Download Geometry - Benchmarks-Curriculum

Geometry Curriculum Updated 2015
North Bergen School District Benchmarks
Grade: 9, 10, 11, 12
Subject: Geometry
First Marking Period
Students will master the following topics:
1. Understanding the Essentials of Geometry
 Identify points, Lines, and Planes
 Use Segments and Congruency
 Use Midpoint And Distance Formulas
 Measure and Classify Angles
 Describe Angle Pair Relations
HSG-MG.A.3, HSG-CO.A.1
HSG-MG.A.3, HSG-CO.A.1
HSG-MG.A.3, HSG-CO.A.1
HSG.MG.A.3, HSGCO.A.1
HSG-CO.1, HSG-CO.9
2. Use Reasoning and Proofs
 Use Inductive Reasoning. Apply Deductive Reasoning
HSG-CO.1, HSG-CO.9, HSG-CO.10, HSG-CO.11
 Analyze Conditional Statements
HSG-CO.A.1
 Use Postulates and Diagrams (Postulates)
HSG-CO.9, HSG-CO.10, HSG-CO.11
 Reason Using Properties from Algebra HSG-MG.A.3
 Prove Statements about Segments and Angles HSG-CO.A.9
 Prove Angle Pair Relationships
HSG-CO.A.9
3. Investigate Parallel and Perpendicular Lines
 Identify Pairs of Lines and Angles
HSG-MG.A.3, HSG-CO.A.1
 Use Parallel Lines and Transversals
HSG-MG.A.3, HSG-CO.A.9
 Prove Lines are Parallel
HSG-CO.A.9, HSG-CO.A.12
 Find and Use Slopes of Lines
HSG-CO.A.9, HSG-GPE.B.5
 Write and Graph Equations of Lines
HSG-GPE.B.5
 Prove Theorems about Perpendicular Lines
HSG-CO.A.9, HSG-GPE.B.5
Geometry Curriculum Updated 2015
Second Marking Period
Students will master the following topics:
4. Prove Congruent Triangles
 Apply Triangle Sum Properties HSG-MG.A.3, HSG-CO.A.10
 Apply Congruence and Triangles
HSG-MG.A.3, HSG-SRT.B.5
 Prove Triangles Congruent by SSS
HSG-MG.A.3, HSG-CO.A.12, HSG-SRT.B.5
 Prove Triangles Congruent by SAS and HL
HSG-MG.A.3, HSG-SRT.B.5
 Prove Triangles Congruent by ASA and AAS HSG-SRT.B.5
 Use Congruent Triangles
HSG-CO.A.12, HSG-SRT.B.5
 Use Isosceles and Equilateral Triangles
HSG-MG.A.3, HSG-CO.A.10
 Midsegment Theorem and Coordinate Proof
HSG-MG.A.3, HSG-CO.A.10, HSG-GPE.B.4
 Use Perpendicular Bisectors
HSG-C.A.3, HSG-CO.A.9, HSG-CO.A.10,
HSG-CO.A.12
 Use Angle Bisectors of Triangle
HSG-C.A.3, HSG-CO.A.10, HSG-CO.A.12, HSG-GPE.B.4
5. Investigate the Relationships Within Triangles
 Use Medians and Altitudes
HSG-CO.A.10, HSG-CO.A.12, HSG-GPE.B.4
 Use Inequalities in a Triangle
HSG-MG.A.3, HSG-CO.A.10
 Inequalities in Two Triangles and Indirect Proof HSG-CO.A.10
6. Apply the Properties and Theorems of Similarity
 Classify Polygons
HSG-GMD.4, HSG-MG.1
 Use Similar Polygons
HSG-SRT.B.5
 Prove Triangles Similar by AA HSG-SRT.A.3, HSG-SRT.B.5
 Prove Triangles Similar by SSS and SAS
HSG-MG.A.3, HSG-SRT.B.4, HSG-SRT.B.5
Geometry Curriculum Updated 2015
Third Marking Period
Students will be able to:
7. Use Proportionality Theorems
HSG-CO.A.12, HSG-SRT.B.4, HSG-SRT.B.5
8. Apply Properties of Right Triangles and Trigonometry
 Apply the Pythagorean Theorem
HSG-SRT.C.8
 Use the Converse of the Pythagorean Theorem
HSG-SRT.C.8
 Use Similar Right Triangles
HSG-MG.A.3, HSG-SRT.B.4, HSG-SRT.B.5
 Special Right Triangles
HSG-SRT.B.4, HSG-SRT.B.5, HSG-SRT.C.8
 Apply the Tangent Ratio
HSG-SRT.C.6, HSG-SRT.C.8
 Apply the Sine and Cosine Ratios
HSG-SRT.C.6, HSG-SRT.C.7, HSG-SRT.C.8, HSG-SRT.D.9
 Solve Right Triangles
HSG-MG.A.3, HSG-SRT.C.8
9. Understand and Apply Properties of Quadrilaterals
 Find Angle Measures in Polygons
HSG-MG.A.1
 Use Properties of Parallelograms
HSG-MG.A.1, HSG-CO.A.11, HSG-GPE.B.4
 Show that a Quadrilateral is a Parallelogram
HSG-MG.A.1, HSG-CO.A.11, HSG-CO.A.12, HSG-GPE.B.4
 Properties of Rhombuses, Rectangles, and Squares
HSG-MG.A.1, HSG-MG.A.3, HSG-CO.A.11, HSG-GPE.B.4
 Use Properties of Trapezoids and Kites HSG-MG.A.1, HSG-GPE.B.4
 Identify Special Quadrilaterals
HSG-CO.A.11, HSG-GPE.B.4
Geometry Curriculum Updated 2015
Fourth Marking Period
Students will be able to:
10. Apply Properties of Transformations
 Relate Transformation and Congruence HSG-CO.A.2, HSG-CO.A.6, HSG-CO.A.7
 Perform Congruence Transformations
HSG-MG.A.3, HSG-CO.A.2, HSG-CO.A.5, HSG-CO.A.6
 Relate Transformations and Similarity HSG-SRT.1, HSG-SRT.1, HSG-C.1
 Perform Similarity Transformation
HSG-CO.2, HSG-SRT.1, HSG-GPE.4
 Translate Figures and Use Vectors
HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6
 Perform Reflections
HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6
 Perform Rotations
HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6
 Apply Compositions of Transformations
HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6
 Identify Symmetry
HSG-SO.3
 Identify and Perform Dilation HSG-SRT.1
11. Understand and Apply Properties of Circles
 Use Properties of Tangents
HSG-CO.1, HSG-C.1, HSG-C.4(+)
 Find Arc Measures
HSG-CO.1
 Apply Properties of Chords
HSG-CO.12, HSG-C.2, HSG-C.3
 Use Inscribed Angles and Polygons
HSG-C.2, HSG-C.3, HSG-C.4(+), HSG-C.5
 Apply Other Angle Relationships in Circles
HSG-C.2, HSG-C.5
 Find Segment lengths in Circles HSG-C.2
Geometry Curriculum Updated 2015
Domain: Congruence
Cluster: Experiment with transformations in the plane
Standards: (G.CO.1-5)
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
point, line, distance along a line, and distance around a circular arc.
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions
that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to
those that do not (e.g., translation versus horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and
line segments.
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.
Essential Questions
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Enduring Understandings

Why do you need to
know, understand, and
speak in geometric
terms to function in
everyday life?
How does a
transformation change
the image of a figure?
Why do we need to
know that?

Comprehending and
applying basic geometric
terms are important for
future theorems and
postulates
An understanding of
transformations can be
applied in real-life
situations.
Activities, Investigation, and Student Experiences
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Create vocabulary flash cards
Students will draw corresponding examples with proper
notation for given terms.
Given a figure, such as a rectangle, on the coordinate
plane, students will model each transformation
individually.
Students will model basic transformations’
compositions.
Students will investigate transformations using
tessellations.
Interactive Example for distance.
Sample Lessons, Examples, Class work, Homework,
Worksheet
Geometry Curriculum Updated 2015
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Content Statements
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21st Century Life and Careers
Students should define 9.1 Personal Financial Literacy 9.2
basic geometric terms Career Awareness, Exploration, and
Preparation
Students should
identify and draw the
three basic
transformations
Students can represent a
figure’s size without
changing its shape.
Students will change a
figure’s position
without changing its
size and shape.
Assessments
Sample Lessons and Examples
SMART board lessons and examples.
Differentiation
(ELLs, Special Education, Gifted and Talented)
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Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Geometry Curriculum Updated 2015
Task 1:
Each figure below is part of a capital letter in the English
alphabet. To find the whole letter, combine the figure with its
image for the appropriate rotation or reflection. What letter
corresponds to each figure? What transformation produces each
letter?
Answer:
Social Studies:
Map reading
Parkway exits are also ride markers
Parking/Driving (parallel lines and parking)
Career and Life Skills:
Carpentry (perpendicular objects)
Geometry Curriculum Updated 2015
Task 2:
Copy the graph shown below. On the same set of axes, graph the image of
MNOP for a dilation with center (0, 0) and scale factor
2. Use coordinate geometry and the definition of similar
polygons to prove that MNOP is similar to its image.
Answer:
Geometry Curriculum Updated 2015
Geometry Curriculum Updated 2015
Teacher Resources 2012
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Student Participation

Questioning
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Quizzes (Teacher Given and Self Quizzes)
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Benchmark/Test (See Attached)
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Homework
Equipment Needed:
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SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
Teacher Resources:
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Math Open Reference Website:
http://www.mathopenref.com/coorddist.html
Pennsylvania Department of Education Standards
Aligned System:
http://www.pdesas.org/module/content/resources/18110/
view.ashx
Click here for Resource Folder.
http://my.hrw.com/index.jsp
http://www.khanacademy.com
http://prc.parcconline.org
Math Series
Geometry Curriculum Updated 2015
Domain: Congruence
Cluster: Understand congruence in terms of rigid motions
Standards: (G.CO.6-8)
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.
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.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid
motions.
Essential Questions
Enduring Understandings

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What types of motion
in the plane maintain
the congruence of a
figure?

