Download Geometry Mathematics Curriculum Guide

Geometry Mathematics Curriculum Guide
Unit 7: Circles
2014 – 2015
Time Frame: 10 Days
Primary Focus
Students continue to expand their understanding of geometry by exploring geometric relationships pertaining to circles. As was the case in algebra 1 and earlier in geometry,
attributes of circles observed at earlier grades will now be looked at more precisely through proof. Many of the geometric relationships and properties proven in geometry,
particularly those related to triangle congruence and triangle similarity, will be applied in these proofs. In this unit, students will begin their exploration of analytic geometry,
where the use of coordinates helps to connect algebra and geometry and provides students with additional analysis and problem solving techniques. Here, students will build on
their understanding of distance in coordinate systems and draw on their growing command of algebra to discover the connection between the graph of a circle in the coordinate
plane and an algebraic equation. Along with physical models, dynamic geometry environments will provide students with tools for investigating, experimenting with,
conjecturing about, and modeling geometric phenomena related to circles.
Common Core State Standards for Mathematical Practice
Standards for Mathematical Practice
MP4 - Model with mathematics.
MP5 - Use appropriate tools strategically.
MP6 - Attend to precision.
MP7 - Look for and make use of structure.
MP8 - Look for and express regularity in repeated reasoning.
Unit 7
How It Applies to this Topic…
Use a variety of methods to model, represent, and solve real-world problems.
Select and use appropriate tools to best model/solve problems.
Transform figures efficiently and accurately and label them appropriately.
Use patterns or structure to make sense of mathematics and connect prior knowledge to similar
situations and extend to new situations.
Generalize the process to create a shortcut which may lead to developing rules or creating a
formula.
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2014 – 2015
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
Perform constructions related to a circle and is able to justify each step in a construction when solving for geometric problems.
UNDERSTANDINGS
Students will understand that…
Meaning Goals
ESSENTIAL QUESTIONS
• All circles are similar.
• The geometric relationships that come from proving triangles congruent or from
proving triangles similar may be used to prove relationships between geometric objects.
• Different relationships among inscribed angles, radii, and chords of a circle, and
between the angles of a quadrilateral inscribed in a circle are provable
Using previously proven relationships between geometric objects.
• A circle drawn in the coordinate plane can be represented by an algebraic equation
that is dependent upon the coordinates of the center of the circle and the radius of the
circle.
• The relationship between the length of the arc of a circle, the central angle of the circle
that intercepts this arc, and the radius of the circle.
• What are the different relationships among inscribed angles, radii, and chords of a
circle, and of the angles of a quadrilateral inscribed in a circle?
• What is the relationship between the length of the arc of a circle, the central angle of
the circle that intercepts this arc, and the radius of the circle?
• What is the area of a sector of a circle?
• Given the coordinates of the center of the circle and the radius of that circle, what is
the equation of the circle?
• Given an equation for a circle drawn in the coordinate plane, what are the coordinates
of the center of the circle and the radius of the circle?
Acquisition Goals
Students will know and will be skilled at…
Identify (and name) radii, diameters, and chords.
Prove that all circles are similar (by comparing ratios of radii and circumference of different circles and proving that a constant scale factor exists).
Identify tangent lines, inscribed angles, and central angles.
Apply the relationship between two congruent inscribed angles and their intercepted arcs.
Identify (and apply) that an inscribed angle is one half of the measure of the intercepted arc or central angle that intersects the same arc.
Identify (and apply) that a circumscribed angle is the supplement of the central angle with the same intercepted arc.
Prove that an inscribed angle in a semi-circle is a right angle.
Describe (and apply) how the radius of a circle is perpendicular to the tangent line at the point of tangency.
Construct the inscribed and circumscribed circles of a triangle.
Construct the orthocenter, circumcenter, incenter, and centroid of a triangle.
Prove that the opposite angles of a quadrilateral inscribed in a circle are supplementary.
Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius.
Unit 7
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Geometry Mathematics Curriculum Guide
2014 – 2015
Define the radian measure of the angle as the constant of proportionality.
Derive the formula for the area of a sector.
Find the arc length and area of sectors.
Derive the equation of a circle given the center and radius using the Pythagorean Theorem.
Complete the square to find the center and radius of a circle given by an equation.
