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Transcript
Geometry Unit 8 –Conic Sections – Circles and Parabolas
U9Enduring understanding (Big Idea): Students will understand that concepts related to circles and conic sections are applicable in real
world scenarios as they explore properties of tangent lines, represent circles as equations based on the center and radius, calculate the measures of
central angles, inscribed angles, and their intercepted arc measures and lengths.
Essential Questions:
1.
2.
3.
4.
5.
6.
Discuss what it means to “go off on a tangent”?
How do you find the equation of a circle in a coordinate plane?
When lines intersect a circle or intersect within a circle, how do you find the measure of resulting angles, arcs, and segments?
How can you prove relationships between angles and arcs in a circle?
What is the intersection of a cone and a plane parallel to a line along side of the cone?
How can you derive the equation for a parabola, given a focus and directrix?
BY THE END OF THIS UNIT:
Students will know…
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
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
Properties of tangent lines as it relates to a circle
Concepts of chords, arcs, and angle measures as it relates to a circle
Arc Length and Segment Lengths as it relates to circles
Equations of a Circle and Parabola
Conic Sections
Vocabulary: arc measure, arc length, inscribed angle, intercepted arc,
chord, point of tangency, tangent line (tangent to a circle), secant,
standard form of the equation of a circle, conic sections, directrix, focus,
parabola, ellipse, hyperbola
Unit Resources
Learning Task:click on Circle Formulas – download file, print, and copy
http://www.mathworksheetsgo.com/sheets/geometry/circles/circle-formulagraphic-organizer.php
Performance Task:Have students view the power point presentation. In
writing, allow students to describe how well the presentation reflects what was
learned in class. Be sure to include what concepts were discussed and which
were left out.www.btinternet.com/~mathsanswers/CircleTheorems.ppt
Unit Review Game:Jeopardy Review
Gamehttp://www.superteachertools.com/jeopardyx/jeopardy-review-gameconvert.php?gamefile=../jeopardy/usergames/May201221/jeopardy1337974033.
txt
Students will be able to…






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Identify a tangent and use properties of tangent as it relates to a circle
Compute chord, arc, and angle measures
Find arc length given the arc’s central angle and the circle’s diameter or
radius
Find lengths of segments related to circles and its intersecting lines
Write the equation of a circle given its center and radius
Identify conic sections
Write the equation of a parabola given its directrix and focus.
Mathematical Practices in Focus:
1.
2.
4.
6.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Model with mathematics.
Attend to precision.
NOTE: For Unit Resources, the Performance Task Activity can
also be a Project.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
CORE CONTENT
Cluster Title: Find arc lengths and areas of sectors of circles
Standard: 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
Concepts and Skills to Master


Identify major and minor arcs and semicircles
Find the measure of a central angle and its intercepted arc
 Compute the circumference of a circle and arc length (i.e. distances along circular paths)
SUPPORTS FOR TEACHERS
Critical Background Knowledge


Circumference of a Circle
Exact Circumference (leave your answer in terms of pi)
 Congruent circles have congruent radii
Academic Vocabulary
circle, center, diameter, radius, congruent circles, central angle, semicircle, minor arc, major arc, adjacent arcs, intercepted arc, circumference, pi,
concentric circles, arc length, congruent arcs, exact circumference
Suggested Instructional Strategies
 Be sure to highlight for students that an arc is measured by the
central angle that defines it. The central angle captures within
its rays the intercepted arcs.
 Error Prevention: Students may benefit from tracing the cited
arc(s) of the figure(s) with colored pencils
 Explain to students that as it relates to standard G.C.5, the
length of an arc can be found by multiplying the ratio of the
arc’s measure to 360 degrees by the circle’s circumference.
 Students often confuse arc measure with arc length. Be sure to
note that one is measured in degrees and the other is measured
in units.
Resources

Textbook Correlation: 10-6 Circles and Arcs
 Online Teacher Resource Center: www.pearsonsuccessnet.com
Activities, Games, and Puzzles (10-6 Circles and Arcs Crossword)

Commonly Confused: Arc Measure & Arc Length
Bright storm Video – use the link below
www.brightstorm.com/math/geometry/.../arc-length/
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
Sample Formative Assessment Tasks
Skill-based task
Problem Task
Find the arc measure and arc length of each darkened arc.
Leave your answer in terms of π.
1.
2.
3.
Task: It is 5:00. What is the measure of
theminor arc formed by the hands of an
analog clock hanging on a classroom wall?
What is the arc length if the radius of the
clock is 6 inches?
Sketch a wall clock to support your answer.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
CORE CONTENT
Cluster Title: Understand and apply theorems about circles
Standard: G.C.2Identify 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.
Concepts and Skills to Master




Tangent Lines
Chord and Arc Measures
Central and Inscribed Angles
Angle Measures and Segment Lengths
SUPPORTS FOR TEACHERS
Critical Background Knowledge






