Download Hackettstown HACKETTSTOWN, NEW JERSEY Geometry 9

Hackettstown
HACKETTSTOWN, NEW JERSEY
Geometry
9-11
CURRICULUM GUIDE
FINAL DRAFT
August 2015
Mr. David C. Mango, Superintendent
Ms. Nadia Inskeep, Director of Curriculum & Instruction
Developed by:
Michelle DeFilippis
Suzanne Sloan
This curriculum may be modified through varying techniques,
strategies and materials, as per an individual student’s
Individualized Education Plan (IEP).
Approved by the Hackettstown Board of Education
At the regular meeting held on
8/19/2015
And
Aligned with the New Jersey Core Curriculum Content Standards
And Common Core Content Standards
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Table of Contents
Philosophy and Rationale: 3
Mission Statement: 3
Units: 3-19
NJ Content Standards: 20
21st Century Skills: 20
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Philosophy and Rationale
The mission of the Hackettstown Mathematics Department is to design and implement a mathematics curriculum
that stresses the key ideas identified in the Common Core State Standards. This goal is to be achieved by
continually returning to the organizing principles of mathematics and the laws of algebraic and geometric
structure to reinforce those concepts.
Through a variety of real world scenarios and applications, our students will be challenged to think
critically, apply, evaluate, and communicate the ideas and concepts presented in each course as it relates to the
field of mathematics. Students will also be challenged to analyze their performance in an effort to become
independent problem solvers.
Our students’ performance will be assessed using formative techniques such as homework, quizzes, and
tests. Additionally, our students will be assessed using state and national assessment tools based upon the branch
of mathematics being studied.
Mission Statement
Building on Tradition and success, the mission of the Hackettstown School District is to educate and inspire
students through school, family and community partnerships so that all become positive, contributing members
of a global society, with a life-long commitment to learning.
Unit Rationale: This unit introduces various topics in the study of geometry and focuses on the areas of
visualization, reasoning, and measurement.
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Stage 1: Desired Results
Topic: Tools of Geometry
Unit: 1
Common Core State Standards
G.CO.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
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an
angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular
bisector of a line segment; and constructing
G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
Essential Questions
Enduring Understandings
What are the building blocks of geometry?
Definitions establish meanings and remove
How can you describe the attributes of a segment or angle? possible misunderstandings.
Attributes such as length, area, volume, and angle
measure are measurable.
Knowledge and Skills:
By the end of the unit, students will be able:
to understand basic terms and postulates of geometry
to name and use points, lines, and planes
to find and compare lengths of segments and measures of angles (based on knowledge from 8th grade)
to identify special angle pairs and use their relationships to find angle measures (based on knowledge from 7th
grade)
to find the midpoint of a segment
to make basic constructions using a straightedge and a compass
to use physical models to test conjectures
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
understand basic terms and postulates of geometry
name and use points, lines, and planes
find and compare lengths of segments and measures of angles
identify special angle pairs and use their relationships to find angle measures
find the midpoint of a segment
make basic constructions using a straightedge and a compass
use physical models to test conjectures
Assessment Methods:
Formative:
Assignment: A) Construct, measure, and label a line segment and an angle.
B) Is it possible to construct a triangle with side measures of 5", 6", and 12'? Use the geometric tools as
appropriate. Write a short paragraph explaining your response to the question.
Summative: Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Construct triangle ABC with A(4, 7), B(0, 0), and C(8, 1).
a. Which sides are congruent? How do you know?
b. Construct the bisector of angle B. Mark the intersection of the ray and AC as D.
c. What do you notice about AD and CD?
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Stage 3: Learning Plan
DIFFERENTIATION: based on quiz results, group students homogeneously to complete constructions that are
differentiated by difficulty.
Information Literacy
 Access and Evaluate Information
 Access information efficiently (time) and effectively (sources)
 Evaluate information critically and competently
 Activity: Students will read about an antiquated system of measurement, explain its limitations and
compare it to the American Standard System as well as the Metric System.
 Activity: Students will read directions to a buried treasure and interpret the directions to locate it using
the tools of geometry.
