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Geometry, Quarter 4, Unit 4.3 Circles, Part 2 Overview Number of instruction days: 8–10 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Use a variety of methods of proof, both formal and informal, to justify conjectures about properties and attributes of circles and their component parts. 2 Reason abstractly and quantitatively. Use a diagram of a circle as an aid to solve problems with other circles. Construct inscribed and circumscribed triangles and tangents to a circle. Create representations of angles and arcs and their measures. Use similarity to prove 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. Attend to the meaning of the radian as a unit of measure. Derive the formula for the area of a sector and apply it to real-world problems. 3 Construct viable arguments and critique the reasoning of others. Use relationships among lines and angles in circles to justify conclusions, communicate to others, and respond to the arguments of others. Critique the arguments of others by comparing the effectiveness of two plausible arguments, distinguishing correct logic or reasoning from that which is flawed, and, if there is a flaw in an argument, explain what it is. 5 Use appropriate tools strategically. Providence Public Schools Use technology, patty paper, and compasses for constructions and to solve real-world problems. D-120 Circles, Part 2 (8–10 days) Geometry, Quarter 4, Unit 4.3 Version 5 Essential Questions How can you justify conjectures about properties and attributes of circles and their component parts? What kind of real-world problems can you solve using the formula for the area of a tangent? How does our knowledge of circles help us in determining measurement of angles, arcs, and lines relating to circles? What are the relationships among the angles and lines or line segments that can be drawn in or about a circle, and what are their implications? What are the relationships between inscribed and circumscribed polygons and circles? Why is the concept of proof important in geometry involving relationships in circles? What is the difference between the length of an arc and the measure of an arc? Standards Common Core State Standards for Mathematical Content Geometry Circles G-C Understand and apply theorems about circles 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 [Radian introduced only as unit of measure] 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. Providence Public Schools D-121 Geometry, Quarter 4, Unit 4.3 Circles, Part 2 (8–10 days) Version 5 Common Core State Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a D-122 Providence Public Schools Circles, Part 2 (8–10 days) Geometry, Quarter 4, Unit 4.3 Version 5 website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning Students begin in kindergarten with the identification and description of shapes, including circles. Students develop the concept of fractional degree measurement in a circle in fourth grade. In fifth grade, students extend this knowledge to the coordinate plane. In Grade 8, students experimented with rigid motion through transformations that preserved distance and angle measures, and they described the effect of dilations on two-dimensional figures using coordinates. Conic circles are studied in Algebra I. Current Learning In this unit, students justify conjectures about properties and attributes of circles and their component parts. They construct figures related to circles, such as inscribed and circumscribed triangles as well as tangents to circles. Use of construction tools is a fluency in Geometry. Work with circles is a critical area in the CCSS at this level. This is the first time students are introduced to radians as a unit of measure. Students prove relationships among arcs and angles. They solve real-world problems using the formula for area of a sector. All standards addressed in this unit are classified as additional content by the PARCC Model Frameworks for Mathematics. Future Learning In Algebra II, basic circle concepts will be extended to the unit circle. The concept of circles will be extended to hyperbolas and parabolas in Precalculus. Careers such as graphic design, science, automotive technology, plumbing, HVAC technology, and engineering use these concepts. Additional Findings Understanding the process of proofs is challenging for students. Research shows “Alternative approaches, helping students expand their understanding of the nature of proof, may be more successful (Driscoll, 1983). Such approaches may include cooperative investigations in which students make conjectures and resolve conflicts by presenting arguments and evidence, prove nonobvious statements, and formulate hypotheses to prove (Fawcett, 1938; Human & Nel, 1989). (A Research Companion to Principles and Standards for School Mathematics, p. 168) Providence Public