Download Geometry Curriculum - Poudre School District

Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Geometry Standards Aligned With the Geometry PARCC Assessment
Performance Based Assessment (PBA/MYA) and End Of Year (EOY)
Cluster
Standard
PBA/MYA
Unit 1: Congruence, Proof, and Constructions
HS.G.CO.A
HS.G.CO.A.1
X
Experiment with transformations in the plane.
HS.G.CO.A.2
X
HS.G.CO.B
Understand congruence in terms of rigid motions.
HS.G.CO.C
Prove Geometric theorems.
HS.G.CO.D
Make geometric constructions.
HS.G.CO.A.3
X
HS.G.CO.B.6
X
HS.G.CO.A.4
HS.G.CO.A.5
HS.G.CO.B.7
HS.G.CO.B.8
HS.G.CO.C.9
HS.G.CO.C.10
HS.G.CO.C.11
HS.G.CO.D.12
HS.G.CO.D.13
Unit 2: Similarity, Proof, and Trigonometry
HS.G.SRT.A
Understand similarity in terms of similarity
transformations.
HS.G.SRT.B
Prove theorems involving similarity.
HS.G.SRT.C
Define trigonometric ratios and solve problems
involving right triangles.
HS.G.SRT.D
Apply trigonometry to general triangles.
X
X
X
X
HS.G.SRT.C.7
HS.G.SRT.C.8
HS.G.SRT.D.9
HS.G.SRT.D.10
X
X
X
HS.G.SRT.C.6
X
X
X
HS.G.SRT.B.4
HS.G.SRT.B.5
X
X
X
HS.G.SRT.A.3
X
X
HS.G.SRT.A.1
HS.G.SRT.A.2
EOY
X
X
X
X
X
X
X
X
X
X
X
X
X
HS.G.SRT.D.11
Page | 1
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Geometry Standards Aligned With the Geometry PARCC Assessment
Performance Based Assessment (PBA/MYA) and End Of Year (EOY)
Cluster
Standard
PBA/MYA
Unit 3: Connecting Algebra and Geometry through Coordinates
HS.G.GPE.B
HS.G.GPE.B.4
X
Use coordinates to prove simple geometric theorems.
HS.G.GPE.B.5
X
HS.G.GPE.B.6
HS.G.GPE.B.7
HS.G.GPE.A
Translate between the geometric description and the
HS.G.GPE.A.2
equation for a conic section.
Unit 4: Circles With and Without Coordinates
HS.G.MG.A
HS.G.MG.A.1
Apply geometric concepts in modeling situations.
HS.G.C.A
HS.G.C.A.1
Understand and apply theorems about circles.
HS.G.C.A.2
HS.G.C.A.3
HS.G.C.A.4
HS.G.C.B
HS.G.C.B.5
Find arc lengths and areas of sectors of circles.
HS.G.GPE.A
Translate between the geometric description and the
HS.G.GPE.A.1
equation for a conic section.
HS.G.GPE.B
Use coordinates to prove simple geometric theorems
HS.G.GPE.B.4
algebraically.
Unit 5: Extending to Three Dimensions
HS.G.MG.A
HS.G.MG.A.1
Apply geometric concepts in modeling situations.
HS.G.MG.A.2
HS.G.GMD.A
Explain volume formulas and use them to solve
problems.
HS.G.GMD.B
Visualize the relation between two-dimensional and
three-dimensional objects.
HS.G.MG.A.3
HS.G.GMD.A.1
HS.G.GMD.A.3
HS.G.GMD.B.4
X
X
EOY
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Page | 2
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Geometry Standards Aligned With the Geometry PARCC Assessment
Performance Based Assessment (PBA/MYA) and End Of Year (EOY)
Cluster
Standard
Unit 6: Applications of Probability
HS.S.CP.A
HS.S.CP.A.1
Understand independence and conditional probability
HS.S.CP.A.2
and use them to interpret data.
HS.S.CP.A.3
HS.S.CP.B
Use the rules of probability to compute probabilities of
compound events in a uniform probability.
HS.S.MD.B.6
Use probability to evaluate outcomes of decisions.
PBA/MYA
EOY
HS.S.CP.A.4
HS.S.CP.A.5
HS.S.CP.B.6
HS.S.CP.B.7
HS.S.CP.B.8
HS.S.CP.B.9
HS.S.MD.B.6
HS.S.MD.B.7
Page | 3
Curriculum Guide 2014-2015
High School Geometry
Course Overview
Poudre School District
The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric
experiences from the middle grades. Students explore more complex geometric situations and
deepen their explanations of geometric relationships, moving towards formal mathematical
arguments. Important differences exist between this Geometry course and the historical approach
taken in Geometry classes. For example, transformations are emphasized early in this course. Close
attention should be paid to the introductory content for the Geometry conceptual category found in
the high school CCSS. The Mathematical Practice Standards apply throughout each course and,
together with the content standards, prescribe that students experience mathematics as a coherent,
useful, and logical subject that makes use of their ability to make sense of problem situations. The
critical areas, organized into six units are as follows.
Unit 1: Congruence, Proof, and Constructions
In previous grades, students were asked to draw triangles based on given measurements. They also
have prior experience with rigid motion: translations, reflections, and rotations and have used
these to develop notions about what it means for two objects to be congruent. In this unit, students
establish triangle congruence criteria, based on analyses of rigid motions and formal constructions.
They use triangle congruence as a familiar foundation for the development of formal proof.
Students prove theorem – using a variety of formats – and solve problems about triangles,
quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and
explain why they work.
Students will be able to…
• Experiment with transformations in the plane.
HS.G.CO.A.1, HS.G.CO.A.2, HS.G.CO.A.3, HS.G.CO.A.4, HS.G.CO.A.5
• Understand congruence in terms of rigid motions.
HS.G.CO.B.6, HS.G.CO.B.7, HS.G.CO.B.8
• Prove geometric theorems.
HS.G.CO.C.9, HS.G.CO.C.10, HS.G.CO.C.11
• Make geometric constructions.
HS.G.CO.D.12, HS.G.CO.D.13
Page | 4
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 2: Similarity, Proof, and Trigonometry
Students apply their earlier experience with dilations and proportional reasoning to build a formal
understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve
problems, and apply similarity in right triangles to understand right triangle trigonometry, with
particular attention to special right triangles and the Pythagorean Theorem. Students develop the
Laws of Sines and Cosines in order to find missing measures of general (not necessarily right)
triangles. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2,
or infinitely many triangles.
Students will be able to…
• Understand similarity in terms of similarity transformations.
HS.G.SRT.A.1, HS.G.SRT.A.2, HS.G.SRT.A.3
• Prove theorems involving similarity.
HS.G.SRT.B.4, HS.G.SRT.B.5
• Define trigonometric ratios and solve problems involving right triangles.
HS.G.SRT.C.6, HS.G.SRT.C7, HS.G.SRT.C.8
• Apply trigonometry to general triangles.
HS.G.SRT.D.9, HS.G.SRT.D.10, HS.G.SRT.D.11
Unit 3: Connecting Algebra and Geometry through Coordinates
Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a
rectangular coordinate system to verify geometric relationships, including properties of special
triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students continue their
study of quadratics by connecting the geometric and algebraic definitions of the parabola.
Students will be able to…
• Use coordinates to prove simple geometric theorems.
HS.G.GPE.B.4, HS.G.GPE.B.5, HS.G.GPE.B.6, HS.G.GPE.B.7
• Translate between the geometric description and the equation for a conic section.
HS.G.GPE.A.2
Page | 5
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 4: Circles With and Without Coordinates
In this unit, students prove basic theorems about circles, with particular attention to perpendicular
and inscribed angles, in order to see symmetry in circles and as an application of triangle
congruence criteria. They study relationships among segments on chords, secants, and tangents as
an application of similarity. In the Cartesian coordinate system, students use the distance formula
to write the equation of a circle when given the radius and the coordinates of its center. Given an
equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving
quadratic equations to determine intersections between lines and circles or parabolas and between
two circles.
Students will be able to…
• Apply geometric concepts in modeling situations.
HS.G.MG.A.1
• Understand and apply theorems about circles.
HS.G.C.A.1, HS.G.C.A.2, HS.G.C.A.3, HS.G.C.A.4
• Find arc lengths and areas of sectors of circles.
HS.G.C.B.5
• Translate between the geometric description and the equation for a conic section.
HS.G.GPE.A.1
• Use coordinates to prove simple geometric theorems algebraically.
HS.G.GPE.B.4
Unit 5: Extending to Three Dimensions
Students’ experience with two-dimensional and three-dimensional objects is extended to include
informal explanations of circumference, area and volume formulas. Additionally, students apply
their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result
of rotating a two-dimensional object about a line.
Students will be able to…
• Apply geometric concepts in modeling situations.
HS.G.MG.A.1, HS.G.MG.A.2, HS.G.MG.A.3
• Explain volume formulas and use them to solve problems.
HS.G.GMD.A.1, HS.G.GMD.A.3
• Visualize the relation between two-dimensional and three-dimensional objects.
HS.G.GMD.B.4
Page | 6
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 6: Applications of Probability
Building on probability concepts that began in the middle grades, students use the languages of set
theory to expand their ability to compute and interpret theoretical and experimental probabilities
for compound events, attending to mutually exclusive events, independent events, and conditional
probability. Students should make use of geometric probability models wherever possible. They use
probability to make informed decisions.
Students will be able to…
• Understand independence and conditional probability and use them to interpret data.
HS.S.CP.A.1, HS.S.CP.A.2, HS.S.CP.A.3, HS.S.CP.A.4, HS.S.CP.A.5
• Use the rules of probability to compute probabilities of compound events in a uniform
probability.
