image, pre-image, scale factor, center, * Dilation
and similar figures as they relate to transformations. Center
G.SRT.1 Verify experimentally the properties of
Scale Factor
dilations given by a
I can identify a dilation stating its scale factor and
center and a scale factor.
Identify a dilation stating its scale factor and center. center.
a. A dilation takes a line not passing through the (Knowledge)
center of the dilation
I can verify experimentally that a dilated image is
to a parallel line, and leaves a line passing
Verify experimentally that a dilated image is similar similar to its pre-image by showing congruent
through the center unchanged.
to its pre-image by showing congruent
corresponding angles and proportional sides.
b. The dilation of a line segment is longer or
corresponding angles and proportional sides.
shorter in the ratio given by the scale factor.
(Reasoning)
I can verify experimentally that a dilation takes a line
not passing through the enter of the dilation to a
Verify experimentally that a dilation takes a line not parallel line by showing that the lines are parallel.
passing through the center of the dilation to a
parallel line by showing the lines are parallel.
I can verify experimentally that dilation leaves a line
(Reasoning)
passing through the center of the dilation
unchanged by showing that it is the same line.
Verify experimentally that dilation leaves a line
passing through the center of the dilation
unchanged by showing that it is the same line.
(Reasoning)
G.SRT.1 Standard (continued)
Verify experimentally that the dilation of a line
segment is longer or shorter in the ratio given by
the scale factor. (Reasoning)
3
*Bolded word
Use Marzano 6 Step Process
I can explain that the scale factor represents how
many times longer or shorter a dilated line segment
is than its pre-image.
I can verify experimentally that the dilation of a line
segment is longer or shorter in the ratio given by the
scale factor.
13
Vocabulary
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 2: Similarity, Proof, and Trigonometry
3
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.
Unit 2: Similarity, Proof, and Trigonometry
G.SRT.3 Use the properties of similarity
transformations to establish the
AA criterion for two triangles to be similar.
3
Unit 2: Similarity, Proof, and Trigonometry
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.
3
*Bolded word
Use Marzano 6 Step Process
Deconstructed Standards
By using similarity transformations, explain that
triangles are similar if all pairs of corresponding
angles are congruent and all corresponding pairs of
sides are proportional. (Knowledge)
Given two figures, decide if they are similar by
using the definition of similarity in terms of
similarity transformations. (Reasoning)
I Can
14
Vocabulary
I can use similarity transformations to explain that * Similarity Transformation
triangles are similar if all pairs of corresponding
angles are congruent and all corresponding pairs of
sides are proportional.
I can decide if two figures are similar by using the
definition of similarity in term sof similarity
transformations.
Recall the properties of similarity transformations. I can recall the properties of similarity of
(Knowledge)
transformations.
* Similarity Transformation
Establish the AA criterion for similarity of triangles
by extending the properties of similarity
transformations to the general case of any two
similar triangles. (Reasoning)
I can establish the AA criterion for similarity of
triangles by extending the properties for similarity
transformations to the general case of any two
similar triangles.
Names the sides of right triangles as related to an
acute angle. (Knowledge)
Recognize that if two right triangles have a pair of
acute, congruent angles that the triangles are
similar. (Knowledge)
I can name the sides of right triangles as they relate Opposite Leg
to an acute angle.
Adjacent Leg
Hypotenuse
I recognize that if two right triangles have a pair of Trigonometric Ratios
acute, congruent angles, then the triangles are
similar.
Compare common ratios for similar right triangles
and develop a relationship between the ratio and
the acute angle leading to the trigonometry ratios.
(Reasoning)
I can compare common ratios for similar right
triangles and develop a relationship between the
ratio and the acute angle leading to the
trigonometric ratios.
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 2: Similarity, Proof, and Trigonometry
3
Deconstructed Standards
I Can
Use the relationship between the sine and cosine of I can use the relationship between the sine and
complementary angles. (Knowledge)
cosine of complementary angles.
