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Foundations of Geometry – Unit 5 – Circles Enduring understanding (Big Idea): Students will identify parts and properties of the circle as well as understand U9that concepts related to circles involves many aspects of geometry including lines, segments, angles, and arcs. Essential Questions: 1. 2. 3. 4. What are the line segments and angles that are related to circles? When lines intersect a circle or intersect within a circle, how do you find the measure of resulting angles, arcs, and segments? How can the distance formula (Pythagorean Theorem) be extended to derive the equation of a circle? How can patty paper and/or constructions be used to further explore the definition and properties of the circle? BY THE END OF THIS UNIT: Students will know… The line segments and angles that are related to the circle How to calculate measures of central and inscribed angles and arcs Concepts of chords, arcs, and angle measures as it relates to a circle How to compute an arc’s length as well compare it to arc measure How to extend the distance formula (Pythagorean Theorem) to derive the equation of a circle Vocabulary: center, diameter, radius, chord, central angles, arc Students will be able to… Identify the center, radius, diameter, chord, arc, and sector of a circle Identify a tangent and use properties of tangent as it relates to a circle Compute arc and angle measures Find arc length given the arc’s central angle and the circle’s diameter or radius Write the equation of a circle Use paper folding or constructions to explore circle concepts- measure, arc length, inscribed angle, intercepted arc, point of tangency, tangent line (tangent to a circle), standard form of the equation of a circle Unit Resources Learning Task: If possible, take students to the computer lab and have them view the power point presentation with the following URL: www.btinternet.com/~mathsanswers/CircleTheorems.ppt Have students submit a written description of how well the presentation reflects what was learned in class. 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. Performance Task: Use Patty Paper Geometry Book by Michael Serra Assign and/or Model: Investigation Set 7 – pages 105-120 Unit Review Game: Jeopardy http://www.superteachertools.com/jeopardyx/jeopardy-review-gameconvert.php?gamefile=../jeopardy/usergames/May201221/jeopardy1337974033 .txt 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. Foundations of Geometry – Unit 5 – Circles CORE CONTENT Cluster Title: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. Standard: 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Concepts and Skills to Master Accurately communicate parts of a circle using appropriate mathematical language Understand why the ratio of circumference to diameter can be expressed as pi, Π Identify major and minor arcs and semicircles Compute the circumference and area of a circle Solve mathematical and real-life problems involving circles SUPPORTS FOR TEACHERS Critical Background Knowledge Formulas for finding the circumference and area of the circle The value of pi and how to leave answers to problems in terms of pi (Ex: exact circumference) Understand the relationship between the radius and diameter of a circle Academic Vocabulary circle, center, diameter, radius, chord, semicircle, major arc, minor arc, central angle, circumference, pi, exact circumference Suggested Instructional Strategies Resources Assess student knowledge of this standard using a pretest reviewing lessons 1-8 and 10-6. Modify core content based on student results. Use Lesson 1-8 and/or 10-6 from the Foundations of Geometry or Geometry book as a supplemental tool if needed. Have students complete the Circumference Ratio Geometry Activity as an exploration. (See Problem Textbook Correlation Online Teacher Resource Center: www.pearsonsuccessnet.com 1-8 Perimeter, Circumference, and Area (Use Content Relevant to Circles only) 10-6 Circles and Arcs (Use 10-6 ELL Support and 10-6 Activities, Games, and Puzzles) Online Practice: Have students review the parts of the circle by sketching the drawings Task) Tell students that the circumference of a circle can be thought of as the perimeter of a circle in order to investigate the meaning pi. from this site. Then, read the review questions and answer the exercises for immediate feedback. http://www.mathgoodies.com/lessons/vol2/geometry.html Patty Paper Geometry Book by Michael Serra Finding the Center of a Circle p.105 or p.115 or use this link (p.105 only): http://www.pflugervilleisd.net/curriculum/math/documents/Circles_properties.p df 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. Foundations of Geometry – Unit 5 – Circles Sample Formative Assessment Tasks Skill-based task Problem Task Geometry Activity Circumference and Area Find the circumference and area of each circle. Leave your answer in terms of π. Objective: Discover the special relationship that exists between the circumference of a circle and its diameter. I. Gather Data and Analyze - Collect ten round objects. A. Measure the circumference and diameter of each object using a millimeter measuring tape. Record the measures in a table like the one below. C B. Compute the value of to the nearest hundredth for each d object. Record the result in the fourth column of the table. (Note to teacher: Each ratio should be near 3.1) Algebra Find the value of the variable. Object Circumference = C diameter = d C d 1. 