HS.S.CP.B.6, HS.S.CP.B.7, HS.S.CP.B.8, HS.S.CP.B.9
• Use probability to evaluate outcomes of decisions.
HS.S.MD.B.6, HS.S.MD.B.7
Page | 7
Curriculum Guide 2014-2015
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Poudre School District
In previous grades, students were asked to draw triangles based on given measurements. They also
have prior experience with rigid motion: translations, reflections, and rotations and have used
these to develop notions about what it means for two objects to be congruent. In this unit, students
establish triangle congruence criteria, based on analyses of rigid motions and formal constructions.
They use triangle congruence as a familiar foundation for the development of formal proof.
Students prove theorem – using a variety of formats – and solve problems about triangles,
quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and
explain why they work.
Sub-Unit A:
Transformations
HS.G.CO.A.1
HS.G.CO.D.12
HS.G.CO.D.13
HS.G.CO.A.4
HS.G.CO.A.2
HS.G.CO.B.6
HS.G.CO.A.5
HS.G.CO.A.3
Sub-Unit B:
Congruence
HS.G.CO.B.7
HS.G.CO.C.9
HS.G.CO.B.8
HS.G.CO.C.11
HS.G.CO.C.10
Page | 8
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.G.CO.A: Experiment with transformations in the plane.
Build on student experience with rigid motion from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations
move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center
through a specified angle.
HS.G.CO.A.1
HS.G.CO.A.1
Lessons
Know precise definitions of angle, circle,
HS.G.CO.A.1
perpendicular line, parallel line, and line
Prentice Hall
segment, based on the undefined notions of
• PH G
point, line, distance along a line, and distance
o Ch 1.3-1.6
around a circular arc.
o Ch 3.3
o Ch 10.6
HS.G.CO.A.2
HS.G.CO.A.2
Represent transformations in the plane using,
Students may use geometry software and/or
HS.G.CO.A.2/HS.G.CO.A.4
e.g., transparencies and geometry software;
manipulatives to model and compare transformations. Prentice Hall
describe transformations as functions that take
• PH G
points in the plane as inputs and give other
o Ch 9.1-9.3
points as outputs. Compare transformations that
o p. 477
preserve distance and angle to those that do not
o p. 483
(e.g., translation versus horizontal stretch).
o p. 490
o Ch 9.5-9.6
HS.G.CO.A.3
HS.G.CO.A.3
Given a rectangle, parallelogram, trapezoid, or
Students may use geometry software and/or
HS.G.CO.A.2
regular polygon, describe the rotations and
manipulatives to model transformations.
Howard County
reflections that carry it onto itself.
• Experiment with Transformations
• Similarity and Congruence
HS.G.CO.A.4
HS.G.CO.A.4
Develop definitions of rotations, reflections, and
Students may use geometry software and/or
HS.G.CO.A.3
translations in terms of angles, circles,
manipulatives to model transformations. Students may Prentice Hall
perpendicular lines, parallel lines, and line
observe patterns and develop definitions of rotations,
• PH G
segments.
reflections, and translations.
o Ch 9.4
Page | 9
Curriculum Guide 2014-2015
Common Core State Standards
HS.G.CO.A.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.
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
HS.G.CO.A.5
Students may use geometry software and/or
manipulatives to model transformations and
demonstrate a sequence of transformations that will
carry a given figure onto another.
Poudre School District
Resources
Lessons (continued)
HS.G.CO.A.3 (continued)
Howard County
• Carrying a Figure onto Itself
HS.G.CO.A.4
Howard County
• Defining Transformations that Create
Congruent Figures
HS.G.CO.A.5
Prentice Hall
• PH G
o Ch 9.6
o p. 514
Tasks
HS.G.CO.A.1/HS.G.CO.4
Inside Mathematics
• Once Upon a Time
HS.G.CO.A.2/HS.G.CO.A.3/HS.G.CO.A.4
Inside Mathematics
• Between the Lines
HS.G.CO.A.4/HS.G.CO.A.5
Inside Mathematics
• The Shape of Things
Page | 10
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.CO.A.5
Howard County
• Transformation Sensation
Illustrative Mathematics
• Reflected Triangles
• Regular Tessellations of the Plane
Activities
HS.G.CO.A.1
NCTM Illuminations
• Shape Cutter
HS.G.CO.A.2
Howard County
• Patty Paper Tutorial Transformations
that Preserve Distance
NCTM Illuminations
• Mirror Tool
HS.G.CO.A.5
NCTM Illuminations
• Algebraic Transformations
Page | 11
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
Poudre School District
Resources
Practice
HS.G.CO.A.1
Prentice Hall
• PH G
o p. 21 #55-60, 65-67 (C)
o p. 571 #59 (A)
Assessments
HS.G.CO.D: Make geometric constructions.
Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the
desired objects.
Some of these constructions are closely related to previous standards and can be introduced in conjunction with them.
HS.G.CO.D.12
HS.G.CO.D.12
Lessons
Make formal geometric constructions with a
Students may use geometric software to make
HS.G.CO.D.12
variety of tools and methods (compass and
geometric constructions.
Prentice Hall
straightedge, string, reflective devices, paper
• PH G
• Construct a triangle given the lengths of two sides
folding, dynamic geometric software, etc.).
o Ch 1.7
and the measure of the angle between the two
Copying a segment; copying an angle; bisecting a
o p. 51
sides.
segment; bisecting an angle; constructing
o p. 303
Construct the circumcenter of a given triangle.
perpendicular lines, including the perpendicular
o Ch 3.8
bisector of a line segment; and constructing a line
o p. 319-320
parallel to a given line through a point not on the
o p. 328
line.
o Ch 6.3-6.4
HS.G.CO.D.13
Construct an equilateral triangle, a square, and a
regular hexagon inscribed in a circle.
HS.G.CO.D.13
Students may use geometric software to make
geometric constructions.
Page | 12
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
Poudre School District
Resources
Lessons (continued)
HS.G.CO.D.12/HS.G.CO.D.13
Howard County
• Investigating Properties of
Parallelograms
MARS Shell Center
• Inscribing and Circumscribing Right
Triangles
HS.G.CO.D.13
Prentice Hall
• PH G
o Ch 6.2
Tasks
HS.G.CO.D.12
Howard County
• First to Finish
• Safe Crossings
Illustrative Mathematics
• Angle Bisection and Midpoints of Line
Segments
• Bisecting an Angle
• Construction of Perpendicular Bisector
• Locating Warehouse
• Reflected Triangles
Page | 13
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.CO.D.12 (continued)
Inside Mathematics
• Once Upon a Time
• Polly Gone
• The Shape of Things
• What's Your Angle?
HS.G.CO.D.12/HS.G.CO.D.13
Illustrative Mathematics
• Placing a Fire Hydrant
Inside Mathematics
• Circular Reasoning
Activities
HS.G.CO.D.12
Howard County
• Compass Tutorial Copying a Segment
or Angle
• Compass Tutorial Constructing Parallel
and Perpendicular Lines
• Patty Paper Construction of Parallel
and Perpendicular Lines Tutorial
• Patty Paper Tutorial Copying a
Segment or Angle
• Sketchpad Constructions of Parallel
and Perpendicular Lines Tutorial
Page | 14
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
Poudre School District
Resources
Activities (continued)
HS.G.CO.D.12/HS.G.CO.D.13
Howard County
• Patty Paper Tutorial Angles in
Parallelogram
• Patty Paper Tutorial Diagonals of
Rectangles and Parallelograms
• Sketchpad Tutorial Parallelogram
Theorems
Practice
Assessments
HS.G.CO.B: Understand congruence in terms of rigid motions.
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 be used to prove other theorems.
HS.G.CO.B.6
HS.G.CO.B.6
Lessons
Use geometric descriptions of rigid motions to
A rigid motion is a transformation of points in space
HS.G.CO.B.6
transform figures and to predict the effect of a
consisting of a sequence of one or more translations,
Prentice Hall
given rigid motion on a given figure; given two
reflections, and/or rotations. Rigid motions are
• PH G
figures, use the definition of congruence in terms assumed to preserve distances and angle measures.
o p. 504
of rigid motions to decide if they are congruent.
o p. 513-514
Students may use geometric software to explore the
o p. 517
effects of rigid motion on a figure(s).
o p. 522
Howard County
• Congruent Figures
Page | 15
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit A: Transformations
Start of Quarter 1 – October 3, 2014
Explanations/Examples
Poudre School District
Resources
Tasks
HS.G.CO.B.6
Illustrative Mathematics
• Building a Tile Pattern by Reflecting
Hexagons
• Building a Tile Pattern by Reflecting
Octagons
• Reflections and Equilateral Triangles I
• Reflections and Equilateral Triangles II
• Symmetries of a Circle
Activities
Practice
Assessments
Page | 16
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 1: Congruence, Proof, and Constructions
Sub-Unit B: Congruence
October 6, 2014 – December 6, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.G.CO.B: Understand congruence in terms of rigid motions.
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.
HS.G.CO.B.7
HS.G.CO.B.7
Lessons
Use the definition of congruence in terms of rigid A rigid motion is a transformation of points in space
HS.G.CO.B.7
motions to show that two triangles are congruent consisting of a sequence of one or more translations,
MARS Shell Center
if and only if corresponding pairs of sides and
reflections, and/or rotations. Rigid motions are
• Analyzing Congruence Proofs
corresponding pairs of angles are congruent.
assumed to preserve distances and angle measures.
