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1 Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Coach, Mathematics teache lesson followed by the coesponding student lesson. We ae confident that using this lesson will help you achieve you assessment pepaation goals fo you entie class. The Common Coe Coach, Mathematics pogam is based on the philosophy that mathematical skills ae built on concepts. Math, maybe moe than any othe school subject, builds fom concept to concept, one on top of the othe, ove seveal yeas. When students undestand concepts and how they connect to skills, they ae bette equipped to solve poblems that they encounte in the eal wold. This pogam is 100% aligned to the Common Coe State Standads and povides a set of lessons fo each of the five CCSS domains, with each lesson aligning to one o moe standad togethe, lessons cove all the domain s standads. Concept Lessons begin with an undelying concept that connects diectly to the skill o skills taught in that lesson. Students will use a fou-step poblem-solving pocess Read, Plan, Solve, Check to appoach any mathematical poblem. Inteactive questions follow examples and ask students to discuss a topic, model a situation, ty to solve a poblem on thei own, o check thei wok. With this instuctional ancho, you can implement the Common Coe State Standads with confidence. We ae happy to povide you this complimentay sample and would love to know what you think. Once you have ead though this lesson, do what you do best pesent it to you students. Then, don t foget to complete a quick suvey by going to Regads, Tiumph Leaning Join the convesation about Common Coe today by visiting commoncoe.com, the place whee teaches, paents, and expets come togethe to shae best pactices and pactical infomation fo successfully implementing Common Coe standads in the classoom. 136 Madison Avenue New Yok, NY p: f :

2 1 LESSON Aea and Cicumfeence of Cicles Leaning Objectives Students will deive the fomula fo finding the aea of a cicle by dividing a cicle into conguent wedges, eaanging them, and using what they know about the cicumfeence; students will use fomulas to calculate the cicumfeence and aea of a cicle. Students will use fomulas to find unknown lengths in cicles and use those lengths to find the cicumfeence o aea of a cicle; students solve eal-wold poblems that involve finding the aea and cicumfeence of cicles. Common Coe State Standad 7.G.4 Know the fomulas fo the aea and cicumfeence of a cicle and use them to solve poblems; give an infomal deivation of the elationship between the cicumfeence and aea of a cicle. Vocabulay aea the numbe of squae units inside a figue cicumfeence the distance aound a cicle adius the distance fom the cente of a cicle to any point on the cicle Mateials piece of sting fo each student (at least 1 centimetes long) ule scissos (optional) Math Tool: Cicumfeence and Aea Fomulas (Cicles) Math Tool: Spinnes Befoe the Lesson Review the fact that the cicumfeence of a cicle is the distance aound it. Hand each student a piece of sting. Each student should tun to Math Tool: Spinnes and place the sting so that it completely coves the outside of one of the cicula spinnes. Students can use thei finges to keep tack of the distance aound the outside of the cicle o they can Undestand Connect To help develop conceptual undestanding of the aea of a cicle, students can divide a cicle into smalle and smalle conguent wedges. Step 1 shows dividing a cicle into eight conguent wedges. Be sue to point out to students that the use scissos to cut the sting so it is exactly as long as the distance aound one of the cicles. Students should then pull the sting staight and measue its length. Explain that this length, appoximately 11 centimetes, is about equal to the cicumfeence of the cicle. Tell students that the diamete of the cicula spinne is 3.5 centimetes. Ask them to use the fomula fo finding the cicumfeence of a cicle and an appoximation fo p to estimate the cicumfeence of the cicle: C 5 pd cm. Notice that students found the same, o simila, estimates when they used a piece of sting as when they used the fomula. Segue into the lesson, explaining that now that students have eviewed what the cicumfeence of a cicle is, they can apply that knowledge to help them detemine a fomula fo finding the aea of a cicle. distance aound half of the cicle is equal to half of the cicumfeence, o: 1 3 C 5 1 (p) 5 p units. Students can use the diagam in steps and 3 to visualize how the wedges can be aanged to look 64