Schools D-123 Geometry, Quarter 4, Unit 4.3 Circles, Part 2 (8–10 days) Version 5 Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students’ attainment of the mathematics within the unit. Use definitions, properties, and theorems to identify and describe relationships among: inscribed angles, circumscribed angles, radii, chords, central angles. Use definitions, properties, and theorems to prove properties of angles for a quadrilateral inscribed in a circle. Make the following constructions: inscribed and circumscribed circles of a triangle. Find the arc length of a circle. Derive and apply the formula for area of a sector. Identify and apply the radian measure of the angle as the constant of proportionality. D-124 Providence Public Schools Circles, Part 2 (8–10 days) Geometry, Quarter 4, Unit 4.3 Version 5 Instruction Learning Objectives Students will be able to: Identify key parts of circles (including central angles, major arcs, minor arcs, and semicircles) and determine their angular measures to solve problems. Determine arc lengths and compare arc lengths to arc measures in problem situations. Investigate how formulas for circumference and area of a circle relate to finding the area of a sector and define radian measure. Investigate and apply relationships between arcs and chords to solve problems. Use a variety of methods of proof, both informal and formal, to verify conjectures about relationships between arcs and chords. Investigate and apply properties of inscribed angles in circles, including properties of inscribed polygons. Use a variety of methods of proof, both informal and formal, to verify conjectures about inscribed angles and polygons. Review and demonstrate knowledge of important concepts and procedures related to circles. Resources Geometry, Glencoe McGraw-Hill, 2010, Student/Teacher Editions Sections 10-2 through 10-5(pp. 692 - 725) Section 11-3 (pp. 782 – 788) http://connected.mcgraw-hill.com/connected/login.do: Glencoe McGraw-Hill Online Chapter 10 Resource Masters (pp. 11 – 35) Teaching with Manipulatives Using Overhead Manipulatives – Locating the Center of a Circle (p. 138) Using Overhead Manipulatives – Investigating Inscribed Angles (p. 140) Geometry Lab – Inscribed Angles (pp. 141 - 143) Providence Public Schools D-125 Geometry, Quarter 4, Unit 4.3 Circles, Part 2 (8–10 days) Version 5 Using Overhead Manipulatives – Constructing a Circle to Inscribe a Triangle (p. 144) Using Overhead Manipulatives – Inscribing a Circle in a Triangle (p. 147) Interactive Classroom CD (PowerPoint Presentations) Teacher Works CD-ROM Exam View Assessment Suite Algebra 2, Glencoe McGraw-Hill, 2010, Student/Teacher Editions (see the Supplementary Materials Section of this binder) Section 13-2 (pp. 819 – 820) Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery section below for specific recommendations. Materials Compasses, rulers with centimeters and inches, patty paper, dynamic software, TI-Nspire graphing calculator, protractors, scissors, straightedge (or index cards) for construction, white paper, white boards with coordinate grid (optional), dry erase markers (optional), string Instructional Considerations Key Vocabulary intercepted arc arc measure arc length radian tangent Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: inscribed, circumscribed, construction, circle, radius, diameter, circumference, equilateral, quadrilateral, triangle, chord, and similarity. In this unit, students investigate and apply many theorems that relate to circles and lines within and about circles. Because it is a long unit, use a variety of strategies to maintain student interest and involvement, such as the ones suggested here and in the resource materials. A kinesthetic model of a circle can be used as a cue (another cues, questions, and advance organizers tool) and to help students organize the information in this chapter. The Central Angles and Arcs student D-126 Providence Public Schools Circles, Part 2 (8–10 days) Geometry, Quarter 4, Unit 4.3 Version 5 worksheet later in this unit can serve as such an activity to accompany Section 10-2. Ask students guiding questions as they work. Consider developing similar activities with cardboard circles for other concepts and theorems in this unit. The Teaching with Manipulatives activities provided in the resource section also provide additional kinesthetic activities for enriching student’s experiences with circle. Students work cooperatively in small groups to investigate the geometric concepts. Refer to Section 13-2 in Algebra 2 to define a radian. Have students concretely arrive at an understanding of 1 radian by using the following nonlinguistic representation strategy, which is a kinesthetic representation: 1. Trace circles of varying sizes. 2. Fold the circles into quadrants to locate the centers. 3. Using string, determine the diameter of one of the circles and cut the string to that length. 