Tasks
Congruence of triangles
HS.G.CO.B.7
Two triangles are said to be congruent if one can be
Illustrative Mathematics
exactly superimposed on the other by a rigid motion,
• Are the Triangles Congruent?
and the congruence theorems specify the conditions
under which this can occur.
Inside Mathematics
HS.G.CO.B.8
HS.G.CO.B.8
• Between the Lines
Explain how the criteria for triangle congruence
(ASA, SAS, and SSS) follow from the definition of
HS.G.CO.B.8
congruence in terms of rigid motions.
Illustrative Mathematics
• Reflections and Isosceles Triangles
• When Does SSA Work to Determine
Triangle Congruence?
• Why Does ASA Work?
• Why Does SAS Work?
• Why Does SSS Work?
Inside Mathematics
• The Shape of Things
Page | 17
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit B: Congruence
October 6, 2014 – December 6, 2014
Explanations/Examples
Poudre School District
Resources
Activities
HS.G.CO.B.7
NCTM Illuminations
• Congruence Theorems
Practice
Assessments
HS.G.CO.C: Prove geometric theorems.
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. Implementation of F.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as
preparation for G.C.3 in Unit 4.
HS.G.CO.C.9
HS.G.CO.C.9
Lessons
Prove theorems about lines and angles. Theorems Students may use geometric simulations (computer
HS.G.CO.C.9/HS.G.CO.C.10
include: vertical angles are congruent; when a
software or graphing calculator) to explore theorems
Prentice Hall
transversal crosses parallel lines, alternate
about lines and angles.
• PH G
interior angles are congruent and corresponding
o Ch 3.1-3.4
angles are congruent; points on a perpendicular
o Ch 5.1-5.3
bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
HS.G.CO.C.9
Prentice Hall
HS.G.CO.C.10
HS.G.CO.C.10
• PH G
Prove theorems about triangles. Theorems
Students may use geometric simulations (computer
o Ch 2.5
software or graphing calculator) to explore theorems
include: measures of interior angles of a triangle
o p. 126
about triangles.
sum to 180°; base angles of isosceles triangles are
o p. 145
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.
Page | 18
Curriculum Guide 2014-2015
Common Core State Standards
HS.G.CO.C.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.
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit B: Congruence
October 6, 2014 – December 6, 2014
Explanations/Examples
HS.G.CO.C.11
Students may use geometric simulations (computer
software or graphing calculator) to explore theorems
about parallelograms.
Poudre School District
Resources
Lessons (continued)
HS.G.CO.C.10
Prentice Hall
• PH G
o Ch 4.5
o p. 397
o Ch 7.5
HS.G.CO.C.11
Prentice Hall
• PH G
o p. 271
o Ch 6.2
o Ch 6.4
Tasks
HS.G.CO.C.9/HS.G.CO.C.10
Inside Mathematics
• Between the Lines
HS.G.CO.C.9
Illustrative Mathematics
• Points Equidistance from Two Points
in a Plane
• Tangent Lines and the Radius of a
Circle
Page | 19
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit B: Congruence
October 6, 2014 – December 6, 2014
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.CO.C.10/HS.G.CO.C.11
Inside Mathematics
• The Shape of Things
HS.G.CO.C.10
Illustrative Mathematics
• Finding the Area of an Equilateral
Triangle
• Seven Circles I
HS.G.CO.C.11
Illustrative Mathematics
• Congruence of Parallelograms
• Is This a Parallelogram?
• Midpoints of the Sides of a
Parallelogram
• Quadrilaterals
Activities
HS.G.CO.C.9
Howard County
• Patty Paper Tutorial
AlA/Corresponding Angles
• Patty Paper Tutorial Vertical Angles
Proof
• Proof Block Tutorial Vertical Angles
• Sketchpad Tutorial AIA/Corresponding
Angles
• Sketchpad Tutorial Vertical Angles
Proof
Page | 20
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 1: Congruence, Proof, and Constructions
Sub-Unit B: Congruence
October 6, 2014 – December 6, 2014
Explanations/Examples
Poudre School District
Resources
Activities (continued)
HS.G.CO.C.10
Prentice Hall
• PH G
o p. 259
o p. 272
HS.G.CO.C.11
NCTM Illuminations
• Parallelogram Exploration Tool
Practice
Assessments
HS.G.CO.C.9/HS.G.CO.C.10
Inside Mathematics
• Circles in Triangles
HS.G.CO.C.9
Inside Mathematics
• Quadrilaterals
Page | 21
Curriculum Guide 2014-2015
High School Geometry
Unit 2: Similarity, Proof, and Trigonometry
Poudre School District
Students apply their earlier experience with dilations and proportional reasoning to build a formal
understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve
problems, and apply similarity in right triangles to understand right triangle trigonometry, with
particular attention to special right triangles and the Pythagorean Theorem. Students develop the
Laws of Sines and Cosines in order to find missing measures of general (not necessarily right)
triangles. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2,
or infinitely many triangles.
Sub-Unit A:
Similarity
HS.G.SRT.A.1
HS.G.SRT.A.2
HS.G.SRT.A.3
HS.G.SRT.B.4
HS.G.SRT.B.5
Sub-Unit B:
Trigonometry
HS.G.SRT.C.6
HS.G.SRT.C.8
HS.G.SRT.D.10
HS.G.SRT.D.11
HS.G.SRT.C.7
HS.F.SRT.D.9
Page | 22
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit A: Similarity
January 6, 2015 – January 16, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.SRT.A: Understand similarity in terms of similarity transformations.
HS.G.SRT.A.1
HS.G.SRT.A.1
Lessons
Verify experimentally the properties of dilations
A dilation is a transformation that moves each point
HS.G.SRT.A.1/HS.G.SRT.A.2
given by a center and a scale factor:
along the ray through the point emanating from a fixed Prentice Hall
a. A dilation takes a line not passing through
center, and multiplies distances from the center by a
• PH G
the center of the dilation to a parallel line,
common scale factor.
o Ch 9.5
and leaves a line passing through the center
Students may use geometric simulation software to
unchanged.
HS.G.SRT.A.1
b. The dilation of a line segment is longer or
model transformations. Students may observe patterns Howard County
shorter in the ratio given by the scale factor. and verify experimentally the properties of dilations.
• Verifying Dilation
HS.G.SRT.A.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.
HS.G.SRT.A.3
Use the properties of similarity transformations
to establish the AA criterion for two triangles to
be similar.
HS.G.SRT.A.2
A similarity transformation is a rigid motion followed
by a dilation.
Students may use geometric simulation software to
model transformations and demonstrate a sequence of
transformations to show congruence or similarity of
figures.
HS.G.SRT.A.3
HS.G.SRT.A.2/HS.G.SRT.A.3
Prentice Hall
• PH G
o Ch 7.3
MARS Shell Center
• Analyzing Congruence Proofs
HS.G.SRT.A.2
Howard County
• Similar Figures
MARS Shell Center
• Geometry Problems: Circles and
Triangles
Page | 23
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit A: Similarity
January 6, 2015 – January 16, 2015
Explanations/Examples
Poudre School District
Resources
Tasks
HS.G.SRT.A.1
Illustrative Mathematics
• Dilating a Line
Inside Mathematics
• Surrounded and Covered
HS.G.SRT.A.2
Illustrative Mathematics
• Are They Similar?
HS.G.SRT.A.3
Illustrative Mathematics
• Similar triangles
Activities
Practice
Assessments
HS.G.SRT.A.2
Inside Mathematics
• Hopewell Geometry
• Rhombuses
Page | 24
Curriculum Guide 2014-2015
Common Core State Standards
HS.G.SRT.B.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.
HS.G.SRT.B.5
Use congruence and similarity criteria for
triangles to solve problems and to prove
relationships in geometric figures.
High School Geometry
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit A: Similarity
January 6, 2015 – January 16, 2015
Explanations/Examples
HS.G.SRT.B: Prove theorem involving similarity.
HS.G.SRT.B.4
Students may use geometric simulation software to
model transformations and demonstrate a sequence of
transformations to show congruence or similarity of
figures.
HS.G.SRT.B.5
Similarity postulates include SSS, SAS, and AA.
Congruence postulates include SSS, SAS, ASA, AAS, and
H-L.
Students may use geometric simulation software to
model transformations and demonstrate a sequence of
transformations to show congruence or similarity of
figures.
Poudre School District
Resources
Lessons
HS.G.SRT.B.4/HS.G.SRT.B.5
Prentice Hall
• PH G
o Ch 4
o Ch 7.3-7.5
HS.G.SRT.B.4
Prentice Hall
• PH G
o p. 226-227
o p. 258
o Ch 8.1
MARS Shell Center
• Modeling: Rolling Cups
• Proofs of the Pythagorean Theorem
• Solving Geometry Problems:
Floodlights
Tasks
HS.G.SRT.B.4
Illustrative Mathematics
• Joining Two Midpoints of Sides of a
Triangle
Page | 25
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit A: Similarity
January 6, 2015 – January 16, 2015
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.SRT.B.5
Illustrative Mathematics
• Bank Shot
• Congruence of Parallelograms
• Extensions, Bisections and Dissections
in a Rectangle
• Is This a Rectangle?
• Points from Directions
• Unit Squares and Triangles
Activities
Practice
Assessments
HS.G.SRT.B.5
Inside Mathematics
• Hopewell Geometry
• Rhombuses
Page | 26
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit B: Trigonometry
January 20, 2015 – February 13, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.SRT.C: Define trigonometric ratios and solve problems involving right triangles.
HS.G.SRT.C.6
HS.G.SRT.C.6
Lessons
Understand that by similarity, side
Students may use applets to explore the range of values of the
HS.G.SRT.C
ratios in right triangles are properties trigonometric ratios as θ ranges from 0 to 90 degrees.