2. 3. … 10. Word Problem A Ferris wheel has a 50-m radius. How many kilometers will a passenger travel during a ride if the wheel makes 10 revolutions? Round your answer to the nearest tenth of a kilometer. II. Make a Conjecture Question: What seems to be the relationship between the circumference and the diameter of the circle? (Note to teacher: Student answers should be C ≈ 3.14d) 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. Foundations of Geometry – Unit 5 – Circles CORE CONTENT Cluster Title: Experiment with transformations in the plane (Note: Do not allow cluster title to mislead you. G.CO.1 is a part of the Congruence domain and has broad meaning.) Standard: 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. (Note: Arc length is covered in G.C.5 in the Geometry Curriculum Guide.) Concepts and Skills to Master Find the measure of a central angle and the measure of its intercepted arc Compute the circumference of a circle in terms of pi to compute the distance of an arc Calculate distances along circular paths or part of a circle’s circumference (i.e. arc length - a concept further developed in Geometry) SUPPORTS FOR TEACHERS Critical Background Knowledge Formula for computing the 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’s measure in a circle is the same value as the size of its central angle. 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. Optional: Although the proportion below will be taught in Geometry, you may choose to differentiate by using it for struggling students. arc length = central angle circumference 360 Students often confuse arc measure with arc length. Explain to them that an arc’s measure is expressed in degrees but an arc’s length is measured in units. CONTINUED NEXT PAGE… Resources Textbook Correlation: 10-6 Circles and Arcs (Think About a Plan) www.pearsonsuccessnet.com Online Lesson Plan (aligned with Glencoe Geometry Book-Lesson 10.2): http://cllenz.wmwikis.net/file/view/Angles+and+Arcs+Lesson+Plan.pdf Note: Warm-Up Activity #1 can be used as an investigation. Visual Aids for Core Content Math Open Reference – use the link(s) below: http://www.mathopenref.com/arc.html - An arc of a circle http://www.mathopenref.com/circlecentral.html - Central angle of a circle http://www.mathopenref.com/arclength.html - Length of an arc in a circle 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. Foundations of Geometry – Unit 5 – Circles IMPORTANT!!! Only calculate distances of circular paths that are simple fractional parts of a circle’s circumference (e.g. 1 1 1 , , ... ). 4 6 8 Allow students to demonstrate further mastery by doing this for circles of various radii but DO NOT move to more complex fractional parts of a circle to develop computing arc length. This concept will be further taught Geometry. The concepts and skills to master in this standard of the unit can be introduced to students at a level 1 or a level 2 competency in order to better promote conceptual mastery in Geometry. Sample Formative Assessment Tasks Skill-based task Problem Task Note: This task is the same as the skill-based task in the Geometry Curriculum Guide Unit 8, Standard G.C.5. If repeated in Geometry, Foundations of Geometry students should show increased ability. Note: This task is very similar to the problem task in the Geometry Curriculum Guide Unit 8, Standard G.C.5. If repeated in Geometry, Foundations of Geometry students should show increased ability when working the problem. Find the arc measure and arc length of each darkened arc. An analog clock hanging on a classroom wall shows that the time is 3:00 pm in the afternoon. Answer the following: Leave your answer in terms of π. 1. 2. 3. 1. What is the measure of arc formed by the hands of the analog clock hanging on a classroom wall? 2. Is the arc a major or minor arc? How do you know? Sketch a wall clock with the time 3pm to support your answer 3. Lastly, what is the arc length if the radius of the clock is 4 inches? 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. Foundations of Geometry – Unit 5 – Circles CORE CONTENT Cluster Title: Understand and apply theorems about circles. Standard: 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. 