Download Geometry 2015 - Shore Regional High School

Course Title: Geometry
Content Area: Mathematics
Grade Level(s): 10
Course Description: Throughout this Geometry course, students will investigate the properties of geometric figures and solids. Congruence,
similarity, and measurement are among the topics to be examined for both two-dimensional figures and three-dimensional solids. Analytic
geometry will relate numerical coordinates to geometric points, thus allowing a deeper analysis of geometric properties. In this course,
students will develop spatial sense through experiences that enable them to recognize, visualize, categorize, represent, and transform
geometric shapes.
Curriculum Writer(s): Nicole Mazza/Carol A. Burkley
Date Created: 6/29/15
Date Approved by Board of Education: October of 2015
Pacing Guide
Unit 1A: Foundations of Geometry
Unit 1B: Reasoning
Unit 1C: Congruence
Unit 1D: Polygons and Quadrilaterals
Unit 1E: Transformations
Unit 1F: Area and Perimeter
Unit 2: Similarity
Unit 3: Triangles
Unit 4: Circles
Unit 5: Spatial Reasoning
11-15 days
5-7 days
8-10 days
6-8 days
4-6 days
4-6 days
6-7 days
11-15 days
8-10 days
Unit 6: Probability
3-5 days
2-4 days
Unit 1(A) - Foundations of Geometry
Unit Summary: Unit 1 will focus on triangle congruence conditions and establish their usage using analysis of rigid motion and formal
constructions. Various formats will be used to prove theorems about angles, lines, triangles, and other polygons.
Interdisciplinary Connections/Content Area Integrations Including TechnologyEngineering: Adjusting the position of a surveying instrument on a tripod.
Geography: Find the distance between two points on a map.
Physical Education: Parallel bars in gymnastics.
Physics: The angle that sunlight hits a rain drop to create a prism.
Biology: The patterns on a snake are of intersecting lines.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.CO.A1
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.
CCSS.MATH.CONTENT.HSG.CO.A3
Given a rectangle. parallelogram, trapezoid, or regular polygon,
describe the rotations and reflections that carry it onto itself.
CCSS.MATH.CONTENT.HSG.CO.A4
Develop definitions of rotations, reflections, and translations in
terms of angles, circles, perpendicular lines, parallel lines, and
line segments.
CCSS.MATH.CONTENT.HSG.CO.A5
Summative Assessments:
Homework, Quizzes, Tests, Projects, Written Assignments, Posters,
Models, Research Assignments, Mid-Term Exam
Formative Assessments:
Do-nows, Partner/Group Work, Homework, Open notes quiz,, Board Work,
Mini white boards, Exit tickets, Review games, Teacher observations
Enduring Understandings:
Students will understand that:
*Points, lines, and planes are used to create plane objects.
*Transformations of figures in a coordinate plane can be described
verbally and visually.
*Coordinate geometry is a way to describe and measure geometric figures
on a coordinate plane.
*The distance formula gives the measure of a segment connecting two
points.
*Equations can be used to describe the relationship between measures of
geometric figures.
*Calculating area, perimeter, and circumference are formula processes.
*When a transversal crosses parallel lines, different pairs of congruent
angles are formed.
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.
*Parallel lines define specific angle-pair relationships.
*Relationships between lines (parallel, perpendicular, or neither) are
based on their slopes.
*The equation of a line can be determined from points and/or slopes.
Essential Questions:
*What are points, lines, and planes, and why are they considered
undefinable?
*How are points, lines, and planes used to create both plane figures and
solid objects?
*What are transformations, and what do they do to geometric objects and
figures?
*How do you compute the area and perimeter of polygons using
coordinate formulas?
*What angle relationships are created when a transversal intersects
parallel lines?
Instructional Outcomes:
Students will be able to:
*Define points, lines, and planes and use them to create both plane and
solid objects.
*Use a combination of translations and reflections to transform geometric
objects.
*Compute the area and perimeter of polygons using coordinate formulas.
*Calculate the slope of a line between 2 points.
