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1 Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational 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. Common Coe Suppot Coach was built fom the goundup to povide students with scaffolded suppot fo content equied by the Common Coe State Standads fo Mathematics (CCSSM). The pogam featues 0 lessons which focus on 0 citical skills essential to mastey of Common Coe State Standads. These skills wee chosen because they ae citical components of the mathematical pogessions that connect ealie standads to this gade level and fom this gade level to the next. While each lesson connects to pio foundational skills and concepts, it can be viewed as an independent unit of instuction. In this way, the 0 lessons allow teaches to diffeentiate instuction accoding to the equiements of each student. Key to the philosophy behind Common Coe Suppot Coach is the ecognition that math skills and concepts ae pat of a pogession that begins ealy in students lives and continues beyond thei cuent gade level with inceased complexity and depth. Fo students, achieving tue undestanding of CCSSM at any gade level means mastey of pio content that connects to this gade and mastey of content that connects within the gade. Often, students who cannot cope with a specific pat of thei gade s cuiculum ae missing one o moe undestandings that would allow mastey. Common Coe Suppot Coach supplies the missing pieces. 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 By doing so, you will be enteed into ou quately affle fo one of five Ameican Expess $100 gift cads. 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 LESSON 15 Solving Poblems with Cicles Plug In Radius, Diamete, and Cicumfeence of a Cicle FOUNDATIONAL UNDERSTANDING ONLEVEL TARGET Plug In Radius, Diamete, and Cicumfeence of a Cicle Student Edition. pp powe up Aea of a Cicle eady to go Solving Poblems with Cicles OBJECTIVES CONCEPTS AND SKILLS VOCABULARY Calculate the diamete of a cicle given the adius and vice vesa. Calculate cicumfeence given the diamete o the adius. Calculate the aea of a cicle given the adius o the cicumfeence. Solve poblems using fomulas fo the cicumfeence and aea of a cicle. Recognize the popeties of a cicle (adius, diamete, and cicumfeence) and the elationship between them. CCSS: 7.G.4 Know the fomula fo the aea of a cicle. Give an infomal deivation of the elationship between the cicumfeence and aea of a cicle. CCSS: 7.G.4 Know the fomulas fo the aea and cicumfeence of a cicle, and use them to solve poblems. CCSS: 7.G.4 adius diamete cicumfeence pi aea of a cicle MATERIALS Math Tool: Cicle Fomulas, p. A1 (Student Edition p. 47) Pape stips (suggested) Scissos (suggested) Sting (suggested) ENGLISH LANGUAGE LEARNERS MP5 MP6 ELL students may need exta suppot fo undestanding the elationship between adius and diamete. Have students cut a pape stip the same length as the diamete of a cicle and anothe stip the same length as the adius. Tell them to fold the diamete stip in half. Have them compae the diamete to the adius and see that the length of two adii is the same length as the diamete. Build Backgound Talk to students about easons to find the cicumfeence of a cicle in eal life. Fo example, Gina has a ing with a diamete of 16 mm, and he ing finge has a cicumfeence of 5 mm. Will the ing fit on he finge? Explain that finding the cicumfeence of the ing is one way to answe that question. MP4 Have students discuss additional examples of eal situations that involve finding the adius, diamete, o cicumfeence of a cicle. Tell students they will wok on finding the adius, diamete, and cicumfeence of cicles. Intoduce and Model Intoduce Concepts and Vocabulay Guide students though the methods fo finding adius, diamete, and cicumfeence. Use Wods to Know to claify thei undestanding of vocabulay. MP6 Have students explain the elationship between adius, diamete, and cicumfeence. Suppot Discussion MP1 Have patnes discuss biefly befoe goup discussion. As needed, diect students to Wods to Know and the elationship between the adius and the diamete. Pompt: What ae the fomulas fo finding cicumfeence? Sentence State: I can substitute adius into the fomula that uses diamete by Lesson Tiumph Leaning, LLC