Students will know that rigid
transformations preserve size
and shape or distance and
angles, within the concept of
congruency.
Based on rigid motion,
students will be able to
develop and explain the
triangle congruency based on
the following criteria: ASA,
SSS, and SAS.
Activities, Investigation, and Student Experiences
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Content Statements
21st Century Life and Careers
Given two triangles, prove they are congruent based on
congruency definitions and theorems.
Given three points, students will prove that these are the
vertices of an isosceles triangle, translate this figure 5
units to the right, and prove that the final image is
congruent to the pre-image.
Each student to make a quilt block using congruent right
triangles with specific measure. Then combine all
students work to create a class quilt.
Sample Lessons, Examples, software for triangle
congruence.
Discovery Activity - Angle Sum in Triangles Proof using
Rotation and a Parallel Line
Interactive Congruent Triangles Activity
SMART board lessons and examples
Differentiation
Geometry Curriculum Updated 2015
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Review identification
and definitions of three
basic transformations
including isometries.
Students will apply the
criteria for figure
congruence, i.e.
corresponding parts of
congruent triangles.
Assessments
(ELLs, Special Education, Gifted and Talented)
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

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Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Geometry Curriculum Updated 2015

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (See Attached)

Homework
Equipment Needed:
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











SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
Writing: Math Journal
Teacher Resources:
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Pennsylvania Department of Education Standards
Aligned System:
http://www.pdesas.org/module/content/resources/4890/vi
ew.ashx
Texas Instruments:
http://education.ti.com/calculators/downloads/US/Activit
ies/Detail?id=7381&ref=%2fcalculators%2fdownloads%
2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022
%26sa%3d5024%26t%3d5056
IES Inc. ( International Education Software ):
http://www.ies.co.jp/math/products/geo1/applets/cong/co
ng.html
http://my.hrw.com/index.jsp
http://www.khanacademy.com
http://prc.parcconline.org
Math Series
Geometry Curriculum Updated 2015
Domain: Congruence
Cluster: Prove geometric theorems
Standards: (G.CO.9-11)
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel
lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line
segment are exactly those equidistant from the segment’s endpoints.
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.
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.
Essential Questions
Enduring Understandings
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
What is a geometric
proof?
Why do we have to
prove statements that
have already been
proven by
mathematicians of the
past?
Content Statements

Student will understand that
practicing geometric proofs
teaches the logic of
deductive reasoning.
Activities, Investigation, and Student Experiences
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
21st Century Life and Careers
Students will be able to 9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
apply definitions,
Preparation
theorems, postulates
and properties about
vertical angles, parallel
lines, perpendicular

Students should know that a theorem is different from a
postulate because it is a formula or statement deduced
from a chain of reasoning, or a series of other proofs
already accepted. A postulate, on the other hand, is an
assumed truth based on rational geometric principles.
Different students prefer different types of proofs. Some
take a liking to paragraph proofs so they can provide a
stream-of-consciousness rant and have it be an
acceptable answer. Others prefer the two-column proof
so that all their arguments are clearly laid out in front of
them. Be lenient as to the format of the proof at first, but
make sure your students know what the point of a proof
actually is and what their goal is in writing one.
Knowing basic information and definitions about lines
and angles is crucial, since they'll be the building blocks
of your students' arguments. For instance, many
Geometry Curriculum Updated 2015
lines, bisectors,
triangles, and
parallelograms to
develop and justify a
geometric proof.
Assessments
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Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (Click here for Resources
Folder.)

Homework
theorems concerning lines and angles will make use of
the fact that a straight line is 180°. Vertical angles come
to mind.
 You can let students know that once one proof is worked
out, they can use the result in future proofs. Proofs are
meant to be collected and used when tackling difficult
problems. So we can use the vertical angle proof to
prove another theorem about alternate interior angles.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-co-9.html)
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Students should know the basic definitions that come
with triangles and how to classify them based on angles
and sides. When they can use the words "equiangular"
and "isosceles" in everyday conversation, you'll know
you've done a good job.
Students should also be comfortable with the angles of a
triangle, both interior and exterior. They should know
that all the interior angles of a triangle add up to 180°,
and they should know how to prove it. It's better to
introduce these concepts to them by using concepts they
should already know, like parallel lines and transversals.
But that's just the tip of the triangular iceberg. There's
way more inside triangles than just three interior angles.
For instance, we can fill a triangle with medians, line
segments that join the vertices of a triangle to the
midpoints of its opposite sides. We can also connect the
midpoints of each side in the triangle to form a similar
triangle that's half the size of the original one.
Finally, students shouldn't get lost with all the theorems
and postulates. They all build on each other, and it's best
to keep track of these proofs and postulates so that
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Geometry Curriculum Updated 2015
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Geometry Curriculum Updated 2015
students don't get confused. Also, students should know that using
proofs and theorems they've already learned isn't cheating; it's
applying the skills they've learned, and it's highly encouraged.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-co-10.html)
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Students should learn how to link what they know in a
logical string to prove or disprove an argument. They
should have already done this with lines and triangles, so
the next logical step would be to head right into the
world of parallelograms.
In order to prove theorems about parallelograms,
students might want to know what a parallelogram is.
(Spoiler alert: it's a quadrilateral with opposite sides that
are parallel.) From there, students will want to come up
with an argument to prove, whether it's that opposite
sides of a parallelogram are congruent, opposite angles
are congruent, or that the diagonals bisect each other.
You wouldn't order a grilled cheese just to sit there and
stare at it. (Marveling at the beauty of the perfect grilled
cheese is excusable and even understandable, but not
tasting its delicious cheesy goodness is near
blasphemous.) Just the same, students didn't learn about
parallel lines, transversals, congruent triangles, and
complementary angles for nothing. They should use the
knowledge they already have and apply it to
parallelograms.
If students are struggling, tell them that pictures are
always an excellent way to start proofs. That way, they
should at least be able to get off the ground to start with.
11 | P a ge
Geometry Curriculum Updated 2015
Refreshing their memory about the theorems and
definitions they'll be using might be helpful as well.
Also, using the two-column proof format might help
students organize their thoughts better, at least in the
beginning. Those paragraph proofs can get messy to the
point of uselessness.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-co-11.html)
Differentiation
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Writing: Math Journal
12 | P a ge
Geometry Curriculum Updated 2015
Equipment Needed:













SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
Teacher Resources:
 Math Warehouse Website:
http://www.mathwarehouse.com/geometry/angle/interact
ive-transveral-angles.php
 Click here for Resources Folder.
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series
13 | P a ge
Geometry Curriculum Updated 2015
Domain: Congruence
Cluster: Make geometric constructions
Standards: (G.CO.12-13)
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.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Essential Questions
Enduring Understandings
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What can geometric
constructions teach
you?
How are constructions
different from sketches
and drawings?
Do you need geometric
constructions to erect a
building or other
structure?
Content Statements
 Students will be able to
use tools, e.g. ruler,
compass, protractor,
straightedge, etc. to
copy, bisect, and
construct line segments,
angles, circles and
polygons.
Students will understand
that geometric constructions
are evident and necessary in
the surrounding
environment.
Activities, Investigation, and Student Experiences
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21st Century Life and Careers
9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
Preparation

The standard itself lists a few examples of both the tools
students might be presented with and the tasks they
should be able to perform. However, it takes more than
fancy tools to make these constructions. Students should
keep definitions, properties, and theorems about line
segments, rays, angles, and parallel and perpendicular
lines safe in their back pockets in order to support their
drawings.
Students will benefit from plenty of opportunities to
practice each skill. Even more helpful will be different
contexts for each one. For example, students might first
construct an angle bisector on a single angle, then the
angle bisectors for all three angles of a triangle to show
that they are concurrent, and finally explore whether the
angle bisectors for other polygons are also concurrent.
Once they have mastered using a compass and
straightedge for a particular construction, students will
benefit from exploring using other tools and methods,
such as paper folding or using a mirror, to complete the
same task. This will reinforce their understanding of the
rules of the shape they're working with.
13 | P a ge
Geometry Curriculum Updated 2015
14 | P a ge
Geometry Curriculum Updated 2015
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (See Attached)

Homework

Valentine Construction Assessment (with
Rubric); Click for Resource Folder.
Equipment Needed:
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
SMART board
Projector
Paper and pencil
TI-84 (plus software)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards



Students are expected to construct equilateral triangles,
squares, and circle-inscribed regular hexagons. This
means they need to know the theorems and rules about
these shapes, specifically that each shape is composed of
congruent sides and congruent angles.
Students also need to know how to construct congruent
line segments and congruent angles. They'll need to be
able to draw perpendicular lines, too.
It will be beneficial for students to see demonstrations of
how to construct each shape, particularly noting that the
side length of the hexagon-in-a-circle is equal to the
radius of the circle.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-co-12.html and
http://www.shmoop.com/common-core-standards/ccss-hs-g-co13.html)
Teacher Resources:
 Texas Instruments:
http://education.ti.com/calculators/downloads/US/Activit
ies/Detail?id=13208&ref=%2fcalculators%2fdownloads
%2fUS%2fActivities%2fSearch%2fSubject%3fs%3d502
2%26sa%3d5024%26t%3d5049
 Click for Resource Folder.
 Illuminations - National Council of Teachers of
Mathematics:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=22
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series

14 | P a ge
Geometry Curriculum Updated 2015
Domain: Similarity, Right Triangles and Trigonometry
Cluster: Understand similarity in terms of similarity transformations
Standards: (G.SRT.1-3)
1. Verify experimentally the properties of dilations given by a center and a scale factor.
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the
center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
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.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Essential Questions



How do you recognize
similarity in figures?
What are the effects of
performing dilations on
geometric figures?
How are
transformations and
similar figures related?
Content Statements
Enduring Understandings
Activities, Investigation, and Student Experiences


Students will be able to
perform dilations and
predict the effects of the
dilation

Students will use
compositions of rigid
motions and dilations to help
understand the properties of
similarity.
21st Century Life and Careers
Complete the interactive activities in the following site
and write a reflection based on what you learned.
http://bit.ly/xa7x4d
Differentiation
15 | P a ge
Geometry Curriculum Updated 2015


Students will be able to 9.1 Personal Financial Literacy 9.2
perform dilations given Career Awareness, Exploration, and
Preparation
a center and a scale
factor
Students will identify
similarity
transformations and
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
16 | P a ge
Geometry Curriculum Updated 2015
verify properties of
similarity.
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test

Homework
Equipment Needed:













SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
Interdisciplinary Connections
Photo Editing
Architecture – Blue Prints on actual buildings
Teacher Resources:








http://www.hcstonline.org/hcccmath/math%20912/geometry/g.srt.1-3/resources/G.SRT.13.Similarity%20and%20Dilations.pdf
http://www.shmoop.com/common-core-standards/ccsshs-g-srt-1.html
http://www.shmoop.com/common-core-standards/ccsshs-g-srt-2.html
http://www.shmoop.com/common-core-standards/ccsshs-g-srt-3.html
http://my.hrw.com/index.jsp
http://www.khanacademy.com
http://prc.parcconline.org
Math Series
17 | P a ge
Geometry Curriculum Updated 2015
Domain: Similarity, Right Triangles and Trigonometry
Cluster: Understand similarity in terms of similarity transformations
Standards: (G.SRT.4-5)
4. Prove theorems about triangles.
5. Use congruence and similarity criteria for triangles to solve problems to prove relationships in geometric figures.
Essential Questions



How do you use
proportions to find
side lengths in
similar figures?
How do you show
that two triangles
are similar?
How do you
identify
corresponding parts
of similar triangles?
Content Statements
 Students will form
proportions based
on know lengths of
corresponding
sides.
 Students will use
the AA similarity
postulate and the
Enduring Understandings


Activities, Investigation, and Student Experiences

Students will be able to
show that two triangles
are similar knowing the
relationship between
only two or three pairs of
corresponding parts.

Students will be able to
apply similar triangles to
real-life situations.

21st Century Life and Careers
9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
Preparation

Students should know that in triangles and in life, fair
doesn't necessarily mean equal. The pieces of the triangle
are proportional, but that doesn't always mean congruent.
They each get the right amount according to their own
proportional needs.
As for the Pythagorean Theorem, well, that's not
implicitly about splitting things up, but if we intend to
prove it using similar triangles (which we do), then we
will be explicitly splitting things up. We're talking about
the fact that to prove similarity, we should look at ratios
of side lengths rather than the lengths themselves.
Students will probably want to keep in mind the
importance of being similar. Not to each other—though
they will certainly be on top of the mismatched socks
trend anyway—but the similarity between triangles that
are, by definition, similar.
If they forget the basics of triangle similarity, give them
a quick review of corresponding angles, corresponding
sides, congruence transformations, and similarity
transformations. These are pretty much the fundamental
blocks with which students can build proofs about
triangles.
17 | P a ge
Geometry Curriculum Updated 2015
18 | P a ge
Geometry Curriculum Updated 2015

SAS and SSS
similarity theorems.
Students will be
able to prove that a
line parallel to one
side of a triangle
divides the other
two proportionally.
Assessments

Student Participation

Questioning

Self-Quizzes

Lesson Quizzes (See attached)

Unit Test
Equipment Needed:

Furthermore, students should definitely be able to name
the three tests for identifying similar triangles: AA, SSS,
and SAS. Remind them that when we're talking about
similarity, those S's are more about the right proportions
than they are about equality.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-srt-4.html)


The criteria to determine whether triangles are congruent
or similar include proportionality or congruency of side
lengths and congruency of angles. Fortunately, students
don't have to check all three sides and all three angles
every time they want to test for congruency or similarity.
That would make the process rather inefficient.
There are several ways to combine the side length
proportionality or congruency and angle congruency to
check for triangle congruency or similarity. They're
various combinations and permutations of the letters S
(for side) and A (for angle) and students should already
be familiar with them. In fact, they should have mastered
these congruency versus similarity tests for triangles.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-srt-5.html)
Teacher Resources:
18 | P a ge
Geometry Curriculum Updated 2015


Computer Lab / Projector

SMART board








http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_1.
html
http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_2.
html
http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_3.
html
http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_4.
html
http://www.khanacademy.org/video/similaritypostulates?playlist=Geometry
http://my.hrw.com/index.jsp
http://www.khanacademy.com
http://prc.parcconline.org
Math Series
19 | P a ge
Geometry Curriculum Updated 2015
Domain: Similarity, Right Triangles and Trigonometry
Cluster: Define trigonometric ratios and solve problems involving right triangles
Standards: (G.SRT.6-8)
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.
7. Explain and use the relationship between the sine and cosine of complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Essential Questions
Enduring Understandings



How do you find a side
length or angle measure
in a right triangle?
How do trigonometric
ratios relate to similar
right triangles?
Content Statements

Students will understand
how to use trigonometric
ratios to form proportions
Students will understand
how to apply sine, cosine
and tangent ratios and
Pythagorean theorem to
solve problems with right
triangles
21st Century Life and Careers
Activities, Investigation, and Student Experiences






Interactive examples, PowerPoint lessons, regular level
and challenge level problems, sketchpad activities, and
hands on activities
TI-84 Activity
Videos – Trig. Ratio Introduction, Proof of Pythagorean
Theorem with Similarity, Special Right Triangles
SMART Notebook File – Examples and Problems
Hypsometer Activity
PowerPoint lessons, Interactive Examples, Worksheets –
Sine, Cosine, Tangent
Differentiation
20 | P a ge
Geometry Curriculum Updated 2015



9.1 Personal Financial Literacy 9.2
Students will be able to Career Awareness, Exploration, and
Preparation
explore the sine of an
acute angle and the
cosine of its
complement and
determine their
relationship.
Students will examine
sine, cosine and tangent
ratios.
The student will solve
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
21 | P a ge
Geometry Curriculum Updated 2015
real-world problems
involving right triangles
by using the
Pythagorean Theorem,
properties of special
right triangles, and right
triangle trigonometry.
Assessments

Student Participation

Questioning

Self-Quizzes
o http://teachers.henrico.k12.va.us/math/igo/07
RightTriangles/7-2PythagoreanThm/7-2t.htm
Interdisciplinary Connections
Indirect measurement as related to various fields of student
Astronomy
o http://teachers.henrico.k12.va.us/math/igo/07
RightTriangles/7-3SpecialRtTriangles/7-3.htm
o http://teachers.henrico.k12.va.us/math/igo/07
RightTriangles/7-4Trigonometry/7-4t.htm

Lesson Quizzes (Click here for Resource Folder.)