*Prove lines parallel (or not parallel) based on their slopes.
*Prove lines perpendicular (or not perpendicular) based on their slopes.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board
notes/presentations), Board work, Investigation activities,
Partner/Group work, Real World
Connections, Individual practice, Flipped classroom, Exit tickets, Review
activities and games, Mini presentations, Question and Answer - using
wait
time, Use of manipulatives, Daily quiz
Curriculum Development Resources:
May include, but not limited to:
*Textbook: Geometry - Prentice Hall Mathematics
*Textbook resource materials
*Teacher prepared guided notes
*Daily Do-Nows
*Graphing Calculator
*GeoGebra
*Garnet Valley School District Geometry Curriculum
*Internet resources/websites
Notes/Comments:
Unit 1(B) - Geometric Reasoning
Unit Summary: Unit 1 will focus on triangle congruence conditions and
establish their usage using analysis of rigid motion and formal
constructions. Various formats will be used to prove theorems about
angles, lines, triangles, and other polygons.
Interdisciplinary Connections/Content Area Integrations Including Technology*Home Economics: Scissors form vertical angles.
*Construction/Architecture/Engineering: Perpendicular and parallel lines.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically.
CCSS.MATH.CONTENT.HSG.CO.C.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.
Summative Assessments:
Homework, Quizzes, Tests, Projects, Written Assignments, Posters,
Models, Research Assignments, Mid-Term Exam
Formative Assessments:
Do-nows, Partner/Group Work, Homework, Open notes quiz,,
Board Work, Mini white boards, Exit tickets, Review games,
Teacher observations
Enduring Understandings:
Students will understand that:
*The measures of angles created by intersecting lines are related.
*One counterexample can prove that a statement is false.
*Arguments are built by linking definitions, conditional
statements, properties, and previously proven theorems.
Essential Questions:
*What is the difference between inductive and deductive
reasoning?
*What is a conditional statement?
*How can we use inductive and deductive reasoning to make and
verify conjectures?
Instructional Outcomes:
Students will be able to:
*Draw conclusions based on fact.
*Identify the hypothesis and conclusion of a conditional statement.
*Draw conclusions based on patterns.
*Use the substitution property when needing to show that a
quantity can be used in place of an equal quantity.
*Use the addition, subtraction, multiplication, and division
properties of equality when needing to show that performing the
same operation to two equal quantities results in two equal
quantities.
*Use the reflexive property when needing to show that any
quantity is equal to itself.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board
notes/presentations), Board work, Investigation activities,
Partner/Group work, Real World
Connections, Individual practice, Flipped classroom, Exit tickets,
Review activities and games, Mini presentations, Question and
Answer - using wait
time, Use of manipulatives, Daily quiz
Curriculum Development Resources:
May include, but not limited to:
*Textbook: Geometry - Prentice Hall Mathematics
*Textbook resource materials
*Teacher prepared guided notes
*Daily Do-Nows
*Graphing Calculator
*GeoGebra
*Garnet Valley School District Geometry Curriculum
*Internet resources/websites
Notes/Comments:
Unit 1 (C) - Triangle Congruence
Unit Summary: Unit 1 will focus on triangle congruence conditions
and establish their usage using analysis of rigid motion and formal
constructions. Various formats will be used to prove theorems
about angles, lines, triangles, and other polygons.
Interdisciplinary Connections/Content Area Integrations Including Technology*Sports: Cheerleaders use triangular shapes for support of their formations.
*Art/Jewelry Making: Triangular shapes are used in the creation of artistic pieces.
*Construction/Architecture/Engineering: Structural support beams often use the triangular shape.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.CO.C.10
Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180 degrees; 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.
CCSS.MATH.CONTENT.HSG.CO.B.6
Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a 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.
CCSS.MATH.CONTENT.HSG.CO.B.8
Explain how the criteria for triangle congruence (ASA, SAS and SSS)
follow from the definition of congruence in terms of rigid motions.
