Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of geometry wikipedia , lookup
Lie sphere geometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Analytic geometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
Problem of Apollonius wikipedia , lookup
Line (geometry) wikipedia , lookup
Conic section wikipedia , lookup
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… 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… 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. 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. 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 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/ 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. Geometry Unit 8 –Conic Sections – Circles and Parabolas in units. 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. 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. 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 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. Geometry Unit 8 –Conic Sections – Circles and Parabolas 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 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. 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. IfmC =42°,whatismE? 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) 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. 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) 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. Geometry Unit 8 –Conic Sections – Circles and Parabolas 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. Sample Formative Assessment Tasks Skill-based task 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 1. Suppose you know the center of a circle and a point on the circle. How do you determine the equation of the circle? 1. center(2,3);radius = 5 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. (x4)2+(y–3)3 = 16 Problem-based task What is the standard equation of each circle? 2. center(0,1);radius = Equation of Circle Interactive Applet http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php 4. (x+7)2+y2 = 10 Teacher Created Argumentation Task (W1-MP3&6) THINK ABOUT A PLAN: Find the circumference and area of the circle whose equation is (x – 9)2 + (y – 3)2 = 64. Leave your answers in terms of pi. Include in your answer the following: What essential information do you need? What formulas will you use? (Taken from PH Geometry, p.802 #44) 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. 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 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. Geometry Unit 8 –Conic Sections – Circles and Parabolas 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 Sample Formative Assessment Tasks 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. Geometry Unit 8 –Conic Sections – Circles and Parabolas 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 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. 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 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. 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 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.