G.SRT.7 Explain and use the relationship
between the sine and cosine of complementary Explain how the sine and cosine of complementary
angles.
angles are related to each other. (Reasoning)
I can explain how the sine and cosine of
complementary angles are related to each other.
Unit 2: Similarity, Proof, and Trigonometry
G.SRT.8 Use trigonometric ratios and the
Pythagorean Theorem to solve
right triangles in applied problems.★
Recognize which methods could be used to solve
right triangles in applied problems. (Knowledge)
I can recognize which methods could be used to
solve right triangles in applied problems.
Solve for an unknown angle or side of a right
triangle using sine, cosine, and tangent.
(Knowledge)
I can solve for an unknown angle or side of a right
triangle using sine, cosine, and tangent.
Apply right triangle trigonometric ratios and the
Pythagorean Theorem to solve right triangles in
applied problems. (Reasoning)
3
Describe a topographical grid system. (Knowledge) I can describe a topographical grid system.
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 topographic grid systems based on ratios).*
Apply geometric methods to solve design problems
(e.g., designing an object or structure to satisfy
I can apply geometric methods to solve design
physical constraints or minimize cost; working with problems.
topographic grid systems based on ratios).
(Reasoning)
*Bolded word
Use Marzano 6 Step Process
Vocabulary
Sine
Cosine
Complementary Angles
* Angle of Elevation
* Angle of Depression
Pythagorean Theorem
I can apply right triangle trigonometric ratios and
the Pythagorean Theorem to solve right triangles in
applied problems.
Unit 2: Similarity, Proof, and Trigonometry
3
15
From Appendix A: Focus on situations well modeled
by trigonometric ratios for acute angles.
I can identify situations well-modeled by
(Reasoning)
trigonometric ratios for acute angles.
* Angle of Elevation
* Angle of Depression
Constraints
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 2: Similarity, Proof, and Trigonometry
Deconstructed Standards
Recall right triangle trigonometry to solve
mathematical problems (Knowledge)
I Can
I can recall right triangle trigonometry to solve
mathematical problems.
16
Vocabulary
* Auxiliary Line
G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C)
for the area of a triangle by drawing an auxiliary Derive the formula A = 1/2 ab sin(C) for the area of I can derive the formula A = 1/2 ab sin(C) for the
line from a vertex perpendicular to the opposite a triangle by drawing an auxiliary line from a vertex area of a triangle by drawing an auxiliary line from
side.
perpendicular to the opposite side. (Reasoning)
the vertex perpendicular to the opposite side.
3
Unit 2: Similarity, Proof, and Trigonometry
Use the Laws of Sines and Cosines this to find
missing angles or side length measurements.
G.SRT.10 (+) Prove the Laws of Sines and Cosines (Knowledge)
and use them to solve
problems.
Prove the Law of Sines (Reasoning)
I can use the Law of Sines and Cosines to find
missing angles or side length measurements.
I can prove the Law of Sines.
I can prove the Law of Cosines.
Prove the Law of Cosines (Reasoning)
3
*Bolded word
Use Marzano 6 Step Process
I can recognize when the Law of Sines or Law of
Recognize when the Law of Sines or Law of Cosines Cosines can be applied to a problem.
can be applied to a problem and solve problems in
context using them. (Reasoning)
I can extend the general case of the Laws of Sines
and Cosines to obtuse angles.
From Appendix A: With respect to the general case
of Laws of Sines and Cosines, the definition of sine
and cosine must be extended to obtuse angles.
* Law of Sines
* Law of Cosines
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 2: Similarity, Proof, and Trigonometry
3
Deconstructed Standards
I Can
Vocabulary
Determine from given measurements in right and I can determine whether it is appropriate to use the * Law of Sines
non-right triangles whether it is appropriate to use Law of Sines or Cosines given the measurements in a * Law of Cosines
the Law of Sines or Cosines. (Knowledge)
right or non-right triangle.