CCSS.MATH.CONTENT.HSG.CO.B.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.
Summative Assessments:
Homework, Quizzes, Tests, Projects, Written Assignments, Posters,
Models, Research Assignments, Mid-Term Exam
Formative Assessments:
Do-nows, Partner/Group Work, Homework, Open notes quiz,, Board
Work, Mini white boards, Exit tickets, Review games, Teacher
observations
Enduring Understandings:
Students will understand that:
*Triangles can be classified based on their sides and/or angles.
*Congruence of figures can be proven using corresponding parts’
congruence criteria.
*Once two triangles are proven to be congruent, it can be
concluded that their remaining corresponding parts are also
congruent.
*The interior angles of figures have specific sums based upon the
number of their sides.
Essential Questions:
*How can congruent triangles be used to prove properties of
isosceles triangles, midsegments, and medians?
*How are coordinates used to prove simple geometric theorems
algebraically?
*How are the criteria for triangle congruence (AAS, ASA, SAS, SSS,
and HL) used with the definition of congruence in terms of rigid
motions?
Instructional Outcomes:
Students will be able to:
*Prove triangles are congruent using corresponding parts
congruence criteria: ASA, SSS, AAS, SAS, and HL.
*Prove and interpret the following theorems**The measures of interior angles of a triangle have the sum of
180 degrees.
**The 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 its length.
**The medians of a triangle meet at a point.
*Prove that geometric figures, other than triangles, can be
congruent.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board
notes/presentations), Board work, Investigation activities,
Partner/Group work, Real World
Connections, Individual practice, Flipped classroom, Exit tickets,
Review activities and games, Mini presentations, Question and
Answer - using wait
time, Use of manipulatives, Daily quiz
Curriculum Development Resources:
May include, but not limited to:
*Textbook: Geometry - Prentice Hall Mathematics
*Textbook resource materials
*Teacher prepared guided notes
*Daily Do-Nows
*Graphing Calculator
*GeoGebra
*Garnet Valley School District Geometry Curriculum
*Internet resources/websites
Notes/Comments:
Unit 1 (D) - Polygons and Quadrilaterals
Unit Summary: Unit 1 will focus on triangle congruence conditions
and establish their usage using analysis of rigid motion and formal
constructions. Various formats will be used to prove theorems
about angles, lines, triangles, and other polygons.
Interdisciplinary Connections/Content Area Integrations Including Technology*Photography: Photographs can be proportionally enlarged or reduced in size.
*Art: The use by artists to change the proportions of a picture/painting.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals.
CCSS.MATH.CONTENT.HSG.GPE.B.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).
CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically.
Summative Assessments:
Homework, Quizzes, Tests, Projects, Written Assignments, Posters, Models, Research Assignments, Mid-Term Exam
Formative Assessments:
Do-nows, Partner/Group Work, Homework, Open notes quiz,, Board Work, Mini white boards, Exit tickets, Review games, Teacher
observations
Enduring Understandings:
Students will understand that:
*Special quadrilaterals have additional properties to those of parallelograms.
*Congruent triangles can be used to prove some properties of special quadrilaterals.
*Relationships between segments and angles in quadrilaterals and parallelograms can be expressed using equations.
Essential Questions:
*How are the opposite sides of a parallelogram related?
*What properties are true of all quadrilaterals?
*What properties are true of all parallelograms?
*What properties are true of all rectangles?
*What properties are true of all rhombi?
*What relationship do the diagonals of a parallelogram have?
Instructional Outcomes:
Students will be able to:
*Classify polygons based upon their sides and angles.
*Find and use measures of interior and exterior angles of polygons.