Unit 7
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2014 – 2015
Stage 1 Established Goals: Common Core State Standards for Mathematics
Cluster: Standard(s): Understand and apply theorems about circles.
G.C.1 Prove that all circles are similar.
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.
Find arc lengths and areas of sectors of circles.
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.
2008 Geometry Standard connection: G.3.H & I, G.6.A
Unit 7
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Geometry Mathematics Curriculum Guide
Explanations, Examples, and Comments
G.C.1
Using the fact that the ratio of diameter to circumference is the same for circles, prove that all circles are
similar.
Prove that all circles are similar by showing that for a dilation centered at the center of a circle, the preimage
and the image have equal central angle measures.
Students may use geometric simulation software to model transformations and demonstrate a sequence of
transformations to show congruence or similarity of figures.
G.C.2
Identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.
Describe the relationship between a central angle and the arc it intercepts.
Describe the relationship between an inscribed angle and the arc it intercepts.
Describe the relationship between a circumscribed angle and the arcs it intercepts.
Recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right
angle.
Recognize that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G.C.3
Define the terms inscribed, circumscribed, angle bisector, and perpendicular bisector.
Construct the inscribed circle whose center is the point of intersection of the angle bisectors (the incenter).
Construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors of
each side of the triangle (the circumcenter).
Apply the Arc Addition Postulate to solve for missing arc measures.
Prove that opposite angles in an inscribed quadrilateral are supplementary.
Using definitions, properties, and theorems, prove properties of angles for a quadrilateral inscribed in a circle.
Students may use geometric simulation software to make geometric constructions.
Unit 7
Clover Park School District 2015-2016
2014 – 2015
Stage 3
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of
the materials below best meet student instructional needs.
All materials listed may not be needed.
Holt Geometry Lesson 11-1 Lines that Intersect Circles
Holt Geometry Lesson 11-2 Arcs and Chords
Holt Geometry Lesson 11-4 Inscribed Angles
Holt Geometry Lesson 11-3 Sector Area and Arc Length
Holt Geometry Lesson 11-4 Inscribed Angles
Holt Geometry Lesson 11-5 Angle Relationships in Circles
Holt Geometry Lesson 11-6 Segment Relationships in
Circles
Supplemental Materials
Discovering Geometry 1.7 Circles
Discovering Geometry 6.1 Tangent Properties
Discovering Geometry 6.2 Chord Properties
Discovering Geometry 6.3 Arcs and Angles
Discovering Geometry 6.4 Proving Circle Conjectures
Discovering Geometry 6.5 The Circumference/Diameter
Ratio
Discovering Geometry 6.7 Arc Length
Discovering Geometry Exploration Intersecting Lines
Through A Circle
Discovering Geometry 8.6 Any Way You Slice It (Sector
Area)
Discovering Geometry 13.6 Circle Proofs
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Geometry Mathematics Curriculum Guide
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Use the image below to match the term to the appropriate component of the circle. A is the center of the
circle, and ⃖����⃗
𝐷𝐷 and ⃖���⃗
𝐶𝐶 are tangent to circle A. (G.C.2)
����
a) Central Angle
1) 𝐵𝐵
b) Tangent
2) ∠BDC
����
c) Circumscribed
3) 𝐴𝐴
Angle
d) Chord
4) ∠BAC
�
e) Minor arc
5) 𝐷𝐷𝐷
�
f) radius
6) 𝐵𝐵
⃖���⃗
g) Major arc
7) 𝐶𝐶
h) Inscribed angle
8) ∠DEC
1b) Write an equation that puts the m∠BDC, in terms of m∠BAC.
1c) Which two angles in the image can you be certain are congruent?
•
���� where B is at (0, 0) and C is at (0, 10). Write the equation
∆𝐴𝐴𝐴 is a right triangle with hypotenuse 𝐵𝐵
that describes the circle that circumscribes ∆𝐴𝐴𝐴
Cluster: Standard(s): Experiment with transformations in the plane.
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.
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Geometry Mathematics Curriculum Guide
2014 – 2015
G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
2008 Geometry Standard connection: G.3.I
Explanations, Examples, and Comments
Stage 3
Given Circle A with diameter (BC) ̅ construct quadrilateral BDCE such that it is a quadrilateral inscribed in circle MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of
A, and a regular polygon. (G.CO.13)
the materials below best meet student instructional needs.
