<p>In this lesson, students will learn (or review) the parts of a circle. They will learn calculating the circumference and area of a circle using formulas. Students notice patterns in changing areas and circumferences to determine the relationship between increased radius or diameter and increased circumference and area. Students will apply circle concepts to a wide range of real-world problems. Students will:&nbsp;<br>- identify the parts of a circle.&nbsp;<br>- estimate and calculate the circumference of a circle based on its radius or diameter.&nbsp;<br>- estimate and calculate the area of a circle based on its radius or diameter.&nbsp;<br>- estimate the radius or diameter of a circle based on its area or circumference.&nbsp;<br>- recognize patterns in the circumference and area of a circle as the radius increases.&nbsp;<br>- solve real-world problems involving calculations with circles.</p>

<p>- How can we use the relationship between area and volume to draw, construct, model, and represent real scenarios, and/or solve problems of area and volume?</p>

<p>- Area: The number of square units contained within a closed figure.&nbsp;<br>- Central Angle: An angle formed by two radii in a circle with its vertex at the center.&nbsp;<br>- Chord: A line segment with both endpoints on the circumference of a circle.&nbsp;<br>- Circle: The set of all points equal distance from a given point, called the center.&nbsp;<br>- Circumference: The number of linear units around a circle.&nbsp;<br>- Diameter: A chord that goes through the center of a circle.&nbsp;<br>- pi: The ratio of the circumference and diameter of a circle, estimated to be 3.14.&nbsp;<br>- Radius: The distance from the center to any point on the circle.&nbsp;<br>- Sector: A part of the interior of a circle defined by two radii and the corresponding arc.</p>

<p>- student copies of Vocabulary Journal pages, copied back to back (M-7-6_Vocabulary Journal)<br>- student copies and one transparency or enlargement of the Circle diagram (M-7-6-1_Circle Diagram)<br>- copies of the Circle Terms cards, cut apart, enough sets to give one to each group of 3-4 students; each set stored in an envelope or small baggie (M-7-6-1_Circle Terms)<br>- Your copy of the Circle Terms Reference sheet (M-7-6-1_Circle Terms Reference)<br>- flat circular objects or paper circles of various sizes for students to measure (or a variety of everyday objects such as cardboard or plastic container lids such as oatmeal or coffee can lids, or tape rolls), enough for each pair or small group of students; or, paper circle cut-outs of various sizes<br>- yarn or string cut into 12-18 inch pieces (1 for each student or pair of students)<br>- index cards, any size, enough for each pair or small group<br>- chart paper<br>- markers<br>- student copies of Partner Circles Practice (M-7-6-1_Partner Circles Practice and KEY)<br>- 5–12 copies of the Circle Application Cards (M-7-6-1_Circle Application Cards and KEY), cut cards apart, plan for one to three cards per group<br>- rulers (with inch and centimeter markings)<br>- Optional Activity—Discovering Pi and Circumference (M-7-6-1_Optional Activity—Discovering Pi and Circumference)<br>- student copies of the Lesson 1 Exit Ticket (M-7-6-1_Exit Ticket Lesson 1 and KEY)<br>- student copies of Dividing It Up (M-7-6-1_Dividing It Up and KEY)</p>

<p>- Checkpoint 1: Before proceeding to circle calculations, assess students comprehension of parts of a circle.&nbsp;<br>- Checkpoint 2: Before moving on to real-world applications, assess students' abilities to understand and use circle formulas to determine the circumference and area of a circle.&nbsp;<br>- Assess student comprehension throughout work time and presentations of the Partner Circles Practice and Circle Application Cards activities.&nbsp;<br>- Exit Ticket Lesson 1 is a direct and individual assessment of student understanding that will be used at the end of the class.&nbsp;<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction<br>W: Students are asked to recall prior knowledge about circles. The terms used to describe parts of a circle are discussed. The lesson begins with a discussion of where students can find circles in the real world and what calculations they may need to make regarding circles.<br>H: Demonstrate how to draw a circle using a ruler with a center point on a board or chart paper and encourage students to draw their own. Discuss the radius, diameter, circumference, and area in relation to your sample circle.&nbsp;<br>E: Students use string and rulers to measure circles. The relationships among radius, diameter, circumference, and area are investigated. Several problems are solved as a class. Students work in pairs or groups to complete the Partner Circles Practice and Circle Application Card activities. By the end of the class, students have solved problems requiring circle calculations and presented their strategies and solutions.