Work Creatively with Others
 Develop, implement and communicate new ideas to others effectively
 Be open and responsive to new and diverse perspectives; incorporate group input and feedback into the
work
 Demonstrate originality and inventiveness in work and understand the real world limits to adopting new
ideas
 View failure as an opportunity to learn; understand that creativity and innovation is a long-term, cyclical
process of small successes and frequent mistakes
 Activity: Explore angle pairs with a partner using a clock as a base to gauge understanding by being able
to successfully estimate the time shown by the various tasks.
Time Allotment: 13 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: This unit focuses on lines and their relationships and the student’s ability to differentiate
between characteristics of various line pairs.
Stage 1: Desired Results
Unit: 2
Topic: Linear Relationships
Common Core State Standards
G.CO.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.
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an
angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.GPE.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).
Essential Questions
Enduring Understandings
Who was Euclid and what did he/she do in mathematics?
The special angles formed by parallel lines and a
How do you prove that two lines are parallel or
transversal are congruent, supplementary, or both.
perpendicular?
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You can use certain angle pairs to determine
whether lines are parallel.
You can graph a line and write its equation when
you know certain facts about the line, such as its
slope and a point on the line.
Knowledge and Skills:
By the end of the unit, students will be able:
to know who Euclid was and to understand his contribution to the development of geometry
to identify angles formed by two lines and a transversal (based on knowledge from 8th grade)
to prove theorems about parallel lines
to use properties of parallel lines to find angle measures (based on knowledge from 8th grade)
to use parallel lines to prove a theorem about triangles
to determine whether two lines are parallel
to relate slope to parallel and perpendicular lines
to find the measures of the angles of a triangle (based on knowledge from 8th grade)
to identify and measure interior and exterior angles of a triangle (based on knowledge from 8th grade)
to use triangles to find the sum of the measures of the interior and exterior angles of polygons (based on
knowledge from 8th grade)
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
know who Euclid was and to understand his contribution to the development of geometry
identify angles formed by two lines and a transversal
prove theorems about parallel lines
use properties of parallel lines to find angle measures
use parallel lines to prove a theorem about triangles
determine whether two lines are parallel
relate slope to parallel and perpendicular lines
find the measures of the angles of a triangle
identify and measure interior and exterior angles of a triangle
use triangles to find the sum of the measures of the interior and exterior angles of polygons
Assessment Methods:
Formative: Complete the following using a straight edge, a compass, and a pencil or using paper
folding: a)construct the line parallel to a given line and through a given point not on the line; b) construct a
quadrilateral with one pair of parallel sides; c)construct the perpendicular to a given line through a point not on
the line.
Summative: Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Exploring Spherical Geometry
In spherical geometry, the curved surface of a sphere is studied. A line is a great circle. A great circle is the
intersection of a sphere and a plane that contains the center of the sphere. 1) You can use latitude and longitude
to identify positions on the Earth. Look at the markings on a globe to answer the following: a) If you 'slice' the
globe with a plane at each latitude, do any of the slices contain the center of the globe? b) If you 'slice' the globe
with a plane at each longitude, do any of the slices contain the center of the globe? c) which latitudes, if any,
suggest great circles? Which longitudes, if any, suggest great circles? Explain your responses.
2) Hold a string taut between any two points on a tennis ball (sphere). The string forms a 'segment' that is part of
a great circle. Connect three such segments to form a triangle. Trace your triangle on the tennis ball. a) what is
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the sum of the angle measures in your triangle? b) Try this experiment several times to explain how the Triangle
Angle-Sum Theorem in Euclidean geometry differs from your results in spherical geometry.
Stage 3: Learning Plan
Produce Results
Demonstrate additional attributes associated with producing high quality products including the abilities to:
 Work positively and ethically
 Manage time and projects effectively
 Multi-task
 Participate actively, as well as be reliable and punctual
 Present oneself professionally and with proper etiquette
 Collaborate and cooperate effectively with teams
 Respect and appreciate team diversity
 Be accountable for results
 Sample activity: Students will complete a walk-around activity in pairs. Students will solve equations
and search the room for the answer. If students complete all of the equations correctly and check their
answers, they will arrive back at their original problem.
Interact Effectively with Others
 Know when it is appropriate to listen and when to speak
 Conduct themselves in a respectable, professional manner
 Activity: Student pairs will work together to create a diagram of parallel and perpendicular roads to
answer a relationship question.
