Geometry Unit 8 –Conic Sections – Circles and Parabolas
U9-
BY THE END OF THIS UNIT:
CORE CONTENT
Cluster Title: Find arc lengths and areas of sectors of circlesStandard: 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/
Sample Formative Assessment Tasks
Skill-based task
Find the arc measure and arc length of each darkened arc.
Leave your answer in terms of π.
1. 2. 3.
/ Problem Task
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.
CORE CONTENT
Cluster Title: Understand and apply theorems about circlesStandard: 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
· 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. / Resources
· 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
Sample Formative Assessment Tasks
Skill-based task
Refer to Cabovefor Exercises 1–3. Segmentis tangent to C.
1. IfDE =4andCE =8,whatistheradius?
2. IfDE =8andEF =4,whatistheradius?
3. IfmÐC =42°,whatismÐE? / 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.
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)
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
· 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. / Resources
· 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
Sample Formative Assessment Tasks
Skill-based task
What is the standard equation of each circle?
1. center(2,3);radius = 5
2. center(0,-1);radius =
What is the center and radius of each circle?
3. (x-4)2+(y–3)3 = 16 4. (x+7)2+y2 = 10 / 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?
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?
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:princemath.wikispaces.com/file/view/parabolas.ppt+parabola+and+its+equations+powerpoint&hl=en&gl=us&pid=bl&srcid=ADGEESh5fKhyjqpZxcMuqaQOU5kouLHLYDR4TuYHy5eWBU8yqGviMzQqb_iESTO7MRFVXhc3mKlAOn-c0nbIFTkIgQggy6EXbwLGEzz1vJAfGo1wYmUlIynOQgDtEreV1tKGzC4yU9RT&sig=AHIEtbRUvWX5ZWLTFr68Jn4HTR3RP-aRLQ
Sample Formative Assessment Tasks
Skill-based task
Write an equation of a parabola with vertex at the origin and the given focus.
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.
/ Problem Task
Task 1:
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?
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
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
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.