<br>R: Student groups are encouraged to change their solutions as necessary during work time and class discussions. You and your student peers ask questions to help partners and presenters identify inaccuracies in their thinking and correct them. Students are encouraged to write down additional methods to their solutions as other groups offer them in order to develop their own problem solving skills and get a knowledge that different approaches to problems can often be effective.&nbsp;<br>E: Students are asked to share prior knowledge of circle terms and formulas at the start of the lesson. Throughout class discussions, student work time, and presentations, you can conduct informal assessments of student understanding through observation. Checkpoint 1 questions are used between learning circle terms and calculating circumference and area. Checkpoint 2 questions are used before the real-world applications activity to ensure mastery of the basic skills required to progress through the class. An Exit Ticket is completed at the end of the class to help with the selection of Extension activities for remediation or enrichment.&nbsp;<br>T: Use the Extension ideas to personalize the lesson to meet the requirements of your students. The small group exercise is appropriate for students who require further assistance, while the expansion can be used for students who demonstrate proficiency. Additional exercises are suggested for classroom stations and the use of technology.<br>O: In the first activity, students learn and identify appropriate vocabulary terms for the parts of a circle. Students learn the value of pi. The relationships between radius and diameter, diameter and circumference, as well as radius and area, are emphasized. Students have multiple opportunities to practice circle calculations. Once students have demonstrated mastery with the calculations, they are expected to apply their knowledge to real-world problems using circle calculations and present their solution(s) in class.</p>

<p><strong>Note:</strong> You should use your discretion when determining which vocabulary words are necessary to teach and which are not.&nbsp;<br><br>Before students arrive, draw a large circle on the board or chart paper, or use the circle diagram transparency (M-7-6-1_Circle Diagram). Cut apart and place the circle terms in a container: <i>arc (major), arc (minor), center, central angle, chord, circle, diameter, radius, secant, sector, semi-circle, tangent</i> (M-7-6-1_Circle Terms). Also, include the vocabulary terms area and perimeter on the front board. Begin class by having students share what they already know about circles. Provide each student with two or three vocabulary journal pages to use during the unit (M-7-6_Vocabulary Journal).<br><br><strong>"Raise your hand if you can describe the word </strong><i><strong>circle</strong></i><strong> in your own words to the class."</strong> Call on one or more students to share their ideas. Emphasize accurate concepts while clarifying misconceptions.<br><br><strong>"If we combine the ideas we just discussed, we may conclude that the definition of a circle is the set of all points that are the same distance apart from a specific point, which is the center of the circle. Today, we will examine at several elements of circles, starting with the definition of their parts and moving toward calculations and problem solving."</strong><br><br><strong>Parts of a Circle Activity</strong><br><br>Distribute one circle diagram to each student (M-7-6-1_Circle Diagram). Put students in small groups of three or four. Give each group a bag or envelope with circle terms (M-7-6-1_Circle Terms).<br><br><strong>"I will give you a few minutes to locate and place the term labels for several parts of a circle on your circle diagram with your group. Place the terms you find from your envelope on only one of the circles in your group. If you are able to complete all of the terms before the time limit, raise your hand to have me check your work. If you are having problems understanding some of the terms, we can help you in 3 or 4 minutes when we go over all of them."</strong>&nbsp;<br><br>Walk around the room and note which terms students are struggling with to help guide the class discussion. Separate the terms in your container into two piles: those most students are familiar with and those they are unfamiliar with. Return those that the majority of students are familiar with to the container.&nbsp;<br><br>After 3 to 4 minutes, direct the student's attention back to the front of the classroom.&nbsp;<br><br><strong>"While I walk around the room, it appears that most groups were already familiar with the terms _____, ______, ______, ..."</strong><br><br><strong>"Let's quickly review those. As we go, each of you should put these labels on your own circle, including the symbols used to symbolize it and any notes that will help you remember it."</strong><br><br>Draw a term from your container. <strong>"I would like someone to explain this term to the class."</strong> Either choose a person or group at random to describe the term, or have students volunteer.