Think Creatively
 Use a wide range of idea creation techniques (such as brainstorming)
 Create new and worthwhile ideas (both incremental and radical concepts)
 Elaborate, refine, analyze and evaluate their own ideas in order to improve and maximize creative efforts
 Activity: Construct a path to connect a trail to a parking lot. Determine the cost of the path and if there is
a more cost effective path.
Time Allotment: 11 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Making a miniature, a model, or other construction requires skill and knowledge of how to
find measures of similar figures. Students will develop an understanding of similar triangles and other polygons
and will use their knowledge to solve problems.
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Stage 1: Desired Results
Topic: Congruence
Unit: 3
Common Core State Standards
G.CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if
and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motions.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Essential Questions
Enduring Understandings
What is congruence?
Congruent triangles have specific characteristics.
How do you classify triangles?
Two triangles can be proven to be congruent
How does coordinate geometry allow you to find
without having to show that all corresponding
relationships in triangles?
parts are congruent.
How do you show that two triangles are congruent?
Angles and sides of particular triangles have
How do you identify corresponding parts of congruent
special relationships.
triangles?
What makes shapes alike and different?
Knowledge and Skills:
By the end of the unit, students will be able:
to recognize congruent figures and their corresponding parts
to prove triangles congruent
to apply corresponding parts of congruent triangles to triangle proofs
to prove right triangles congruent
to use and apply properties of isosceles and equilateral triangles
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
recognize congruent figures and their corresponding parts
prove triangles congruent
apply corresponding parts of congruent triangles to triangle proofs
prove right triangles congruent
use and apply properties of isosceles and equilateral triangles
Assessment Methods:
Formative:
The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments 2 cm and 8 cm long. Find
the length of the altitude to the hypotenuse. Use a ruler to make an accurate drawing of the right triangle.
Describe how you drew the triangle above.
Summative: Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
Critical Thinking and Problem Solving
Reason Effectively
 Use various types of reasoning (inductive, deductive, etc.) as appropriate to the situation
 Activity: Station activity to construct triangles and develop conjectures
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Learning and Innovation Skills
Creativity and Innovation
Think Creatively
 Use a wide range of idea creation techniques (such as brainstorming)
 Create new and worthwhile ideas (both incremental and radical concepts)
 Elaborate, refine, analyze and evaluate their own ideas in order to improve and maximize creative efforts
 Activity: Writing task to relate AAS congruence to ASA congruence.
Time Allotment: 11 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Students will build on their understanding of angles and triangles using visualization
techniques and constructions.
Stage 1: Desired Results
Unit: 4
Topic: Triangle Relationships
Content Standards
G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to
180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle
is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the
other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity
Essential Questions
Enduring Understandings
How does coordinate geometry allow you to find
There are two special relationships between a
relationships in triangles?
midsegment and the third side of the triangle.
How do you solve problems that involve measurements of
For any triangle, certain sets of lines are always
triangles?
concurrent.
The angles and sides of a triangle have special
relationships that involve inequalities.
Knowledge and Skills:
Students will be able:
to use properties of midsegments to solve problems
to use the Triangle Inequality Theorem
to use and apply properties of perpendicular bisectors, angle bisectors, altitudes, and medians to solve problems
to find the four points of concurrency in a triangle
to formulate conjectures about the relationships between the angles of a triangle and the lengths of the sides
to use the Side Splitter Theorem and the Triangle Angle Bisector Theorem
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
use properties of midsegments to solve problems
use the Triangle Inequality Theorem
use and apply properties of perpendicular bisectors, angle bisectors, altitudes, and medians to solve problems
find the four points of concurrency in a triangle
formulate conjectures about the relationships between the angles of a triangle and the lengths of the sides
use the Side Splitter Theorem and the Triangle Angle Bisector Theorem
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Assessment Methods:
Formative: Complete the following using a straight edge, a compass, and a pencil or using paper
folding: a)construct the line parallel to a given line and through a given point not on the line; b) construct a
quadrilateral with one pair of parallel sides; c)construct the perpendicular to a given line through a point not on
the line.
Summative: Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
Financial, Economic, Business and Entrepreneurial Literacy
Using entrepreneurial skills to enhance workplace productivity
Activity: Use the perpendicular bisector to place a t shirt stand equidistant from two attractions, and then 3
attractions.
