# Geometric Relationships - Saginaw Valley State University ```Unit Design
For
Geometric Relationships
Developed by
Jacqueline Isaacson
Marvin L. Winans Academy of Performing Arts
2012
UBD Unit Design Worksheet / Saginaw Valley State University
1
Understanding by Design
Unit Design Worksheet
Unit Title: Geometric Relationships
Topic: Circles
Subject/Course: Geometry
Staff Name: Jacqueline Isaacson
Stage 1 - Desired Results
Established Goals (Common Core State Standards):
1. 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.
2. 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.
3. 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.
4. G-C.1 Prove that all circles are similar.
5. 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.
6. G-GPE.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
7. 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).★
8. G-MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to
satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Understandings:
Students will understand
1. how to recognize an angle, a circle,
perpendicular lines, parallel lines, and a line
segment when given a picture in a plane.
2. what an inscribed circle looks like, the different
parts of the circle, and how those parts relate to
the shape they are inscribed with.
3. what a theorem is, what we use to prove it, why
it needs to be proven, and the formats used for
UBD Unit Design Worksheet / Saginaw Valley State University
Essential Questions:
1. What are the definitions of basic shapes of
geometry?
2. What is the relationship among an inscribed
3. How do you prove theorems about lines and
angles? ( vertical angles, transversal crosses,
alt. interior angles)
2
writing proofs.
4. similar circles and be able to write a proof to
demonstrate their understanding.
5. that the arc length intercepted by an angle is
6. how algebra and geometry are related.
7. how to use and apply a variety of shapes to real
life.
8. how to find a solution to a problem and then
clearly explain why it would solve it.
UBD Unit Design Worksheet / Saginaw Valley State University
4. What makes a circle similar to another circle?
5. Is an arc intercepted by an angle is
proportional to the radius and how do we
derive the formula of the sector that it creates?
6. What is the use of the Pythagorean Theorem
in relation to a circle and how do we use it to
solve for missing parts?
7. How do you measure shapes and describe
their properties?
8. What types of problems in life could
geometry help me solve?
3
Students will know:
Students will be able to:
1. how to recognize an angle, a circle,
perpendicular lines, parallel lines, and a line
segment when given a picture in a plane.
2. what an inscribed circle looks like. They
should be able to tell me the different parts of
the circle and how they relate to the shape they
are inscribed with.
3. what a theorem is and what we use to prove it.
Why it needs to be proven. (to fully understand
how things work together.) They will know the
formats we write a proof in.
4. what similar circles means and be able to write
me a proof showing full understanding of why.
5. how to be able to put into their own words
using similarity the fact that the arc length
intercepted by an angle is proportional to the
radius. Then they would be able to give the
formula for the area of a sector.
1. describe an angle, circle, perpendicular lines,
parallel lines, and line segment.
2. explain how angles, circles, perpendicular
lines, parallel lines, and line segments relate.
3. show how geometric figures are used in
inscribed circles.
4. identify and use geometric shapes within a
circle (inscribed) to solve for missing values.
5. write a clear proof of a theorem and explain it
to their peers in their own words.
6. produce mathematical problems involving
similarity, arc length, and area of a sector.
They will be able to explain in their own
words.
7. solve for missing sides of a right triangle with
algebra, the Pythagorean Theorem, and
completing the square in relation to a circle.
6. that algebra relates to geometry in many ways.
You want them to be able to tell me how and
give examples. You want them to be able to tell
me what the Pythagorean and comp. sq. does
and how it relates to a circle.
8. explain how they use geometric shapes at
home, at school, during sports, etc.
7. how they would/do use a variety of shapes in
real life.
10. design situations where geometric theories
would help.
9. apply concepts of geometry to situation in real
life to solve problems.
8. how to find a solution to a problem and then
clearly explain why it would solve it.
Unit Enduring Understanding:
Unit Question:
Students will know how all geometric figures relate to
circles and real life.
How do all geometric figures relate to circles and real
life?
Stage 2 - Assessment Evidence
Goal: Your task is to show your knowledge of geometric shapes relating to circles by creating a blueprint of a
new performing arts school and proposing the new building to board members.