Students will use understanding of congruent triangles and right triangles to prove statements about tangent lines.
Prior knowledge of a circle and its common features are needed: center, radius, diameter, chord, arc.
Triangle Angle Sum Theorem
Pythagorean Theorem
Perimeter of Polygons
Congruence
Academic Vocabulary
tangent to a circle, point of tangency, inscribed circles, chord, arc, semicircle, inscribed angles, circumscribed polygons, secant
Suggested Instructional Strategies
Resources




Students sometimes get confused identifying segments of a
circle. Have students create a vocabulary sheet that includes
definitions and diagrams of each type of segment.
Students sometimes get confused identifying central and
inscribed angles and, therefore, use the wrong formula to
compute angle measures. Perhaps making a connection that
a central angle has its vertex in the center of the circle will
help students distinguish between the two.
Paper folding activities offer students a good way to develop
key concepts related to central angles, chords, and arcs.
Have students to organize all the theorems taught in sections
12.1 to 12.4 in an effort to increase learning.

Cluster Review
http://library.thinkquest.org/20991/geo/circles.html
 Circle Concept Interactive Math Site
http://www.mathopenref.com/chordsintersecting.html
(Explore circle concept by scrolling down and clicking from the
selection on the bottom left of the screen)

Concept Byte Exploration Activity: p.770 - Paper Folding With Circles
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
Sample Formative Assessment Tasks
Skill-based task
Problem Task
Reasoning Challenge
Is the statement true or false? If it is true, give a convincing argument. If
it is false, give a counterexample.
Refer to
Cabovefor Exercises 1–3. Segment
1. IfDE =4andCE =8,whatistheradius?
2. IfDE =8andEF =4,whatistheradius?
3. IfmC =42°,whatismE?
is tangent to
C.
1. If two angles inscribed in a circle are congruent, then they intercept
the same arc.
2. If an inscribed angle is a right angle, then it is inscribed in a
semicircle.
3. A circle can always be circumscribed about a quadrilateral whose
opposite angles are supplementary.
(See Teacher Edition – Chapter 12 p.786 #35-37 for answers)
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
CORE CONTENT
Cluster Title: Equations of Circles – Translate between the geometric description and the equation for a conic section.
Standard: G.GPE.1Derive the equation of a circle given a center and radius using the Pythagorean Theorem: complete the square to find the
center and radius of a circle given by an equation.
Concepts and Skills to Master


Write the equation of a circle and apply it given a graph or a circle’s center and radius.
Find the center and radius of a circle using the coordinate plane or the general form of the equation of a circle.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Distance Formula
 Sketching graphs on a coordinate plane (x-y axis).
Academic Vocabulary
standard form of an equation of a circle, center of a circle on the coordinate plane (h, k), radius (r)
Suggested Instructional Strategies
Resources

Arrange students into pairs of mixed abilities. On the
board, draw a circle on a coordinate plane. One student
will write an equation of the circle using the center and the
radius, and the other student will use the center and one
point. Tell them to share their equations and discuss any
discrepancies. You may vary this activity by having one
student draw a circle on a coordinate plane and the other
write the equation. The drawings can be done on graph
paper in a page protector so that the paper can be cleaned
and reused.

Emphasize that writing the equation for a circle in standard
form makes it easier to identify the center (h, k).

Remind students to take the square root of the value r2 in
order to find the radius.
 Equation of Circle Interactive Applet
http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php

Equations of Circles Powerpoint
(Including Completing the Square)
www.mathxtc.com/Downloads/MeasureGeo/files/Circles.ppt

Online Teacher Resource Center
www.pearsonsuccessnet.com- Geometry
Dynamic Activity 12-5: Circles in the Coordinate Plane

Completing the square is not covered in the Pearson Geometry text.
However, online resources from Chapter 10 of the Pearson Algebra 2
text can be used as a resource to teach or review completing the
square.
www.pearsonsuccessnet.com - Algebra 2 p.633 – Problem 4
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
Sample Formative Assessment Tasks
Skill-based task
Problem-based task
1. Suppose you know the center of a circle and a point on the circle. How
do you determine the equation of the circle?
What is the standard equation of each circle?
1. center(2,3);radius = 5
2. center(0,1);radius =
2. A student says that the center of a circle with equation:
(x – 2)2 + (y + 3)2 = 16 is (-2, 3). What is the student’s error? How should
the equation read in order to make the student correct?
7
What is the center and radius of each circle?
3. (x4)2+(y–3)3 = 16
4. (x+7)2+y2 = 10
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
CORE CONTENT
Cluster Title: Conic sections and the Parabola – Translate between the geometric description and the equation for a conic section.
Standard: G.GPE.2 – Derive the equation of a parabola given a focus and directrix.
Concepts and Skills to Master



Identify and graph the conic sections
Identify lines of symmetry and the domain and range once given the graph of a conic section
Write the equation of a parabola and graph it
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Domain and Range
 Graphing on a Coordinate Plane
 Lines of Symmetry
Academic Vocabulary
conic sections (parabolas, circles, ellipses, and hyperbolas), lines of symmetry, focus, directrix, focal length
Suggested Instructional Strategies

At this point, do not make graphing the conic sections a
more difficult task by having students solve for x and or y.
Instead, simply have student graph conic sections using a
table of values that range from -5 to +5; substituting for
whichever variable is easier. [Note: If you have a
classroom set of graphing calculators, you may want
students to practice solving for y in order to use the equation
editor and table of values.] (Also, note that more emphasis will be
placed on conic sections in further math courses.)