3 LESSON 15 B Plug In Evey cicle has a cente, a adius (), a diamete (d), and a cicumfeence (C). The length of the diamete, d, of a cicle is two times the length of the adius,. d If 5 3 cm, then d cm. The length of the adius is half the length of the diamete! So 5 d 4, o d_. Wods to adius Know a segment fom the cente of a cicle to any point on it, epesented by Solving Poblems with Cicles Radius, Diamete, and Cicumfeence of a Cicle diamete a segment acoss a cicle though its cente, epesented by d The cicumfeence is the peimete of a cicle. Multiply pi ( ) by the diamete, d, to find the cicumfeence. can appoximated by If d 5 6 cm, then C cm. I see! I use when I find the cicumfeence because 3.14 is the appoximate value of pi. cicumfeence the distance aound a cicle, epesented by C C 5 d o C d C d pi the atio of the cicumfeence of a cicle to its diamete, C4d 3.14 You can use the adius to appoximate the cicumfeence of a cicle. Find the cicumfeence of the cicle. 1 Wite the fomula fo the cicumfeence using the diamete. Descibe the elationship between diamete and adius. 3 Wite a cicumfeence fomula using. 4 Plug in 3.14 fo and the length of the adius in the cicumfeence fomula. 5 Multiply. Wite the appoximate cicumfeence. PRACTICE 15 Solving Poblems with Cicles 4.5 m C 5 d The length of the diamete is times the length of the adius. d 5 C 5 d o I emembe! The symbol C means is C appoximately equal to. C 8.6 m The cicumfeence is appoximately 8.6 metes. Cicle Fomulas can be found on p. 47. Rose measues the adius of he bicycle wheel to be 14 inches. She odes a new tie fo the wheel by finding its cicumfeence. Howeve, she calculates its value using 3 fo instead of What should she expect when he tie aives? Possible esponse: Since she is using a smalle and less accuate appoximation fo, the tie will be too small fo the wheel. Find the length of the adius o diamete when given the othe measue. 1 If d 5 8, then 5 4. If 5 15, then d Lesson 15 How can you find the cicumfeence of a cicle if you know only the adius? Since the diamete is twice the adius, you can multiply the adius by and then multiply the poduct by 3.14, o. A You can use multiplication to find the diamete of a cicle. Find the diamete of the cicle. cm 1 Wite the fomula fo the diamete of a cicle. d 5 Plug in the length of the adius in the fomula and d cm multiply. The diamete measues 4 cm. 3 Wite the length of the diamete. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Find the cicumfeence of the cicle. Use 3.14 fo. Show you wok. Check students wok ft d 35 in. 7 m in. 6.8 ft m 144 Lesson Model Application A Explain that since the diamete of a cicle is twice the length of the adius, the adius is half the length of the diamete. B Have students use the elationship between adius and diamete to ceate diffeent fomulas fo finding cicumfeence. Suppot Discussion MP Have patnes discuss biefly befoe goup discussion. Pompt: How ae the values fo cicumfeence diffeent when you use 3 and 3.14 fo pi? Sentence State: The values ae... spotlight on MATHEMATICAL LANGUAGE MP6 Suppot students in using mathematical language as they wok: The adius is epesented by. The diamete passes though the cente of a cicle. How do you find the cicumfeence of a cicle? Pactice and Assess Ask students to complete pactice items 1 5 on page 145 independently o in pais. Monito ongoing wok. Obseve whethe students coectly use the elationship between adius and diamete, and use the coect fomula to find cicumfeence. Use the chat below as needed to addess any difficulties. Obsevation Students use the incoect fomula fo cicumfeence. 014 Tiumph Leaning, LLC Action Have students cut a piece of sting the same length as the adius o diamete. Mak a stating point along the cicumfeence. Move the piece of sting endtoend along the cicumfeence. Have students count the numbe of adii ( 6) o diametes ( 3) it takes to tavel the cicumfeence. Students can use this infomation to wite an expession fo cicumfeence. Solving Poblems with Cicles 115