G.SRT.11 (+) Understand and apply the Law of
Sines and the Law of
Cosines to find unknown measurements in right Apply the Law of Sines and the Law of Cosines to
I can apply the Law of Sines and Law of Cosines to
and non-right triangles
find unknown measurements in right and non-right find unknown measurements in right and non-right
(e.g., surveying problems, resultant forces).
triangles (e.g., surveying problems, resultant
triangles.
forces). (Reasoning)
I can extend the definition of sine and cosine to
From Appendix A: With respect to the general case obtuse angles with respect to the general case of
of the Laws of Sines and Cosines, the definition of the Laws of Sines and Cosines.
sine and cosine must be extended to obtuse angles.
Unit 4: Connecting Algebra and Geometry
Through Coordinates
Use the coordinates of the vertices of a polygon to I can use the coordinates of the vertices of a polygon
find the necessary dimensions for finding the
to find the necessary dimensions for finding the
perimeter (i.e., the distance between vertices).
perimeter.
G.GPE.7 Use coordinates to compute perimeters (Knowledge)
of polygons and areas
of triangles and rectangles, e.g., using the
Use the coordinates of the vertices of a triangle to I can use the coordinates of the vertices of a triangle
distance formula.★
find the necessary dimensions (base, height) for
or rectangle to find the necessary dimensions for
finding the area (i.e., the distance between vertices finding the area.
by counting, distance formula, Pythagorean
Theorem, etc.). (Knowledge)
3
17
From Appendix A: G.GPE.7 provides practice
with the distance formula and its connection
with the Pythagorean theorem
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). (Knowledge)
Formulate a model of figures in contextual
problems to compute area and/or perimeter.
(Reasoning)
.
*Bolded word
Use Marzano 6 Step Process
I can formulate a model of figures in contextual
problems to compute area and/or perimeter.
* Coordinate Geometry
Perimeter
Area Formulas
Vertices
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 2: Similarity, Proof, and Trigonometry
G.MG.1 Use geometric shapes, their measures,
and their properties to
Given a real world object, classify the object as a
describe objects (e.g., modeling a tree trunk or a known geometric shape – use this to solve
human torso as a
problems in context. (Reasoning)
cylinder).*
From Appendix A: Focus on situations well modeled
by trigonometric ratios for acute angles.
Vocabulary
I Can
Deconstructed Standards
Use measures and properties of geometric shapes
to describe real world objects. (Knowledge)
18
I can use measures and properties of geometric
shapes to describe real world objects.
* Oblique Polyhedron
I can classify real world objects as a known
geometric shape and use this to solve problems in
context.
I can identify situations well-modeled by
trigonometric ratios for acute angles.
4
Unit 2: Similarity, Proof, and Trigonometry
Define density. (Knowledge)
I can define density.
G.MG.2 Apply concepts of density based on area Apply concepts of density based on area and
I can apply concepts of density based on area and
and volume in
volume to model real-life situations (e.g., persons volume to model real-life situations.
modeling situations (e.g., persons per square
per square mile, BTUs per cubic foot). (Reasoning)
mile, BTUs per cubic
foot).*
4
*Bolded word
Use Marzano 6 Step Process
Density
Volume
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 3: Extending to Three Dimensions
4
Deconstructed Standards
Recognize cross-sections of solids as twodimensional shapes
I Can
I can decompose volume formulas into area
formulas using cross-sections.
19
Vocabulary
Volume
Circumference
G.GMD.1 Give an informal argument for the
Cylinder
formulas for the
Recognize formulas for area and circumference of a I can recognize cross sections of solids as twoPyramid
circumference of a circle, area of a circle,
circle and volume of a cylinder, pyramid, and cone. dimensional shapes.
Cone
volume of a cylinder, pyramid,
* Dissection
and cone. Use dissection arguments, Cavalieri’s Recognize Cavalieri's principle.