*Use properties of parallelograms and special quadrilaterals to solve problems.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/Group work,
Real World
Connections, Individual practice, Flipped classroom, Exit tickets, Review activities and games, Mini presentations, Question and Answer using wait time, Use of manipulatives, Daily quiz
Curriculum Development Resources:
May include, but not limited to:
*Textbook: Geometry - Prentice Hall Mathematics
*Textbook resource materials
*Teacher prepared guided notes
*Daily Do-Nows
*Graphing Calculator
*GeoGebra
*Garnet Valley School District Geometry Curriculum
*Internet resources/websites
Notes/Comments:
Unit 1(E) - Transformational Geometry
Unit Summary: Unit 1 will focus on triangle congruence conditions and establish their usage using analysis of rigid motion and formal
constructions. Various formats will be used to prove theorems about angles, lines, triangles, and other polygons.
Interdisciplinary Connections/Content Area Integrations Including Technology:*Music/Performance Arts: Translations are used in creating performance arrangements.
*Biology: Diatoms are examples of symmetry in nature.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a 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.
CCSS.MATH.CONTENT.HSG.CO.A3
Given a rectangle. parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that
carry it onto itself.
CCSS.MATH.CONTENT.HSG.CO.A4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
CCSS.MATH.CONTENT.HSG.CO.A5
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.
CCSS.MATH.CONTENT.HSG.MG.A.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).
Summative Assessments:
Homework, Quizzes, Tests, Projects, Written Assignments, Posters, Models, Research Assignments, Mid-Term Exam
Formative Assessments:
Do-nows, Partner/Group Work, Homework, Open notes quiz, Board Work, Mini white boards, Exit tickets, Review games, Teacher
observations
Enduring Understandings:
Students will understand that:
*In an isometry, the pre-image and image are congruent.
*A dilation is a transformation that changes the size of a figure, but not its shape.
*A tessellation is a repeating pattern that covers a plane.
Essential Questions:
*How do you know when a transformation is a reflection? A rotation? A translation? A dilation?
*How do you locate a figure’s line of symmetry?
*How is a rotation different than a reflection?
Instructional Outcomes:
Students will be able to:
*Identify and draw reflections.
*Identify and draw translations.
*Identify and draw dilations.
*Identify and draw lines of symmetry.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/Group work,
Real World
Connections, Individual practice, Flipped classroom, Exit tickets, Review activities and games, Mini presentations, Question and Answer using wait
time, Use of manipulatives, Daily quiz
Curriculum Development Resources:
May include, but not limited to:
*Textbook: Geometry - Prentice Hall Mathematics
*Textbook resource materials
*Teacher prepared guided notes
*Daily Do-Nows
*Graphing Calculator
*GeoGebra
*Garnet Valley School District Geometry Curriculum
*Internet resources/websites
Notes/Comments:
Unit 1(F) - Area, Circumference, and Perimeter
Unit Summary: Unit 1 will focus on triangle congruence conditions and establish their usage using analysis of rigid motion and formal
constructions. Various formats will be used to prove theorems about angles, lines, triangles, and other polygons.
Interdisciplinary Connections/Content Area Integrations Including Technology*Physics: The circumference of the path traveled by an object in a circular motion is used to calculate the linear velocity of the object.
*Architecture/Construction/Design: Computation of areas and/or perimeters of spaces or objects.
*Art: Framing a picture uses both area and perimeter formulas.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.
CCSS.MATH.CONTENT.HSG.CO.A1
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.
CCSS.MATH.CONTENT.HSG.MG.A.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).
Summative Assessments:
Homework, Quizzes, Tests, Projects, Written Assignments, Posters, Models, Research Assignments, Mid-Term Exam
Formative Assessments:
Do-nows, Partner/Group Work, Homework, Open notes quiz, Board Work, Mini white boards, Exit tickets, Review games, Teacher
observations
Enduring Understandings:
Students will understand that:
*Analyzing data and applying the principles of probability enable us to make informed decisions and conclusions.
*The ratio of the circumference of a circle to its diameter is always a constant (pi).
Essential Questions:
*How can you find the area of a figure?
*How can you find the shaded area of a figure?
*How can you find the perimeter of a figure?
Instructional Outcomes:
Students will be able to:
*Apply formulas for the areas and perimeters of triangles and special quadrilaterals.
*Apply formulas for the area and circumference of circles.