Creativity and Innovation
Work Creatively with Others
 Demonstrate originality and inventiveness in work and understand the real world limits to adopting new
ideas
 Activity: Locate a home that meets given requirement
Time Allotment: 13 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Trigonometry began as a computational component of geometry. Students will learn that
trigonometry depends on angle measurement and quantities determined by the measure of an angle. . In order
to understand trigonometry, students must be familiar with and calculate using the Pythagorean Theorem.
Stage 1: Desired Results
Unit: 5
Topic: Right Triangles and Trigonometry
Common Core State Standards
+F.TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions
to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
+F.TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and
π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of
their values for x, where x is any real number.
+F.TF.B.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate
the solutions using technology, and interpret them in terms of the context.*
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the
other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle,
leading to definitions of trigonometric ratios for acute angles.
G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
+G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from
a vertex perpendicular to the opposite side.
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Essential Questions
How can I use the Pythagorean Theorem to determine side
length in right triangles?
What is important about the study of right triangle
trigonometry?
What are angles of elevation and depression and how are
they used?
Enduring Understandings
Right triangles have constant ratios of of side
lengths.
Construction, surveying, and aeronautics use
applications of trigonometry involving indirect
measurement.
Knowledge and Skills:
Students will be able:
to find and use similarities in right triangles
to use the Pythagorean Theorem and its converse (based on knowledge from 8th grade)
to use the properties of special right triangles
to use trigonometric ratios to determine side lengths and angle measures in right triangles
to use angle of elevation and angle of depression in trigonometric ratios
to create the unit circle in degrees based on special right triangles
to use the unit circle to solve problems in degrees
to find area of regular polygons using trigonometry
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
find and use similarities in right triangles
use the Pythagorean Theorem and its converse
use the properties of special right triangles
use trigonometric ratios to determine side lengths and angle measures in right triangles use angle of elevation
and angle of depression in trigonometric ratios
create the unit circle in degrees based on special right triangles
use the unit circle to solve problems in degrees
find area of regular polygons using trigonometry
Assessment Methods:
Formative: TASK A: You are creating a design for your website. The design consists of a 6-inch circle with a
square inside of it. You want the square to be as large as possible and still be inside the circle. Determine the
length of the sides of the square that will fit inside the circle. Explain your solution.
TASK B: Work with a partner to answer this question. Be prepared to explain your solution to the class.
A farmer's conveyor belt carries bales of hay from the ground to the barn loft as shown in the diagram. The
conveyor belt moves at 100 ft/minute. How many seconds does it take for a bale of hay to go from the ground to
the barn loft?
Summative:
Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
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CRITICAL THINKING AND PROBLEM SOLVING
Reason Effectively
 Use various types of reasoning (e.g., inductive, deductive, etc.) as appropriate to the situation
Activity: Draw two triangles of different sizes each with your given angles. Use a protractor. Measure the sides
of each triangle to the nearest tenth of a centimeter. Use a calculator to find the ratios of the lengths of each pair
of corresponding sides. What conclusion can you hypothesize about the relationship of the two triangles?
Use Systems Thinking
 Analyze how parts of a whole interact with each other to produce overall outcomes in complex systems
Activity: Similarity in right triangles
Time Allotment:15 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Students will examine the hierarchy of polygons and quadrilaterals. They will build a
classification system built on the attributes of various shapes.
Stage 1: Desired Results
Unit: 6
Topic: Quadrilaterals and Similarity
Common Core State Standards
G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are
parallelograms with congruent diagonals.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove
that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point
(1, √3) lies on the circle centered at the origin and containing the point (0, 2).
G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using
the distance formula.★
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Essential Questions
Enduring Understandings
How can you classify quadrilaterals?
Parallelograms have special properties regarding
How can you use coordinate geometry to prove
their sides, angles, and diagonals.
relationships?
A quadrilateral can be proven to be a
parallelogram by its sides, angles, and diagonals.
A rhombus, rectangle, square, and trapezoid have
specific properties for identification.
Variables can be used to name the coordinates of a
figure in the coordinate plane.
Proportionality involves a relationship in which
the ratio of two quantities change. Proportions
involve multiplicative rather than additive
comparisons.
Congruent shapes are also similar, but similar
shapes may not be congruent.