UBD Unit Design Worksheet / Saginaw Valley State University
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Role: You are part of a team of architects with a strict budget. You need to create a blueprint and proposal of a
new creative building. Since it is a performing arts school your design should be very avant-garde and utilize
unusual techniques in building structure. Your clients prefer organic shapes and circles to promote a calming,
creative atmosphere. Your job is to create a school building that is sound in structure, creative with the use of
circles, within the budget, and propose this plan to a board of school members.
Audience: Your target audience is the school board, principal, parents, and high school students.
Situation: The challenge involves creating an 18” x 20” blueprint of the new school and a name for the school.
The design must include circles and have a secure structure. Then write a clear proposal explaining why you
used the structure you used with the strict budget of \$5,000.
Product: One 18” x 20” blueprint of the new school. One 4 page essay explaining the blueprint, budget usage,
and structure of the school. One 5 minute speech on why your school is the one to choose for the project.
Standard: Your blue print, paper, and speech will be judged with a rubric.
Key Criteria: (Rubrics, etc.)
Holistic Rubric:
Holistic Rubric
for (Tests)
4
Student
understands the
concept with no
missed steps
and/or confusing
rules.
3
2
1
Student
understands the
concept, but
missed a step
and/or is confusing
learned rules.
Student work has a
lot of errors
showing the
student has not
fully grasped the
concept. They are
mixing up
concepts or rules,
and work lacks
clarity.
Student attempted
the problem, but
clearly does not
understand the
concept.
Building A Structure : Analysis of Created Product: Blue Print
Teacher Name:
Student Name:
________________________________________
CATEGORY
Mathematical
computation
4
3
All instances with Few errors in
geometric
mathematical
computation are
computation.
correct. Example: If
a triangle is
inscribed, then all
UBD Unit Design Worksheet / Saginaw Valley State University
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1
At least 10 errors in More than 10 errors
mathematical
in mathematical
computation.
computation.
5
angles depicted are
correct. All
measurements of
perimeter and area
are correct, etc.
Mathematical
usage of tools
All drawn
geometric shapes
have been drawn
using geometric
construction
techniques and tools
(compass and
straight edge).
Shows a clear
understanding of
how to construct
geometric figures.
80% of drawn
geometric shapes
have been drawn
using geometric
construction
techniques and tools
(compass and
straight edge).
Shows a clear
understanding of
how to construct
geometric figures.
70% of drawn
geometric shapes
have been drawn
using geometric
construction
techniques and tools
(compass and
straight edge).
Shows a surface
level understanding
of how to construct
geometric figures.
60% or below of
drawn geometric
shapes have been
drawn using
geometric
construction
techniques and tools
(compass and
straight edge).
Shows a shallow
understanding of
geometric
construction.
Mathematical
usage of geometry
on blueprint and
in paper.
Plans for the
structure show a
strong
understanding of
how geometric
figures relate to a
circle in the correct
manner. All
mathematical
analysis is correct
and makes sense.
All measurements
are correct.
Plans for the
structure show an
understanding of
how geometric
figures relate to a
circle in the correct
manner. 80% of
mathematical
analysis is correct
and makes sense.
Most measurements
are correct.
Plans for the
structure show little
understanding of
how geometric
figures relate to a
circle in the correct
manner. 70% of
mathematical
analysis is correct
and makes sense.
Many
measurements are
incorrect.
Plans for the
structure show very
little to no
understanding of
how geometric
figures relate to a
circle in the correct
manner. 69% or
below of
mathematical
analysis is correct
and makes sense.
Information
Gathering
Accurate
information taken
from several
sources in a
systematic manner.
Accurate information
taken from a couple
of sources in a
systematic manner.
Accurate
information taken
from a couple of
sources but not
systematically.
Information taken
from only one
source and/or
information not
accurate.
Computations:
Teacher Name:
Student Name:
________________________________________
CATEGORY
4
UBD Unit Design Worksheet / Saginaw Valley State University
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2
1
6
Mathematical
Concepts
Explanation shows
complete
understanding of the
mathematical
concepts used to
solve the
problem(s).