Unit Test
Equipment Needed:
Teacher Resources:
22 | P a ge
Geometry Curriculum Updated 2015
















SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Straws
Paperclips
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad






http://www.shmoop.com/common-core-standards/ccss-hsg-srt-6.html
http://www.shmoop.com/common-core-standards/ccss-hsg-srt-7.html
http://www.shmoop.com/common-core-standards/ccss-hsg-srt-8.html
Henrico County Public Schools:
o http://teachers.henrico.k12.va.us/math/igo/07
RightTriangles/7_2.html
o http://teachers.henrico.k12.va.us/math/igo/07
RightTriangles/7_3.html
o http://teachers.henrico.k12.va.us/math/igo/07
RightTriangles/7_4.html
Texas Instruments:
o http://education.ti.com/calculators/downloads
/US/Activities/Detail?ID=1855&MICROSIT
E=ACTIVITYEXCHANGE
Khan Academy video archive:
o http://www.khanacademy.org/video/basictrigonometry?playlist=Trigonometry
o http://www.khanacademy.org/video/pythagor
ean-theorem-proof-usingsimilarity?playlist=Geometry
o http://www.khanacademy.org/video/30-6090-triangle-side-ratiosproof?playlist=Geometry
o http://www.khanacademy.org/video/45-4590-triangle-side-ratios?playlist=Geometry
Math Warehouse Website:
o http://www.mathwarehouse.com/trigonometr
y/sine-cosine-tangent-home.php
23 | P a ge
Geometry Curriculum Updated 2015
Domain: Similarity, Right Triangles and Trigonometry
Cluster: Apply Trigonometry to General Triangles
Standards: (G.SRT.9-11)
9. (+) Derive the formula A=½ ab sin (C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the
opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
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).
Essential Questions
 Can trigonometry be
used on non-right
triangles?
 How would you find the
missing measurements
of a non-right triangle
given two
measurements?
Content Statements
Enduring Understandings
 Students will be able to use
the Law of Sines and
Cosines to find missing
angles and side lengths in
non-right triangles.
 Use the Laws of Sines and
Cosines solve real-life
problems.
21st Century Life and Careers
Activities, Investigation, and Student Experiences
 Guided Activities – Students will use right triangle
trigonometry and Pythagorean theorem to develop the Law of
Sines and Law of Cosines
 Hands on Activity – Students will explore the ambiguous
SSA case by building a model
 Interactive examples, PowerPoint lessons, and regular level
and challenge level problems
 Geometer’s Sketchpad Activity - Law of Sines Investigation
 TI-84/TI-Nspire Discovery Activities
 Videos - Proof of Law of Sines and Cosines
 SMART Notebook File – Examples and Problems
 Active Inspire File – Flowchart of Law of Sines and Cosines
Procedure
Differentiation
23 | P a ge
Geometry Curriculum Updated 2015

9.1 Personal Financial Literacy 9.2
In a non-right triangle,
students will be able to Career Awareness, Exploration, and
Preparation
draw an auxiliary line
from a vertex,
perpendicular to the
opposite side and derive
the formula, A=½ ab sin
(C), for the area of a
triangle, using the fact
that the height of the
triangle is, h=a sin(C).
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
24 | P a ge
Geometry Curriculum Updated 2015



Students will be able to
use trigonometry and the
relationship among sides
and angles of any
triangle, such as sin(C) =
(h/a), prove the Law of
Sines.
Students will be able to
trigonometry and the
relationship among sides
and angles of any
triangle and the
Pythagorean Theorem to
prove the Law of
Cosines.
Students will be able to
use the Laws of Sines
and Cosines solve
problems.
Assessments
 Student Participation
 Questioning
 Self Quizzes:
o http://teachers.henrico.k12.va.us/math/ito_08/10Addition
alTrig/10les1/lawofsinessq.htm
o http://teachers.henrico.k12.va.us/math/ito_08/10Addition
alTrig/10les2/lawofcosinessq.htm
 Lesson Quiz (Click here for Resource Folder.)
 Benchmark Assessment (Click here for Resource Folder.)
 Unit Test
Interdisciplinary Connections
Career and Life Skills:
Geometry at work, explore how using trigonometry; a surveyor can find the
latitude, longitude and elevation of each by point in the survey.
24 | P a ge
Geometry Curriculum Updated 2015
Equipment Needed:
Teacher Resources:


Promethean Board / Smart Board

Computer lab access with Geometer’s Sketchpad

Projector for interactive examples

Activity Sheets


TI-84/TI-Nspire Calculators & Activity files


Rulers

Scissors

Markers

Card Stock

Paper fasteners

http://www.shmoop.com/common-core-standards/ccss-hs-gsrt-9.html
http://www.shmoop.com/common-core-standards/ccss-hs-gsrt-10.html
http://www.shmoop.com/common-core-standards/ccss-hs-gsrt-11.html
Illuminations - National Council of Teachers of Mathematics:
o http://illuminations.nctm.org/LessonDetail.aspx?ID=
U177
Henrico County Public Schools:

o http://teachers.henrico.k12.va.us/math/ito_08/10Addit
ionalTrig/10-1LawSines.html
o http://teachers.henrico.k12.va.us/math/ito_08/10Addit
ionalTrig/10-2LawCosines.html
Texas Instruments:

o http://education.ti.com/calculators/downloads/US/Acti
vities/Detail?id=5367
o http://education.ti.com/calculators/downloads/US/Acti
vities/Detail?id=9849
Khan Academy Video Archive:

o http://www.khanacademy.org/video/proof--law-ofsines?playlist=Trigonometry
o http://www.khanacademy.org/video/law-ofcosines?playlist=Trigonometry
25 | P a ge
Geometry Curriculum Updated 2015
Domain: Circles
Cluster: Understand and apply theorems about circles
Standards: (G.C.1-4)
1. Prove that all circles are similar.
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.
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a
circle.
4. (+) Construct a tangent line from a point outside a given circle to the circle.
Essential Questions
Enduring Understandings





Why are all circles
similar?
How can you prove
relationships between
angles in a circle?
When lines intersect a
circle, or within a
circle, how do you find
the measures of
resulting angles?
What is the relationship
between inscribed and
circumscribed figures?


Students will understand the
different components of
circles and how they relate to
angle measurements.
Students will understand that
inscribed angles on a diameter
are right angles.
Students will understand that
the radius of a circle is
perpendicular to the tangent
where the radius intersects the
circle.
Activities, Investigation, and Student Experiences




Students should understand that a circle is a closed curve
on a plane. What's special about circles is that all the
points on a circle are equidistant from the center. That
means circles are defined by two details: position (the
center of the circle) and size (the distance from the
center to a point on the circle).
Students should know that position isn't a problem when
we're talking about similarity or congruence
transformations. If we have two congruent figures at
different positions, translating them will easily map one
on top of the other. But what about size?
Students should know that unlike polygons that have
dimensions independent of one another (base and height,
for instance), a circle's size depends only on one
measurement: the radius r. Sure, we can look at the
diameter d or the circumference C or even the area A, but
they all depend on r as well. (Now would be a good time
to tell them that d = 2r and C = 2πr and A = πr2.)
Since all aspects of a circle's size depend on r, we can
26 | P a ge
Geometry Curriculum Updated 2015
Content Statements





21st Century Life and Careers
Students will be able to 9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
use the fact that the
Preparation
ratio of diameter to
circumference is the
same for circles to
prove that all circles are
similar.
Students will be able to
use definitions,
properties, and
theorems, to identify
and describe relations
among inscribed angles,
radii, and chords
(including central,
inscribed, and
circumscribed angles.)
Students will be able to
construct inscribed
circles of a triangle.
Students will be able to
construct circumscribed
circles of a triangle.
Students will be able to
use definitions,
properties, and
theorems to prove
properties of angles for
a quadrilateral inscribed


change the size of any circle simply by dilating the
radius by a constant scale factor. Students should already
know that dilations, whether they're expansions or
contractions, are similarity transformations. We're
changing the size of the circle, but not its shape.
If dilation changes a circle's size and translation can
change its position, we can easily map one circle onto
another using only those two transformations. Since both
dilation and translation are similarity transformations,
you can safely tell students that all circles are similar. In
fact, we don't even need reflections or rotations to help
us out. (Try rotating and reflecting circles and see how
they change. Hint: they don't.)
Students can also use two circles and proportions to
determine similarity. Two similar shapes will have a
constant ratio when corresponding sides are compared.
Circles don't have sides, but they do have radii,
diameters, and circumferences that we can compare. If
students do that, they'll quicky realize that a small circle
with radius r and a larger circle with radius r' are similar
by a constant scale factor.

Also, we could show in the same way that the area ratio
equals the square of the radius ratio.

The circumferences, diameters, and radii of our two
27 | P a ge
Geometry Curriculum Updated 2015
28 | P a ge
Geometry Curriculum Updated 2015



in a circle.
Students will be able to
find the measure of an
inscribed angle.
Students will be able to
find the measure of an
angle formed by a
tangent and a chord.
Students will be able to
construct a tangent line
from a point outside a
given circle to the
circle.
circles are all in proportion, which means they're similar.
Essentially, this states that circles come in a variety of
sizes, but they'll always be the exact same shape: a
circle.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-c-1.html)

Assessments



Student Participation
Questioning
Quizzes (Teacher Given and Self Quizzes):
o http://teachers.henrico.k12.va.us/math/igo/
08Circles/8-1Terminology/8-1t.htm
o http://teachers.henrico.k12.va.us/math/igo/
08Circles/8-3Tangents/8-3t.htm
o http://teachers.henrico.k12.va.us/math/igo/
08Circles/8-4ArcsChords/8-4t.htm


o http://teachers.henrico.k12.va.us/math/igo/
08Circles/8-5AngleFormulas/8-5t.htm
o http://teachers.henrico.k12.va.us/math/igo/
08Circles/8-6SegmentFormulas/8-6t.htm

Benchmark/Test (Click here for resource folder)
http://sites.google.com/site/ppshighschoolmath/ho