Knowledge and Skills:
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Students will be able:
to use relationships among sides, angles, and diagonals of parallelograms
to determine whether a quadrilateral is a parallelogram
to use and apply properties of special quadrilaterals including square, rectangle, rhombus, trapezoid, and kite.
to classify polygons in the coordinate plane using coordinate proofs
to identify and apply similar polygons
to use similarity to find indirect measurements
to write ratios and proportions
to write and solve proportions
to write an extended ratio a:b:c
to find the geometric mean
to know the Means-Extremes Property of proportions
to write similarity statements and identify scale factor
to prove triangles similar
to use similarity to find indirect measurement
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
use relationships among sides, angles, and diagonals of parallelograms
determine whether a quadrilateral is a parallelogram
use and apply properties of special quadrilaterals including square, rectangle, rhombus, trapezoid, and kite.
classify polygons in the coordinate plane using coordinate proofs
identify and apply similar polygons
use similarity to find indirect measurements
write ratios and proportions (based on knowledge from 7th grade)
write and solve proportions (based on knowledge from 7th grade)
write an extended ratio a:b:c
find the geometric mean
know the Means-Extremes Property of proportions
write similarity statements and identify scale factor
prove triangles similar
use similarity to find indirect measurements
Assessment Methods:
Formative:
TASK 1: ABCDEF is a regular hexagon. Use your understanding of geometry to precisely classify
quadrilateral GBHE. How do you know? What are the measures of the interior angles of GBHE? Respond in
writing using proper mathematical terminology.
Summative:
Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
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COMMUNICATION AND COLLABORATION
Collaborate with Others
 Demonstrate ability to work effectively and respectfully with diverse learners
 Assume shared responsibility for collaborative work, and value the individual contributions made by
each team member
Activity: Properties of Special Parallelograms
VISUAL LITERACY
 Demonstrate the ability to interpret, recognize, appreciate, and understand information presented through
visible actions, objects and symbols, natural or man-made2
Activity: Counter examples to false special parallelogram statements to introduce kites and expand on
trapezoids.
Time Allotment: 12 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Students will explore the complexity of circles and use their knowledge to solve problems.
Stage 1: Desired Results
Unit: 7
Topic: Circles
Common Core State Standards
G.CO.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.
G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle
G.C.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.
G.A.2 Derive the equation of a parabola given a focus and directrix.
+G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle
Essential Questions
Enduring Understandings
How can you prove relationships between angles and arcs
You can use information about congruent parts of
in a circle?
a circle to find information about other parts of the
When lines intersect a circle or within a circle, how do you circle.
find the measures of resulting angles, arcs, and segments?
Geometric relationships in circles involve angles,
What geometric relationships can be found in circles?
arcs, and segments.
What is the equation of a parabola?
Angles formed by intersecting lines have a special
relationship to the arcs the intersecting lines
intercept.
There is a relationship between the equation of a
circle and the equation of a parabola.
Knowledge and Skills:
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Students will be able:
to use properties of a tangent to a circle
to use properties of chords in a circle
to find angle measures in circles including central, inscribed, and angles formed by lines
to find segment lengths in circles
to write the equation of a circle
to write the equation of a parabola
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
use properties of a tangent to a circle
use properties of chords in a circle
find angle measures in circles including central, inscribed, and angles formed by lines
find segment lengths in circles
write the equation of a circle
write the equation of a parabola
Assessment Methods:
Formative:
TASK A: A 2-ft-wide circular track for a camera dolly is set up in a studio to film a movie scence. The two
rails of the track form concentric circles. The radius of the inner circle is 8 ft. How much further does a wheel
on the outer rail travel than a wheel on the inner rail of the track in one turn? Include a diagram in your solution.
TASK B: Work with a partner to solve this problem. Be prepared to share your answer in class.
A floating dock sits in a lake. The dock is an 8 ft x 10 ft. rectangle. The bow of a canoe is attached
to one corner of the dock with a 10 ft. rope. Sketch a diagram to show the region in which the bow of the canoe
can travel. Then determine the area of that region. Round your answer to the nearest square foot.