&nbsp;<br><br>Continue this process until you've covered all of the terms that students are already familiar with. Add details to any descriptions that are unclear or incomplete. Also, indicate the symbols used to symbolize each part of a circle. When the students have finished with the terms in the container, use the Circle Terms Reference sheet (M-7-6-1_Circle Terms Reference) to discuss the remaining terms.&nbsp;<br><br><strong>"Are there any questions on the names and symbols we use to describe the parts of a circle?" </strong>Answer any questions that arise.<br><br><i><strong>Checkpoint 1:</strong></i> Cover any display you have which show the circle term names and symbols. Draw a circle on the board or chart paper. Add the circle's parts one at a time. Select students at random to state the name and symbol used to symbolize the part of the circle before adding the next part. Clarify any errors as you proceed. You could also draw a term from your container and have a randomly chosen student come to the board and draw that part of the circle, labeling it.<br><br><strong>"Before we move on to the next section of our lesson, I'd like you to take note of the last two circle terms I've written on the board right here</strong> (point to circumference and area)<strong>. The </strong><i><strong>circumference</strong></i><strong> of a circle is the distance around it. The </strong><i><strong>area</strong></i><strong> of a circle is defined as the space inside it, measured in square units. We'll learn how to use these measurements to solve a variety of problems later in this lesson. We need to begin by learning about the very special value known as </strong><i><strong>pi</strong></i><strong>."</strong><br><br><strong>Calculating Circumference and Area</strong><br><br><strong>"Can someone please review for the class how we can find the circumference of a circle?"</strong> (Use the formula <i>C</i> = π • <i>d</i>.)&nbsp;<br><br><strong>"So, to calculate the circumference, we multiply p by the diameter. Remember, p is simply a number; it is around 3.14."</strong>&nbsp;<br><br><strong>Note: An optional exercise can be used to show students where pi comes from</strong> (M-7-6-1_Optional Activity—Discovering Pi and Circumference).&nbsp;<br><br>Optional example: Ask, <strong>"What is the circumference of a circle when the diameter of the circle is 7 inches? To calculate circumference, multiply diameter by 3.14. Therefore, multiply 7 by 3.14. We get 21.98 inches."</strong><br><br><strong>"How can we use this formula to calculate the diameter if we know the circumference?</strong><br><br><i><strong>C</strong></i><strong> = π • </strong><i><strong>d</strong></i><br><br><strong>Let's get </strong><i><strong>d</strong></i><strong> alone to solve for diameter. To eliminate the p, we apply the inverse operation.&nbsp;</strong><br><br><strong>So, we'll divide each side of the equation by p, and we get:</strong><br><br><i><strong>C</strong></i><strong>/π = </strong><i><strong>d</strong></i><br><br><strong>Divide the circumference by 3.14 to get the diameter."</strong>&nbsp;<br><br><i>Optional example:</i> <strong>"If the circumference of a circle is 65 cm, what is the diameter of the circle? To find the diameter, divide the circumference by 3.14. So let us calculate 65 divided by 3.14. We end up with a diameter of 20.70 cm."</strong><br><br><strong>"The </strong><i><strong>circumference</strong></i><strong> is the distance around a circle. We sometimes need to measure the space inside a circle, often known as the area. We use pi to determine the area in square units. The formula is related to the circumference formula, A = π • </strong><i><strong>r</strong></i><strong>². Notice that for this formula, we must use the radius rather than the diameter."</strong>&nbsp;<br><br><strong>"How can we find the radius if we only know the diameter?"</strong> (<i>diameter divided by 2</i>)&nbsp;<br><br><i>Optional example:</i> <strong>"If the diameter of a circle is 12 feet, what is the radius of the circle? We divide the diameter by 2, yielding a radius of 6 feet."</strong><br><br><strong>"What if we only know the circumference, can we still find the radius to use it for the area?"</strong> (<i>dividing the circumference by 3.14 yields the diameter, which may then be divided by 2 to get the radius</i>)&nbsp;<br><br><strong>"You'll solve some problems like this with a partner in a few minutes. Let's start with a few simple examples."</strong> Practice calculating the circumference and area of two or three different circles.</p><figure class="image"><img style="aspect-ratio:549/273;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_68.png" width="549" height="273"></figure><p><i><strong>Example answers:</strong></i><br><br>1. <i>r = 5cm</i> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2. <i>d = 3.2 in. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</i>3. <i>C = 40.82 cm</i><br>&nbsp; &nbsp;<i>C = 31.4 cm</i> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<i>C = 10.048 in.</i> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<i>d = 13 cm, r = 6.5 cm</i><br>&nbsp; &nbsp;<i>A = 78.5 cm² &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;A = 8.0384 in.