4 powe up Aea of a Cicle FOUNDATIONAL UNDERSTANDING ONLEVEL TARGET Plug In Radius, Diamete, and Cicumfeence of a Cicle powe up Aea of a Cicle Student Edition. pp eady to go Solving Poblems with Cicles OBJECTIVES CONCEPTS AND SKILLS VOCABULARY Calculate the diamete of a cicle given the adius and vice vesa. Calculate cicumfeence given the diamete o the adius. Calculate the aea of a cicle given the adius o the cicumfeence. Solve poblems using fomulas fo the cicumfeence and aea of a cicle. Recognize the popeties of a cicle (adius, diamete, and cicumfeence) and the elationship between them. CCSS: 7.G.4 Know the fomula fo the aea of a cicle. Give an infomal deivation of the elationship between the cicumfeence and aea of a cicle. CCSS: 7.G.4 Know the fomulas fo the aea and cicumfeence of a cicle and use them to solve poblems. CCSS: 7.G.4 adius diamete cicumfeence pi aea of a cicle MATERIALS Math Tool: Cicle Fomulas, p. A1 (Student Edition p. 47) Math Tool: Gid Pape,. p. A4 (Student Edition. p. 53) Gid pape (suggested) english language leanes MP5 Have students daw cicles on gid pape o on Math Tool: Gid Pape to model aea. Have them count the numbe of squae units inside the cicle and wite the numbe unde the cicle, including units. Emphasize the connection between the squae unit label fo aea and the squae units inside the cicle on the gid pape. Build Backgound Talk to students about easons to find the aea of a cicle in eal life. Fo example, a capente has a pipe with a diamete of 10 inches. Will it fit though a cicula hole with an aea of 80 squae inches? Explain that using the fomula fo aea of a cicle can help you find the answe. MP4 Have students discuss additional examples of eal situations that involve finding the aea of a cicle. Tell students they will wok on finding the aea of a cicle given the adius o diamete. Intoduce and Model Intoduce Concepts and Vocabulay Guide students though the infomal deivation of the aea of a cicle. Emphasize that the aea fomula fo a cicle is not the same as that fo a ectangle. Use Wods to Know to claify thei undestanding of vocabulay. MP6 Have students explain the aea of a cicle to a patne. Suppot Discussion MP3 MP6 Have patnes discuss biefly befoe goup discussion. As needed, diect students to Wods to Know and have them compae what an aea and cicumfeence epesent. Pompt: How would you use a dawing to show me what the aea of a cicle is? What about the cicumfeence? Sentence State: The aea of a cicle is Lesson Tiumph Leaning, LLC
5 15 Solving Poblems with Cicles POWER up Aea of a Cicle The cicumfeence of a cicle is elated to the aea of a cicle. This octagon is divided into 8 equivalent tiangles. Its shape is close to, but not exactly, a cicle. If cut apat, the 8 tiangles can be aanged as a paallelogam. If half of the fist tiangle is cut off and moved to the end, the paallelogam becomes a ectangle. The bases of fou tiangles, o about half the cicumfeence of the cicle, make up the length of the ectangle. We can appoximate the length: l 1 ( ) 5. The width of the ectangle is equal to the height, h, of the tiangle. The height, h, is about the same as the adius,, of the cicle, so we can appoximate the width: w. The fomula fo the aea of a ectangle is A 5 lw. If we substitute fo l and fo w, the esult is the fomula fo the aea of a cicle. A 5 3, o. h l b w Wods to aea of a cicle Know the total numbe of squae units that fit inside the cicle; The fomula fo the aea of a cicle is A 5. I see! The aea of a cicle is, o a little moe than thee times the adius squaed! Juanita calculates the cicumfeence and aea fo a cicle with adius cm. She finds they ae both equal to 4. She asks how the distance aound a cicle can be the same as the aea inside a cicle. What would you tell he? Possible esponse: I would tell he that the units ae diffeent. The cicumfeence is in centimetes, and the aea is in squae centimetes. They descibe diffeent A You can use the adius to appoximate the aea of a cicle. things. Find the aea of the cicle. 5 in. 1 Wite the fomula fo the aea of a A 5 cicle. A Plug in 3.14 fo and the length of the adius in the fomula. A Multiply. Wite the appoximate A 78.5 in. aea. The aea is appoximately 78.5 squae inches. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Always check whethe the poblem o diagam B You can use diamete to find the appoximate aea of a cicle. lists the diamete o Find the aea of the cicle. the adius. 