Before students can identify and describe the various
angles in a circle, they should be familiar with what
these angles are. Students should also be familiar with
the concept of arc measurement and how it relates to the
measure of the different kinds of angles. Throw some
chords in there, too, and we aren't talking about your
guitar skills (though we're sure you can rock and roll
with the best of 'em).
Students should know that a central angle is formed by
two radii (where the vertex of the angle is the center of
the circle), an inscribed angle is an angle formed by two
chords (where the vertex of the angle is some point on
the circle), and a circumscribed angle has a vertex
outside the circle and sides that intersect with the circle.
To solidify the relationships between arc measure and
angle measure, we suggest bringing a pizza to class.
Explain that usually, we cut pizzas so that each slice of
pizza has a particular central angle. The amount of crust
(arc measure) depends on the central angle of the pizza
slice.
If you have the whole pizza to yourself, you could even
cut a huge slice from one side to the other, resulting in an
inscribed angle. The measure of that angle will also
affect how much crust you get on your slice.
29 | P a ge
Geometry Curriculum Updated 2015
30 | P a ge
Geometry Curriculum Updated 2015
me/benchmark-assessments/assessment


Homework

Somewhere in between the discussions of angles and
pizza toppings, throw in the formulas for finding the
measures of inscribed and circumscribed angles when
given the central angle or arc measure. Most important
of these is that an inscribed angle's measure is equal to
half the measure of its intercepted arc.
After students have mastered these angles and arc
measures, talk about tangents and secants of circles,
lines that intersect a circle at one and two points,
respectively. It's also important for students to know that
a point on a circle can only have one tangent and that
tangents are always perpendicular to the radius of a
circle.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-c-2.html)

Students should know that an inscribed circle is the
largest circle that can fit on the inside of a triangle, with
the three sides of the triangle tangent to the circle. A
circumscribed circle is one that contains the three
vertices of the triangle. Students should know the
difference between and be able to construct both of these
circles.
29 | P a ge
Geometry Curriculum Updated 2015




Students should understand that while circles have one
defined center, triangles try to outdo them by having four
different centers: the incenter, the circumcenter, the
centroid, and the orthocenter. Imitation is the sincerest
form of flattery and all, but do you think maybe they're
compensating for something?
Students should know the definitions of these centers
and how each one differs from the others. But how do
we get to the true center of the triangle? The world may
never know.
We can draw any triangle by plotting three points on a
circle. Students can take this one point further, plot four
points on a circle, connect them, and make a cyclic
quadrilateral. (They're called that because the vertices
are all on the circle, in kind of a, well, cycle.)
Students be able to prove theorems about cyclic
quadrilaterals, the most important of which is that
opposite angles in a cyclic quadrilateral add to 180°, or
opposite angles in a cyclic quadrilateral are
supplementary.
30 | P a ge
Geometry Curriculum Updated 2015
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-c-3.html)


Equipment Needed:

SMART board
If students don't already know the properties of circles
and tangents before this construction, they should take
away a few main points from it:
o Tangents drawn to a circle are perpendicular to
the circle's radius at the point of tangency.
o Two tangents drawn to a circle from the same
point outside the circle are equal. You can have
students do this construction and measure the
segments from the point of tangency to the
shared point.
o Two tangents drawn to a circle from the same
point outside the circle make an angle that, when
bisected, includes the circles center. Students can
construct the angle bisector and see for
themselves.
o Tangents to a circle at either end of a diameter
are parallel.
Given a circle with center B and a point A outside the
circle, the construction of a tangent to ⊙B that goes
through A is relatively simple. What's important is that
students also understand the properties of circles and
their tangents in order to make these constructions.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-c-4.html)
Teacher Resources:
 Henrico County Public Schools:
 http://teachers.henrico.k12.va.us/math/igo/08Circ
les/8_1.html
31 | P a ge
Geometry Curriculum Updated 2015
32 | P a ge
Geometry Curriculum Updated 2015

Projector

Paper and pencil

Calculator

Compass

Ruler

Protractor

Geometer’s Sketchpad

http://teachers.henrico.k12.va.us/math/igo/08Circ
les/8_3.html
 http://teachers.henrico.k12.va.us/math/igo/08Circ
les/8_4.html
 http://teachers.henrico.k12.va.us/math/igo/08Circ
les/8_5.html
 http://teachers.henrico.k12.va.us/math/igo/08Circ
les/8_6.html
 Math Warehouse:
 http://www.mathwarehouse.com/geometry/circle/
interactive-central-angle-of-circle.php
 http://www.mathwarehouse.com/geometry/circle/
inscribed-angle.html#inscribedAngleDemo
 http://www.mathwarehouse.com/geometry/circle/
tangent-to-circle.php
 Click here for Resource Folder.
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series
32 | P a ge
Geometry Curriculum Updated 2015
Domain: Circles
Cluster: Find arc lengths and areas of sectors of circles
Standards: (G.C.5)
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.
Essential Questions
Enduring Understandings



How is arc length
related to the radius and
central angle of a
circle?
How is the area of a
sector related to the
radius and central angle
of a circle?
Content Statements



Students will understand the
relationship between the arc
length and the radius and
central angle of a circle.
Students will understand the
relationship between the
area of a sector and the
radius and central angle of a
circle.
Activities, Investigation, and Student Experiences


21st Century Life and Careers
Students will be able to 9.1 Personal Financial Literacy 9.2
use similarity to derive Career Awareness, Exploration, and
the fact that the length Preparation
of the arc intercepted
by an angle is
proportional to the
radius and identify the
constant of
proportionality as the
radian measure of the
angle.
Students will be able to
find the arc length of a


Students should first draw a circle with radius r and a
central angle of θ. Visually, it should be clear that the arc
length s depends on both the central angle and the radius.
If we look at the circumference C as one bit arc length,
we can see that its central angle is 360° (or 2π radians).
If C = 2πr, then the arc length should equal the central
angle (in radians) times the radius, or s = θr.
If students aren't as familiar with radians, make sure to
take some time and explain it to them. It can be tricky
going from degrees to radians, but you can use the
circumference equation to translate from one to another.
(We have 2π as our factor but it must come from 360°
somehow. If we say 360k = 2π and solve for k, we have
our conversion factor from degrees to radians. Yippee.)
Students should also be able to derive the formula for the
sector of a circle. Just like we used the C = 2πr to find
arc length, we can use A = πr2 to find the area of a sector.
This time, the 360 degrees translates to π, not 2π. That
means we're dividing the angle by 2, so our formula
should be A = ½θr2.
The key thing to note here is that θ must be in radians.
You can explain to them that a radian is about 57.3° and
that it's the angle at which the ratio of the arc length to
33 | P a ge
Geometry Curriculum Updated 2015


circle.
Students will be able to
use similarity, derive
the formula for the area
of a sector.
Students will be able to
find the area of a sector
in a circle.
the radius is 1:1.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-c-5.html)
Differentiation
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)
o http://teachers.henrico.k12.va.us/math/igo/
08Circles/82AreaCircumference/8_2_circles_arcs_cir
c_area.htm

Benchmark/Test

Homework
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Writing: Math Journal
34 | P a ge
Geometry Curriculum Updated 2015
Equipment Needed:









SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Teacher Resources:

Henrico County Public Schools:
http://teachers.henrico.k12.va.us/math/igo/08Circles/
8_2.html
35 | P a ge
Geometry Curriculum Updated 2015




Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
36 | P a ge
Geometry Curriculum Updated 2015
Domain: Expressing Geometric Properties with Equations
Cluster: Translate between the geometric description and the equation for a conic section
Standards: (G.GPE.1-3)
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center
and radius of a circle given by an equation.
2. Derive the equation of a parabola given a focus and directrix.
3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the
foci is constant.
Essential Questions
Enduring Understandings





How do you find the
equation of a circle in
the coordinate plane?
How do you find the
equation of a parabola
in the coordinate plane?
How do you find the
equation of an ellipse in
the coordinate plane?
How do you find the
equation of a hyperbola
in the coordinate plane?
Content Statements



Students will understand the
differences between the
equations of conic sections.
Students will understand
how the center and the
radius of a circle effect the
equation of a circle.
Students will understand
how the focus and the
directrix of a parabola effect
the equation of a parabola.
Students will understand
how the distances to the foci
of an ellipse effect the
Activities, Investigation, and Student Experiences

Find the equation of a circle centered on point F that has
line segment FG as a radius.
37 | P a ge
Geometry Curriculum Updated 2015





Students will be able to
use the Pythagorean
Theorem to derive the
equation of a circle,
given the center and the
radius.
Students will be able to
complete the square to
find the center and
radius of a circle when
given the general conic
form of a circle.
Students will be able to
derive the equation of a
parabola when given a
focus and directrix of a
parabola.
Students will be able to
identify the vertex,
focus, directrix, and
axis of symmetry and
note that every point on
the parabola is the same
distance from the focus
and the directrix.
Students will be able to
derive the equation of
an ellipse and note that
the sum of the distances
from the foci to any
fixed point on the
ellipse is constant.

(A)(x – 5)2 + (y – 8)2 = 144
equation of an ellipse.
Students will understand
how the distances to the foci
of a hyperbola effect the
equation of a hyperbola.
(B)(x + 5)2 + (y + 8)2 = 144
(C)(x + 5)2 + (y + 8)2 = 169
(D)(x – 5)2 + (y – 8)2 = 169
o Explanation: Here, we have a circle with a center
at (-5, -8) and radius 13. The right side of the
equation must be 13 2 = 169, so (A) and (B) are
incorrect. Since the center is negative for both x
and y variables, the equation should have x + 5
and y + 8 on the left side.
21st Century Life and Careers
9.1 Personal Financial Literacy
9.2 Career Awareness,
Exploration, and Preparation
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-1.html)

Students should know that a focus is a point near a
parabola and a directrix is a line near a parabola. So
what's so special about them? In Mathese, the parabola is
the set of all the points that are the same distance
between two things: a given point and a given line. In
picture form, that would be something like this.
37 | P a ge
Geometry Curriculum Updated 2015



Students will be able to
identify the major and
minor axis of an ellipse.
Students will be able to
derive the equation of a
hyperbola when given
the foci, and note that
the absolute value of
the differences of the
distances from the foci
to a point on the
hyperbola is constant.
Students will be able to
identify the vertices,
center, transverse axis,
conjugate axis, and
asymptotes of a
hyperbola.

Students should know that because the distances are
equal, they find the equation of a parabola by applying
the bulky equation below, where xf and yf are the
coordinates of the focus and either xd or yd can be
substituted with the value of the directrix. The other
value becomes its respective variable (so either xd
becomes x, or yd becomes y).

To help students become more efficient, tell them to
square both sides and then expand them. Then, they can
solve for y to get the equation of the parabola. No sweat
(unless they happen to be on a treadmill or something).
Students can also use the (x – h)2 = 4p(y – k) formula for
a horizontal directrix and (y – k)2 = 4p(x – h) for a
vertical directrix. They should know that (h, k) are the
coordinates of the parabola's vertex and p is the distance
from the focus to the vertex. Remind students that p will
be negative when the focus is either under or to the left
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (See Attached)

Homework

38 | P a ge
Geometry Curriculum Updated 2015
of the directrix.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-2.html)





Focus, vertex, asymptote, transverse, conjugate, standard
position, Pythagorean Theorem—all important geometric
terms that students need to know. On the other hand,
square, simplify, solve for x, and complete the square are
crucial algebraic skills they need to be able to perform.
They should know the equations that describe both
ellipses (which look like elongated circles) and
hyperbolas (which look like a pair of parabolas). For
ellipses or hyperbolas with center (h, k), a major axis of
a, and a minor axis of b, the formulas are:
Students should know that when the major axes are
vertical rather than horizontal, the a2 and b2 values are
switched. Whichever axis is elongated gets the larger
a underneath it.
Chances are students will also want to know the distance
formula. After all, the standard itself features the word
"distances." Lucky for them, the distance formula is still
the same as it always was.
Students should understand that while parabolas have a
single focus, ellipses and hyperbolas have two of them.
39 | P a ge
Geometry Curriculum Updated 2015
(The plural of focus is "foci" so don't let that trip them up.) The
distance from one focus to any point on the ellipse and back to the
other focus will always be constant. The same applies to hyperbolas,
only instead of adding the distances, we subtract them.



Given a set of two foci, students should be able to find
the equation of an ellipse. They can do this by setting up
the equation in the form of two distances calculations
(the distance from one focus to (x, y) and the other focus
to that same point) that equal a constant, or by finding
the values of h, k, a, and b.
It also helps to imagine a right triangle inside the ellipse
with vertices at the center, one of the foci, and at the
vertical vertex. The length of the hypotenuse is half the
constant distance c (the other half is the hypotenuse of
the triangle on the other side), the length of the base
helps find the location of the foci, and the length of the
vertical leg is b (the same b as in the equations above).
Students can also find a by dividing the constant distance
by 2.
You might consider representing the constant distance as
40 | P a ge
Geometry Curriculum Updated 2015


Equipment Needed:






SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
a string loosely connected to two foci. Using the string to
guide your marker, draw the ellipse and explain the
relationship between the values and distances. Bringing
in triangles will help mathematically cement these
relationships. Students should familiarize themselves
with all the intricacies of Ellipses so that they can
actually get around.
The itinerary for Hyperbola City should look about the
same. Students should be able to perform essentially the
same calculations, finding the equation when given the
foci. The only difference is that instead of a sum of two
distances, we're talking about a difference.
If students are struggling, they might benefit from:
o reviewing the algebra used to simplify big
equations, especially completing the square
o considering how the Pythagorean Theorem is
applied here
o focusing on when and where to use addition and
subtraction
o referring to a labeled drawing of the shape in
question
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-2.html)
Teacher Resources:
1. Illuminations - National Council of Teachers of
Mathematics:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L792,
http://illuminations.nctm.org/ActivityDetail.aspx?ID=195
2. Math Warehouse Website:
http://www.mathwarehouse.com/conic-sections/equationsformula-of-conic-sections.php#
41 | P a ge
Geometry Curriculum Updated 2015







Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
42 | P a ge
Geometry Curriculum Updated 2015
Domain: Expressing Geometric Properties with Equations
Cluster: Use coordinates to prove simple geometric theorems algebraically
Standards: (G.GPE.4-7)
4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin
and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use 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).
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*
Essential Questions
Enduring Understandings





How do you use
coordinate geometry to
prove theorems?
What properties of
geometric figures can
be proven using
coordinate geometry?
How can you derive the
distance and midpoint
formulas using
coordinate geometry?
How can you calculate
areas using coordinate
geometry?
Content Statements

Students will understand the
relationship between algebra
and coordinate geometry.
Students will understand the
concept of slope and how it
relates to geometric figures.
Activities, Investigation, and Student Experiences

Students will need to know several major formulas and
equations in order to prove they can hold their own
against Descartes. Here are a few of the most important
equations:
o the distance
formula:
o the slope formula:
o the midpoint
formula:
o the equation of a circle: (x – h)2 + (y – k)2 = r2
o the equation of an
21st Century Life and Careers
ellipse:
43 | P a ge
Geometry Curriculum Updated 2015





Students will be able to 9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
use coordinate
Preparation
geometry to prove
geometric theorems
algebraically.
Students will be able to
use slope to prove lines
are parallel or
perpendicular.
Students will be able to
find equations of lines
based on certain slope
criteria such as finding
the equation of a line
parallel or
perpendicular to a given
line that passes through
a given point.
Students will be able to
find a point on a line
segment that divides the
segment into a given
ratio when given two
points.
Students will be able to
use coordinate
geometry and the
distance formula to find
the perimeters of
polygons and the areas
of triangles and
rectangles.




o the equations of all the rest of the
shapes/functions that can be plotted on the
coordinate plane (parabolas, hyperbolas, etc.)
They'll also need to be pretty fluent in the properties of
geometric shapes. If they don't know that the sum of any
two sides of a triangle is always greater than the
remaining side, or that all four sides of a rhombus are
congruent, they may or may not be in some serious
trouble.
To prove a geometric theorem about a shape on the
coordinate plane, it's not enough for students to know
these properties and formulas. They'll need to be able to
apply them to the coordinate plane as well.
Most statements students need to prove will somehow
involve angles and lengths. For instance, to prove that a
given figure is a square, students could prove that the
slopes of adjacent sides are perpendicular (slope
formula), and that all sides are the exact same length
(distance formula).
There are a great number of possibilities for
algebraically proving the many properties of the many
shapes in the coordinate plane. While students don't have
to know all the ways to prove a particular geometric
theorem, they should be able to come up with at least
one.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-4.html)

Students are expected to know that lines in the same
plane can fall into one of three categories: parallel,
44 | P a ge
Geometry Curriculum Updated 2015
45 | P a ge
Geometry Curriculum Updated 2015
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test

Homework



perpendicular, or neither. Given two lines, they should
be able to prove which of these three categories is
applicable. They should also know how to find the
equation of a line that is parallel or perpendicular to a
given line.
Your students may need a quick refresher on some slope
basics. You know, point your toes together, watch out
for trees, and don't eat yellow snow. That kind of stuff.
A line's slope, of course, is the ratio of rise over run. In
the slope-intercept form of a line's equation, it is the
coefficient on the x term. Positively sloped lines point
up, while negatively sloped ones point down. The greater
the magnitude of the slope, the steeper the line.
Parallel lines, or what second graders would refer to as
"train track lines," have the same slope and never cross
each other. Perpendicular lines, on the other hand, cross
at a right angle and feature slopes that are opposite
reciprocals of each other.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-5.html)


The three main methods that students can use to find the
partition point are as distinct from each other as sitcom
character archetypes. Students should be able to use the
midpoint formula (if the ratio of the parts is 1:1), the
section formula, and the distance formula.
If we're being honest, though, the midpoint formula is
just a special case of the section formula. But it comes
first in the list because it's the easiest (unlike the game
show Pyramid). Just find the averages of the x and y
coordinates to find the midpoint, which gives the
46 | P a ge
Geometry Curriculum Updated 2015
47 | P a ge
Geometry Curriculum Updated 2015
partition point these coordinates.

The section formula is just a fancier version of the
midpoint formula. If a line segment has endpoints (x1, y1)
and (x2, y2), and a partition point P will separate the line
segment into a ratio of m:n, then students should plug the
numbers into the section formula to find the coordinates
of P.

Essentially, the midpoint formula is to finding averages
as the section formula is to finding weighted averages.
Given the endpoints of a line segment, students should
be able to use both formulas to find the midpoint M and
the partition point P at a specified ratio.
Students should also be able to determine the ratio of a
partition using the distance formula. They need to
remember that we are talking about directed line
segments here. So it does matter on which side of the
partition point the bigger segment lies. Remind them to
keep track of which segments they're looking at.