² &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; A = 132.665 cm²</i><br><br>Students will work in pairs to complete the Partner Circles Practice (M-7-6-1_Partner Circles Practice and KEY). Walk around the room, assisting students. Tell them whatever place value you want them to use for rounding their responses. Rounding to the nearest hundredth is suggested. Review the results (M-7-6-1_Partner Circles Practice and KEY).<br><br><i><strong>Checkpoint 2:</strong></i> Randomly select students to answer area and circumference questions, or have them respond in the Think-Pair-Share format to make sure they understand. You could use questions similar to those provided in the Partner Circles Practice activity.</p><figure class="image"><img style="aspect-ratio:550/274;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_69.png" width="550" height="274"></figure><p><i><strong>Checkpoint 2 Answers:</strong></i><br><br>1. <i>r = 8 in.</i> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2. <i>d = 9 cm</i> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3. <i>C = 109.9 ft</i><br>&nbsp; &nbsp;<i>C = 50.24 in. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;C = 28.26 cm &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;d = 35 ft, r = 17.5 ft</i><br>&nbsp; &nbsp;<i>A = 200.96 in.² &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;A = 63.585 cm² &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; A = 961.625 ft²</i><br><br><br>Answer student questions and correct any errors in their thinking.&nbsp;<br><br><strong>Circle Applications Activity and Presentations&nbsp;</strong><br><br>Ask students to give their thoughts on when the area and circumference of a circle are useful in real life. Write down numerous ideas on chart paper. <strong>"The final part of the lesson is to apply what you've learned about circles to real-world problems."</strong> Distribute the problem cards (M-7-6-1_Circle Application Cards and KEY), chart paper, and markers. Depending on your time frame, each group should be assigned one to three problems.<br><br><strong>"You'll be working with ______ other people. You'll have _____ minutes to work. Prepare to explain your strategies and solutions to the class at that time. You may use a calculator, but please show your work. Please round your answers to the nearest _______."&nbsp;</strong><br><br>Encourage groups to correct any logical or calculation problems. Also encourage students audience to use the presented strategies and solutions to reflect on their own work. Allow students to revise and add to their work.&nbsp;<br><br>Additional, optional application problem:&nbsp;<br><br>To answer the questions, use the figure below.</p><figure class="image"><img style="aspect-ratio:264/214;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_70.png" width="264" height="214"></figure><p>1. What is the radius of the circle? (<i>9 in.</i>)<br>2. What is the area of the circle? (<i>81π in. ≈ 254.34 in.²</i>)<br>3. What is the area of the square? (<i>324 in.²</i>)<br>4. What is the area of the shaded parts? (<i>69.66 in.²</i>)<br><br>At the end of the class, have each student complete an Exit Ticket to assess his or her level understanding (M-7-6-1_Exit Ticket Lesson 1 and KEY).<br><br><strong>Extension:</strong><br><br>Discuss how important it is to understand and use the correct vocabulary words while communicating mathematical ideas. During this lesson, students should record the following terms in their Vocabulary Journals: <i>area, central angle, chord, circle, circumference, diameter, major arc, minor arc, pi, radius, sector</i>. Keep a supply of Vocabulary Journal pages on hand so that students can add them as needed. Bring up examples of area and circumference from throughout the school year. Ask students to bring up circle calculation examples from outside of class and analyze the use and meaning in each particular context. Distinguish between identifying lengths like circumference with standard units and areas with square units as they are used throughout the year. Always require students to use appropriate labeling in both verbal and written responses.<br>Students who require more instruction can participate in small groups during an Area Review Activity. Use this activity for the entire class or for small groups of students who may need more practice calculating the circumference and area of a circle.&nbsp;<br>If a computer is available, visit this website:&nbsp;<br><a href="https://www.youtube.com/watch?v=lWDha0wqbcI&amp;feature=related">https://www.youtube.com/watch?v=lWDha0wqbcI&amp;feature=related</a>.&nbsp;<br><br>This 1½ minute song and video clip explains the basic parts of a circle and the formulas for circumference and area. If time allows, play it a second time. This clip may make it easier for visual and musical learners to remember these information.&nbsp;<br><br>Review with students:<br>radius is half the diameter.&nbsp;<br>diameter is 2 times the radius.&nbsp;<br>practice calculating circumference.&nbsp;<br>practice calculating the area.