Explanation shows
substantial
understanding of the
mathematical
concepts used to
solve the problem(s).
Explanation shows
some understanding
of the mathematical
concepts needed to
solve the
problem(s).
Explanation shows
very limited
understanding of the
underlying concepts
needed to solve the
problem(s) OR is
not written.
Mathematical
Reasoning
Uses complex and
refined
mathematical
reasoning.
Uses effective
mathematical
reasoning.
Some evidence of
mathematical
reasoning.
Little evidence of
mathematical
reasoning.
Mathematical
Errors
90-100% of the
steps and solutions
have no
mathematical errors.
Almost all (85-89%)
of the steps and
solutions have no
mathematical errors.
Most (75-84%) of
the steps and
solutions have no
mathematical errors.
More than 75% of
the steps and
solutions have
mathematical errors.
Mathematical
Correct terminology
Terminology and and notation are
Notation
always used,
making it easy to
understand what
was done.
Correct terminology
and notation are
usually used, making
it fairly easy to
understand what was
done.
Correct terminology
and notation are
used, but it is
sometimes not easy
to understand what
was done.
There is little use, or
a lot of
inappropriate use, of
terminology and
notation.
Strategy/
Procedures
Typically uses an
Typically uses an
efficient and
effective strategy to
effective strategy to solve the problem(s).
solve the
problem(s).
Sometimes uses an Rarely uses an
effective strategy to effective strategy to
solve problems, but solve problems.
does not do it
consistently.
Research:
Teacher Name:
Student Name:
________________________________________
UBD Unit Design Worksheet / Saginaw Valley State University
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CATEGORY
4
3
2
1
Research Packet
Research packet is
complete with all
parts filled in.
Research packet is
missing a few parts
but have been
completed since.
Research packet is
vague and missing
parts.
Research is
inaccurate or
unfinished.
Research in
packet
Research is
Research is mostly
accurate, clear, and complete, but has a
follows format of
few inaccuracies.
packet.
Student used
unreliable websites
to find information
or research is
inaccurate.
Student’s research
is incomplete and
unreliable either by
using improper
websites or making
things up.
Oral Presentation Rubric : GRASPS Rubric:
Architecture Project
Teacher Name:
Student Name:
________________________________________
CATEGORY
4
3
2
1
Vocabulary
Uses vocabulary
appropriate for the
audience. Extends
audience vocabulary
by defining words
that might be new to
most of the
audience.
Uses vocabulary
appropriate for the
audience. Includes 12 words that might
be new to most of
the audience, but
does not define them.
Uses vocabulary
appropriate for the
audience. Does not
include any
vocabulary that
might be new to the
audience.
Uses several (5 or
more) words or
phrases that are not
understood by the
audience.
Time-Limit
Presentation is 4-5
minutes long.
Presentation is 3
minutes long.
Presentation is 2
minutes long.
Presentation is less
than 2 minutes OR
more than 5
minutes.
Content
Shows a full
understanding of the
topic, both
geometric and
economic.
Shows a good
understanding of the
topic, both geometric
and economic.
Shows a good
Does not seem to
understanding of
understand the topic
parts of the topic,
very well.
both geometric and
economic.
Evaluates Peers
Fills out peer
evaluation
completely and
always gives scores
based on the
Fills out almost all of
the peer evaluation
and always gives
scores based on the
presentation rather
Fills out most of the
peer evaluation and
always gives scores
based on the
presentation rather
UBD Unit Design Worksheet / Saginaw Valley State University
Fills out most of the
peer evaluation but
scoring appears to
be biased.
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presentation rather
than other factors
(e.g., person is a
close friend).
than other factors
(e.g., person is a
close friend).
than other factors
(e.g., person is a
close friend).
Other Evidence:
Journals, rough drafts, vocabulary activity, quizzes, concept maps, observations, daily assignments, textbook,
worksheets, prompts on backing up mathematical thought.