*Use area and perimeter formulas to solve problems.
*Calculate geometric probabilities.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/Group work,
Real World
Connections, Individual practice, Flipped classroom, Exit tickets, Review activities and games, Mini presentations, Question and Answer using wait
time, Use of manipulatives, Daily quiz
Curriculum Development Resources:
May include, but not limited to:
*Textbook: Geometry - Prentice Hall Mathematics
*Textbook resource materials
*Teacher prepared guided notes
*Daily Do-Nows
*Graphing Calculator
*GeoGebra
*Garnet Valley School District Geometry Curriculum
*Internet resources/websites
Notes/Comments:
Unit 2: Similarity
Unit Summary: Unit 2 will build on a student’s understanding of transformations, including dilations and proportional reasoning, to
develop an understanding of similarity.
Interdisciplinary Connections/Content Area Integrations Including Technology:
Art/Architecture: The Golden Ratio is a ratio prevalent in many ancient structures and works of art.
Engineering: Similar polygons and solids are used to create scale diagrams, blueprints, and models.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.SRT.A.1
Verify experimentally the properties of dilations given by a center and a scale factor
CCSS.MATH.CONTENT.HSG.SRT.A.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if
they are similar; explain using similarity transformations the meaning of similarity for triangles as
the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs
of sides.
CCSS.MATH.CONTENT.HSG.SRT.A.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to
be similar.
CCSS.MATH.CONTENT.HSG.SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
CCSS.MATH.CONTENT.HSA.CED.A.1
Create equations and inequalities in one variable and use them to solve problems.
Summative Assessments:
Homework, Quizzes, Tests, Final exam
Formative Assessments:
Do-nows, Partner/Group work, Homework, Open-notes quiz, 3x summarization, Mini white boards, Board work, Review games, Exit ticket,
Teacher observation
Enduring Understandings:
Students will understand what it means for figures to be similar and how that similarity can be determined. They will also understand the
connection between similarity and transformations in the plane.
Essential Questions:
 How are proportions used to verify similarity between objects?
 How is the concept of similarity applied to scale drawings?
 What are the properties of similar polygons, and how can they be used to find missing measures?
 What information is needed to determine if triangles are similar?
 How can you use a scale factor to determine the image of an object under a dilation?
Instructional Outcomes:
Students will be able to:
 identify similar polygons.
 use proportions to find missing lengths in similar polygons.
 calculate the area and perimeters of similar polygons.
 use properties of transformations to determine similarity.
 apply similarity properties in the coordinate plane.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/group work,
Individual practice, Flipped classroom, Exit tickets, Review activities and games
Curriculum Development Resources:
May include, but are not limited to:
 Text: Geometry - Prentice Hall Mathematics
 Text Resource materials
 Guided notes (teacher prepared)
 GeoGebra
 Garnet Valley School District Geometry Curriculum
Notes/Comments:
Unit 3: Triangles
Unit Summary: Unit 3 will explore the properties of triangles. The concept of similarity will extend to right triangles, and right triangle
trigonometry will be used to find missing measures (both angles and lengths) in right triangles. The Law of Sines and the Law of Cosines
will allow students to find missing measures in any triangles.
Interdisciplinary Connections/Content Area Integrations Including Technology:
Engineering: Trigonometry can be used in constructions and used as a means of measuring.
Physics: Vector problems require the use of trigonometry to solve.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.SRT.B.4
Prove theorems about triangles.
CCSS.MATH.CONTENT.HSG.SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
CCSS.MATH.CONTENT.HSG.SRT.C.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.
CCSS.MATH.CONTENT.HSG.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles.
CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
CCSS.MATH.CONTENT.HSG.SRT.D.11
Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically.
CCSS.MATH.CONTENT.HSG.GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.
CCSS.MATH.CONTENT.HSG.MG.A.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).
CCSS.MATH.CONTENT.HSG.MG.A.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).