4. Fold the string in half and cut for the radius. 5. Select a point on the circle at which to begin and lay the “radius” along the circle. Sketching radii to the beginning point on the circle and to the end point of the “radius” identifies the o degree measure of 1 radian (approximately 57.3° or exactly 180 ). If students continue this “radius string,” laying it completely around the circle and marking off as they go, they should be able to see six complete radians with a bit left over (approximately 0.28 “radius string”). Hence, there are approximately 6.28 radians—or exactly 2 radians—in a circle, no matter what the radius. This discovery activity is essential to student understanding of the connection between degree measure and radian measure. Extend this understanding by applying the concept of similarity and the proportionality of the length of an intercepted arc and the radius. The proofs of Theorem 10.2 in Section 10-3 and Example 3 on page 711 in Section 10-4 provide great opportunities to explore simple, formal proofs. Based on Chapter 10 Resource Masters, page 21 problem 3, students could be given a piece of a paper plate to determine the plate’s original size, radius, and circumference. This provides students with an opportunity to show what they know about the relationships between these measures. Students have learned about x-intercepts and y-intercepts in previous courses. In Section 10-4, they are introduced to the concept of intercepted arc. Be sure that students understand the general concept of the term intercept so that they will be able to access its meaning in this context. Many of the properties of circles can be expressed using algebraic expressions. Students can use technology, such as dynamic geometry software, to collect data about the properties of segment lengths Providence Public Schools D-127 Geometry, Quarter 4, Unit 4.3 Circles, Part 2 (8–10 days) Version 5 and angle measures in a circle and then use that data to generate an algebraic expression describing the property. Alternatively, students can use compasses and straightedges to construct the parts of the circle being investigated and then use rulers and protractors to measure segment lengths and angles. From their data, students can inductively generate the relationships among the different measures. The Lines and Segments That Interact with Circles handout later in this unit is an excellent tool for students to use to summarize their learning prior to an end-of-unit assessment or activity. Use a modified version of problem 49 on page 715 as an extended application assignment. Have students create a logo (for the school, your class, a school team, advisory group, etc.) as indicated in the problem, then have them investigate properties of their logo related to the content of this unit. D-128 Providence Public Schools Circles, Part 2 (8–10 days) Geometry, Quarter 4, Unit 4.3 Version 5 Central Angles and Arcs Record Sheet (accompanies Section 10-2) You will investigate the relationships among central angles, arc measure, and arc length. An arc encompasses the same number of degrees as the central angle whose endpoints define the arc. The arc length is part of the circumference. For this activity, you will use a cardboard circle, Wikki Stix, a metal fastener, a ruler, and a protractor. 1. Place a Wikki Stix across your circle so that it forms a diameter. Make sure it goes across the center of the circle. Use a metal fastener to attach the stick to the circle at the center point. 2. Measure the diameter of the circle. Write the formula for circumference below and calculate the circumference of the circle. Formula for circumference: C = Diameter of the circle = Circumference of the circle = 3. Think of the diameter as two radii. Move one radius to form a central angle and draw a mark to indicate the angle. Form each of the following angles: 45º, 120º, 60º. 4. Refer to the Key Concept on page 695 of your textbook to find the measure of the arc inscribed by each of your angles. Providence Public Schools D-129 Geometry, Quarter 4, Unit 4.3 Circles, Part 2 (8–10 days) Version 5 Lines and Segments That Interact with Circles How are chords, diameters, radii, secants, and tangents similar? How are they different? The first column in the table lists characteristics of lines and segments and their relationship to circles. Think about how each characteristic relates to each type of line or segment and fill in each blank space with Always, Sometimes, or Never. tangent chord diameter radius secant Includes the center of a circle Intersects the circle at exactly one point Has points outside the circle Has one or more endpoint on the circle Has exactly one endpoint on the circle D-130 Providence Public Schools Circles, Part 2 (8–10 days) Geometry, Quarter 4, Unit 4.3 Version 5 Notes Providence Public Schools D-131