Download Unit 4 Circles Geometry ACC - Long Beach Unified School District

Unit 4 Circles
Geometry ACC
Unit Goals – Stage 1
Number of Days: 18 days
1/30/17 – 2/24/17
Unit Description: Strategically choosing their geometry tools, the students continue their study of relationships, this time among angles and line
segments in and around circles. Using the central angle, students discover how to calculate the measure of inscribed angles and arc lengths. The
properties of angles and line segments build to the topics of circumference, diameter, chords, tangents and arc lengths. Students will prove circle
theorems using a variety of formats.
Materials: Straightedge, protractor, compass, patty paper, tape measures, round objects, calculators, Geometry software (ex.: Geogebra)
Standards for Mathematical Practice
Transfer Goals
SMP 1 Make sense of problems and
Students will be able to independently use their learning to…
persevere in solving them.
• Make sense of never-before-seen problems and persevere in solving them.
SMP 2 Reason abstractly and
• Construct viable arguments and critique the reasoning of others.
quantitatively.
Making Meaning
SMP 3 Construct viable arguments
UNDERSTANDINGS
ESSENTIAL QUESTIONS
and critique the reasoning of
Students will understand that…
Students will keep considering…
others.
• The circumference of a circle varies directly
• What are the relationships among the angles and
SMP 4 Model with mathematics.
line segments in and around circles?
with its diameter and π is the constant of
SMP 5 Use appropriate tools
• When making calculations, how and why should
variation.
strategically.
• A ratio can be used, instead of a formula, to
the value of π be incorporated into the problem: i.e.,
SMP 6 Attend to precision.
solve for the length of the arc of a circle.
3.14 or π?
SMP 7 Look for and make use of
Acquisition
structure.
KNOWLEDGE
SKILLS
SMP 8 Look for and express
Students
will
know…
Students will be skilled at and/or be able to…
regularity in repeated
•
The
properties
of
• Construct a line tangent to a circle from a point outside that circle.
reasoning.
tangents,
chords
and
• Solve for angles in a circle given a line tangent to a circle.
Standards for Mathematical Content
central
angles.
•
Solve for segment lengths and angles using information from a chord.
Clusters Addressed
•
The
relationship
between
•
Solve
for an angle given an inscribed angle and its intercepted arc.
[a] G-C.A
Understand and apply
central angles and
• Prove properties of angles for a quadrilateral inscribed in a circle.
theorems about circles.
inscribed angles, and
• Calculate π, the ratio of the circumference of a circle to its diameter.
[a] G-C.B
Find arc lengths and
arcs.
areas of sectors of
• Give an informal argument for the formula for circumference.
• The difference between
circles.
• Calculate rates, times, and distances of motion using the Circumference
arc length and arc
[a] G-GMD.A Explain volume
Conjecture.
measure.
formulas and use them
• Derive using similarity the fact that the length of the arc intercepted by an
• The ratio of the
to solve problems.
angle is proportional to the radius.
circumference of a circle
• Calculate the length of an arc or a radius using the arc length formula.
to the diameter of that
circle.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
1
Posted1/13/17
Unit 4 Circles
Geometry ACC
Assessed Grade Level Standards
Standards for Mathematical Practice
SMP 1
Make sense of problems and persevere in solving them.
SMP 2
Reason abstractly and quantitatively.
SMP 3
Construct viable arguments and critique the reasoning of others.
SMP 4
Model with mathematics.
SMP 5
Use appropriate tools strategically.
SMP 6
Attend to precision.
SMP 7
Look for and make use of structure.
SMP 8
Look for and express regularity in repeated reasoning.
Standards for Mathematical Content
[a]
G-C.A
G-C.2
G-C.3
G-C.4
Understand and apply theorems about circles.
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.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
(+) Construct a tangent line from a point outside a given circle to the circle.
[a]
G-C.B
G-C.5
Find arc lengths and areas of sectors of circles.
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.