BEFORE
DURING
Pre-quizzes on vocabulary
The teacher will test the students on
vocabulary to see what they know
prior to starting the unit.
Vocabulary log
Students will keep a running
vocabulary log for personal
reference.
Brainstorming
The students will brainstorm ideas
for their building before beginning
the GRASPS project. This will
promote critical thinking and
problem solving.
Questioning
The teacher will ask questions as
the students work and learn.
Think/pair/ share
Students will think, pair, and share
give them feedback on what they
will be learning.
AFTER
Posttest GRASPS
GRASPS project on architecture.
Student Review
Students will re-teach parts of the
unit as partners to the class to help
promote learning.
Daily assignments
Daily book and worksheet
assignments.
Drawings
Students will practice their
constructions with small
assignments similar to the
GRASPS assignment to check for
understanding.
Observations
The teacher will monitor the
students’ in-class work and check
for understanding.
Quizzes
The teacher will retest the students
on vocabulary and information to
see if they learned the terms and
learned before.
Describe the assessment/s and state the prompt if
applicable.
□F XS
UBD Unit Design Worksheet / Saginaw Valley State University
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What type of scoring tools will be used for evaluation?
X Analytic rubric
X Holistic rubric
X Criterion rubric
□ Checklist
□ Other
Student Self-Assessment and Reflection:
Students will write a few short journal entries to reflect on their work and demonstrate understanding.
 How do you feel about the unit so far?
 What is the easiest concept for you to understand, what is the most difficult?
 Explain a concept in your own words.
 How does what we learned today relate to real life?
 How did you do this week?
At the end of the project, students will reflect on how much work they have done. They will be expected to give
other groups a grade for their work as well.
 What parts of the project did I contribute to?
 What could I have done more on?
 What did I learn from this project?
 How did the circles make creating the building more difficult?
 How did the budget make creating the building more difficult?
 What other subjects did I have to use to complete this project?
 How will this project and unit help me later on in life?
 Did performing in front of an audience cause me to pay attention to the information more closely?
 Did working with a group of peers help me to understand concepts I didn’t understand before?
Group project peer analysis questionnaire:
 What parts did I contribute to, what parts did the others do?
 Who did not do anything?
 Who helped you the most when it came to understanding the mathematical analysis?
As a group, how did you think you did? Rate your group on a scale of 1-10, with one being the lowest and ten
being the highest.
UBD Unit Design Worksheet / Saginaw Valley State University
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Stage 3 - Learning Plan
Differentiated Instruction:
Level C – 90 points (All activities are required)
1.
2.
3.
Unit vocabulary list (10 points)
“Do Now” unit bell work tracker (2 points a day – 30 points)
Daily homework problem sets (5 points a day – 50 total points)
Level B – 25 points (Activity one is required, choose one of the next two)
1.
2.
3.
Use a compass and protractor to discover how to make inscribed shapes and complete a worksheet. (10
points)
Create a series of 5 mini textbooks showing a variety of proofs of inscribed shapes, angles, similar
circles, and constructions of all the geometric shapes. Must explain in own words. ( 15 points)
Create/complete work packet with a partner. (15 points)
Level A – 56 points
1.
GRASPS project: create a blueprint for a new school with a given budget
a. Research packet with filled in questions and essays ( 8 points)
b. Written notes showing all work of all mathematical computation (20 points)
c. Fully rendered blueprint (16 points)
d. Oral presentation (12 points)
Learning Activities:




The end of the unit project asks students to answer the question of, where are we going? They will have
to rethink what they have learned and re organize the information to create a blueprint of a new school
with a given budget which will deepen understanding of geometry in relation to real life.
Students will use geometric tools to understand the construction of geometric objects and why we
compute the way we do. They will use a variety of worksheets, inquiry based group work, and choices
to tailor to their needs.
The teacher will hook students’ interest by giving them choices and projects that use new tools and
ideas they have not used before. Students will either create a series of 5 mini textbooks with visual aids
that explain the main unit points or complete a work packet with a partner.
At the end of the unit, students will present their GRASPS project, reflect on their project, and peer
evaluate the projects.