1 Find the adius of the cicle. Wite the fomula fo the aea of the cicle. d 8 cm 3 Plug in 3.14 fo and the length of the adius in the fomula. The adius is half the diamete, so 5 4 cm. 4 Multiply. Wite the appoximate aea. A 5 A A A 50.4 cm The aea is appoximately 50.4 squae centimetes. Rob made an eo when he solved fo the aea of the cicle shown. What can you tell him about his wok? What is the coect answe? A A d 6 cm A cm Cicle Fomulas can be found Possible esponse: Rob multiplied the adius by instead on p. 47. of squaing it. The coect answe is A cm. PRACTICE Find the appoximate aea of the cicle. Use 3.14 fo. Show you wok. Check students wok mi d 5 m squae miles 4 7 mm d 13 ft squae millimetes squae metes squae feet Lesson Lesson Model Application A Guide students though finding the aea of a cicle using the adius. Monito that students ae squaing the adius in thei calculations, not just multiplying by. B Have students find the aea of a cicle using the diamete. Monito that students ae dividing the diamete by befoe using the fomula fo aea. Suppot Discussion MP3 Have patnes discuss biefly befoe goup discussion. As needed, diect students to pefom the calculation themselves. COMMON ERRORS Students may multiply the adius by instead of squaing it when calculating the aea. Review how to evaluate an expession that contains an exponent. Stess that the adius is multiplied by itself because the aea fomula has an exponent of. Pompt: How did Rob detemine the numbes that he used to compute the aea? Sentence State: To find the adius, I would... Pactice and Assess Ask students to complete pactice items 1 4 on page 147 independently o in pais. Monito ongoing wok. Obseve whethe students ae finding the aea coectly. Use the chat below as needed to addess any difficulties. Obsevation Students have difficulty judging whethe they ae using the aea fomula coectly. 014 Tiumph Leaning, LLC Action Have students daw cicles on gid pape. Have them count the appoximate numbe of squae units inside the cicle and compae this to the value calculated using the aea fomula. Solving Poblems with Cicles 117
6 eady to go Solving Poblems with Cicles FOUNDATIONAL UNDERSTANDING ONLEVEL TARGET Plug In Radius, Diamete, and Cicumfeence of a Cicle powe up Aea of a Cicle eady to go Solving Poblems with Cicles Student Edition. pp OBJECTIVES CONCEPTS AND SKILLS VOCABULARY Calculate the diamete of a cicle given the adius and vice vesa. Calculate cicumfeence given the diamete o the adius. Calculate the aea of a cicle given the adius o the cicumfeence. Solve poblems using fomulas fo the cicumfeence and aea of a cicle. Recognize the popeties of a cicle (adius, diamete, and cicumfeence) and the elationship between them. CCSS: 7.G.4 Know the fomula fo the aea of a cicle. Give an infomal deivation of the elationship between the cicumfeence and aea of a cicle. CCSS: 7.G.4 Know the fomulas fo the aea and cicumfeence of a cicle and use them to solve poblems. CCSS: 7.G.4 adius diamete cicumfeence pi aea of a cicle MATERIALS Lesson 15 Quiz, Assessment Manual. pp Lesson 15 Quiz Answe Key, Assessment Manual Math Tool: Gid Pape,. p. A5 (Student Edition p. 55) Highlightes in two colos (suggested) Build Backgound Talk to students about easons to use fomulas to find the cicumfeence and aea of cicles in eal life. Fo example, Sonja uns aound a cicula tack with a diamete of 50 yads. How fa does she un in one lap? Explain that using a fomula to find the cicumfeence of the tack is one way to answe that question. MP4 Have students discuss additional examples of eal situations that involve using fomulas to find the cicumfeence and aea of cicles. Tell students they will solve poblems involving cicles using fomulas fo the cicumfeence and aea. spotlight on mathematical PRACTICES MP3..Citiquing Othes Reasoning Help students think about Tina's easoning citically by asking pobing questions: Did Tina use the coect values in the fomula? Did Tina pefom the calculations coectly? Intoduce and Model Intoduce Concepts Guide students though finding the adius, cicumfeence, and aea of a cicle. Emphasize that the adius is used in the fomulas fo both the cicumfeence and the aea. Suppot Discussion MP3 Have patnes discuss biefly befoe goup discussion. As needed, eview the fomula fo aea of a cicle. Pompt: Whee did Tina make an eo? What is the coect answe? Sentence State: To calculate the aea, Tina should have... lesson link Connect to Foundational Undestanding Skills leaned in the Plug In and Powe Up ae efeenced in the Lesson Link. Explain that, as long as the adius o diamete is known, fomulas fo cicumfeence and aea can be used to solve poblems about cicles. 118 Lesson Tiumph Leaning, LLC