I can recognize formulas for area and circumference Cavalieri's Principle
principle, and informal
of a circle and volume of a cylinder, pyramid, and a * Limit
limit arguments.
Decompose volume formulas into area formulas
cone.
using cross-sections.
I can use the techniques of dissection and limit
Apply dissections and limit arguments (e.g.
arguments.
Archimedes' inscription and circumscription of
polygons about a circle and as a component of the I can apply Cavalier's Principle as a component of
informal argument for the formulas for the
the informal argument for the formulas for the
circumference and area of a circle.)
volume of a cylinder, pyramid, and a cone.
Apply Cavalieri's Principle as a component of the
I can make informal arguments for area and volume
informal argument for the formulas for the volume formulas and can make use of the way in which area
of a cylinder, pyramid, and cone.
and volume scale under similarity transformations.
Unit 3: Extending to Three Dimensions
G.GMD.1 (Standard continued)
4
*Bolded word
Use Marzano 6 Step Process
From Appendix A: 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 K^2
times the area of the first. Similarly, volumes of
solid figures scale by K^3 under a similarity
transformations with scale factor K.
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 3: Extending to Three Dimensions
G.GMD.3 Use volume formulas for cylinders,
pyramids, cones, and
spheres to solve problems
Deconstructed Standards
20
Vocabulary
I Can
Utilize the appropriate formula for volume
depending on the figure. (Knowledge)
I can utilize the appropriate formula for volume
depending on the figure.
Use volume formulas for cylinders, pyramids,
cones, and spheres to solve contextual problems.
(Reasoning)
I can use volume formulas for cylinders, pyramids,
cones, and spheres to solve contextual problems.
Sphere
Volume
Cylinders
Pyramids
Cones
I can make informal arguments for area and volume
From Appendix A: Informal arguments for area and formulas using the way area and volume scale under
volume formulas can make use of the way in which similarity transformations.
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.
4
Unit 3: Extending to Three Dimensions
Use strategies to help visualize relationships
between two-dimensional and three dimensional
G.GMD.4 Identify the shapes of two-dimensional objects. (Knowledge)
cross-sections of three-dimensional objects, and
identify three-dimensional objects generated
Relate the shapes of two-dimensional crossby rotations of two-dimensional objects.
sections to their three-dimensional objects.
(Reasoning)
4
I can identify the shapes of two-dimensional cross- * Cross-section
sections of three-dimensional objects, and identify
three-dimensional objects generated by rotations of
two-dimensional objects.
I can use strategies to help visualize relationships
between two-dimensional and three-dimensional
objects.
Discover three-dimensional objects generated by
rotations of two-dimensional objects. (Reasoning) I can relate the shapes of two-dimensional crosssections to their three-dimensional objects.
I can discover three-dimensional objects generated
by rotations of two-dimensional objects.
*Bolded word
Use Marzano 6 Step Process
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 3: Extending to Three Dimensions
4
Deconstructed Standards
Use measures and properties of geometric shapes
to describe real world objects. (Knowledge)
I Can
21
Vocabulary
I can use geometric shapes, their measures, and
their properties to describe objects.
G.MG.1 Use geometric shapes, their measures,
and their properties to
Given a real world object, classify the object as a
describe objects (e.g., modeling a tree trunk or a known geometric shape; use this to solve problems I can be given a real world object, classify the object
human torso as a cylinder).*
in context. (Reasoning)
as a known geometric shape and use this to solve
problems in context.
From Appendix A: Focus on situations that require
relating two- and three-dimensional objects,
determining and using volume, and the
trigonometry of general triangles.
Unit 4: Connecting Algebra and Geometry
Through Coordinates
Define a parabola including the relationship of the
focus and the equation of the directrix to the
parabolic shape. (Knowledge)
I can derive the equation of a parabola given a focus Parabola
and directix.