Students will be equipped with materials and supplies to complete each learning activity.
UBD Unit Design Worksheet / Saginaw Valley State University
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Essential Vocabulary
Angles: two rays that share one end-point.
Area: any particular extent of space or surface; part.
Arc: any unbroken part of the circumference of a circle or other curved line.
Bisector: a line or plane that bisects an angle or line segment.
Center: the middle point, as the point within a circle or sphere equally distant from all points of the
circumference or surface.
Chord: the line segment between two points on a given curve.
Circle: a closed plane curve consisting of all points at a given distance from a point within it called the center.
Congruent: equal parts.
Diameter: a straight line passing through the center of a circle or sphere and meeting the circumference or
surface at each end.
Equidistant: equally distant.
Interior Angles: an angle formed between parallel lines by a third line that intersects them.
Inscribe: to draw or delineate (one figure) within another figure so that the inner lies entirely within the
boundary of the outer, touching it at as many points as possible.
Line Segment: a finite section of a line.
Parallel Line: extending in the same direction, equidistant at all points, and never converging or diverging.
Perimeter: the border or outer boundary of a two-dimensional figure.
Perpendicular Line: meeting a given line or surface at right angles.
Proportion: comparative relation between things or magnitudes as to size, quantity, number, etc.; ratio.
Pythagorean Theorem: the theorem that the square of the hypotenuse of a right triangle is equal to the sum of
the squares of the other two sides.
Radius: a straight line extending from the center of a circle or sphere to the circumference or surface.
Segment: a part cut off from a figure, especially a circular or spherical one, by a line or plane, as a part of a
circular area contained by an arc and its chord or by two parallel lines or planes.
Similar: having the same shape; having corresponding sides proportional and corresponding angles equal.
Vertical Angle: one of two opposite and equal angles formed by the intersection of two lines.
Sequencing the Learning
MONDAY
Pre-quiz:
TUESDAY
Identifying Basic
WEDNESDAY
Intro to circles
UBD Unit Design Worksheet / Saginaw Valley State University
THURSDAY
Manipulation of/
FRIDAY
Manipulation of/
12
Identify what basic
vocabularies are.
Identify what the
basic geometric
shapes are.
geometric shapes
and how they relate
to each other.
Level C: daily
assignment
similarity of circles
similarity of circles
Level C: daily
assignment
Quiz over week
information.
Level C: daily
assignment
Level C: Packet for
the week
Complete basic
math skills i.e.
multiplication,
division, and
solving for variables
Level C: daily
assignment
MONDAY
Basic proofing
knowledge by
brainstorming in
pairs
Level C: daily
assignment
MONDAY
TUESDAY
Proofing of similar
circles
Level B: activity
day
Think-pair-share
activity
Inquiry based work
sheet by drawing
constructions
Level C: daily
assignment
Research on relation
to real life day. How
to research
effectively.
Level C: daily
assignment
Level C: daily
assignment
THURSDAY
FRIDAY
Inscribed circles
Quiz
Level C: daily
assignment
Level C: Packet for
the week
Level C: daily
assignment
TUESDAY
Pythagorean
theorem with
inscribed and non
triangles
MONDAY
WEDNESDAY
WEDNESDAY
THURSDAY
FRIDAY
Level B: choice
assignment
Level B: choice
assignment
Level C: Packet for
the week
Work day
Work day
Level B: choice
assignment due
TUESDAY
WEDNESDAY
THURSDAY
FRIDAY
Level A
Level A
Level A
Level A
Level A
GRASPS work day
GRASPS work day
GRASPS work day
GRASPS work day
GRASPS work day
Research packet due
UBD Unit Design Worksheet / Saginaw Valley State University
Mathematical
computation work
packet due
13
MONDAY
TUESDAY
Level A
Presentations
GRASPS work day
Blueprints due
WEDNESDAY
Presentations
Complete peer
evaluations
UBD Unit Design Worksheet / Saginaw Valley State University
THURSDAY
FRIDAY
Geometric
relationships test
Unit vocabulary
packets due
14
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