7 15 Solving Poblems with Cicles READY TO go You can find the cicumfeence and aea of a cicle using fomulas. Find the cicumfeence and aea of a cicula building with diamete of 16 yads. Use the fomula fo the cicumfeence of a cicle. Substitute the value fo d. C 5 d C C C 50.4 yd The cicumfeence of the cicula building is appoximately 50.4 yads. I emembe! I can use 3.14 fo. And I use the symbol to find an appoximate value. Solving Poblems with Cicles Use the fomula fo the aea of a cicle. Substitute the value fo. A 5 A A A A yd The aea of the cicula building is appoximately squae yads. I have to fist find the adius. Since the diamete is 16, the adius is WORk TOgEThER You can estimate the cicumfeence and aea. Then check you estimate. The adius is 9 cm. The cicumfeence is a little moe than 3 times the diamete, o twice the adius. So the cicumfeence will be about , o 54 cm. The aea is a little moe than 3 times the adius squaed. So the aea will be about , o 43 cm. C 5 A 5 C 5 9 A 5 9 C A C 56.5 cm A cm Cicle Fomulas can be found on p. 47. A You can use the cicumfeence of a cicle to solve poblems. Find the length of fencing needed to enclose the playgound. Use 3.14 fo. I can estimate the cicumfeence and aea to see if my answes ae easonable. 1 cm Lesson 15 Tina made an eo when solving fo the aea of the cicle shown. Identify he eo, and wite the coect answe. How fa off was he answe? d 11 mm A 5 11 A Estimate the cicumfeence. Use the fomula to find the cicumfeence of the playgound. 3 Wite the length of fencing needed. 50 yd Playgound A A mm Possible esponse: Tina squaed the diamete instead of the adius. The coect answe is A mm. He answe was 4 times the coect answe o lesson link off by sq m. Plug In POWER up go! The diamete o the adius can be used to find the cicumfeence of a cicle. d C 5 d o The adius can be used to find the aea of a cicle. 3 ft A I get it! I can use what I leaned about cicles to solve poblems involving cicumfeence and aea. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC The diamete is 50 yd. The cicumfeence is a little moe than 3 times the diamete. So the cicumfeence will be about , o 150 yd. C 5 d C C 157 yd The length of fencing needed to enclose the playgound is appoximately 157 yads. Look at the fomulas fo the cicumfeence and aea of a cicle. How is the fomula fo cicumfeence elated to the fomula fo aea? Possible esponse: They both use pi and adius. The aea of a cicle is half the cicumfeence times the adius. 148 Lesson Wok Togethe Explain that students will fist estimate the cicumfeence and aea of a cicle and then calculate them moe pecisely using fomulas. Emphasize that the estimated values should be close to the calculated values. A Be sue students undestand that fencing goes aound the outside of the playgound, so the cicumfeence of the playgound must be found to detemine how much fencing is needed. Monito that students ae using the coect fomula and that the estimated cicumfeence is close to the calculated value. Suppot Discussion MP MP7 Have patnes discuss biefly befoe goup discussion. As needed, suggest that students think about the definition of p. ENGLISH LANGUAGE LEARNERS MP1 MP6 Have students highlight clue wods in poblems that tell them what value to find. Highlight clue wods fo cicumfeence like length and cicumfeence in one colo and clue wods fo aea like squae yads (o othe units) and aea in a diffeent colo. Pompt: What expession fo p could we substitute into the aea fomula? Sentence State: The expession C d would fist need to be Tiumph Leaning, LLC Solving Poblems with Cicles 119