* Focus
* Directrix
I can define a parabola including the relationship of Axis of Symmetry
the focus and the equation of the directix to the
* Latus Rectum
parabolic shape.
From Appendix A: The directrix should be parallel
to a coordinate axis.
I can understand that the directix should be parallel
to a coordinate axis.
G.GPE.2 Derive the equation of a parabola given
a focus and directrix.
4
*Bolded word
Use Marzano 6 Step Process
Derive the equation of parabola given the focus and I can derive the equation of parabola given the focus
directrix. (Reasoning)
and directix.
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 5: Circles With and Without Coordinates
G.C.1 Prove that all circles are similar.
4
Deconstructed Standards
Recognize when figures are similar. (Two figures
are similar if one is the image of the other under a
transformation from the plane into itself that
multiplies all distances by the same positive scale
factor, k. That is to say, one figure is a dilation of
the other. )
(Knowledge)
I can recognize when figures are similar.
Compare the ratio of the circumference of a circle
to the diameter of the circle. (Reasoning)
I can compare the ratio of the circumference of a
circle to the diameter of the circle.
Discuss, develop and justify this ratio for several
circles. (Reasoning)
Determine that this ratio is constant for all circles.
(Reasoning)
Unit 5: Circles With and Without Coordinates
4
Identify inscribed angles, radii, chords, central
angles, circumscribed angles, diameter, tangent.
G.C.2 Identify and describe relationships among (Knowledge)
inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and
Recognize that inscribed angles on a diameter are
circumscribed angles; inscribed angles on a
right angles. (Knowledge)
diameter are right angles;
the radius of a circle is perpendicular to the
Recognize that radius of a circle is perpendicular to
tangent where the radius
the radius at the point of tangency. (Knowledge)
intersects the circle.
Examine the relationship between central,
inscribed and circumscribed angles by applying
theorems about their measures. (Reasoning)
*Bolded word
Use Marzano 6 Step Process
I Can
22
Vocabulary
Circle
Circumference
Diameter
Ratio
Pi
* Concentric Circles
I can discuss, develop and justify this ratio for
several circles.
I can determine that this ratio is constant for all
circles.
I can examine the relationship between central,
inscribed and circumscribed angles by applying
theorems about their measures.
I can identify inscribed angles, radii, chords, central
angles, circumscribed angles, diameter, tangent.
I can recognize that inscribed angles on a diameter
are right angles.
I can recognize that radius of a circle is
perpendicular to the radius at the point of tangency.
* Inscribed Angle
* Central Angle
Tangent
Circumscribed Angle
* Tangent Line
* Tangent Segment
* Secant Line
* Point of Tangency
* Chord
Diameter
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 5: Circles With and Without Coordinates
G.C.4 (+) Construct a tangent line from a point
outside a given circle to
the circle.
Deconstructed Standards
I Can
Recall vocabulary:
Tangent
Radius
Perpendicular bisector
Midpoint (Knowledge)
I can recall vocabulary: Tangent, Radius,
Perpendicular bisector, and Midpoint.
Identify the center of the circle (Knowledge)
I can identify the center of the circle.
23
Vocabulary
*Common Tangents
Tangent
Radius
Perpendicular Bisector
Midpoint
Center of a Circle
Point of Tangency
Synthesize theorems that apply to circles and
tangents, such as:
I can synthesize theorems that apply to circles and
*Tangents drawn from a common external point
tangents.
are congruent and
*A radius is perpendicular to a tangent at the point
of tangency. (Reasoning)
4
Construct the perpendicular bisector of the line
segment between the center C to the outside point I can construct the perpendicular bisector of the line
P. (Performance)
segment between the center C to the outside point
P.
Construct arcs on circle C from the midpoint Q,
having length of CQ. (Performance)
I can construct arcs on circle C from the midpoint Q,
having length of CQ.
Construct the tangent line. (Performance)
I can construct the tangent line.