[a]
G-GMD.A Explain volume formulas and use them to solve problems.
G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
Key:
[m] = major clusters; [s] = supporting clusters, [a] = additional clusters

Indicates a modeling standard linking mathematics to everyday life, work, and decision-making
CA Indicates a California-only standard
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
2
Posted1/13/17
Unit 4 Circles
Geometry ACC
Evidence of Learning – Stage 2
Assessment Evidence
Unit Assessment
Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
Concepts and skills that may be assessed in Claim1:
[a] G-C.A
Understand and apply theorems about circles.
• Students will identify and describe relationships among inscribed angles, radii and chords.
• Students will identify and explain why inscribed angles on a diameter are right angles.
• Students will identify and explain why the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
• Students will prove properties of angles for a quadrilateral inscribed in a circle.
• Students will construct a tangent line from a point outside a given circle to the circle.
[a] G-C.B
Find arc lengths and areas of sectors of circles.
• Students will derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius.
[a]
•
G-GMD.A Explain volume formulas and use them to solve problems.
Students will give an informal argument for the formula for the circumference of a circle.
Claim 2: Students can solve a range of wellposed problems in pure and applied
mathematics, making productive use of
knowledge and problem-solving strategies.
Standard clusters that may be assessed in
Claim 2:
•
None
Claim 3: The student can clearly and precisely
construct viable arguments to support their own
reasoning and critique the reasoning of others.
Standard clusters that may be assessed in
Claim 3:
•
None
Claim 4: The student can analyze complex,
real-world scenarios and can construct and use
mathematical models to interpret and solve
problems.
Standard clusters that may be assessed in
Claim 4:
•
G-MGD.A
Other Evidence
Formative Assessment Opportunities
•
•
•
•
•
•
•
•
•
Opening Tasks
Informal teacher observations
Checking for understanding using active participation strategies
Exit slips/Summaries
Modeling Lessons (SMP 4)
Tasks
Formative Assessment Lessons (FAL)
Quizzes/Chapter Tests
SBAC Interim Assessment Blocks
Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
3
Posted1/13/17
Unit 4 Circles
Geometry ACC
Learning Plan – Stage 3
Days
1-2
days
Learning Target
I will explore the
components of a
circle by
participating in
the Opening
Task.
Suggested Sequence of Key Learning Events and Instruction
Discovering
Geometry
Expectations
(Activities and
Lessons)
Opening Task – I Am Alive Because of Pi
Students are asked to solve for the diameter of a large circle by
estimating its circumference. Without any formal measuring devices,
they need to be creative with how they approach this challenge. The
teacher is encouraged to allow the students to grapple with this problem
and come up with their own techniques. Students will present their
solutions to the class. Suggestions for the teacher are included in the
lesson.
Curriculum Intranet
Procedural Skills and
Fluency:
• I Am Alive Because of
Pi
This activity serves to motivate and review topics that will be used
throughout Unit 4.
3-4
days
I will explore
properties of
tangents and
chords by…
• Using geometry tools to explore relationships among circles, radii,
and tangent lines. (Geogebra) (SMP 5)
• Applying tangent line properties to real-world problems.
• Constructing tangent lines. (Geogebra) (SMP 5)
• Discovering the properties relating central angles, chords and arcs of
circles. (SMP 8)
• Answering questions such as…
o How does the Tangent Segments Conjecture follow logically
from the Tangent Conjecture and the converse of the Angle
Bisector Conjecture?
o Given a quadrilateral formed by two lines tangent to a circle and
two radii to the lines points of tangency, what figure is formed?
How do you know?
o What is needed to ensure that two arcs have the same size and
shape?
o What can you say about the distance from the center of a circle
if one chord is longer than the other?
o How do you know whether or not arcs are congruent?