(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-6.html)


Students are expected to be able to use the distance
formula to find the distance between two coordinate
points and then apply that information and know-how to
calculate the perimeter and area of various polygons.
They should note that the distance formula is derived
from the Pythagorean theorem. We suggest squaring
Geometry Curriculum Updated 2015
both sides of the distance formula so it's a little easier to see that
both equations set the sum of two squared terms equal to a third
squared term.





Basically, the distance formula assumes that the distance
we're measuring is the hypotenuse of a right triangle.
The base of the triangle is denoted as a, or (x2 – x1). The
height is b, or (y2 – y1). And the hypotenuse, c, is the
distance, D.
As one might expect, the distance formula is pretty
handy for computing the distance between two
coordinate points. Of course, it's best saved for when
those points are not located along the same vertical or
horizontal, because then simple subtraction would be the
most efficient way to find the distance between them.
No, the distance formula is more aptly used when the
points are spaced diagonally.
Students should know that if we compute the distances
between the points around a polygon, then the distances
can be added to find the polygon's perimeter. Students
should know that this works for all polygons.
Area is a little different, since it depends on the actual
shape. Students should know that pretty much all
polygons on the coordinate plane can be split into
rectangles and triangles. As such, students should know
how to calculate the areas of rectangles and triangles.
Adding the areas of the individual pieces should give
them the area of the entire shape.
We recommend giving students a variety of shapes so
they can apply their knowledge and problem-solving
skills rather than automatically plugging points into a
formula. If students are ever confused, they can always
Geometry Curriculum Updated 2015
plot points on the coordinate plane, too.
(Taken from: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-7.html)
Equipment Needed:













SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
Teacher Resources:
 Math Warehouse Website:
http://www.mathwarehouse.com/algebra/linear_equation
/parallel-perpendicular-lines.php
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series
Geometry Curriculum Updated 2015
Domain: Geometric Measurement and Dimension
Cluster: Explain volume formulas and use them to solve problems
Standards: (G.GMD.1-3)
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Essential Questions
Enduring Understandings



Would we be able to
function in everyday
life without these
measurements?
How would we be able
to cook if did not have
volume?


Content Statements
Students will understand
that consistency in the units
of measure is essential in
describing and solving realworld situations.
Students will understand
that the formula and unit
used to calculate the
measurement is based upon
the reason for which the
measurement is calculated.
Students will understand and
apply the formulas for twoand three-dimensional
figures.
21st Century Life and Careers
Activities, Investigation, and Student Experiences






“Popcorn Cylinders” activity. Calculate and compare
volumes.
Interactive examples, PowerPoint lessons, regular level
and challenge level problems, sketchpad activities, and
hands on activities
Video - Cylinder Volume and Surface Area
Video - Solid Geometry Volume
Video – Volume of a Sphere
Sample Lesson Plans and Examples
Differentiation
Geometry Curriculum Updated 2015


Students will be able to 9.1 Personal Financial Literacy 9.2
calculate the following Career Awareness, Exploration, and
formulas: area, volume, Preparation
perimeter, and
circumference.
Students will be able to
informally justify the
formulas for two- and
three-dimensional
figures.
Assessments
1. Self-quizzes
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Career and Life Skills:
Architecture (Historical)
Marketing
Art and Design
Geometry Curriculum Updated 2015
Equipment Needed:

SMART board

Projector

Paper and pencil

Calculator

Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)

Geometer’s Sketchpad
Teacher Resources:
 Henrico County Public Schools:
o http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9_1.html
o http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9_3.html
o http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9_4.html
 Khan Academy video archive:
o http://www.khanacademy.org/video/cylindervolume-and-surface-area?playlist=Geometry
 Khan Academy video archive:
o http://www.khanacademy.org/video/solidgeometry-volume?playlist=Geometry
 Khan Academy video archive:
o http://www.khanacademy.org/video/volume-of-asphere?playlist=Developmental+Math+2
 http://www.shmoop.com/common-core-standards/ccss-hsg-gmd-1.html
 http://www.shmoop.com/common-core-standards/ccss-hsg-gmd-2.html
 http://www.shmoop.com/common-core-standards/ccss-hsg-gmd-3.html
Geometry Curriculum Updated 2015
Domain: Geometric Measurement and Dimension
Cluster: Visualize relationships between two-dimensional and three-dimensional objects
Standards: (G.GMD.4)
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
Essential Questions
Enduring Understandings


How can we use twoand three-dimensional
figures and their
characteristics to
describe real-world
solids and solve
problems?

Content Statements
Students will understand
that there is application in
the relationships of twodimensional and threedimensional objects in
present careers, e.g.
architecture, construction
trades.
Students will understand
how two-dimensional shapes
are associated with threedimensional forms.
21st Century Life and Careers
Activities, Investigation, and Student Experiences

Jared is scheduled for some tests at his doctor’s office
tomorrow. His doctor has instructed him to drink 3 liters
of water today to clear out his system before the tests.
Jared forgot to bring his water bottle to work and was
left in the unfortunate position of having to use the
annoying paper cone cups that are provided by the water
dispenser at his workplace. He measures one of these
cones and finds it to have a diameter of 7cm and a slant
height (measured from the bottom vertex of the cup to
any point on the opening) of 9.1cm. Note: 1 cm 3 =1 ml
o How many of these cones of water must Jared
drink if he typically fills the cone to within 1cm
of the top and he wants to complete his drinking
during the work day?
o Suppose that Jared drinks 25 cones of water
during the day. When he gets home he measures
one of his cylindrical drinking glasses and finds it
to have a diameter of 7cm and a height of 15cm.
If he typically fills his glasses to 2cm from the
top, about how many glasses of water must he
drink before going to bed?
Differentiation
Geometry Curriculum Updated 2015


9.1 Personal Financial Literacy 9.2
Students will identify
Career Awareness, Exploration, and
attributes of two- and
Preparation
three-dimensional
objects and relate them
to each other.
Students will be able
construct threedimensional objects
using two-dimensional
nets and rotations.
Assessments
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Geometry Curriculum Updated 2015

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (Click here for Resource Folder.)

Homework
Equipment Needed:















SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. models, prisms)
Math journal for each student
Glue or tape
Geo-boards
Geometer’s Sketchpad
Writing: Math Journal
Teacher Resources:





http://map.mathshell.org/materials/download.php?fileid=
828
http://www.illustrativemathematics.org/standards/hs
three-dimensional solid shapes (actual models, or
classroom/home objects that resemble solid shapes) or
create paper models using free printables from the
following Web site
(http://www.senteacher.org/worksheet/12/nets.xhtml)
variety of nets copied for students to construct prisms
and pyramids
(http://www.senteacher.org/worksheet/12/nets.xhtml or
http://mathematics.hellam.net/nets.html)
Illuminations – NCTM National Council of Teachers of
Mathematics:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=20
5
Geometry Curriculum Updated 2015
Domain: Modeling with Geometry
Cluster: Apply geometric concepts in modeling situations
Standards: (G.MG.1-3)
1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).*
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*
3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).*
Essential Questions
Enduring Understandings




How do geometric
shapes relate to
everyday objects?
How do we use area
and volume to explore
the concepts of density?
How can geometric
properties be used in
design?
Content Statements
Students will understand the
relationship between
geometry and real life
situations.
Activities, Investigation, and Student Experiences



21st Century Life and Careers
Find a picture of an object of interest to you. Use a
program such as Paint or PowerPoint to create the object
using only geometric shapes
Research the U.S. at night over time. Create a visual with
no less than four graphics showing the changes in
population density based on images from the evening
sky
Use a computer program to investigate fractals and
discuss the apparent limitations of the objects:
http://www.fractovia.org/
Differentiation
Geometry Curriculum Updated 2015


Students will be able to 9.1 Personal Financial Literacy 9.2
use geometric shapes, Career Awareness, Exploration, and
Preparation
their measures, and
their properties to
describe objects.
Students will be able to
use the concept of
density when referring
to situations involving
area and volume
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Geometry Curriculum Updated 2015

models.
Students will be able to
solve design problems
by designing an object
or structure that
satisfies certain
constraints.
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (See Attached)

Homework
Interdisciplinary Connections
Career and Life Skills:
Occupancy Regulations
Fire safety
Engineering
Code Officer/ Building
Inspector
Geometry Curriculum Updated 2015
Equipment Needed:













SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Compass
Protractor
Ruler
Straightedge
Graph Paper
Isometric Dot Paper
Hands-on and virtual two- and three-dimensional
manipulatives (i.e. prisms)
Geo-boards
Geometer’s Sketchpad
Teacher Resources:
 http://www.yummymath.com/tag/g-mg-1/
 Illuminations - National Council of Teachers of
Mathematics:
http://illuminations.nctm.org/LessonDetail.aspx?id=L79
3
 http://www.shmoop.com/common-core-standards/ccsshs-g-mg-1.html
 http://www.shmoop.com/common-core-standards/ccsshs-g-mg-2.html
 http://www.shmoop.com/common-core-standards/ccsshs-g-mg-3.html
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series

Geometry Curriculum Updated 2015
Domain: Statistics and Probability
Cluster: Understand independence and conditional probability and use them to interpret data
Standards: (S-CP.1-5)
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”).
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.
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.
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.
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.
Essential Questions


How can probability be
used to predict the
outcome of an event?
How can the concepts
of probability be
applied to real life
situations?
Enduring Understandings
Activities, Investigation, and Student Experiences

Geometric Probability Problem
 Have students determine the probability that a point
randomly chosen in a partially shaded square lands in
each of the shaded regions.
 Students should make use of geometric probability
models wherever possible. They use probability to make
informed decisions.
o In your teams, make conjectures about what it

Students will understand
how to organize data and
identify sample space using
frequency tables.
Students will understand
how to measure the
likelihood that an event will
occur.
Geometry Curriculum Updated 2015
Content Statements






Students will be able to
define a sample space
and events within the
sample space.
Students will be able to
identify subsets from
sample space when
given defined events,
including unions,
intersections and
complements of events.
Students will be able to
identify two events as
independent or not.
Students will be able to
explain properties of
Independence and
Conditional
Probabilities in context
and simple English.
Students will be able to
define and calculate
conditional
probabilities.
Students will be able to
use the Multiplication
Principal to decide if
two events are
independent and to
calculate conditional

Students will understand
how the outcome of one
event can affect the
probability of another event.
means to find the area of the shaded region and
how you might determine the shaded area for
each square.
o When ready, in your teams determine the
probability that a randomly chosen point in the
square lies in the shaded region.
o In your teams, try multiple approaches and be
sure to ask the question, does our conjecture
make sense?”
Conditional Probability Problem
 In the following questions, identify two ways to group a
sample (labels for a two-way frequency table) that could
answer the question you are given.
o Do third grade girls prefer chocolate or vanilla
pudding when compared to boys?
o Do kiwi birds live longer than sea turtles?
o Do guavas prefer to have rubber bands, paper
clips, or paint balls shot at them?
o Is there a relationship between high school
students that score well in math and those who
brush their teeth twice daily?
o Is there a relationship between people who snore
at night and people who enjoy strawberry
shortcake?
Geometry Curriculum Updated 2015


probabilities.
Students will be able to
construct and interpret
two-way frequency
tables of data for two
categorical variables
and calculate
probabilities from the
table. They will also be
able to use probabilities
from the table to
evaluate independence
of two variables.
Students will be able to
recognize and explain
the concepts of
independence and
conditional probability
in everyday situations.
21st Century Life and Careers
9.1 Personal Financial Literacy
9.2 Career Awareness, Exploration,
and Preparation
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test

Homework
Differentiation
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
History: Political party comparison
English: Venn Diagrams describing character traits and literature
Technology: Computers and programs such as Excelto create Venn
Diagrams
57 | P a ge
Geometry Curriculum Updated 2015
Equipment Needed:



SMART board
Projector
Paper and pencil
Teacher Resources:
 http://learnzillion.com/lessons/2323-determine-theprobability-of-an-event-and-its-complement
 http://www.mathsisfun.com/pascals-triangle.html
58 | P a ge
Geometry Curriculum Updated 2015


Calculator (Graphing and Scientific)
Ruler









5858
http://www.shmoop.com/common-core-standards/ccsshs-s-cp-1.html
http://www.shmoop.com/common-core-standards/ccsshs-s-cp-2.html
http://www.shmoop.com/common-core-standards/ccsshs-s-cp-3.html
http://www.shmoop.com/common-core-standards/ccsshs-s-cp-4.html
http://www.shmoop.com/common-core-standards/ccsshs-s-cp-5.html
http://my.hrw.com/index.jsp
http://www.khanacademy.com
http://prc.parcconline.org
Math Series
Geometry Curriculum Updated 2015
Domain: Statistics and Probability
Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model
Standards: (S.CP.6-9)
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in
terms of the model.
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.
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.
9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Essential Questions
Enduring Understandings




What is the difference
between combinations
and permutations?
How do you decide
when to use
combinations and
permutations?
In what type of
situations would
compound probability
be used?
Content Statements

5959
Students will be able to
calculate conditional
probabilities using the
definition: ―the
conditional probability
of A given B as the


Students will understand
that the probability of
compound events can be
found by using the
probability of each part of
the compound event.
Students will understand
how to use tables, tree
diagrams, and formulas to
find conditional probability.
Students will understand
when to use permutations or
combinations.
Activities, Investigation, and Student Experiences





Find the expected payoff for a game of chance. For
example, find the expected winnings from a state lottery
ticket or a game at a fast-food restaurant.
Evaluate and compare strategies on the basis of expected
values. For example, compare a high-deductible versus a
low-deductible automobile insurance policy using
various, but reasonable, chances of having a minor or a
major accident.
Modeling Conditional probabilities:
o http://map.mathshell.org/materials/download.php
?fileid=1215
http://www.youtube.com/watch?v=LIM_PSq0LeQ
https://www.khanacademy.org/math/trigonometry/prob_
comb/dependent_events_precalc/v/independent-events-1
Geometry Curriculum Updated 2015





6060
fraction of B’s
outcomes that also
belong to A and
interpret the probability
in context.
Students will be able to
identify two events as
disjoint (mutually
exclusive).
Students will be able to
calculate probabilities
using the Addition Rule
and interpret the
probability in context.
Students will be able to
calculate probabilities
using the General
Multiplication Rule and
interpret in context.
Students will be able to
identify situations as
appropriate for use of a
permutation or
combination to
calculate probabilities.
Students will be able to
use permutations and
combinations in
conjunction with other
probability methods to
calculate probabilities
of compound events
21st Century Life and Careers
9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
Preparation
Differentiation
(ELLs, Special Education, Gifted and Talented)









Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Physical Education and Health: Probabilities in sports
Geometry Curriculum Updated 2015
and to solve problems.
Assessments
6161

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test (Click here for Resource Folder.)

Homework
Geometry Curriculum Updated 2015
Equipment Needed:





6262
SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Ruler
Teacher Resources:
 http://geometrycommoncore.com/index.html
 http://www.mathsisfun.com/pascals-triangle.html
 http://www.shmoop.com/common-corestandards/ccss-hs-s-cp-6.html
 http://learnzillion.com/lessons/2576-understandconditional-probability
 http://mathforum.org/library/drmath/sets/high_perms
_combs.html
 http://www.shmoop.com/common-corestandards/ccss-hs-s-cp-6.html
 http://www.shmoop.com/common-corestandards/ccss-hs-s-cp-7.html
 http://www.shmoop.com/common-corestandards/ccss-hs-s-cp-8.html
 http://www.shmoop.com/common-corestandards/ccss-hs-s-cp-9.html
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series

Geometry Curriculum Updated 2015
Domain: Statistics and Probability
Cluster: Use probability to evaluate outcomes of decisions
Standards: (S.MD.6-7)
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at
a fast- food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
Essential Questions
Enduring Understandings



What does it mean for
an event to be random?
How can probability be
used to make choices
and to help make
decisions based on prior
experience?
Content Statements


6363

Students will understand the
concept of randomness.
Students will understand
how to use probabilities in
decision making.
Activities, Investigation, and Student Experiences

21st Century Life and Careers
Students will be able to 9.1 Personal Financial Literacy 9.2
Career Awareness, Exploration, and
set up a probability
Preparation
distribution for a
random variable
representing payoff
values in a game of
chance.
Students will be able to
make decisions based

Come up with an example or a decision that needs to be
made, and then propose ways by which it can be made.
Are the decisions fair from a probability standpoint? Let
the students decide after numerous tries to ensure
randomness. Are the decisions fair from a moral
standpoint (for instance, randomly deciding who fails the
class based on a coin toss)?
o Toss a fair die
o Drop a fair die
o Toss an unfair die (but don't tell them about it)
o Select a randomly chosen card from a "fresh"
deck of 52
o Flip a coin
o Use lots or straws
o Use a spinning top
o Use a random number generator
Discuss the strengths and weaknesses, randomness or
bias, fairness or unfairness of each method from a
Geometry Curriculum Updated 2015

on expected values
Students will be able to
use expected values to
compare long term
benefits of several
situations.
mathematical standpoint. Discuss methods to undermine
or enhance the fairness of each method, and when such
methods will be useful in making decisions.
Differentiation
Assessments

Student Participation

Questioning

Quizzes (Teacher Given and Self Quizzes)

Benchmark/Test

Homework
(ELLs, Special Education, Gifted and Talented)


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
Extended time
Supplemental Materials, i. e. translation dictionaries
Preferential seating
Learning Centers
Cooperative learning grouping
Independent Study
Teacher & Peer tutoring
Modified assignments
Differentiated instruction
Interdisciplinary Connections
Physical Education: Game decisions
Technology: Excel/graphing calculators to create/ analyze number
generators
6464
Geometry Curriculum Updated 2015
Equipment Needed:

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6565
SMART board
Projector
Paper and pencil
Calculator (Graphing and Scientific)
Ruler
Teacher Resources:
 http://ccssmath.org/?page_id=2400
 http://www.shodor.org/interactivate/activities/Adjustable
Spinner/
 http://www.shodor.org/interactivate/activities/Advanced
MontyHall/
 http://www.shmoop.com/common-core-standards/ccsshs-s-md-6.html
 http://www.shmoop.com/common-core-standards/ccsshs-s-md-7.html
 http://my.hrw.com/index.jsp
 http://www.khanacademy.com
 http://prc.parcconline.org
 Math Series