&nbsp;<br><br>Next, if computers are available, direct students to any or all of the following sites for more practice with circle terms and calculations with immediate feedback:<br><br>Interactive practice for parts of a circle&nbsp;<br><a href="http://www.ixl.com/math/practice/grade-7-parts-of-a-circle">http://www.ixl.com/math/practice/grade-7-parts-of-a-circle</a>&nbsp;<br><br>Interactive circle word problems&nbsp;<br><a href="http://www.ixl.com/math/practice/grade-7-circle-word-problems">http://www.ixl.com/math/practice/grade-7-circle-word-problems</a>&nbsp;<br><br>Interactive circumference and area questions&nbsp;<br><a href="http://www.ixl.com/math/practice/grade-7-circles-calculate-area-circumference-radius-and-diameter">http://www.ixl.com/math/practice/grade-7-circles-calculate-area-circumference-radius-and-diameter</a>&nbsp;<br><br>If computers are not available, use index cards to write down 5-10 circle vocabulary, circumference, and area questions (using whole number values for radius and diameter). Give a card to each student in the small group. Students should record their work and answers on a piece of paper, then rotate cards. Repeat the steps. The number of practice cards that students complete will be determined by the amount of time available and the students' level of skill. Students who finish before the group is ready to rotate the cards may receive an additional card from you. Work one-on-one with students who are having trouble using the formulas to calculate.&nbsp;<br><br><strong>Station: Exploring Patterns of Change:</strong> Allow students to investigate how the circumference and area of various circles relate as the radius changes. Have students choose a radius for a circle and calculate its diameter, area, and circumference. They should then double the radius and recalculate the diameter, circumference, and area. Repeat these steps to triple and cut the radius in half. Ask students to make observations on the relationships between the scale used to increase or decrease the radius and the corresponding increase or decrease in the diameter, circumference and radius. The pattern of change they should notice is that diameter and circumference increase or decrease by the same scale factor as the radius, whereas the area will increase by the square of the scale factor.&nbsp;<br><br><strong>Technology: Create Your Own Applications Activity:</strong> Students can work individually or in groups. Provide paper, pencils, markers, and computer access. Students will create at least five real-world problems involving the area and circumference of circles. Allow students to use the computer or media center to investigate circle topics and circle measures in real-life situations. Students will provide a product format of their choice that includes their questions, such as a worksheet, quiz, or trivia game. Students should include an answer key as well. Allow students to use a website like <a href="http://www.calculatorsoup.com/calculators/geometry-plane/circle.php">http://www.calculatorsoup.com/calculators/geometry-plane/circle.php</a> to help them do initial calculations or double-check their calculations. This website allows students to:&nbsp;<br>enter the radius to get both area and circumference.&nbsp;<br>enter area to get both the circumference and radius.&nbsp;<br>enter circumference to get both area and radius.&nbsp;<br>If time allows, have students or groups share completed products (game, quiz, or worksheet). Have the student or group who created the product answer any questions the user may have and score the results with the answer key they created. After students have exchanged and graded each other's work, allow them to engage in a discussion about the reasoning, errors, and alternative possible methods of calculating the results.<br><br><strong>Expansion: Dividing It Up!:</strong>&nbsp;</p><p>Tell students, <strong>"You will need to apply additional problem-solving strategies for this activity. One new strategy to examine is finding the area of a sector. Remember that a sector is a part of the inner area of a circle. It is bounded by two radii and an arc. Use the central angle to calculate what fractional part of the area is in the sector. Consider the length of the arc on the rim of the circle. The arc will be a fractional part of the circumference proportional to the size of the central angle, just as the sector's area. Can somebody explain how we would go about starting a problem involving arc length or the area of a sector?"</strong> (<i>To find the area of a sector, first calculate the area of the entire circle, or discover the circumference of an arc problem. If you know the center angle for the sector, you may calculate the fraction or percent of the whole circle you are working with based on the full 360˚. To determine the area of the needed sector, multiply the entire area by this fraction or percent. Do the same for the fractional part of the circumference of the arc.</i>)&nbsp;<br>Give students a copy of the Dividing It Up! worksheet (M-7-6-1_Dividing It Up and KEY). Allow students to work independently or in small groups.&nbsp;<br><br>If time permits, have students create one or more real-world problems requiring arc and/or sector calculations. Ask them to include labeled illustrations and instructions.</p>