Summative:
Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
Hackettstown
PRODUCTIVITY AND ACCOUNTABILITY
Manage Projects
 Set and meet goals, even in the face of obstacles and competing pressures
 Prioritize, plan, and manage work to achieve the intended result
 Activity: "Pizza eating Contest"
SCIENTIFIC AND NUMERICAL LITERACY
 Demonstrate the ability to evaluate the quality of scientific and numerical information on the basis of its
sources and the methods used to generate it
 Demonstrate the capacity to pose and evaluate scientific arguments based on evidence and to apply
conclusions from such arguments appropriately
 Demonstrate ability to reason with numbers and other mathematical concepts
 Activity: Develop the inscribed angle theorem by following a set of directions.
INITIATIVE AND SELF-DIRECTION
Work Independently
 Monitor, define, prioritize, and complete tasks without direct oversight
 Activity: on student's own time, verify length relationships found by the intersection of a tangent and a
secant, two secants, or two chords
Time Allotment: 13 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Students will gain experience in drawing, explaining, and sketching transformations of shapes,
including translations, reflections, rotations, and scaling. Tessellations provide an engaging, artistic way to
explore geometry.
Stage 1: Desired Results
Unit: 8
Topic: Transformations
Common Core State Standards
G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs. Compare
transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections
that carry it onto itself.
G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using,
e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a
given figure onto another.
G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given
rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions
to decide if they are congruent.
G.SRT.1ab 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.
Hackettstown
G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if
they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality
of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be
similar.
Essential Questions
Enduring Understandings
How can coordinates be used to describe and analyze
Shapes may have reflection (line) symmetry or
geometric objects?
rotation symmetry or neither.
How can transformations be described mathematically?
An object's location on a plane or in space can be
described quantitatively. The position of any
point on a surface can be specified by two
numbers. Computations with these numbers
allow us to describe and measure geometric
objects.
Knowledge and Skills:
Students will be able:
to translate, rotate, reflect, and dilate figures (based on knowledge from 8th grade)
to write mathematical rules to describe transformations
to graph transformations on the coordinate plane (based on knowledge from 8th grade)
to identify symmetry in a figure, including line symmetry and rotational symmetrry
to perform composite transformations
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
translate, rotate, reflect, and dilate figures
write mathematical rules to describe transformations
graph transformations on the coordinate plane
identify symmetry in a figure, including line symmetry and rotational symmetry
perform composite transformations
Assessment Methods:
Formative:
Graph A (5, 2). Graph B, the image of A for a 90-degree rotation about the origin O. Graph C, the image of A
for a 180-degree rotation about O. Graph D, the image of A for a 270-degree rotation about O. What type of
quadrilateral is ABCD
Summative:
Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
Hackettstown
CRITICAL THINKING AND PROBLEM SOLVING
Use Systems Thinking
 Analyze how parts of a whole interact with each other to produce overall outcomes in complex systems
 Activity: Work with a partner to incorporate the review of reflections with a refresher in linear equations.
CRITICAL THINKING AND PROBLEM SOLVING
Solve Problems
 Solve different kinds of non-familiar problems in both conventional and innovative ways
 Activity: Rotations in astronomy. Rotate each star a given amount of degrees, then innovate an
equivalent solution by grouping the stars first and then rotating the whole group.
CREATIVITY AND INNOVATION
Think Creatively
 Use a wide range of idea creation techniques (such as brainstorming)
 Create new and worthwhile ideas (both incremental and radical concepts)
 Elaborate, refine, analyze, and evaluate ideas in order to improve and maximize creative efforts
 Demonstrate imagination and curiosity
 Activity: Explain why it would make sense to categorize reflections and glide reflections as odd
isometries and translations and rotations as even isometries.
CREATIVITY AND INNOVATION
Work Creatively with Others
 Develop, implement, and communicate new ideas to others effectively
 Be open and responsive to new and diverse perspectives; incorporate group input and feedback into the
work
 Demonstrate originality and inventiveness in work and understand the real world limits to adopting new
ideas
 View failure as an opportunity to learn; understand that creativity and innovation is a long-term, cyclical
process of small successes and frequent mistakes
 Activity: Twisted Domino
FLEXIBILITY AND ADAPTABILITY
Adapt to Change
 Adapt to varied roles, job responsibilities, schedules, and contexts
 Work effectively in a climate of ambiguity and changing priorities
Be Flexible
 Incorporate feedback effectively
 Activity: Four in a Row
Time Allotment: 11 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Unit Rationale: Students will calculate the lateral, surface area, and volume of various shapes. They will see
that surface area is greater than lateral area. They will derive formulas for measuring 3-dimensional shapes and
use reasoning and logic to analyze calculations as applied to problem solving. They will apply the concepts of
ratio and proportion to various measurements of shapes.