Unit 5: Circles With and Without Coordinates
4
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 constant of
proportionality; derive
the formula for the area of a sector
*Bolded word
Use Marzano 6 Step Process
Recall how to find the area and circumference of a I can recall how to find the area and circumference
circle. (Knowledge)
of a circle.
Explain that 1° = Π/180 radians (Knowledge)
I can explain that 1° = Π/180 radians
Recall from G.C.1, that all circles are similar.
(Knowledge)
I can recall that all circles are similar.
Determine the constant of proportionality (scale
factor). (Knowledge)
I can determine the constant of proportionality
(scale factor).
* Major Arc
* Minor Arc
* Intercepted Arc
* Radian
Constant of Proportionality
Scale Factor
Central Angle
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Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 5: Circles With and Without Coordinates
G.C.5 (Standard continued)
Deconstructed Standards
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 (Reasoning)
Vocabulary
I can justify the radii of any two circles and the arc
lengths determined by congruent central angles are
proportional.
Verify that the constant of a proportion is the same
as the radian measure, Θ, of the given central
I can verify that the constant of a proportion is the
angle. Conclude s = r Θ (Reasoning)
same as the radian measure, Θ, of the given central
angle.
From Appendix A: 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.
4
Unit 5: Circles With and Without Coordinates
4
I Can
24
Define a circle. (Knowledge)
G.GPE.1 Derive the equation of a circle of given Use Pythagorean Theorem. (Knowledge)
center and radius using
the Pythagorean Theorem; complete the square Complete the square of a quadratic equation.
to find the center and
(Knowledge)
radius of a circle given by an equation.
Derive equation of a circle using the Pythagorean
From Appendix A: Emphasize the similarity of all Theorem – given coordinates of the center and
circles. Note that by similarity of sectors with
length of the radius. (Reasoning)
the same central angle, arc lengths are
proportional to the radius. Use this as a basis for Determine the center and radius by completing the
introducing radian as a unit of measure. It is not square. (Reasoning)
intended that it be applied to the development
of circular trigonometry in this course.
*Bolded word
Use Marzano 6 Step Process
I can define a circle.
I can use Pythagorean Theorem.
I can complete the square of a quadratic equation.
I can derive equation of a circle using the
Pythagorean Theorem – given coordinates of the
center and length of the radius.
I can determine the center and radius by completing
the square.
Completing the Square
Pythagorean Theorem
Quadratic Equation
Equation of a Circle
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Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 5: Circles With and Without Coordinates
Deconstructed Standards
Use measures and properties of geometric shapes
to describe real world objects.
(Knowledge)
I Can
I can use measures and properties of geometric
shapes to describe real world objects.
G.MG.1 Use geometric shapes, their measures,
and their properties to
Given a real world object, I can classify the object as
describe objects (e.g., modeling a tree trunk or a Given a real world object, classify the object as a
a known geometric shape - use this to solve
human torso as a
known geometric shape - use this to solve problems problems in context.
cylinder).*
in context. (Reasoning)
4
From Appendix A: Focus on situations in which the
analysis of circles is required. (Reasoning)
Unit 6: Applications of Probability
4
S.CP.1 Describe events as subsets of a sample
space (the set of
outcomes) using characteristics (or categories)
of the outcomes, or
as unions, intersections, or complements of
other events (“or,” “and,”
“not”).
Unit 6: Applications of Probability
4
S.CP.2 Understand that two events A and B are
independent if the
probability of A and B occurring together is the
product of their
probabilities, and use this characterization to
determine if they are independent.
