<p>Students will investigate the properties of arcs and angles within a circle. Students will:&nbsp;<br>- create a foundation of vocabulary and theorems about arcs and angles of circles.<br>- solve problems dealing with the circumference of a circle.&nbsp;<br>- calculate the degree of an arc formed by central angles.&nbsp;<br>- calculate the arc length.&nbsp;<br>- examine the relationships between arcs and chords.&nbsp;<br>- solve problems associated with inscribed angles on circles.&nbsp;<br>- use algebra to answer problems involving the preceding objectives.</p>

<p>- What are the different characteristics of circles, and how may they be utilized to solve problems?</p>

<p>- <strong>Arc:</strong> A segment of a curve; in circles, the continuous part between two points on the circle.<br>- <strong>Arc Length:</strong> The measure of the distance along a curve of a circle.<br>- <strong>Arc Measurement:</strong> A representation that is equal to the degree measure of the central angle that forms the arc.<br>- <strong>Center:</strong> The point within a circle equally distant from all points along the circle.<br>- <strong>Central Angle:</strong> Of a circle; an angle whose vertex is the center and whose sides are the radii of the circle.<br>- <strong>Chord:</strong> A line segment whose endpoints are on a circle.<br>- <strong>Circle:</strong> A circle is the locus of all points in a plane equidistant from a given point called the center.<br>- <strong>Circumference:</strong> The distance around a circle.<br>- <strong>Diameter:</strong> A line segment that has endpoints on a circle and passes through the center of the circle.<br>- <strong>Inscribed Angle:</strong> An angle whose vertex is on the circle and whose sides are chords of the circle.<br>- <strong>Inscribed Polygon:</strong> A polygon surrounded by a circle, where each vertex falls on the circle.<br>- <strong>Major Arc:</strong> An arc that measures between 180 degrees and 360 degrees.<br>- <strong>Minor Arc:</strong> An arc that measures less than 180 degrees.<br>- <strong>Radius:</strong> A line segment that has one endpoint on a circle and the other endpoint at the center of the circle.<br>- <strong>Semicircle:</strong> An arc of a circle whose endpoints are the endpoints of a diameter.</p>

<p><span style="color:rgb(0,0,0);">- </span><a href="https://docs.google.com/spreadsheets/d/1JFhoHg1GtriW0TVIlps5zPJNOzmfzNjV/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><u>Concept Builder worksheet</u></a><span style="color:rgb(0,0,0);">&nbsp;</span><br><span style="color:rgb(0,0,0);">- </span><a href="https://docs.google.com/presentation/d/1l1rj70WPrDulo25cL-GIudjauZKqt0mL/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><u>Lesson 1 PowerPoint presentation</u></a><span style="color:rgb(0,0,0);">&nbsp;</span><br>- Printout of slides 4, 6, 7, 12, 13, 15, and 16 for students from the Lesson 1 PowerPoint presentation<br>- Arc and Chord Examples (M-G-6-1_Arc and Chord Examples and KEY)<br>- Inscribed Angles Examples handout (M-G-6-1_Inscribed Angles Examples)<br>- Chord Extension Problem (M-G-6-1_Chord Extension Problem and KEY)<br>- Angles and Arcs Mixed Review (M-G-6-1_Angles and Arcs Mixed Review and KEY)</p>

<p>- Reteach using selected student errors from guided practice examples.&nbsp;<br>- Take note of the students' responses to questions in the PowerPoint presentation.&nbsp;<br>- Assess students' success in the Circumference Activity, Chord Extension Problem, and Angles and Arcs Mixed Review worksheets:&nbsp;<br>• &nbsp; &nbsp; &nbsp; &nbsp;Do the dimensions used correspond to those given?&nbsp;<br>• &nbsp; &nbsp; &nbsp; &nbsp;Is every operation appropriate?&nbsp;<br>• &nbsp; &nbsp; &nbsp; &nbsp;Is the computation accurate?<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson teaches students about circles and their geometric properties. Students are already familiar with circles as a shape at this point in the geometry curriculum, but as is done with all other geometric shapes, they begin to learn about the more specific characteristics of a circle. A geometry course is intended to cover shapes, their many properties, and applications. This lesson begins to meet this purpose by covering arcs, angles, circumference, and chords of circles.&nbsp;<br>H: Using a PowerPoint presentation during instruction helps students stay engaged by providing visual and organizational support for the topic. Students can explore the lesson concepts in a variety of situations by utilizing the lesson extensions and materials offered in the Related Resources section. Students are more likely to stay engaged when guided note sheets are used instead of having students draw each example in a notebook. You successfully hold their attention.&nbsp;<br>E: The guided notes worksheets and PowerPoint presentation aim to provide all students with necessary tools for success. The circumference activity can demonstrate how to calculate circumference outside of math class. When the lesson is broken down into more manageable portions, students can stay focused on the topic at hand. When students have acquired all of the necessary skills, they can apply them to the Mixed Review worksheet provided in the lesson.&nbsp;<br>R: Allow students to reflect on the information delivered during the class. Reflection allows students to identify whether they have any questions about the topics. Also, when providing examples for the class, give students brief time to reflect on the information offered and make their own decisions about what to do at each step. This helps them to correct any misunderstandings once the correct processes have been depicted. Giving students the opportunity to practice a few of the examples independently throughout the class allows them to reflect, revisit, and revise their cognitive processes. Your comments is also important in helping students identify where they need to improve their thought processes.&nbsp;<br>E: Extension exercises allow students to demonstrate their comprehension of the lesson's ideas. It is also critical to monitor students throughout classroom discussions, guided practice, and questioning, and to ensure that you provide feedback to them. This feedback assists them in self-evaluation of their work and decide when they need to ask for help.&nbsp;<br>T: This lesson offers adaptable activities to meet the demands of different classrooms settings. The guided note handouts can help students stay organized. Many of the enrichment activities give students who are exceeding the benchmarks the opportunity to broaden their knowledge base. Activities can also be tailored to the needs of partners, groups, or individuals in any classroom.&nbsp;<br>O: This lesson includes a PowerPoint presentation to help students arrange their learning experiences. It allows them to follow along and understand exactly what and where to write important information. Furthermore, the guided notes worksheets assist students in organizing their examples in a format that will be more effective when they get to problems demanding independent effort, which are also included in this lesson.<br>&nbsp;</p>

<p><span style="color:rgb(0,0,0);">The lesson is divided into several sections that cover angles and arcs of circles. Students will be working with circumference, central angles, arc length, diameter, chords, and inscribed angles.&nbsp;</span></p><p><span style="color:rgb(0,0,0);">Before starting the instructional section of this unit, give each student the </span><a href="https://docs.google.com/spreadsheets/d/1JFhoHg1GtriW0TVIlps5zPJNOzmfzNjV/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><u>Concept Builder worksheet</u></a><span style="color:rgb(0,0,0);">. This chart will be used with students to note important definitions, formulas, and theorems throughout the unit. This document will assist students in organizing important information so that it is easily accessible when working through the problems in this course. It has been preformatted with adequate boxes for all important terms, as well as extra space for any changes or additions. The document is also an excellent review and study tool for assessments throughout and after the unit.&nbsp;</span></p><p><span style="color:rgb(0,0,0);">The instructional portions of this lesson will be given via a PowerPoint presentation, so you will not have to spend time drawing each diagram and writing out each definition that appears in the lesson. All examples and concepts are included in the </span><a href="https://docs.google.com/presentation/d/1l1rj70WPrDulo25cL-GIudjauZKqt0mL/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><u>PowerPoint presentation</u></a><span style="color:rgb(0,0,0);">. Your goal is to walk through the presentation, discuss topics as they come along, allow students to record information, and demonstrate the concepts. If whiteboard technology is not accessible, other options include printing presentations into overhead transparencies or drawing examples on the board.&nbsp;</span></p><p>To begin the lesson, ensure that each student has the Concept Builder worksheet and then open the Lesson 1 PowerPoint presentation. To introduce the unit, present the first slide: <strong>"The unit focuses on circles and their geometric properties. The first lesson on circles will cover the angles and arcs that exist in circles."</strong></p><p>Note: To assist students and save time when drawing so many shapes, it may be beneficial to print out the example slides prior to the class so that students may simply follow along with the work as presented.</p><p><strong>Part 1</strong></p><p>Slide 2 shows how students will fill out the Concept Builder worksheet as the lesson progresses. The checkmark indicates that students should include a concept in their document. Instead of simply writing down the words, discuss and clarify these concepts with students so that they understand. Also, allow students time to record the information. Walk around the room to ensure that students are engaged in the assignment.</p><figure class="image"><img style="aspect-ratio:260/243;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_31.png" width="260" height="243"></figure><p>Slides 3 and 4 establish important terms necessary for comprehending this lesson (radius, chord, and diameter). Discuss these terms with students as they record the concepts on the Concept Builder worksheets.</p><figure class="image"><img style="aspect-ratio:582/266;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_29.png" width="582" height="266"></figure><p>Slides 5, 6, and 7 describe the circumference of a circle and how to calculate this value. Discuss the formula with students and allow them time to record it on the Concept Builder worksheets. After discussing the formula, present examples 1-5 to the class.</p><figure class="image"><img style="aspect-ratio:244/208;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_30.png" width="244" height="208"></figure><p>Example 1:<i> C = 2π(5) → 10π or 31.4 units</i></p><p>This is a good point at which to discuss with students about how to format their answers. Answers can be expressed in terms of pi, calculated with a calculator's pi button, or calculated using the 3.14 pi approximation. Instruct students to carefully study the directions for the chosen format and to pay attention to how choices are written on multiple-choice questions. Many standardized exam questions require answers in terms of p, although many real-world problems require decimal or fraction form for measurement-friendly solutions.</p><p>Example 2: <i>C = π(12) or C = 2π(6) → 12π or 37.7 units</i><br><br>Explain to them that there are two ways to solve this problem. Students can utilize the diameter formula, which is provided, or the fact that the radius is always half the diameter, which is expressed as 2pr. It is important to note that in either way, students receive the same answer.</p><p>Example 3: <i>C = π(10) or C = 2π(5) → C = 10π or 31.41 units</i></p><p>Point out to the students that sometimes more information is provided than is required to solve a problem. The sides labeled 6 and 8 (shown on Example 3, slide 6) are not required to calculate the circumference.</p><figure class="image"><img style="aspect-ratio:325/266;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_32.png" width="325" height="266"></figure><p>Examples 4 and 5 show the algebraic side of the circumference formula. These examples demonstrate that, while the circumference of a circle is known, students may need to calculate the radius or diameter to solve the problem.</p><p>Example 4: <i>112.6 = 2πr, divide 112.6 by 2π and r = 17.9</i></p><p>Example 5: <i>80 = πd, divide by π and d = 25.46</i></p><figure class="image"><img style="aspect-ratio:346/316;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_33.png" width="346" height="316"></figure><p>Using slides 8, 9, 10, and 11, demonstrate to the class the ideas and definitions of central angles, arc measurements, and semicircles. Allow students time to record information and ask questions as needed.</p><figure class="image"><img style="aspect-ratio:499/276;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_34.png" width="499" height="276"></figure><figure class="image"><img style="aspect-ratio:274/239;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_35.png" width="274" height="239"></figure><figure class="image"><img style="aspect-ratio:376/260;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_36.png" width="376" height="260"></figure><p>Slides 12 and 13 provide model examples for the information presented in slides 8-11. Model these instances for students. Show students the arc notation that is commonly used. Arc AB =<span style="background-color:rgb(255,255,255);color:rgb(8,42,61);"> </span><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_208.png" width="28" height="26"></p><p>Slide 12:</p><p>1) Arc AB = 104° (the related central angle APB is 104°).</p><p>2) Arc BC = 76° (because AC is the diameter, therefore arc ABC must be 180°, and if arc AB is 104°, then arc BC equals 180° - 104°.)</p><p>3) Arc ADB = 256° (the entire circle is 360°, thus subtract the unneeded portion arc AB, 360° - 104° = 256°)</p><p>4) Arc ADC = 180° (because AC is the diameter, the angle is 180°)</p><figure class="image"><img style="aspect-ratio:229/282;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_37.png" width="229" height="282"></figure><p>Slide 13: These examples use principles from slides 8-11, but they apply algebra to situations rather than concrete values.</p><p>1) <i>x</i> = 9 (since \(\overline{WT}\) is the diameter, therefore <i>m∠WZV + m∠VZU + m∠UZT</i> = 180°; so solve for <i>x</i>).</p><p>2) Arc YT = 117° (since x = 9, <i>m∠YZT</i> = 13(9) <i>→</i>117°, so arc YT = 117°)</p><figure class="image"><img style="aspect-ratio:251/354;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_38.png" width="251" height="354"></figure><p>Slide 14 presents the concept of arc length rather than arc measurement (in degrees). Present the formulas to students so that they can enter the data into their Concept Builder. Then take the time to explain to students where this formula came from. <strong>"The arc length is a portion of the total distance around the circle, which is why the arc length is presented as a fraction of the circumference in the formula. Furthermore, because the central angle associated with the specified arc represents a portion of the entire circle of 360°, we can consider it a fraction of the complete circle. Because they are all on the same circle, these two fractions are in proportion with one another."</strong> Formula 2 differs from Formula 1 in that Formula 2 solves for arc length, saving a step in the computation process.</p><figure class="image"><img style="aspect-ratio:313/267;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_39.png" width="313" height="267"></figure><p>Slides 15 and 16 show examples that should be modeled for students.</p><p>Example 1: <i>Arc AB = 7.42 cm</i></p><p>Example 2: <i>Arc EF = 9.08 in</i>. (Possibly have students try this one on their own before going through it with the class. This gives students time to think on the process and see if they have any questions.)</p><p>Example 3: <i>Arc AB = 10.47 ft.</i> (This example depicts the situation in terms of diameter rather than radius. To complete the problem, students can find the radius by taking half of the diameter, or use pd in the formula instead of 2πr.)</p><figure class="image"><img style="aspect-ratio:572/254;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_40.png" width="572" height="254"></figure><p><strong>Part 2</strong></p><p>Prior to displaying slide 17, ask students to recall the definition of a chord from earlier in the lesson. This should be recorded on their Concept Builder worksheets. Allow students to think independently before asking a student to remind the class of the definition.</p><p><i>Option:</i> If you start a class period at this point in the lesson, you might use this as an anticipatory set or warm-up question.</p><p>Present slides 17-19 to the class, allowing students to enter theorems on their Concept Builder worksheets and process what they are writing. Make sure you explain the theorems instead of simply asking students to copy them.</p><figure class="image"><img style="aspect-ratio:329/368;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_41.png" width="329" height="368"></figure><figure class="image"><img style="aspect-ratio:328/371;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_42.png" width="328" height="371"></figure><figure class="image"><img style="aspect-ratio:338/301;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_43.png" width="338" height="301"></figure><p>Slides 20–26 go through examples pertaining to the theorems presented above. Give students the document Arc and Chord Examples (M-G-6-1_Arc and Chord Examples and KEY). This handout allows students to follow the examples on their own papers while you go through them on the board. This will save time because students will not have to sketch each shape for every example. <strong>Note:</strong> Each example gives a problem to be answered; before answering the problem, students are asked which theorem they will apply. This is an important problem-solving method to explain to students so that they can decide which rules they will use to solve the problem before they solve it. This enables them to identify the direction they should take in the problem-solving process. Allow students to think through this section independently before selecting students to explain why they made their choices.</p><p><strong>Part 3:</strong></p><p>Slide 27 provides important definitions and theorems about inscribed angles. Present this knowledge to students, allowing them time to record the theorems on their Concept Builder worksheet and process their writing. Make sure you explain the definitions and theorems to students rather than simply asking them to copy them.</p><p>Slides 28-30 offer instances that relate to the theorems presented on slide 27. Distribute the Inscribed Angles Examples worksheet (M-G-6-1_Inscribed Angles Examples). Have students record their work on this page as you demonstrate the technique on the board. This saves time because students will not have to sketch each shape for every example.</p><p>Example 1: <i>Angle B (∠B) = 38°, arc BC = 112°</i></p><p>Example 2:<i> x = 12; arc MN = 42°</i></p><p>Example 3: <i>Arc WZY = 194°</i></p><p><strong>Extension:</strong></p><p>Chord Extension Problem: After Part 2 of this lesson, use the Chord Extension Problem (M-G-6-1_Chord Extension Problem and KEY) to help students apply what they've learned. This problem combines the theorems from the class and requires students to solve a variety of difficulties. This could be utilized as an independent practice project, homework assignment, or collaborative task.</p><p>Mixed Review: Use the Angles and Arcs Mixed Review worksheet (M-G-6-1_Angles and Arcs Mixed Review and KEY) to help students practice the concepts covered in this lesson. The worksheet can be utilized for test review, independent/homework practice at the end of the lesson, or as a partner/group activity. Follow up with students about their worksheet performance to clear up any misconceptions they may have before assessing them for the lesson.</p><p><i>Option:</i> Determine whether you want students to be able to refer to their Concept Builder paper while going through the review worksheet or to attempt to complete it without it.</p><p>Before you teach students how to compute the measure of an inscribed angle and its intercepted arc, let them experiment with the applet at <a href="http://www.ies.co.jp/math/java/geo/enshup/enshup.html">http://www.ies.co.jp/math/java/geo/enshup/enshup.html</a> to discover the relationship. (The inscribed angle is half the size of the intercepted arc/central angle.) You can show the image to students and have them write their opinions individually, or you can have students work independently in a computer lab to research the website.</p>