Prentice Hall
of the angles in the triangle, leading to
• PH G
definitions of trigonometric ratios for
o Ch 8
hypotenuse
acute angles.
opposite of θ
HS.G.SRT.C.6
θ
Prentice Hall
Adjacent to
• PH G
θ
o p. 431-432
hypotenuse
opposite
o p. 438
cosecant of θ = csc θ =
sine of θ = sin θ =
opposite
hypotenuse
adjacent
hypotenuse
opposite
tangent of θ = tan θ =
adjacent
cosine of θ = cos θ =
HS.G.SRT.C.7
Explain and use the relationship
between the sine and cosine of
complementary angles.
hypotenuse
adjacent
adjacent
cotangent of θ = cot θ =
opposite
secant of θ = sec θ =
HS.G.SRT.C.7
Geometric simulation software, applets, and graphing calculators can be
used to explore the relationship between sine and cosine.
MARS Shell Center
• Calculating Volumes of
Compound Objects
• Geometry Problems: Circles and
Triangles
• Inscribing and Circumscribing
Right Triangles
• Solving Quadratic Equations:
Cutting Corners
HS.G.SRT.C.8
Prentice Hall
• PH G
o p. 444
o p. 451
o Ch 10.5
Page | 27
Curriculum Guide 2014-2015
Common Core State Standards
HS.G.SRT.C.8
Use trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems. ★
High School Geometry
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit B: Trigonometry
January 20, 2015 – February 13, 2015
Explanations/Examples
HS.G.SRT.C.8
Students may use graphing calculators or programs, tables,
spreadsheets, or computer algebra systems to solve right triangle
problems.
• Find the height of a tree to the nearest tenth if the angle of elevation
of the sun is 28° and the shadow of the tree is 50 ft.
Poudre School District
Resources
Tasks
HS.G.SRT.C.6
Illustrative Mathematics
• Finding the Area of an
Equilateral Triangle
• Mt. Whitney to Death Valley
• Seven Circles I
HS.G.SRT.C.8
Illustrative Mathematics
• Coins in a Circular Pattern
• Neglecting the Curvature of the
Earth
• Setting Up Sprinklers
• Seven Circles III
• Shortest Line Segment from a
Point P to a Line L
Activities
Practice
Assessments
HS.G.SRT.C.6
Inside Mathematics
• Hopewell Geometry
Page | 28
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 2: Similarity, Proof, and Trigonometry
Sub-Unit B: Trigonometry
January 20, 2015 – February 13, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.SRT.D: Apply trigonometry to general triangles.
With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles.
HS.G.SRT.D.9
HS.G.SRT.D.9
Lessons
1
HS.G.SRT.D.9
Derive the formula 𝐴𝐴 = 2 𝑎𝑎𝑎𝑎 sin(𝐶𝐶)
Prentice Hall
for the area of a triangle by drawing
• PH G
an auxiliary line from a vertex
o Ch 10.5
perpendicular to the opposite side.
• Example 2
HS.G.SRT.D.10
Prove the Laws of Sines and Cosines
and use them to solve problems.
HS.G.SRT.D.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).
HS.G.SRT.D.10
HS.G.SRT.D.11
Tara wants to fix the location of a mountain by taking measurements
from two positions 3 miles apart. From the first position, the angle
between the mountain and the second position is 78o. From the
second position, the angle between the mountain and the first
position is 53o. How can Tara determine the distance of the mountain
from each position, and what is the distance from each position?
•
Tasks
HS.G.SRT.D.10/HS.G.SRT.D.11
Illustrative Mathematics
• Seven Circles III
HS.G.SRT.D.11
Inside Mathematics
• Piece it Together
• The Shape of Things
• What's Your Angle?
Activities
HS.G.SRT.D.10/HS.G.SRT.D.11
Prentice Hall
• PH G
o p. 565
Practice
Assessments
Page | 29
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 3: Connecting Algebra and Geometry through Coordinates
Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a
rectangular coordinate system to verify geometric relationships, including properties of special
triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students continue their
study of quadratics by connecting the geometric and algebraic definitions of the parabola.
HS.G.CPE.B.6
HS.G.GPE.B.4
HS.G.GPE.B.5
HS.G.GPE.B.7
HS.G.GPE.A.2
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 3: Connecting Algebra and Geometry through Coordinates
February 17, 2015 – February 27, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.GPE.B: Use coordinates to prove simple geometric theorem algebraically.
This unit has a close connection with the next unit. For example, a curriculum might merge G.GPE.1 and the Unit 4 treatment of G.GPE.4 with the
standards in the unit. Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using
similar right triangles.
Relate wok on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra 1 involving systems of equations having no solution or infinitely
many solutions.
G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean Theorem.
HS.G.GPE.B.4
HS.G.GPE.B.4
Lessons
Use coordinates to prove simple geometric
Students may use geometric simulation software to
HS.G.GPE.B.4
theorems algebraically. For example, prove or
model figures and prove simple geometric theorems.
Prentice Hall
disprove that a figure defined by four given points Use slope and distance formula to verify the polygon
• PH G
formed by connecting the points
in the coordinate plane is a rectangle; prove or
o Ch 1.8
(-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram.
o p. 60
disprove that the point �1, √3� lies on the circle
o Ch 6.6-6.7
centered at the origin and containing the point
(0, 2).
HS.G.GPE.B.5
HS.G.GPE.B.5
HS.G.GPE.B.5
Prentice
Hall
Prove the slope criteria for parallel and
Lines can be horizontal, vertical, or neither.
•
PH
G
perpendicular lines and use them to solve
o Ch 3.7
geometric problems (e.g., find the equation of a
Students may use a variety of different methods to
line parallel or perpendicular to a given line that
construct a parallel or perpendicular line to a given
MARS Shell Center
passes through a given point).
line and calculate the slopes to compare the
• Finding Equations of Parallel and
relationships.
Perpendicular Lines
HS.G.GPE.B.6
HE.G.GPE.B.6
Find the point on a directed line segment
Students may use geometric simulation software to
HS.G.GPE.B.6
between two given points that partitions the
model figures or line segments.
Howard County
• Given A(3, 2) and B(6, 11),
segment in a given ratio.
• Partitioning a Segment
o Find the point that divides the line segment AB
two-thirds of the way from A to B.
The point two-thirds of the way from A to B has Tasks
HS.G.GPE.B.4/HS.G.GPE.B.5
x-coordinate two-thirds of the way from 3 to 6
Illustrative Mathematics
and y coordinate two-thirds of the way from 2
• Unit Squares and Triangles
to 11.
Page | 31
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 3: Connecting Algebra and Geometry through Coordinates
February 17, 2015 – February 27, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.GPE.B.6 (continued)
HS.G.GPE.B.6 (continued)
Tasks (continued)
So, (5, 8) is the point that is two-thirds from
HS.G.GPE.B.4
point A to point B.
Howard County
• Pooltastic
o Find the midpoint of line segment AB.
Illustrative Mathematics
HS.G.GPE.B.7
HS.G.GPE.B.7
• Is this a Rectangle
Use coordinates to compute perimeters of
Students may use geometric simulation software to
polygons and areas of triangles and rectangles,
model figures.
HS.G.GPE.B.5
Illustrative
Mathematics
e.g., using the distance formula. ★
• A Midpoint Miracle
• Triangles Inscribed in a Circle
Activities
Practice
HS.G.GPE.B.4/HS.G.GPE.B.7
Shmoop
• Proving Geometric Shapes via
Coordinate Grid Worksheet
HS.G.GPE.B.4
Prentice Hall
• PH G
o p. 326 #22-24
Page | 32
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 3: Connecting Algebra and Geometry through Coordinates
February 17, 2015 – February 27, 2015
Common Core State Standards
Explanations/Examples
Resources
Practice (continued)
HS.G.GPE.B.7
Prentice Hall
• PH G
o Ch 10.1-10.3
• p. 538 #29-33
• p. 453 #25-27
• p. 550 #37
Assessments
HS.G.GPE.A: Translate between the geometric description and the equation for a conic section.
The directrix should be parallel to a coordinate axis.
HS.G.GPE.A.2
HS.G.GPE.A.2
Lessons
Derive the equation of a parabola given a focus
Students may use geometric simulation software to
Tasks
and directrix.
explore parabolas.
Write and graph an equation for a parabola with focus
(2, 3) and directrix y = 1.
Activities
Practice
Assessments
Page | 33
Curriculum Guide 2014-2015
High School Geometry
Unit 4: Circles With and Without Coordinates
Poudre School District
In this unit, students prove basic theorems about circles, with particular attention to perpendicular
and inscribed angles, in order to see symmetry in circles and as an application of triangle
congruence criteria. They study relationships among segments on chords, secants, and tangents as
an application of similarity. In the Cartesian coordinate system, students use the distance formula
to write the equation of a circle when given the radius and the coordinates of its center. Given an
equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving
quadratic equations to determine intersections between lines and circles or parabolas and between
two circles.
HS.G.MG.A.1
HS.G.C.A.1
HS.G.GPE.A.1
HS.G.C.B.5
HS.G.GPE.B.4
HS.G.C.A.3
HS.G.C.A.2
HS.G.C.A.4
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.MG.A: Apply geometric concepts in modeling situations.
Focus on situations in which the analysis of circles is required.
HS.G.MG.A.1
HS.G.MG.A.1
Lessons
Use geometric shapes, their measures, and their
Students may use simulation software and modeling
HS.G.MG.A
properties to describe objects (e.g., modeling a
software to explore which model best describes a set of Prentice Hall
tree trunk or a human torso as a cylinder).
data or situation.