*Bolded word
Use Marzano 6 Step Process
Define unions, intersections and complements of
events. (Knowledge)
Describe events as subsets of a sample space (the
set of outcomes) using characteristics (or
categories) of the outcomes, or as unions,
intersections, or complements of other events
(“or”, “and”, “not”). (Reasoning)
Categorize events as independent or not using the
characterization that two events A and B are
independent when the probability of A and B
occurring together is the product of their
probabilities. (Knowledge)
From Appendix A: Build on work from 2-way tables
from Algebra 1 Unit 3 (S.ID.5) to develop
understanding of conditional probability and
independence. (Knowledge)
I can define unions, intersections and complements Union
of events.
Intersection
Complement
I can describe events as subsets of a sample space Sample Space
using characteristics of the outcomes, or as unions,
intersections, or complements of other events.
I can categorize events as independent or not using Independent
the characterization that two events A and B are
Dependent
independent when the probability of A and B
occurring together is the product of their
probabilities.
25
Vocabulary
Resources
Technology
Resources
Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 6: Applications of Probability
4
S.CP.3 Understand the conditional probability of
A given B as P(A
and B)/P(B), and interpret independence of A
and B as saying that
the conditional probability of A given B is the
same as the probability
of A, and the conditional probability of B given A
is the same as the
probability of B.
Unit 6: Applications of Probability
4
S.CP.4 Construct and interpret two-way
frequency tables of data when
two categories are associated with each object
being classified. Use the two-way table as a
sample space to decide if events are
independent and to approximate conditional
probabilities. For example, collect data from a
random sample of students in your school on
their favorite subject among math, science, and
English. Estimate the probability that a randomly
selected student from your school will favor
science given
that the student is in tenth grade. Do the same
for other subjects and compare the results.
*Bolded word
Use Marzano 6 Step Process
Deconstructed Standards
I Can
Know the conditional probability of A given B as P(A I know the conditional probability of A given B as
and B)/P(B) (Knowledge)
P(A and B)/P(B)
Interpret independence of A and B as saying that
the conditional probability of A given B is the same
as the probability of A, and the conditional
probability of B given A is the same as the
probability of B. (Knowledge)
Vocabulary
Conditional Probability
I can interpret independence of A and B as saying
that the conditional probability of A given B is the
same as the probability of A, and the conditional
probability of B given A is the same as the
probability of B.
Use the two-way table as a sample space to decide I can use the two-way table as a sample space to
if events are independent and to approximate
decide if events are independent and to
conditional probabilities.
approximate conditional probabilities.
(Knowledge)
I can interpret two-way frequency tables of data
From Appendix A: Build on work with two-way
when two categories are associated with each
tables from Algebra 1 Unit 3 (S.ID.5) to develop
object being classified.
understanding of conditional probability and
independence.
Interpret two-way frequency tables of data when
two categories are associated with each object
being classified. (For example, collect data from a
random sample of students in your school on their
favorite subject among math, science, and English.
Estimate the probability that a randomly selected
student from your school will favor science given
that the student is in 10th grade. Do the same for
other subjects and compare the results.)
(Reasoning)
26
Sample Space
Two-Way Frequency Table
Resources
Technology
Resources
Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 6: Applications of Probability
4
S.CP.5 Recognize and explain the concepts of
conditional probability
and independence in everyday language and
everyday situations. For
example, compare the chance of having lung
cancer if you are a smoker
with the chance of being a smoker if you have
lung cancer.
Deconstructed Standards
Recognize the concepts of conditional probability
and independence in everyday language and
everyday situations.
(Knowledge)
I Can
27
Vocabulary
I can recognize the concepts of conditional
Theoretical Probability
probability and independence in everyday language Conditional Probability
and everyday situations.
Experimental Probability
Explain the concepts of conditional probability and I can explain the concepts of conditional probability
independence in everyday language and everyday and independence in everyday language and
situations. (For example, compare the chance of
everyday situations.
having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.)
(Reasoning)
Unit 6: Applications of Probability
4
Find the conditional probability of A given B as the I can find the conditional probability of A given B as Conditional Probability
fraction of B’s outcomes that also belong to A.
the fraction of B’s outcomes that also belong to A. Compound Event
S.CP.6 Find the conditional probability of A given (Knowledge)
B as the fraction of B’s
I can interpret the answer in terms of the model.
outcomes that also belong to A, and interpret
Interpret the answer in terms of the model.
the answer in terms of the
(Reasoning)
model.