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
4
•
•
Lesson 6-1
Lesson 6-2
Conceptual
Understanding:
• NCTM Illuminations:
Folding Circles:
Exploring Circle
Theorems
• Tangent to a Circle
from a Point
• Open Middle: Longest
Chord in a Circle
Procedural Skills and
Fluency:
• Khan Academy:
Constructing a Line
Tangent to a Circle
• MathOpenRef: Tangent
Posted1/13/17
Unit 4 Circles
Geometry ACC
Learning Plan – Stage 3
Days
Learning Target
•
•
•
•
4-5
days
2-3
days
I will develop
properties of
circles by…
I will give
informal
arguments for
circle formulas
by…
Suggested Sequence of Key Learning Events and Instruction
Discovering
Geometry
Expectations
(Activities and
Lessons)
Discovering relationships between central angles, inscribed angles
• Lesson 6-3
and arcs. (Geogebra)
• Lesson 6-4
Proving properties of inscribed angles and polygons. (SMP 3)
• Lesson 13-6
Proving circle conjectures. (SMP 3)
Answering questions such as…
o What happens to the intercepted arc as you rotate the angle
about a vertex, keeping the angle’s measure constant?
o Use the Inscribed Angle Conjecture to explain why the other
conjectures are true.
o What if the vertex of the angle is in the interior of the circle (or
the exterior of the circle), but is not at the center?
o How does the measure of the angle relate to the measure of the
intercepted arc?
o What do you think about as you are beginning to structure a
proof?
o How do you prove all three cases of the Inscribed Angle
Conjecture?
o What are some examples of postulates that could be used to
prove circle properties?
• Exploring and calculating the ratio of the circumference of a circle to
the length of its diameter. (SMP 8)
• Applying direct variation to the relationship between circumference
and diameter.
• Using pi to in real-world applications.
• Answering questions such as…
o Why would anyone work with π as a symbol rather than
immediately approximating it with a decimal?
o Why is dimensional analysis an important help when working
with real-world applications?
o Using the Circumference Conjecture, what can you solve for?
• Lesson 6-5
• Lesson 6-6
Curriculum Intranet
Conceptual
Understanding:
• Right Triangles
Inscribed in Circles I
• Right Triangles
Inscribed in Circles II
• Opposite Angles in a
Cyclic Quadrilateral
• Mutually Tangent
Circles
Procedural Skills and
Fluency:
• MathOpenRef:
Inscribed Angles
• MathOpenRef: Interior
Angles of an Inscribed
Quadrilateral
Conceptual
Understanding:
• NCTM Illuminations:
Computing Pi
• NCTM Illuminations:
The Ratio of
Circumference to
Diameter
• Desmos: Measuring
Circles
Application:
• 3-Act Lesson: Rotonda
West, Florida
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
5
Posted1/13/17
Unit 4 Circles
Geometry ACC
Learning Plan – Stage 3
Days
Learning Target
2-3
days
I will check my
understanding of
sectors and
ratios by
participating in
the FAL.
Suggested Sequence of Key Learning Events and Instruction
Discovering
Geometry
Expectations
(Activities and
Lessons)
FORMATIVE ASSESSMENT LESSON
•
1-2
days
I will understand
how to solve for
the length of an
arc by…
•
•
Discovering a formula for finding the length of the arc of a circle.
(SMP 8)
Calculating arc lengths.
Answering questions such as…
o Where does the Arc Length Conjecture come from?
o How is arc length different from arc measure?
• Lesson 6-7
• Incorporating the Standards for Mathematical Practice (SMPs) along
with the content standards to review the unit.
2
days
I will prepare for
the unit
assessment on
circles by…
1-2
days
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Curriculum Intranet
FAL: Pied - Sectors,
Ratios, and
Circumferences
Conceptual
Understanding:
• FAL: Calculating Arcs
and Sectors of Circles
Procedural Skills and
Fluency:
• Math Open Ref: Arc
Length
Procedural Skills and
Fluency:
• MathOpenRef: Area of
a Sector
Application:
• 3-Act Lesson: Lucky
Cow
Unit Assessment (LBUSD Math Intranet, Assessment)
6
Posted1/13/17