• PH G
o Ch 1.2
o Ch 11.1
MARS Shell Center
• Rolling Cups
Tasks
Activities
Practice
Assessments
HS.G.C.A.1
Prove that all circles are similar.
HS.G.C.A: Understand and apply theorems about circles.
HS.G.C.A.1
Students may use geometric simulation software to
model transformations and demonstrate a sequence of
transformations to show congruence or similarity of
figures.
Lessons
HS.G.C.A.1
Howard County
• Lesson Similar Circles
Page | 35
Curriculum Guide 2014-2015
Common Core State Standards
HS.G.C.A.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.
High School Geometry
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Explanations/Examples
HS.G.C.A.2
• Given the circle below with radius of 10 and chord
length of 12, find the distance from the chord to the
center of the circle.
•
HS.G.C.A.3
Construct the inscribed and circumscribed circles
of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
HS.G.C.A.4
Construct a tangent line from a point outside a
given circle to the circle.
Find the unknown length in the picture below.
HS.G.C.A.3
Students may use geometric simulation software to
make geometric constructions.
HS.G.C.A.4
Students may use geometric simulation software to
make geometric constructions.
Poudre School District
Resources
Lessons (continued)
HS.G.C.A.2/HS.G.C.A.4
Prentice Hall
• PH G
o p. 677-678
HS.G.C.A.2
Prentice Hall
• PH G
o Ch 10.6
o Ch 12.1-12.4
o p. 669
Howard County
• Angles in a Circle
• Inscribed and Central Angles
NCTM Illuminations
• Power of Points
HS.G.C.A.3
Prentice Hall
• PH G
o Ch 5.3
o p. 271
HS.G.C.A.4
Prentice Hall
• PH G
o Ch 12.1
o p. 693
Page | 36
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Explanations/Examples
Poudre School District
Resources
Tasks
HS.G.C.A.1
Illustrative Mathematics
• Similar Circles
HS.G.C.A.2
Illustrative Mathematics
• Right Triangles Inscribed in Circles 1
• Right Triangles Inscribed in Circles 2
• Tangent Lines and the Radius of a
Circle
Inside Mathematics
• What's Your Angle: Level E
HS.G.C.A.3
Howard County
• Task Your Trip to Paris
Illustrative Mathematics
• Circles and Triangles
• Inscribing a Circle in a Triangle I
• Inscribing a Circle in a Triangle II
• Inscribing a Triangle in a Circle
• Locating a Warehouse
• Placing a Fire Hydrant
Page | 37
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.C.A.4
Illustrative Mathematics
• Neglecting the Curvature of the Earth:
How Far Can You See From the Top of
Mt. Whitney
• Tangent to a Circle at a Point
Activities
Practice
HS.G.C.A.2
Shmoop
• Circles Worksheet 2
HS.G.C.A.3
Shmoop
• Functions Worksheet 3
HS.G.C.A.4
Schmoop
• Circles Worksheet 4
Assessments
HS.G.C.A.2
Inside Mathematics
• Circle and Squares
• Circle in Triangle
Page | 38
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Explanations/Examples
Poudre School District
Resources
Assessments (continued)
HS.G.C.A.3
MARS Shell Center
• Inscribing and Circumscribing Right
Triangles
HS.G.C.B: Find arc lengths and areas of sectors of circles.
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.
HS.G.C.B.5
HS.G.C.B.5
Lessons
Derive using similarity the fact that the length of
Students can use geometric simulation software to
HS.G.C.B
the arc intercepted by an angle is proportional to explore angle and radian measures and derive the
Prentice Hall
the radius, and define the radian measure of the
formula for the area of a sector.
• PH G
angle as the constant of proportionality; derive
o Ch 10.4
the formula for the area of a sector.
o Ch 10.6-10.7
MARS Shell Center
• Sectors of Circles
Tasks
HS.G.C.B
Illustrative Mathematics
• Mutually Tangent Circles
• Setting Up Sprinklers
• Two Wheels and a Belt
Activities
Page | 39
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Explanations/Examples
Poudre School District
Resources
Practice
HS.G.C.B
Prentice Hall
• PH G
o p. 557 #35
Assessments
HS.G.GPE.A: Translate between the geometric description and the equation for a conic section.
HS.G.GPE.A.1
HS.G.GPE.A.1
Lessons
Derive the equation of a circle of given center and Students may use geometric simulation software to
HS.G.GPE.A
radius using the Pythagorean Theorem; complete explore the connection between circles and the
Prentice Hall
the square to find the center and radius of a circle Pythagorean Theorem.
• PH A2
• Write an equation for a circle with a radius of 2
given by an equation.
o Ch 10.3
units and center at (1, 3).
• PH G
• Write an equation for a circle given that the
o Ch 12.5
endpoints of the diameter are (-2, 7) and (4, -8).
• Find the center and radius of the circle
Tasks
4x2 + 4y2 - 4x + 2y – 1 = 0.
HS.G.GPE.A
Illustrative Mathematics
• Explaining the Equation of a Circle
• Slopes and Circles
Activities
Practice
Assessments
Page | 40
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 4: Circles With and Without Coordinates
March 2, 2015 – March 27, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.GPE.B: Use coordinates to prove simple geometric theorems algebraically.
Include simple proofs involving circles.
HS.G.GPE.B.4
HS.G.GPE.B.4
Lessons
Use coordinates to prove simple geometric
Students may use geometric simulation software to
HS.G.GPE.B
theorems algebraically. For example, prove or
model figures and prove simple geometric theorems.
Prentice Hall
disprove that a figure defined by four given points Use slope and distance formula to verify the polygon
• PH G
in the coordinate plane is a rectangle; prove or
formed by connecting the points
o Ch 6.1
(-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram.
o Ch 6.7
disprove that the point �1, √3�lies on the circle
centered at the origin and containing the point
MARS Shell Center
(0, 2).
• Equations of Circles 1
NCTM Illuminations
• Pi Line
Tasks
Activities
Practice
HS.G.GPE.B
Prentice Hall
• PH G
o p. 309 #13-18
o p. 311 #64
Assessments
Page | 41
Curriculum Guide 2014-2015
High School Geometry
Unit 5: Extending to Three Dimensions
Poudre School District
Students’ experience with two-dimensional and three-dimensional objects is extended to include
informal explanations of circumference, area and volume formulas. Additionally, students apply
their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result
of rotating a two-dimensional object about a line.
HS.G.MG.A.1
HS.G.GMD.B.4
HS.G.GMD.A.1
HS.G.GMD.A.3
HS.G.MG.A.3
HS.G.MG.A.2
Page | 42
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 5: Extending to Three Dimensions
March 30, 2015 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.MG.A: Apply geometric concepts in modeling situations.
Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general
triangles.
HS.G.MG.A.1
HS.G.MG.A.1
Lessons
Use geometric shapes, their measures, and their
Students may use simulation software and modeling
HS.G.MG.A.1/HS.G.MG.A.3
properties to describe objects (e.g., modeling a
software to explore which model best describes a set of MARS Shell Center
tree trunk or a human torso as a cylinder).
data or situation.
• Modeling: Rolling Cups
• Solving Quadratic Equations: Cutting
HS.G.MG.A.2
HS.G.MG.A.2
Corners
Apply concepts of density based on area and
Students may use simulation software and modeling
volume in modeling situations (e.g., persons per
software to explore which model best describes a set of HS.G.MG.A.1
square mile, BTUs per cubic foot).
data or situation.
Prentice Hall
• PH G
HS.G.MG.A.3
HS.G.MG.A.3
o Ch 11.1
Apply geometric methods to solve design
Students may use simulation software and modeling
o p. 605
problems (e.g., designing an object or structure to software to explore which model best describes a set of
data or situation.
satisfy physical constraints or minimize cost;
HS.G.MG.A.3
working with typographic grid systems based on
Prentice Hall
ratios).
• PH G
o p. 657
Tasks
HS.G.MG.A
Illustrative Mathematics
• Coins in a Circular Pattern
Page | 43
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 5: Extending to Three Dimensions
March 30, 2015 – April 24, 2015
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.MG.A.1/HS.G.MG.A.2
Illustrative Mathematics
• How Many Cells are in the Human
Body?
• How Many Leaves on a Tree?
• How Many Leaves on a Tree?
(Version 2)
• How Thick is a Soda Can I?
• How Thick is a Soda Can II?
HS.G.MG.A.1/HS.G.MG.A.3
Illustrative Mathematics
• Eratosthenes and the Circumference of
the Earth
• Paper Clip
• Regular Tessellations of the Plane
• Running Around a Track I
• Running Around a Track II
Page | 44
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 5: Extending to Three Dimensions
March 30, 2015 – April 24, 2015
Explanations/Examples
Poudre School District
Resources
Tasks (continued)
HS.G.MG.A.1
Illustrative Mathematics
• Global Positioning System II
• Hexagonal Pattern of Beehives
• Seven Circles III
• Solar Eclipse
• Tennis Balls in a Can
• The Lighthouse Problem
• Tilt of Earth's Axis and the Four
Seasons
• Toilet Roll
• Use Cavalieri’s Principle to
Compare Aquarium Volumes
Activities
HS.G.MG.A
NCTM Illuminations
• Tube Viewer Simulation
HS.G.MG.A.1
NCTM Illuminations
• Soccer Problem
Practice
HS.G.MG.A.2
Prentice Hall
• PH G
o p. 642 #43
Assessments
Page | 45
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 5: Extending to Three Dimensions
March 30, 2015 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.GMD.B: Visualize the relation between two-dimensional and three-dimensional objects.