Summative Assessments:
Homework, Quizzes, Tests, Final exam
Formative Assessments:
Do-nows, Partner/Group work, Homework, Open-notes quiz, 3x summarization, Mini white boards, Board work, Review games, Exit ticket,
Teacher observation
Enduring Understandings:
Students will understand side, angle, and segment relationships in different types of triangles.
Essential Questions:
 What is the relationship between sides and angles in a triangle?
 How are the properties of midsegments, medians, altitudes, angle bisectors, and perpendicular bisectors useful?
 What are the properties of isosceles and equilateral triangles, and how can they be applied?
 What information is needed to solve a triangle?
 How is it determined which trigonometric ratio will be used to find a missing length or angle measure in a right triangle?
Instructional Outcomes:
Students will be able to:
 identify different types of triangles and apply the properties of each.
 identify different segments in triangles and apply the properties of each.
 find missing segment lengths and angle measures in right triangles by applying trigonometric ratios.
 solve triangles using right triangle trigonometry, the Law of Sines, and/or the Law of Cosines.
 apply the Pythagorean Theorem to find missing sides in right triangles.
 classify triangles using the converse of the Pythagorean Theorem.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/group work,
Individual practice, Flipped classroom, Exit tickets, Review activities and games
Curriculum Development Resources:
May include, but are not limited to:
 Text: Geometry - Prentice Hall Mathematics
 Text Resource materials
 Guided notes (teacher prepared)
 GeoGebra
 Garnet Valley School District Geometry Curriculum
Notes/Comments:
Unit 4: Circles
Unit Summary: Unit 4 will focus on the properties of circles and their applications. Unit 4 will also explore segments and angles in/on
circles.
Interdisciplinary Connections/Content Area Integrations Including Technology:
Economics: Circle charts are often used to represent data.
Art: Circles are often used in various types of art.
Biology: Circles and angles are often related to vision issues.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.C.A.1
Prove that all circles are similar.
CCSS.MATH.CONTENT.HSG.C.A.2
Identify and describe relationships among inscribed angles, radii, and chords.
CCSS.MATH.CONTENT.HSG.C.A.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles
for a quadrilateral inscribed in a circle.
CCSS.MATH.CONTENT.HSG.C.B.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.
CCSS.MATH.CONTENT.HSG.GPE.A.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.
CCSS.MATH.CONTENT.HSG.GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.*
CCSS.MATH.CONTENT.HSG.MG.A.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).
CCSS.MATH.CONTENT.HSG.MG.A.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).
CCSS.MATH.CONTENT.HSA.REI.D.10
Understand that the graph of an equation in two variables is the set of all its solutions plotted in
the coordinate plane, often forming a curve (which could be a line).
Summative Assessments:
Homework, Quizzes, Tests, Final exam
Formative Assessments:
Do-nows, Partner/Group work, Homework, Open-notes quiz, 3x summarization, Mini white boards, Board work, Review games, Exit ticket,
Teacher observation
Enduring Understandings:
Students will understand that segment lengths are dependent on the location of the segments in/on a circle, and angle measures are
dependent on the types of segments forming the angles and the location of the angles themselves.
Essential Questions:
 How do tangents, chords, and secants differ? How are they similar?
 How can the measure of a central angle be calculated? How can the measure of an inscribed angle be calculated?
 How can the formula for arc length and sector area be derived?
 What information is needed to determine the equation of a circle?
 How do the different values of h, k, and r in the equation of a circle change the graph of a circle?
Instructional Outcomes:
Students will be able to:
 identify segments in circles (radii, diameters, tangents, chords, secants).
 identify angles (central, inscribed).
 determine angle and arc measures.
 determine segment lengths.
 determine arc lengths.
 determine sector areas.
 graph circles in the coordinate plane given the equation of the circle.
 determine the equation of a circle based on various given information.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/group work,
Individual practice, Flipped classroom, Exit tickets, Review activities and games
Curriculum Development Resources:
May include, but are not limited to:
 Text: Geometry - Prentice Hall Mathematics
 Text Resource materials
 Guided notes (teacher prepared)
 GeoGebra
 Garnet Valley School District Geometry Curriculum
Notes/Comments:
Unit 5: Spatial Reasoning
Unit Summary: Unit 5 will expand on the knowledge of two-dimensional objects in order to explore properties of three-dimensional
objects.