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 Prior knowledge of a circle and its parts, 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 Have students use a circle puzzle as a fun way to test mathematical and conceptual knowledge of core content; i.e. determining angle measurements using central angles, inscribed angles, arcs, angles between tangents and chords and their relationships (see first two resources at right) 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. Students may benefit from tracing intercepted arcs from central and/or inscribed angles with colored pencils or highlighters. Paper folding activities offer students a good way to develop key concepts related to tangents, central angles, chords, and arcs. Resources Textbook Correlation: Lessons 12.1 – 12.4 Circle Puzzle http://www.fayette.k12.il.us/isbe/mathematics/stageJ/math9BJ.pdf Circle Puzzle – Book: Patty Paper Geometry by Michael Serra p.123 #2 and p.124 #3. Concept Byte Exploration Activity: p.770 - Paper Folding With Circles Group Work: Chord Properties Using Patty Paper On the CMS secondary wiki resources for Unit 5– you will have access to download this file. (File name: Chord Patty Paper) http://secondarymath.cmswiki.wikispaces.net/Geometry More Paper Folding Activities Patty Paper Geometry Book by Michael Serra Use pages 107 - 114 or use this link for an online copy of the pages: http://www.pflugervilleisd.net/curriculum/math/documents/Circles_properti es.pdf 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. Foundations of Geometry – Unit 5 – Circles Sample Formative Assessment Tasks - Note: Tasks are the same in Geometry Curriculum Guide Unit 8, Standard G.C.2. If repeated in Geometry, Foundations of Geometry students should show increased ability when working the problems. 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 C above for Exercises 1–3. Segment 1. If DE = 4 and CE = 8, what is the radius? 2. If DE = 8 and EF = 4, what is the radius? 3. If mC = 42°, what is mE? 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) Teacher Created Argumentation Task (W1-MP3&6) Academic Tool: THINK ABOUT A PLAN – www.pearsonsuccessnet.com - Use Online Resource 12-3 Think About a Plan -OROn the CMS secondary wiki under Unit 5 resources – you will have access to download this file. (File name: 12-3 TAAP) http://secondarymath.cmswiki.wikispaces.net/Geometry Argumentation Task Objective: Students will find values of inscribed angles by drawing points on the given circle, describe how the measure of an inscribed angle is related to its intercepted arc, and explain how theorems involving relationships among arcs, tangents, and chords are used to find missing angles. 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. Foundations of Geometry – Unit 5 – Circles CORE CONTENT Cluster Title: Translate between the geometric description and the equation for a conic section Standard: G.GPE.1 Derive 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 Prove the standard equation of the circle using the distance formula Write the equation of a circle using a problem or graph that gives a circle’s center and radius and a circle’s center and point on the circle Find the center and radius of a circle when given a graph on the coordinate plane or the standard equation of a circle Graph a circle on the coordinate plane given its equation SUPPORTS FOR TEACHERS Critical Background Knowledge Distance Formula Pythagorean Theorem Sketching/plotting graphs on the 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 Review the definition of a circle as a set of points whose distance from a fixed point (center) is constant (radius) Have students use the distance formula to derive the equation of a circle given a center and a radius. Begin with the case where the center is the origin. (Optional: Teachers may extend this suggestion by first using the Pythagorean theorem to derive the distance formula.) Emphasize that writing the equation for a circle in standard form makes it easier to identify the center (h, k). Remind students to use the opposites of h and k from the equation. Remind students to take the square root of the value r2 in order to find the radius. Investigate practical applications of circles. Resources Textbook Correlation: Lesson 12.5 How do I find the equation of a circle? EQUATION FOR A CIRCLE – YouTube video – 6min20secs http://www.youtube.com/watch?v=HjN9TTRrQiA (EQUATION OF A CIRCLE – brightstorm video via YouTube – 2min34sec) http://www.youtube.com/watch?v=Sl0VeTcL-s4 Equations of Circles Interactive Applet http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php Online Teacher Resource Center www.pearsonsuccessnet.com - Geometry Dynamic Activity 12-5: Circles in the Coordinate Plane 12-5 Activities, Games, and Puzzles (classroom game) 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. Foundations of Geometry – Unit 5 – Circles Sample Formative Assessment Tasks Skill-based task Problem-based task 1. Write the standard equation of the circle with center (0, 0) and radius 1. Reasoning - Describe the graph of x2 + y2 = r2 when r = 0? 2. Think About a Plan – Find the circumference and area of a circle whose equation is: (x – 9)2 + (y – 3)2 = 64. of 3. Also sketch the graph. ______ 2. Write an equation of a circle with diameter AB if A (3, 0) and B (7, 6). Leave your answer in terms of pi. Find the center and radius of each circle? 3. (x + 4)2 + (y – 1)2 = 16 4. (x – 8)2 + y2 = 9 What essential information do you need? What formulas will you use? 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.