HS.G.GMD.B.4
HS.G.GMD.B.4
Lessons
Identify the shapes of two-dimensional crossStudents may use geometric simulation software to
HS.G.GMD.B
sections of three-dimensional objects, and
model figures and create cross sectional views.
Prentice Hall
Identify the shape of the vertical, horizontal, and other
identify three-dimensional objects generated by
• PH G
cross sections of a cylinder.
rotations of two-dimensional objects.
o Ch 11.1
o Ch 11.4-11.6
MARS Shell Center
• 2D Representations of 3D Objects
• Modeling: Rolling Cups
Tasks
HS.G.GMD.B
Illustrative Mathematics
• Global Positioning System I
• Global Positioning System II
• Tennis Balls in a Can
Activities
Practice
Assessments
Page | 46
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 5: Extending to Three Dimensions
March 30, 2015 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.G.GMD.A: Explain volume formulas and use them to solve problems.
Informal arguments for area and volume formulas can make use of the way in which area an volume scale under similarity transformation: when
one figure in the plane results from another by applying a similarity transformation with scale fact k, its area is k2 times the area of the first.
Similarity, volumes of solid figures scale by k3 under a similarity transformation with scale factor k.
HS.G.GMD.A.1
HS.G.GMD.A.1
Lessons
Give an informal argument for the formulas for
Cavalieri’s principle is if two solids have the same
HS.G.GMD.A
the circumference of a circle, area of a circle,
height and the same cross-sectional area at every level, Prentice Hall
volume of a cylinder, pyramid, and cone. Use
then they have the same volume.
• PH G
dissection arguments, Cavalieri’s principle, and
o Ch 11.4-11.6
informal limit arguments.
HS.G.GMD.A.1
HS.G.GMD.A.3
HS.G.GMD.A.3
Prentice Hall
Use volume formulas for cylinders, pyramids,
Missing measures can include but are not limited to
• PH G
slant
height,
altitude,
height,
diagonal
of
a
prism,
edge
o Ch 1.9
cones, and spheres to solve problems. ★
length, and radius.
o Ch 10.6-10.7
o p. 575
o p. 581
MARS Shell Center
• Equations of Circles 2
HS.G.GMD.A.3
MARS Shell Center
• Calculating Volumes of Compound
Objects
• Evaluating Statements About
Enlargements (2D and 3D)
Page | 47
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 5: Extending to Three Dimensions
March 30, 2015 – April 24, 2015
Explanations/Examples
Poudre School District
Resources
Tasks
HS.G.GMD.A.3
Illustrative Mathematics
• Centerpiece
• Doctor's Appointment
Activities
HS.G.GMD.A.3
NCTM Illuminations
• Dynamic Paper
• Geometric Solids
Practice
Assessments
Page | 48
Curriculum Guide 2014-2015
High School Geometry
Unit 6: Applications of Probability
Poudre School District
Building on probability concepts that began in the middle grades, students use the languages of set
theory to expand their ability to compute and interpret theoretical and experimental probabilities
for compound events, attending to mutually exclusive events, independent events, and conditional
probability. Students should make use of geometric probability models wherever possible. They use
probability to make informed decisions.
HS.S.CP.A.1
HS.S.CP.A.2
HS.S.CP.B.7
HS.S.CP.A.3
HS.S.CP.B.8
HS.S.CP.A.5
HS.S.CP.B.6
HS.S.CP.A.4
HS.S.CP.B.9
HS.S.MD.B.6
HS.S.MD.B.7
Page | 49
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.S.CP.A: Understand the independence and conditional probability and use them to interpret data.
Build on work with two-way tables from Algebra 1 Units 3 (S.ID.5) to develop understanding of conditional probability and independence.
HS.S.CP.A.1
HS.S.CP.A.1
Lessons
Describe events as subsets of a sample space (the Intersection: The intersection of two sets A and B is
HS.S.CP.A.1/HS.S.CP.A.2
set of outcomes) using characteristics (or
the set of elements that are common to both set A and
Prentice Hall
categories) of the outcomes, or as unions,
set B. It is denoted by A ∩ B and is read ‘A intersection
• PH A2
intersections, or complements of other events
B’.
o Ch 9.7
(“or,” “and,” “not”).
a. A ∩ B in the diagram is {1, 5}
HS.S.CP.A.1/HS.S.CP.A.4
b. this means: BOTH/AND
Prentice Hall
• PH A2
o Ch 12.1
U
A
2
3
1
5
B
4
7
8
Union: The union of two sets A and B is the set of
elements, which are in A or in B or in both. It is
denoted by A ∪ B and is read ‘A union B’.
a. A ∪ B in the diagram is {1, 2, 3, 4, 5, 7}
b. this means: EITHER/OR/ANY
c. could be both
Complement: The complement of the set A ∪B is the
set of elements that are members of the universal set U
but are not in A ∪B. It is denoted by (A ∪ B )’
(A ∪ B )’ in the diagram is {8}
HS.S.CP.A.1
NCTM Illuminations
• Exploration with Chance
HS.S.CP.A.2
Interactivate
• Replacement Probability
HS.S.CP.A.3/HS.S.CP.A.5
Prentice Hall
• PH A2
o Ch 12.2
HS.S.CP.A.3
Interactivate
• Conditional Probability and
Simultaneous Events
Page | 50
Curriculum Guide 2014-2015
Common Core State Standards
HS.S.CP.A.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.
High School Geometry
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
HS.S.CP.A.2
HS.S.CP.A.3
HS.S.CP.A.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.
HS.S.CP.A.4
Construct and interpret two-way frequency
tables of data when two categories are associated
with each object being classified. Use the twoway 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.
HS.S.CP.A.4
Students may use spreadsheets, graphing calculators,
and simulations to create frequency tables and conduct
analyses to determine if events are independent or
determine approximate conditional probabilities.
Poudre School District
Resources
Lessons (continued)
HS.S.CP.A.3 (continued)
MARS Shell Center
• Medical Testing
• Modeling Conditional Probabilities 1
• Modeling Conditional Probabilities 2
HS.S.CP.A.4
Stat Trek
• Two Way Tables
HS.S.CP.A.5
Alabama Learning Exchange
• Dartboard Probability
Tasks
HS.S.CP.A.1
Illustrative Mathematics
• Return to Fred's Fun Factory
• The Titanic I
Inside Mathematics
• Friends That You Can Count On
• Got Your Number
• Party Time
• Rod Trains
Math Forum @ Drexel
• Monty Hall Problem
Page | 51
Curriculum Guide 2014-2015
Common Core State Standards
HS.S.CP.A.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.
High School Geometry
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
HS.S.CP.A.5
• What is the probability of drawing a heart from a
standard deck of cards on a second draw, given that
a heart was drawn on the first draw and not
replaced? Are these events independent or
dependent?
• At Johnson Middle School, the probability that a
student takes computer science and French is
0.062. The probability that a student takes
computer science is 0.43. What is the probability
that a student takes French given that the student
is taking computer science?
Poudre School District
Resources
Tasks (continued)
HS.S.CP.A.2
Illustrative Mathematics
• Cards and Independence
• Rain and Lightning
• The Titanic II
HS.S.CP.A.3
Illustrative Mathematics
• Lucky Envelopes
Math Forum @ Drexel
• Boy or Girl
HS.S.CP.A.4
Illustrative Mathematics
• How Do you Get to School
• The Titanic 3
HS.S.CP.A.5
Illustrative Mathematics
• Breakfast Before School
• But Mango is My Favorite...
Activities
HS.S.CP.A.1
Interactivate
• Venn Diagram Shape Sorter
Page | 52
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
Poudre School District
Resources
Activities (continued)
HS.S.CP.A.1 (continued)
NCTM Illuminations
• Venn Diagram Applet
HS.S.CP.A.3
Cut-the-Knot
• Conditional Probability and
Independent Events
Practice
HS.S.CP.A.2
Shmoop
• Conditional Probability Worksheet 2
HS.S.CP.A.3
Shmoop
• Conditional Probability Worksheet
3
HS.S.CP.A.4
Shmoop
• Conditional Probability Worksheet
4
HS.S.CP.A.5
Shmoop
• Conditional Probability Worksheet 5
Page | 53
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
Poudre School District
Resources
Assessments
HS.S.CP.A.1
Inside Mathematics
• Marble Game
HS.S.CP.B: Use the rules of probability to compute probabilities of compound events in a uniform probability model.
HS.S.CP.B.6
HS.S.CP.B.6
Lessons
Find the conditional probability of A given B as
Students could use graphing calculators, simulations,
HS.S.CP.B.6
the fraction of B’s outcomes that also belong to A, or applets to model probability experiments and
Prentice Hall
and interpret the answer in terms of the model.
interpret the outcomes.
• PH A2
o Ch 12.2
HS.S.CP.B.7
Apply the Addition Rule, P(A or B) =
P(A) + P(B) – P(A and B), and interpret the
answer in terms of the model.
HS.S.CP.B.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.
HS.S.CP.B.7
Students could use graphing calculators, simulations,
or applets to model probability experiments and
interpret the outcomes.
In a math class of 32 students, 18 are boys and 14 are
girls. On a unit test, 5 boys and 7 girls made an A grade.
If a student is chosen at random from the class, what is
the probability of choosing a girl or an A student?
HS.S.CP.B.8
Students could use graphing calculators, simulations,
or applets to model probability experiments and
interpret the outcomes.