Interdisciplinary Connections/Content Area Integrations Including Technology:
Art: Properties of three-dimensional objects can be used to create 3-dimensional art pieces (ceramics, modeling, etc…).
Science: Volumes are important in creating chemical reactions.
Architecture/Design: Planners and builders must have knowledge of surface areas.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
CCSS.MATH.CONTENT.HSG.GMD.B.4
Identify the shapes of two-dimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects generated by rotations of twodimensional objects.
CCSS.MATH.CONTENT.HSG.MG.A.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).
CCSS.MATH.CONTENT.HSG.MG.A.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)
CCSS.MATH.CONTENT.HSF.IF.C.8
Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
Summative Assessments:
Homework, Quizzes, Tests, Edible Solids project, Final exam
Formative Assessments:
Do-nows, Partner/Group work, Homework, Open-notes quiz, 3x summarization, Mini white boards, Board work, Review games, Exit ticket,
Teacher observation
Enduring Understandings:
Students will understand that area, surface area, lateral area, and volume have many real-life applications. They will also recognize that many
polygons and polyhedrons have common features based on their common characteristics.
Essential Questions:
 How can surface area and volume be calculated for composite solids?
 How does changing a dimension of a solid affect its surface area and volume?
 How are the volumes and surface areas of similar solids related?
Instructional Outcomes:
Students will be able to:
 classify polyhedrons and other solids.
 determine the surface area, lateral area, and volumes of three-dimensional solids (including composite solids).
 determine the surface area, lateral area, and volumes of similar solids.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/group work,
Individual practice, Flipped classroom, Exit tickets, Review activities and games
Curriculum Development Resources:
May include, but are not limited to:
 Text: Geometry - Prentice Hall Mathematics
 Text Resource materials
 Guided notes (teacher prepared)
 GeoGebra
 Garnet Valley School District Geometry Curriculum
Notes/Comments:
Unit 6: Probability (optional, if time permits)
Unit Summary: Unit 6 will focus on basic probabilities and geometric probabilities.
Interdisciplinary Connections/Content Area Integrations Including Technology:
Meteorology: Meteorologists have to predict the likelihood of a weather event.
Biology: The effectiveness of medicines and treatments are often given in terms of probabilities.
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.HSG.MG.A.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).
CCSS.MATH.CONTENT.HSS.CP.A.2
Understand that two events A and B are independent if the probability of A and B
Occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.
CCSS.MATH.CONTENT.HSS.CP.A.5
Recognize and explain the concepts of conditional probability and independence
in everyday language and everyday situations.
CCSS.MATH.CONTENT.HSS.CP.B.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.
Summative Assessments:
Homework, Quizzes, Tests, Final exam
Formative Assessments:
Do-nows, Partner/Group work, Homework, Open-notes quiz, 3x summarization, Mini white boards, Board work, Review games, Exit ticket,
Teacher observation
Enduring Understandings:
Students will understand the concept of probability and its relationship to geometric figures and solids
Essential Questions:
 How is probability calculated?
 How is geometric probability calculated?
 What is the difference between independent events and mutually exclusive events?
Instructional Outcomes:
Students will be able to:
 compute basic probabilities.


compute geometric probabilities.
identify independent events.
Suggested Learning Activities (Including differentiated instruction):
Do nows, Homework review, Direct instruction (board notes/presentations), Board work, Investigation activities, Partner/group work,
Individual practice, Flipped classroom, Exit tickets, Review activities and games
Curriculum Development Resources:
May include, but are not limited to:
 Text: Geometry - Prentice Hall Mathematics
 Text Resource materials
 Guided notes (teacher prepared)
 GeoGebra
 Garnet Valley School District Geometry Curriculum
Notes/Comments: