<p>Students solve problems that involve angle measurement and the intersection of secants, tangents, and/or chords. Students will:&nbsp;<br>- calculate the angles and/or solve for the unknowns when two secants intersect inside a circle.<br>- compute the angle and/or solve for the unknowns when a secant and tangent intersect at a point of tangency.<br>- calculate angle measurements and/or solve for unknowns when two secants, two tangents, or a secant and tangent intersect outside of a circle.</p>

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

<p>- <strong>Angle:</strong> In geometry, the inclination to each other (divergence) of two straight lines.<br>- <strong>Angle formed by a chord and a tangent:</strong> The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.<br>- <strong>Angle formed by a secant and a tangent:</strong> The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs.<br>- <strong>Angle formed by two chords:</strong> The measure of an angle formed by two intersecting chords is one-half the sum of the measures of the area intercepted by it and its vertical angle.<br>- <strong>Angle formed by two secants:</strong> The measure of an angle formed by two secants intersecting outside the circle is half the difference of the area intercepted by it.<br>- <strong>Chord:</strong> A line segment whose endpoints are on a circle.<br>- <strong>Inscribed Angle:</strong> An angle in the interior of the curve formed by two chords which intersect on the curve. In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.<br>- <strong>Secant (of a circle):</strong> A line that intersects a circle in exactly two points.<br>- <strong>Tangent (of a circle):</strong> A line that touches a circle in exactly one point.</p>

<p>- Handout and files for technology explorations (see Related Resources section at end of lesson)<br>- <a href="https://docs.google.com/spreadsheets/d/10C2a1KN76YnyQ5W0oSjpn0-twLGI_HDY/edit?usp=drive_link&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><span style="color:#1155cc;"><u>Concept Builder worksheet</u></span></a><br>-<a href="https://docs.google.com/presentation/d/1Qbti2BMGJIM2NBgKVJ7NUJWFqLDiMXtA/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><span style="color:#1155cc;"><u> Lesson 2 PowerPoint presentation</u></span></a><br>- Printout of slides 9–14 for students from the Lesson 2 PowerPoint presentation<br>- Circle Angle Relationships Summary (M-G-6-2_Circle Angle Relationships Summary and KEY)<br>- Secant and Tangent Extension Problem (M-G-6-2_Secant and Tangent Extension Problem and KEY)<br>- Secants and Tangents Independent Practice (M-G-6-2_Secants and Tangents Independent Practice and KEY)</p>

<p>- Check student responses to the question "What is the relationship between a chord and a secant?" on slide 3 for the correct correlation of line and line segment.&nbsp;<br>- Examine students' performance on the Theorem Summary Review Activity, the Secants and Tangents Independent Practice worksheet, and the Secant and Tangent Extension Problem to ensure that circle components correspond correctly.&nbsp;<br>- Examine students' performance on the writing application task, looking for original instances. Explanations should include language that indicates comprehension of the theorems rather than just restating them.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction<br>W: In Lesson 2, students continue their investigation of circles by exploring the angle relationships within circles. They follow a logical progression from the topics covered in Lesson 1 to a better grasp of circles and their properties. Students are given the opportunity to demonstrate their comprehension through varied activities and independent practice.&nbsp;<br>H: Lesson 2 promotes student participation through technology-based activities. Introducing the lesson with a Technology Exploration helps to hook and hold students' interest in the lesson topic. It also allows students to take personal responsibility for their learning.&nbsp;<br>E: This lesson aims to teach students how to use similar concepts independently and apply their knowledge in new contexts. The PowerPoint presentation and handouts will help students stay organized and on track throughout the class. Extensions and critical-thinking activities help students make sense of their learning and apply it to future learning contexts.&nbsp;<br>R: During the class, students engage in critical thinking activities to reflect on their learning. During the lesson, students get the opportunity to modify their thought processes using examples and guided-practice problems. The proposed questions help students evaluate their thought processes and revise their thinking. Finally, an independent assignment allows students to revisit the topic and demonstrate their understanding. Your feedback is crucial in helping students identify areas where they need to improve their thought processes.&nbsp;<br>E: Students demonstrate their knowledge of concepts throughout the lesson. Students are given opportunity to conduct self-evaluations to identify areas where they have questions or need to make modifications. Lesson 2 also allows students to demonstrate their learning through a writing assignment. Formative assessments allow students to demonstrate their comprehension, and instructor feedback helps students conduct meaningful self-evaluations.&nbsp;<br>T: This lesson offers two techniques for teaching the concepts. Choose the approach that best meets the demands of your classroom. Extension activities are designed to meet the demands of a wide range of classroom settings. The PowerPoint presentation and the Concept Builder worksheet allow students to gain experience with a range of learning techniques and can be useful when collaborating with different teachers/aides in the classroom.&nbsp;<br>O: The Technology Explorations option allows students to progress from investigative to independent learning, with minimum direct teaching. When using this strategy, make careful to help any students who miss out on some of the information. If proper checking for knowledge does not occur, students may have gaps in their understanding that will show up quickly when they attempt to do independent activities. Option 2 of the lesson uses a direct-instruction style to teaching, but allows students to practice and apply their knowledge through the same critical-thinking and extension activities.</p>

<p><span style="color:rgb(0,0,0);"><strong>Option 1&nbsp;</strong></span></p><p><span style="color:rgb(0,0,0);">Introduce the ideas for Lesson 2 using Technology Explorations 1, 2, and/or 3 from the Related Resources section. These activities are exploratory in nature, guiding students to uncover the important theorems and rules covered in this lesson. After students have completed these activities, use the lesson presentation in Option 2 to fill in any learning gaps and ensure that students have obtained the necessary information to work independently.</span></p><p><span style="color:rgb(0,0,0);"><strong>Option 2&nbsp;</strong></span></p><p><span style="color:rgb(0,0,0);">This instruction option skips the introduction activities and instead focuses on direct concept instruction. Make sure the students have their </span><a href="https://docs.google.com/spreadsheets/d/10C2a1KN76YnyQ5W0oSjpn0-twLGI_HDY/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><u>Concept Builder worksheets</u></a><span style="color:rgb(0,0,0);">. Students will use this chart to keep track of significant definitions, formulas, and theorems throughout the unit. This document will assist students in organizing crucial information so that it is easier to access while working on the problems. The document will also serve as an excellent review and study tool for assessments taken during and after the unit.&nbsp;</span></p><p><span style="color:rgb(0,0,0);">The instructional portions of this lesson should be delivered using the </span><a href="https://docs.google.com/presentation/d/1Qbti2BMGJIM2NBgKVJ7NUJWFqLDiMXtA/edit?usp=sharing&amp;ouid=116344346769586180073&amp;rtpof=true&amp;sd=true"><u>PowerPoint presentation</u></a><span style="color:rgb(0,0,0);"> rather than drawing each graphic and writing out each definition as they appear in the lesson. All examples and concepts are previously included in the Lesson 2 PowerPoint. So you may walk through the PowerPoint, explain topics as they arise, allow students time to record information, and demonstrate concepts for them. If whiteboard technology is not available, other options include printing presentations on overhead transparencies or drawing examples on the board.&nbsp;</span></p><p>When students have completed their Concept Builder worksheets, access the Lesson 2 PowerPoint. Use the first slide to introduce the lesson. <strong>"Today, we will continue our study of circles and their properties. This lesson focuses on angles and arc measurements formed when tangents and secants intersect a circle."</strong></p><p>Lesson 2 begins with slide 2, which introduces two key vocabulary terms. Display these for students to complete on their Concept Builder worksheets.</p><figure class="image"><img style="aspect-ratio:297/266;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_44.png" width="297" height="266"></figure><p>After the class analyzes the definitions on slide 2, slide 3 presents a critical-thinking question for students to consider and then answer individually. <strong>"What is the relationship between a chord and a secant?"</strong></p><p>Ask students to write their opinions on a sheet of paper. After 2 to 4 minutes, lead a class discussion about their ideas. (Possible responses: <i>They are the same thing, but a chord is a segment, whereas a secant is a line. As previously stated, a segment is a portion of a line, so a chord is simply the inner portion of the secant. As a result, each secant contains a chord.</i>)</p><p>Slides 4-7 introduce the important theorems of the lesson. Allow students time to copy these on the Concept Builder worksheet while you explain the theorems.</p><p><strong>Note:</strong> Slide 5 prompts students to speculate on how to apply this theorem to the other side of a circle. This question is designed to ensure that students can apply the concept to all circles (since not all circles will have their secants and tangents drawn in the same orientation as the original theorem). Students should calculate <i>m</i>∠2 = \(1 \over 2\) (<i>m</i> arc ACB).</p><p><strong>Note:</strong> Slide 6 asks students to draw the three situations indicated in the theorem. Students can do this on a piece of paper. The idea is for students to translate the verbal description into a visual representation. Thus, when students see slide 7, they can apply the rules and formulas to their drawings. This provides them various viewpoints on the rules.</p><figure class="image"><img style="aspect-ratio:461/287;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_45.png" width="461" height="287"></figure><figure class="image"><img style="aspect-ratio:366/317;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_46.png" width="366" height="317"></figure><figure class="image"><img style="aspect-ratio:305/264;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_47.png" width="305" height="264"></figure><figure class="image"><img style="aspect-ratio:401/339;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_48.png" width="401" height="339"></figure><p><strong>Activity: Theorem Summary Review</strong></p><p>Provide each student with a Circle Angle Relationships Summary worksheet (M-G-6-2_Circle Angle Relationships Summary and KEY). The purpose of this activity is for students to develop their own summaries of the theorems covered in this unit. Students should collaborate with a partner to review their learning. Tell students to use their Concept Builder worksheet to look for theorems/concepts they may require. During the activity, walk around and monitor student discussions and summaries, then direct students in the right way. After the student discussions have concluded, gather the class for a large group discussion about what they discovered.</p><p>Instruct students to include any missing information in their summaries. Inform students that just because their circles have varied labels and orientations does not mean they are incorrect (M-G-6-2_Circle Angle Relationships Summary and KEY).</p><p>Continue Lesson 2 with slides 9-11. The presentations provide examples of how to apply the theorems to solve problems. Print slides 9-11 so that students can follow along as you model. Students should follow along while you demonstrate the procedures and related computations on the board.</p><p>The solutions to the examples are as follows:</p><p>Slide 9: Example 1: <i>97°</i></p><p>Slide 10: Example 2: <i>248°</i></p><p>Slide 11: Example 3: <i>150°</i></p><figure class="image"><img style="aspect-ratio:376/353;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_49.png" width="376" height="353"></figure><figure class="image"><img style="aspect-ratio:597/326;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_50.png" width="597" height="326"></figure><p>Slides 12-14 demonstrate the application of algebra in solving problems similar to those in slides 9-11. Give students a printout of slides 12-14 so they can follow along while you model. To correctly complete the tasks, students will need to utilize theorems and algebra skills. Option: Have students work in groups or pairs to complete the problems, then discuss the solutions as a class. If you choose this option, make sure to walk around to each group to clean up any inaccuracies.</p><p>Slide 12: Example 4: <i>x = </i>14</p><figure class="image"><img style="aspect-ratio:208/156;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="208" height="156"></figure><figure class="image"><img style="aspect-ratio:550/293;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_52.png" width="550" height="293"></figure><p>Slide 13: Example 5: <i>A </i>= 60°</p><figure class="image"><img style="aspect-ratio:206/173;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_54.png" width="206" height="173"></figure><p>Slide 14: Example 6: <i>x </i>= 23</p><figure class="image"><img style="aspect-ratio:202/136;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_55.png" width="202" height="136"></figure><figure class="image"><img style="aspect-ratio:295/352;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_56.png" width="295" height="352"></figure><p><strong>Extension:</strong></p><p>Writing application: Ask students to choose one of the three theorems presented in Lesson 2 and explain it in their own words, as if they were explaining the theorem to a peer who is unfamiliar with the concept. This might serve as an admission or exit ticket.&nbsp;<br>Problems: Ask students to solve the Secant and Tangent Extension Problem handout (M-G-6-2_Secant and Tangent Extension Problem and KEY). This problem uses numerous theorems in one diagram and demands students to apply the rules in an unfamiliar context.&nbsp;<br>Independent practice: Use the Secants and Tangents Independent Practice handout (M-G-6-2_Secants and Tangents Independent Practice and KEY) to allow students to apply lesson concepts on their own. Use it as a homework assignment or for independent work in class.</p>