Learnist
• Bayes Theorem Links
HS.S.CP.B.7
Learnist
• Webpage Links to Addition Rule
HS.S.CP.B.8
Learnist
• Webpage Links to Multiplication Rule
HS.S.CP.B.9
Prentice Hall
• PH A1
o Ch 12.7-12.8
Page | 54
Curriculum Guide 2014-2015
Common Core State Standards
HS.S.CP.B.9
Use permutations and combinations to compute
probabilities of compound events and solve
problems.
High School Geometry
Poudre School District
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
Resources
HS.S.CP.B.9
Lessons (continued)
Students may use calculators or computers to
HS.S.CP.B.9 (continued)
determine sample spaces and probabilities.
Math Forum @ Drexel
• Choosing Computer Disks
• You and two friends go to the grocery store and
each buys a soda. If there are five different kinds of
• Combinations for License Plates
soda, and each friend is equally likely to buy each
• Random Card Shuffling Probabilities
variety, what is the probability that no one buys the • Seating People in a Row
same kind?
• Telephone Number Possibility
• Ways to List 1 Through 10 Out of
Order
• Winning UK Lottery
Tasks
HS.S.CP.B.7
Illustrative Mathematics
• Coffee at Mom's Diner
HS.S.CP.B.9
Illustrative Mathematics
• Alex Mel and Chelsea Play a Game
• Random Walk 3
• Random Walk 4
Activities
HS.S.CP.B.6
Interactivate
• Fire
HS.S.CP.B.8
Interactivate
• Crazy Choices Game
Page | 55
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
Poudre School District
Resources
Activities (continued)
HS.S.CP.B.9
Better Explained
• Navigating a Grid Using Combinations
and Permutations
IXL
• Combinations Online Quiz
• Permutation and Combination
Notation Quiz
• Permutations Online Quiz
Practice
HS.S.CP.B.6
Shmoop
• Functions Worksheet 6
HS.S.CP.B.7
Shmoop
• Functions Worksheet 7
HS.S.CP.B.8
Shmoop
• Conditional Probability Worksheet 8
HS.S.CP.B.9
Shmoop
• Conditional Probability Worksheet 9
Assessments
Page | 56
Curriculum Guide 2014-2015
High School Geometry
Poudre School District
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.S.MD.B: Use probability to evaluate outcomes of decisions.
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) required an understanding of probability concepts.
HS.S.MD.B.6
HS.S.MD.B.6
Lessons
Use probabilities to make fair decisions (e.g.,
Students may use graphing calculators or programs,
HS.S.MD.B.6
drawing by lots, using a random number
spreadsheets, or computer algebra systems to model
Prentice Hall
generator).
and interpret parameters in linear, quadratic or
• PH G
exponential functions.
o Ch 10.8
HS.S.MD.B.7
HS.S.MD.B.7
Analyze decisions and strategies using
Students may use graphing calculators or programs,
Learnist
probability concepts (e.g., product testing,
spreadsheets, or computer algebra systems to model
• Videos and Web Links for Game
medical testing, pulling a hockey goalie at the end and interpret parameters in linear, quadratic or
Fairness
of a game).
exponential functions.
Math Forum @ Drexel
• Drawing Prizes
• Lottery Strategy and Odds of Winning
• Randomness of a Shuffled Deck
• Selecting a Student Council
HS.S.MD.B.7
Learnist
• Videos and Web links for Decision
Making with Probability
MARS Shell Center
• Evaluating Statements About
Probability
Tools for the Common Core Standards
• Medical Testing Details Page 19
Page | 57
Curriculum Guide 2014-2015
Common Core State Standards
High School Geometry
Unit 6: Applications of Probability
April 27, 2015 - May 15, 2015
Explanations/Examples
Poudre School District
Resources
Tasks
HS.S.MD.B.7
Illustrative Mathematics
• Fred's Fun Factory
Activities
HS.S.MD.B.6
Interactivate
• Adjustable Spinner
HS.S.MD.B.7
Interactivate
• The Game of Life
Practice
Assessments
HS.S.MD.B.6
Shmoop
• Quiz on How to Make Fair Decisions
Page | 58
Performance Level Descriptors – Geometry
Congruence
Transformations
HS.G.CO.B.6
HS.G.CO.C
Similarity
HS.G.SRT.A.1a
HS.G.SRT.A.1b
HS.G.SRT.A.2
HS.G.SRT.B.5
Similarity in
Trigonometry
HS.G.SRT.C.6
HS.G.SRT.C.7-2
HS.G.SRT.C.8
Geometry: Sub-Claim A
The student solves problems involving the Major Content for the grade/course with connections to the
Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Determines and uses
appropriate geometric
theorems and properties of
rigid motions, lines, angles,
triangles and parallelograms
to solve non-routine
problems and prove
statements about angle
measurement, triangles,
distance, line properties and
congruence.
Uses transformations and
congruence and similarity
criteria for triangles and to
prove relationships among
composite geometric
figures and to solve multistep problems.
Determines and uses
appropriate geometric
theorems and properties of
rigid motions, lines, angles,
triangles and parallelograms
to solve routine problems
and prove statements about
angle measurement,
triangles, distance, line
properties and congruence.
Uses given geometric
theorems and properties of
rigid motions, lines, angles,
triangles and parallelograms
to solve routine problems
and prove statements
about angle measurement,
triangles, distance, line
properties and congruence.
Uses given geometric
theorems and properties of
rigid motions, lines, angles,
triangles and parallelograms
to solve routine problems
and reason about angle
measurement, triangles,
distance, line properties and
congruence.
Uses transformations and
congruence and similarity
criteria for triangles to
prove relationships among
geometric figures and to
solve problems.
Uses transformations to
determine relationships
among geometric figures
and to solve problems.
Identifies transformation
relationships in geometric
figures.
Uses trigonometric ratios,
the Pythagorean Theorem
and the relationship
between sine and cosine to
solve right triangles in
applied non-routine
problems.
Uses trigonometric ratios,
the Pythagorean Theorem
and the relationship
between sine and cosine to
solve right triangles in
applied problems.
Uses trigonometric ratios,
the Pythagorean Theorem
and the relationship
between sine and cosine to
solve right triangles in
applied problems.
Uses trigonometric ratios
and the Pythagorean
Theorem to determine the
unknown side lengths and
angle measurements of a
right triangle.
Uses similarity
transformations with right
Uses similarity
transformations with right
triangles to define
Page | 59
Performance Level Descriptors – Geometry
Geometry: Sub-Claim A
The student solves problems involving the Major Content for the grade/course with connections to the
Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Modeling and
Applying
HS.G.SRT.C.7-2
HS.G.SRT.C.8
HS.G.GPE.B.6
G-Int.1
triangles to define
trigonometric ratios for
acute angles.
trigonometric ratios for
acute angles.
Uses geometric
relationships in the
coordinate plane to solve
problems involving area,
perimeter and ratios of
lengths.
Uses geometric
relationships in the
coordinate plane to solve
problems involving area,
perimeter and ratios of
lengths.
Uses geometric
relationships in the
coordinate plane to solve
problems involving area,
perimeter and ratios of
lengths.
Applies geometric concepts
and trigonometric ratios to
describe, model and solve
applied problems (including
design problems) related to
the Pythagorean theorem,
density, geometric shapes,
their measures and
properties.
Applies geometric concepts
and trigonometric ratios to
describe, model and solve
applied problems related to
the Pythagorean theorem,
density, geometric shapes,
their measures and
properties.
Applies geometric concepts
to describe, model and
solve applied problems
related to the Pythagorean
theorem, geometric shapes,
their measures and
properties.
Uses provided geometric
relationships in the
coordinate plane to solve
problems involving area and
perimeter.
Applies geometric concepts
to describe, model and
solve applied problems
related to the Pythagorean
theorem, geometric shapes,
their measures and
properties.
Page | 60
Performance Level Descriptors – Geometry
Transformations
HS.G.CO.A.1
HS.G.CO.A.3
HS.G.CO.A.5
Geometric
Constructions
HS.G.CO.D
Geometry: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Given a figure and a
sequence of
transformations, draws the
transformed figure.
Uses precise geometric
terminology to specify more
than one sequence of
transformations that will
carry a figure onto itself or
another.
Makes geometric
constructions: copying a
segment, copying an angle,
bisecting an angle, bisecting
a segment, including the
perpendicular bisector of a
line segment.
Given a line and a point not
on the line, uses a variety of
tools and methods to
construct perpendicular and
parallel lines, equilateral
triangles, squares and
regular hexagons inscribed
in circles to prove
geometric theorems.
Given a figure and a
transformation, draws the
transformed figure.
Given a figure and a
transformation, draws the
transformed figure.
Uses precise geometric
terminology to specify a
sequence of
transformations that will
carry a figure onto itself or
another.
Specifies a sequence of
transformations that will
carry a figure onto another.
Makes geometric
constructions: copying a
segment, copying an angle,
bisecting an angle, bisecting
a segment, including the
perpendicular bisector of a
line segment.
Makes geometric
constructions: copying a
segment, copying an angle,
bisecting an angle, bisecting
a segment, including the
perpendicular bisector of a
line segment.
Given a line and a point not
on the line, uses a variety of
tools and methods to
construct perpendicular and
parallel lines, equilateral
triangles, squares and
regular hexagons inscribed
in circles.
Given a line and a point not
on the line, constructs
perpendicular and parallel
lines.
Given a figure and a
transformation, draws the
transformed figure.
Makes basic geometric
constructions: copying a
segment, copying an angle,
bisecting an angle, bisecting
a segment, including the
perpendicular bisector of a
line segment.