Be sure that students understand that a conic section is
simply the intersection of a plane and a cone. (Use
resource: Conic Sections Explained as a teaching aid if
needed.)
Resources
 Textbook Correlation: Algebra II Textbook
10-1 Exploring Conic Sections (www.pearsonsuccessnet.com)
10-2 Parabolas (www.pearsonsuccessnet.com)
 Conic Sections Explained
http://math2.org/math/algebra/conics.htm
 Parabolas and Their Equations Powerpoint
https://docs.google.com/viewer?a=v&q=cache:epOo8GEPeOIJ:p
rincemath.wikispaces.com/file/view/parabolas.ppt+parabola+and
+its+equations+powerpoint&hl=en&gl=us&pid=bl&srcid=ADGEE
Sh5fKhyjqpZxcMuqaQOU5kouLHLYDR4TuYHy5eWBU8yqGviM
zQqb_iESTO7MRFVXhc3mKlAOnc0nbIFTkIgQggy6EXbwLGEzz1vJAfGo1wYmUlIynOQgDtEreV1t
KGzC4yU9RT&sig=AHIEtbRUvWX5ZWLTFr68Jn4HTR3RPaRLQ
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
Sample Formative Assessment Tasks
Skill-based task
Problem Task
Write an equation of a parabola with vertex at the origin and the
given focus.
Task 1:
1.focus at(2,0)2. focusat(0,4)
3. Write an equation of a parabola with vertex at the origin and the
givendirectrix, x = 3.
4. Identify the vertex, the focus, and the directrix of the
parabola with the given equation. Then sketch the graph of the
parabola.
x –
Error AnalysisOne student identifies four types of conic sections. Another says
there are only three types (hyperbola, circle, and ellipse). Who is correct? Explain
how conic sections are found.
Task 2:
ReasoningA student wants to graph a circle with the equation x2 + y2 = 25.
What points could he use to determine a sketch of the graph?
1
2
 y + 1  2
4
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
CORE CONTENT
Cluster Title: Understand and apply theorems about circles (i.e. Circle Similarity)
Standard: G.C.1 Prove that all circles are similar.
Concepts and Skills to Master

Prove Similarity in Circles
SUPPORTS FOR TEACHERS – NOTE:This concept is not in the textbook and limited information appropriate for HS students is available online.
Critical Background Knowledge


Definition of Similarity
Applications with Circle Formulas and Right Triangles
Academic Vocabulary
similarity of circles
Suggested Instructional Strategies




Recall: being similar means having corresponding congruent
angles but proportional corresponding sides. See Online
Resource A.
In general, two figures are similar if there is a set of
transformations that will move one figure exactly covering the
other. To view proof, see Online Resource B.
To prove any two circles are similar, only a translation (slide)
and dilation (enlargement or reduction) are necessary. Using
the differences in the center coordinates to determine the
translation and determining the quotient of the radii for the
dilation can always do this. For further explanation, see Online
Resource C.
Problem Task:Take students to the lab if possible to view the
you-tube video that teaches the lesson on circle similarity. If
students do not have access to the site, save the link elsewhere
so that students can view it – or make it a homework
assignment.
(Honor and IB Classes only) If the video is used for Standard
classes, teacher explanation and modeling is necessary.
Resources
 Textbook Correlation: none
 Online Resource A
Core Challenge – Standard G.C.1 – Prove all circles are similar.
Click on ‘download file’.
http://app.corechallenge.org/learningobjects/7878

Online Resource B – All Circles are Similar Examples.pdf
www.cpm.org/pdfs/state_supplements/Similar_Circles.pdf
 Online Resource C – YouTube Video –
All circles are similar demonstration
http://www.youtube.com/watch?v=jTvlvLFZQPY
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 8 –Conic Sections – Circles and Parabolas
Sample Formative Assessment Tasks
Skill-based task
Problem Task(see suggested instructional strategies – item 4)
Use the link below to view the 32min 20 sec you-tube video that discusses
circle similarity. Take notes during the video. After viewing the video,
complete a written assignment, documenting what you have learned.
Link: http://www.youtube.com/watch?v=2QOj02EKDTE
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.