Stage 1: Desired Results
Hackettstown
Unit: 9
Topic: Surface area and volume
Common Core State Standards
G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
G.GMD.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.
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree
trunk or a human torso as a cylinder).★
+G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures.
Essential Questions
Enduring Understandings
What is surface area? What is volume?
Three dimensional figures can be analyzed by
How do you find surface area and volume of solids?
connecting properties of the real objects with
How do you find surface area and volume of similar solids? drawings of these objects.
How can ratio be used compare similar solids?
Length, area, volume, and angles are measurable
attributes of geometric figures.
Similarity among geometric figures can be
established.
Knowledge and Skills:
Students will be able:
to find the area of kites, rhombi, trapezoids, and regular polygons
to find the measures of central angles and arcs
to find arc length
to find area of sectors and segments of a circle
to define polyhedron
to accurately sketch 3-dimensional shapes and identify parts
to find the surface area of prisms, cylinders, cones, pyramids, and spheres
to find the volume of prisms, cylinders, cones, pyramids, and spheres (based on knowledge from 7th and 8th
grade)
to use formulas appropriately when analyzing volume and surface area of 3-dimensional figures
to find the ratios between similar figures and their perimeters, areas, surface areas, and volume
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
find the area of kites, rhombi, trapezoids, and regular polygons
find the measures of central angles and arcs
find arc length
find area of sectors and segments of a circle
define polyhedron
accurately sketch 3-dimensional shapes and identify parts
find the surface area of prisms, cylinders, cones, pyramids, and spheres
find the volume of prisms, cylinders, cones, pyramids, and spheres
use formulas appropriately when analyzing volume and surface area of 3-dimensional figures
find the ratios between similar figures and their perimeters, areas, surface areas, and volume
Hackettstown
Assessment Methods:
Formative:
Complete the following table. Each figure must be accurately sketched or drawn. Present your table on a 12" x
18" piece of construction paper and be prepared to share your work in class.
Type of Figure
Sketch of Figure
Formula for Surface Area
Formula for Volume
Cylinder
Cone
Square Pyramid
Triangular Pyramid
Rectangular Prism
Hexagonal Prism
Sphere
B. Explain in writing how you can justify the formula for the area of a kite (rhombus) using the area of a
triangle.
Summative:
Unit Skills Assessment
Other Evidence and Student Self-Assessment:
Homework checks, quizzes, Journal entries, exit slips
Stage 3: Learning Plan
INITIATIVE AND SELF-DIRECTION
Manage Goals and Time
 Set goals with tangible and intangible success criteria
 Balance tactical (short-term) and strategic (long-term) goals
 Utilize time and manage workload efficiently
 Activity: develop formulas for volume of a cone and a pyramid based on physical relationships to
corresponding cylinders and prisms.
Time Allotment: 16 class periods
Resources:
Student Materials: textbook, notebook, calculator, pencil
Technology: smartboard, Graphing calculator
Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources
Teaching Resources: compass, ruler, graph paper, protractor
Hackettstown
New Jersey Core Curriculum and Common Core Content Standards
http://www.state.nj.us/education/cccs/
Integration of 21st Century Theme(s)
The following websites are sources for the following 21st Century Themes and Skills:
http://www.nj.gov/education/code/current/title6a/chap8.pdf
http://www.p21.org/about-us/p21-framework .
http://www.state.nj.us/education/cccs/standards/9/index.html
21st Century Interdisciplinary Themes (into core subjects)
• Global Awareness
• Financial, Economic, Business and Entrepreneurial Literacy
• Civic Literacy
• Health Literacy
• Environmental Literacy
Learning and Innovation Skills
• Creativity and Innovation
• Critical Thinking and Problem Solving
• Communication and Collaboration
Information, Media and Technology Skills
• Information Literacy
• Media Literacy
• ICT (Information, Communications and Technology) Literacy
Life and Career Skills
• Flexibility and Adaptability
• Initiative and Self-Direction
• Social and Cross-Cultural Skills
• Productivity and Accountability
• Leadership and Responsibility
Integration of Digital Tools
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Classroom computers/laptops
Technology Lab
FM system
Other software programs