Page | 61
Performance Level Descriptors – Geometry
Applying
Geometric
Properties and
Theorems
HS.G.C.A.Int.1
HS.G.C.B.Int.1
HS.G.GPE.A.1-2
Geometric
Formulas
HS.G.GMD.A.1
HS.G.GMD.A.3
HS.G.GMD.A.4
Geometry: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Applies properties and
theorems of angles,
segments and arcs in circles
to solve problems, model
relationships and formulate
generalizations.
Completes the square to
find the center and radius of
a circle given by an
equation.
Applies properties and
theorems of angles,
segments and arcs in circles
to solve problems and
model relationships.
Completes the square to
find the center and radius of
a circle given by an
equation.
Uses volume formulas to
solve mathematical and
contextual problems that
involve cylinders, pyramids,
cones and spheres.
Uses volume formulas to
solve mathematical and
contextual problems that
involve cylinders, pyramids,
cones and spheres.
Uses dissection arguments,
Cavalieri’s principle and
informal limit arguments to
support the formula for the
circumference of a circle,
area of a circle, volume of a
cylinder, pyramid and cone.
Gives an informal argument
for the formula for the
circumference of a circle,
area of a circle and volume
of a cylinder, including
dissection arguments.
Identifies the shapes of twodimensional cross-sections
of three-dimensional
Identifies the shapes of twodimensional cross-sections
of three-dimensional
objects and identifies
Applies properties and
theorems of angles,
segments and arcs in circles
to solve problems.
Applies provided properties
and theorems of angles,
segments and arcs in circles
to solve problems.
Completes the square to
find the center and radius
of a circle given by an
equation.
Using formulas, determines
the volume of cylinders,
pyramids, cones and
spheres.
Using formulas, determines
the volume of cylinders,
pyramids, cones and
spheres.
Gives an informal argument
for the formula for the
circumference of a circle
and area of a circle,
including dissection
arguments.
Identifies the shapes of twodimensional cross-sections
of three-dimensional
objects.
Identifies the shapes of twodimensional cross-sections
of three-dimensional
objects.
Page | 62
Performance Level Descriptors – Geometry
Geometry: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
objects and identifies threedimensional objects
generated by rotations of
two-dimensional objects.
three-dimensional objects
generated by rotations of
two-dimensional objects.
Page | 63
Performance Level Descriptors – Geometry
Geometry: Sub-Claim C
The student expresses course-level appropriate mathematical reasoning by constructing viable arguments,
critiquing the reasoning of others and/or attending to precision when making mathematical statements.
Level 5: Distinguished
Level 4: Strong
Level 3: Moderate
Level 2: Partial Command
Command
Command
Command
Reasoning
HS.C.13.1
HS.C.13.2
HS.C.13.3
HS.C.14.1
HS.C.14.2
HS.C.14.3
HS.C.14.5
HS.C.14.6
HS.C.15.14
HS.C.18.2
Clearly constructs and
communicates a complete
response based on:
Clearly constructs and
communicates a complete
response based on:
Constructs and
communicates a response
based on:
•
•
•
•
•
•
a chain of reasoning to
justify or refute
algebraic and/or
geometric propositions
or conjectures
geometric reasoning in
a coordinate setting, OR
a response to a multistep problem,
•
•
by:
by:
•
•
a chain of reasoning to
justify or refute
algebraic and/or
geometric propositions
or conjectures
geometric reasoning in
a coordinate setting, OR
a response to a multistep problem,
using a logical approach
based on a conjecture
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
providing an efficient
and logical progression
of steps or chain of
reasoning with
appropriate justification
•
•
•
•
a chain of reasoning to
justify or refute
algebraic and/or
geometric propositions
or conjectures
geometric reasoning in
a coordinate setting, OR
a response to a multistep problem,
by :
by:
using a logical approach
based on a conjecture
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
providing a logical
progression of steps or
chain of reasoning with
appropriate
justification
•
•
•
•
Constructs and
communicates an
incomplete response based
on:
• a chain of reasoning to
justify or refute
algebraic and/or
geometric propositions
or conjectures
• geometric reasoning in
a coordinate setting, OR
• a response to a multistep problem,
using a logical approach
based on a conjecture
and/or stated
assumptions
providing a logical, but
incomplete,
progression of steps or
chain of reasoning
performing minor
calculation errors
using some grade-level
vocabulary, symbols
•
•
•
•
using an approach
based on a conjecture
and/or stated or faulty
assumptions
providing an incomplete
or illogical chain of
reasoning, or
progression of steps
making an intrusive
calculation error
using limited gradelevel vocabulary,
Page | 64
Performance Level Descriptors – Geometry
Geometry: Sub-Claim C
The student expresses course-level appropriate mathematical reasoning by constructing viable arguments,
critiquing the reasoning of others and/or attending to precision when making mathematical statements.
Level 5: Distinguished
Level 4: Strong
Level 3: Moderate
Level 2: Partial Command
Command
Command
Command
and labels
symbols and labels
• performing precise
• performing precise
• providing a partial
• providing a partial
calculations
calculations
justification of a
justification of a
• using correct grade• using correct grade•
•
•
level vocabulary,
symbols and labels
providing a justification
of a conclusion
determining whether
an argument or
conclusion is
generalizable
evaluating, interpreting
and critiquing the
validity and efficiency
of others’ responses,
approaches and
reasoning – utilizing
mathematical
connections (when
appropriate) – and
providing a counterexample where
applicable
•
•
level vocabulary,
symbols and labels
providing a justification
of a conclusion
evaluating, interpreting
and critiquing the
validity of others’
responses, approaches
and reasoning –
utilizing mathematical
connections (when
appropriate)
•
conclusion based on
own calculations
evaluating the validity
of others’ approaches
and conclusions
conclusion based on
own calculations
Page | 65
Performance Level Descriptors – Geometry
Modeling
HS.D.1-2
HS.D.2-1
HS.D.2-2
HS.D.2-11
HS.D.3-2
HS.D.3-4
Geometry: Sub-Claim D
The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying
knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,
knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the
Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning
abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure
and/or looking for and expressing regularity in repeated reasoning.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Devises and enacts a plan to
apply mathematics in
solving problems arising in
everyday life, society and
the workplace by:
Devises and enacts a plan to
apply mathematics in
solving problems arising in
everyday life, society and
the workplace by:
Devises and enacts a plan to
apply mathematics in
solving problems arising in
everyday life, society and
the workplace by:
Devises a plan to apply
mathematics in solving
problems arising in
everyday life, society and
the workplace by:
•
•
•
•
•
•
•
•
using stated
assumptions and
making assumptions
and approximations to
simplify a real-world
situation (includes
micro-models)
mapping relationships
between important
quantities
selecting appropriate
tools to create models
analyzing relationships
mathematically
between important
quantities to draw
conclusion
analyzing and/or
•
•
•
using stated
assumptions and
making assumptions
and approximations to
simplify a real-world
situation (includes
micro-models)
mapping relationships
between important
quantities
selecting appropriate
tools to create models
analyzing relationships
mathematically
between important
quantities to draw
conclusions
•
•
•
using stated
assumptions and
approximations to
simplify a real-world
situation
illustrating
relationships between
important quantities
using provided tools to
create models
analyzing relationships
mathematically
between important
quantities to draw
conclusions
•
•
•
•
•
using stated
assumptions and
approximations to
simplify a real-world
situation
identifying important
quantities
using provided tools to
create models
analyzing relationships
mathematically to draw
conclusions
writing an algebraic
expression or equation
to describe a situation
applying proportional
reasoning and
percentages
Page | 66
Performance Level Descriptors – Geometry
Geometry: Sub-Claim D
The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying
knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,
knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the
Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning
abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure
and/or looking for and expressing regularity in repeated reasoning.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
creating constraints,
• interpreting
• interpreting
• applying common
•
•
•
•
•
•
relationships and goals
interpreting
mathematical results in
the context of the
situation
reflecting on whether
the results make sense
improving the model if
it has not served its
purpose
writing a complete,
clear and correct
algebraic expression or
equation to describe a
situation
applying proportional
reasoning and
percentages justifying
and defending models
which lead to a
conclusion
applying geometric
principles and theorems
•
•
•
•
•
•
mathematical results in
the context of the
situation
reflecting on whether
the results make sense
improving the model if
it has not served its
purpose
writing a complete,
clear and correct
algebraic expression or
equation to describe a
situation
applying proportional
reasoning and
percentages
applying geometric
principles and theorems
writing and using
functions in any form to
describe how one
quantity of interest
depends on another
•
•
•
•
•
•
•
•
mathematical results in
a simplified context
reflecting on whether
the results make sense
modifying the model if
it has not served its
purpose
writing an algebraic
expression or equation
to describe a situation
applying proportional
reasoning and
percentages
applying geometric
principles and theorems
writing and using
functions to describe
how one quantity of
interest depends on
another
using statistics
using reasonable
estimates of known
•
•
•
geometric principles
and theorems
using functions to
describe how one
quantity of interest
depends on another
using statistics
using estimates of
known quantities in a
chain of reasoning that
yields an estimate of an
unknown quantity
Page | 67
Performance Level Descriptors – Geometry
Geometry: Sub-Claim D
The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying
knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,
knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the
Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning
abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure
and/or looking for and expressing regularity in repeated reasoning.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
quantities in a chain of
• writing and using
• using statistics
reasoning that yields an
functions in any form to • using reasonable
•
•
describe how one
quantity of interest
depends on another
using statistics
using reasonable
estimates of known
quantities in a chain of
reasoning that yields an
estimate of an unknown
quantity
estimates of known
quantities in a chain of
reasoning that yields an
estimate of an unknown
quantity
estimate of an unknown
quantity
Page | 68