All materials listed may not be needed.
List of constructions in HOLT is available on A87
http://www.mathsisfun.com/geometry/constructions.html
(A good list of basic constructions walkthroughs)
Additional Resources for Constructions
Construction Resource
What is the relationship between segment CB and segment DE?
Assuming the length of CB is 2 units, what is the area of quadrilateral BDCE?
What is the angle relationship between angle CEB and angle CDB. Prove your answer using inscribed angles.
Construct an equilateral triangle with the given side length, and then circumscribe a circle around the
triangle.
Given an equilateral triangle inscribed in a circle, describe the process necessary to construct a hexagon inside
the same circle.
Define a circle using the term point, and distance.
Cluster: Standard(s): Translate between the geometric description and the equation for a conic section
G-GPE.1 Derive the equation of a circle given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by
Unit 7
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2014 – 2015
an equation.
Cluster: Standard(s): Use coordinates to prove simple geometric theorems algebraically
G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. (For example, prove or disprove that the point (1, √ ) lies on the circle centered at the
origin and containing the point (0, 2).)
2008 Geometry Standard connection: G.4.D
Explanations, Examples, and Comments
Identify the center and radius of a circle given its equation.
Draw a right triangle with a horizontal leg, a vertical leg, and the radius of a circle as its hypotenuse.
Use the Pythagorean Theorem, the coordinates of a circle’s center, and the circle’s radius to write the
equation of the circle.
Convert an equation of a circle in general (quadratic) form to standard form by completing the square.
Identify the center and radius of a circle given its equation.
Students may use geometric simulation software to explore the connection between circles and the
Pythagorean Theorem.
EXAMPLES:
𝑥 2 + 8𝑥 + 𝑦 2 + 10𝑦 = 8 Use the equation to describe the center, and radius of the circle described.
Stage 3
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of
the materials below best meet student instructional needs.
All materials listed may not be needed.
Holt Geometry Lesson 11-7 Circles in the Coordinate Plane
Supplemental Resources
Discovering Geometry 9.6 Circles and the Pythagorean
Theorem
MVP Course 2 Module 8 Lessons 1-3
𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎 (1, −2) 𝑤𝑤𝑤ℎ 𝑎 𝑟𝑟𝑟𝑟𝑟𝑟 𝑜𝑜 12 𝑢𝑢𝑢𝑢𝑢.
𝑊𝑊𝑊𝑊𝑊 𝑡ℎ𝑒 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓 𝑡ℎ𝑒 𝑐𝑐𝑐𝑐𝑐𝑐.
Verify that the distance, formula and the standard form equation for a circle are equivalent.
(0, 13) 𝑎𝑎𝑎 (12, −5) are an equal distance from the origin, prove it.
𝐴 (2, 3) 𝑎𝑎𝑎 𝐵 (−2, 0) are the center, and endpoint of a radius of a circle. List a point on the perimeter of
circle A, and a point on the perimeter of circle B.
Unit 7
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
Evaluative Criteria/Assessment Level
Descriptors (ALDs):
Claim 1 Clusters:
2014 – 2015
Stage 2 - Evidence
Sample Assessment Evidence
Concepts and Procedures
NONE
Claim 2 Clusters:
Problem Solving
NONE
Claim 3 Clusters:
Communicating Reasoning
NONE
Go here for Sample SBAC items
Go here for more information about the Achievement Level Descriptors for Mathematics:
Unit 7
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2014 – 2015
Stage 3 – Learning Plan: Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
LEARNING ACTIVITIES:
Recommended Sequence for the unit
NOTES:
Triangle Centers may be taught during this unit.
Holt Geometry Chapter 11 Lesson 1-7
Coordinate Geometry with circles should
introduce the circle equation (𝑥 − ℎ)2 +
(𝑦 − 𝑘)2 = 𝑟 2 and show its connection to the
Pythagorean Theorem and Distance Formula.
Daily Lesson Components
Learning Target
Warm-up
Activities
• Whole Group:
• Small Group/Guided/Collaborative/Independent:
• Whole Group:
Checking for Understanding (before, during and after):
Assessments
Unit 7
Clover Park School District 2015-2016
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