3 like a paallelogam and then eaanged to look like a ectangle. Step 3 shows how the fomula fo finding the aea of a cicle can be deived using the fomula fo the aea of a ectangle and the length (p units) and width ( units) of the ectangle. To connect the concept to pocedual undestanding, show how the fomulas fo finding the cicumfeence and aea of a cicle can be applied to find the aea of cicle O. As you eview steps and 3, explain that the symbol means is appoximately equal to. Ask: Why is that symbol used in steps and 3? Students should state that wheneve 3.14 is used fo p, the measue being calculated is only appoximately equal to the tue measue. So, the symbol is needed. DISCUSS MP Students should state that the fomula C 5 pd could also be used to find the cicumfeence of cicle O. The fomula C 5 p can be ewitten as C 5? p. Since d 5, d can be substituted fo. The fomula then becomes C 5 d? p o C 5 pd. Example EXAMPLE This example diffes fom the pevious poblem because it equies multiple steps. Students must fist substitute known values into the fomula fo finding the cicumfeence of a cicle and solve fo. Only when they know the value fo can the appoximate aea of the cicle be calculated. TRY MP5 This poblem diffes fom the poblem in the example because instead of giving an appoximate cicumfeence, the cicumfeence is given in tems of p. Show how to calculate that the adius is 14 units. Wok may vay. Possible wok: C 5 p 8p 5 p Poblem Solving This is a multistep poblem that applies the aea fomulas to a eal-wold context. PLAN Then subtact the aea of the smalle (shaded) cicle fom the aea of the lage (white) cicle. SOLVE The lage cicle has 5 7. A 7 3 (7) A 154 in. The smalle cicle has d 5 7, so 5 7. A A 38 1 in. Subtact to find the aea of the placemat that is not coveed by the coaste in. CHECK Pactice Does the smalle cicle take up moe than o less than 1 of the lage cicle? less than If the faction of the lage cicle that is shaded is less than 1, then I would expect that the aea of the smalle cicle would be less than half of the aea of the lage cicle The aea of the smalle cicle is less than the aea of the lage cicle. Since the aea being subtacted fom the lage cicle s aea is less than half its aea, I would expect my answe to be geate than half the aea of the lage cicle. My answe is geate than half the aea of the lage cicle, so my answe is easonable. The aea of the placemat that is not coveed by the coaste is about in. As students ae woking, pay special attention to poblem 6. Because the diamete is a multiple of 7, encouage students to use 7 fo p. when calculating this cicumfeence. Students may use Math Tool: Cicumfeence and Aea Fomulas (Cicles) fo efeence if they need to. Fo answes, see page 11. Domain 4 65

4 1 LESSON Aea and Cicumfeence of Cicles UNDERSTAND The cicumfeence, C, is the distance aound a cicle. It can be found using the fomula C 5 p, whee is a adius of the cicle and p is an iational numbe appoximately equal to 3.14 o 7. The aea of a cicle is the total numbe of squae units that fit inside the cicle. Explain how you can use what you know about the cicumfeence of a cicle to detemine the fomula fo finding its aea. 1 Cut a cicle into eight conguent wedges, as shown. The distance fom the cente to any point on the cicle measues units. The distance aound the entie cicle measues p units. So, the distance aound half the cicle is equal to half of that, o p units. Visualize eassembling the wedges to look like a paallelogam. Fist, eaange them to look like a paallelogam that has a length of appoximately p units. 3 Divide one wedge in half. Move that half so the diagam esembles a ectangle. If you continue to divide the cicle into conguent wedges and the numbe of wedges appoaches infinity, the esulting figue would look moe and moe like a ectangle. Its length would be p units and its width would be units. So: A of cicle 5 length 3 width 5 p? 5 p The aea of a cicle is found by the fomula: A 5 p. 118 Domain 4: Geomety

5 Connect Find the appoximate cicumfeence and aea of cicle O. 10 cm O 1 Detemine the length of the adius of cicle O. 3 The line segment labeled 10 cm has endpoints on the cicle and passes though the cente, point O. The line segment is a diamete of the cicle. The adius of a cicle is equal to half its diamete, so: cm Use the fomula A 5 p to find the appoximate aea. Use the fomula C 5 p to find the appoximate cicumfeence. Use 3.14 fo p. C 5 p C < C < 31.4 cm Note: You ae using an appoximation fo p, so the cicumfeence you found, 31.4 cm, is also an appoximation. A 5 p A < A < A < 78.5 cm Again, 78.5 cm is only an appoximation of the aea because you used an appoximate value fo p. The appoximate cicumfeence is 31.4 centimetes, and the appoximate aea is 78.5 squae centimetes. DISCUSS Could you also use the fomula C 5 pd to find the cicumfeence of cicle O? Explain. Lesson 1: Aea and Cicumfeence of Cicles 119