Unit 6: Applications of Probability
Use the Additional Rule, P(A or B) = P(A) + P(B) –
P(A and B) (Knowledge)
S.CP.7 Apply the Addition Rule, P(A or B) = P(A) +
P(B) – P(A and B),
Interpret the answer in terms of the model.
and interpret the answer in terms of the model. (Reasoning)
4
*Bolded word
Use Marzano 6 Step Process
I can use the Additional Rule, P(A or B) = P(A) + P(B) Addition Rule of Probability
– P(A and B)
I can interpret the answer in terms of the model.
Resources
Technology
Resources
Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 6: Applications of Probability
4
S.CP.8 (+) Apply the general Multiplication Rule
in a uniform probability
model, P(A and B) = P(A)P(B|A) = P(B)P(A|B),
and interpret the answer
in terms of the model.
Unit 6: Applications of Probability
S.CP.9 (+) Use permutations and combinations
to compute probabilities
of compound events and solve problems.
4
Unit 6: Applications of Probability
S.MD.6 (+) Use probabilities to make fair
decisions (e.g., drawing by lots,
using a random number generator).
4
*Bolded word
Use Marzano 6 Step Process
Deconstructed Standards
I Can
Use the multiplication rule with correct notation.
(Knowledge)
I can use the multiplication rule with correct
notation.
Apply the general Multiplication Rule in a uniform
probability model P(A and B) = P(A)P(B|A) =
P(B)P(A|B). (Reasoning)
I can apply the general Multiplication Rule in a
uniform probability model P(A and B) = P(A)P(B|A) =
P(B)P(A|B).
Interpret the answer in terms of the model.
(Reasoning)
I can interpret the answer in terms of the model.
Identify situations that are permutations and those I can identify situations that are permutations and
that are combinations.
those that are combinations.
(Knowledge)
Use permutations and combinations to compute
probabilities of compound events and solve
problems. (Reasoning)
I can use permutations and combinations to
compute probabilities of compound events and
solve problems.
Compute Theoretical and Experimental
Probabilities. (Knowledge)
I can compute theoretical and experimental
probabilities.
Use probabilities to make fair decisions (e.g.
drawing by lots, using a random number
generator.) (Reasoning)
I can use probabilities to make fair decisions.
From Appendix A: This unit sets the stage for work
in Algebra II, where the ideas of statistical inference
are introduced. Evaluating the risks associated with
conclusions drawn from sample data (i.e.
incomplete information) requires an understanding
of probability concepts.
28
Vocabulary
Multiplication Rule of Probability
Permutations
Combinations
Compound Event
Theoretical Probability
Experimental Probability
Resources
Technology
Resources
Assessments
Geometry
Curriculum Mapping
8/2/2011
Quarter
Geometry Standards
Unit 6: Applications of Probability
S.MD.7 (+) Analyze decisions and strategies
using probability concepts
(e.g., product testing, medical testing, pulling a
hockey goalie at the end
of a game).
4
*Bolded word
Use Marzano 6 Step Process
Deconstructed Standards
Recall prior understandings of probability.
(Knowledge)
Analyze decisions and strategies using probability
concepts (e.g., product testing, medical testing,
pulling a hockey goalie at the end of a game.)
(Reasoning)
From Appendix A: This unit sets the stage for work
in Algebra II, where the ideas of statistical inference
are introduced. Evaluating the risks associated with
conclusions drawn from sample data (i.e.
incomplete information) requires an understanding
of probability concepts.
I Can
I can recall prior understandings of probability.
I can analyze decisions and strategies using
probability concepts.
29
Vocabulary
Resources
Technology
Resources
Assessments