6 EXAMPLE The cicumfeence of a cicle is appoximately 50.4 metes. What is its appoximate aea? 1 Substitute known values into the equation fo the cicumfeence of a cicle. Substitute 50.4 fo C and 3.14 fo p. C 5 p 50.4 < Solve fo < < < < 3 Substitute that value fo in the aea fomula. A 5 p A < A < The appoximate aea of the cicle is squae metes. TRY The cicumfeence of a cicle is 8p inches. What is its adius? Show you wok. 10 Domain 4: Geomety

7 ead Poblem Solving A cicula coaste, with a diamete of 7 inches, is positioned on a white placemat. The placemat is a cicle whose adius is equal to the diamete of the coaste. About how many squae inches of the placemat ae not coveed by the coaste? 7 in. plan Find the aea of both cicles. Then subtact the aea of the cicle fom the aea of the cicle. solve Find the appoximate aea of each cicle. Use 7 fo p since the adius is a multiple of 7. The lage cicle has 5. The smalle cicle has d 5, so 5. A of lage cicle 5 p A of smalle cicle 5 p A < 7 3 ( ) A < 7 3 ( ) A < in. A < in. Subtact to find the aea of the placemat that is not coveed by the coaste. 5 in. check Look at the diagam. Does the smalle cicle take up moe than o less than 1 of the lage cicle? If the faction of the lage cicle that is shaded is than 1, then I would expect that the aea of the smalle cicle would be than half of the aea of the lage cicle The aea of the smalle cicle is than the aea of the lage cicle. Since the aea being subtacted fom the lage cicle s aea is than half its aea, I would expect my answe to be than half the aea of the lage cicle. My answe is than half the aea of the lage cicle, so my answe easonable. The aea of the placemat that is not coveed by the coaste is about in.. Lesson 1: Aea and Cicumfeence of Cicles 11

8 Pactice A adius,, o diamete, d, of a cicle is given. Find the othe measue. 1. If d 5 4, 5.. If 5 4, d If d 5 3, 5. Find the appoximate cicumfeence of each cicle shown. Show you wok in ft 6. 8 in. HINT Use 3.14 o as an 7 appoximation fo p. Find the appoximate aea of each cicle shown. Show you wok. 7. yd ft 9 m Fill in the blank with an appopiate wod o phase. 10. The adius of a cicle is equal to its diamete. 11. The cicumfeence of a cicle is the aound it. 1. The aea of a cicle is the numbe of that fit inside it. 1 Domain 4: Geomety

9 Choose the best answe. 13. Cicle A has a cicumfeence of 36p metes. What is the adius of cicle A? A. m B. 6 m C. 1 m D. 18 m 14. Cicle O has a cicumfeence of 13 feet. What is the appoximate aea of cicle O? A. 5,544 ft B. 1,386 ft C. 441 ft D ft Solve. 15. Jodan has a cicula flowe bed that is 18 feet in diamete in he yad. Appoximately how many squae feet of he yad is coveed by the flowe bed? 16. A cicula fountain in a pak is 30 feet in diamete. A pak employee will plant maigolds aound it. What is the cicumfeence of the fountain? Flowe bed 18 ft 30 ft 17. Show Two flat, cicula plates ae placed on a cicula tabletop. The diamete of each plate equals the adius of the tabletop. How many squae inches of the tabletop ae not coveed by the plates? What faction of the tabletop is coveed? Show you wok. 18. EXPLAIN This datboad is divided into thee egions. The outemost egion has dak shading, the middle egion is lightly shaded, and the cente egion is unshaded. What is the aea of the egion with the dak shading? Explain how you detemined you answe. 1 inches 1 in. 1 in. 1 in. Lesson 1: Aea and Cicumfeence of Cicles 13