8 READY TO go 15 Solving Poblems with Cicles PRACTICE Use the fomulas to find the cicumfeence and aea of the cicle. Use 3.14 fo. Estimate fist. Then solve. Use 3.14 fo. d 0 km REMEMBER means to multiply a numbe by itself. 3 Ali needs a pool cove fo he cicula pool. What is the cicumfeence of the pool? 1 ft 4 Tom is planning a cicula flowe gaden fo the yad. What aea of the yad will be planted with flowes? 1 C 5 d A 5 C A C 6.8 km A A 314 km 6 ft 11 mi C 5 A 5 C A C A C mi A mi Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Solve. Use 3.14 fo. 5 A spinkle spays wate within a 5mete adius as it spins in a cicle. What is the appoximate aea of lawn that is wateed by the spinkle? 6 A hockey puck is 7.6 centimetes in diamete. What is the cicumfeence of the puck? Estimate: 63 ft; Actual: ft Estimate: 507 ft ; Actual: ft 78.5 m cm Sketch a diagam to make sue you undestand what the poblem is descibing. Make Sense of Poblems Making a diagam can help you Kate keeps a cicula photogaph in a cicula pictue fame. sot though the The photogaph has a diamete of 7 cm. The fame has a diamete infomation given in of 15 cm. Descibe the steps Kate can take to find the total distance a poblem! aound the outside and the inside of the pictue fame. What is the total distance aound the outside and the inside of the pictue fame? cm Possible esponse: 1. Find the cicumfeence of the oute fame.. Find the cicumfeence of the photogaph. 3. Add to find the total distance aound the outside and the inside of the pictue fame. 150 Lesson ADDITIONAL PRACTICE Povide students with additional pactice to model and solve: The face of a clock is 13 inches in diamete. What is the aea of the clock face? A cicula skating ink has a adius of 30 feet. What is the cicumfeence of the ink? Suppot Independent Pactice 1 Remind students to ead the REMEMBER. If needed, have students ead the fomulas out loud to emind them which values to use: C equals (pi) times d. A equals (pi) times squaed. 3 4 What is the poblem asking you to find? What fomula should you use? 5 6 Remind students to ead the avata speech bubble and have them label thei diagam with the measuements given in the poblem. Suppot Discussion MP1 Have patnes discuss biefly befoe goup discussion. As needed, have students model the poblem with a sketch. Pompt: What is the distance aound the outside of the fame equivalent to? What is the total distance aound the inside of the fame? Sentence State: The total distance aound the outside of the pictue fame is... spotlight on mathematical PRACTICES MP1..Solve Poblems Help students make sense of the poblem by guiding them though dawing a diagam of the poblem: If the photogaph has a diamete of 7 cm, then this is the distance aound which pat of the pictue fame? Poblem Solving Model the FouStep Method MP1 MP7 Guide students though the foustep method using thinkaloud stategies. Point out the diamete of the fountain (the gay aea in the cente of the diagam) and the distance fom the fountain to the edge of the pond (the blue aea in the diagam). Think Aloud The fountain is in the middle of the pond, so the aea of the pond is the aea of the pond and fountain minus the aea of the fountain. 10 Lesson Tiumph Leaning, LLC
9 READY TO go 15 Solving Poblems with Cicles PROblEm solving READ PlAn DUCkS AMUCk A fountain, 3 ft in diamete, sits in the cente of a cicula duck pond. The pond measues 3 ft fom the fountain to its edge. What is the aea of the duck pond? What is the poblem asking you to find? the aea of the duck pond 3 f t 3 f t PRACTICE Use the poblemsolving steps to help you. Use 3.14 fo. 1 Bad is eplacing the topsoil in the community gaden. If the aea he is eplacing is in the shape of a half cicle with a diamete of 30 dm, how many squae decimetes of the gaden will he eplace? squae decimetes Be sue to include the units in you answe! CHECKLIST READ PLAN SOLVE CHECK Lesson 15 How can you solve this poblem? Find the diffeence between the aeas of the pond and the fountain. solve ChECk What fomula can you use? A 5 Find the diamete of the d ft pond and fountain. Find the adius of the 5 d ft pond and fountain. Find the aea of the A ft pond and fountain. Find the adius of the fountain. 5 d ft Find the aea of the fountain. A ft Subtact to find the aea of the duck pond. A ft Estimate each solution to see if you answes ae easonable. The aea is a little moe than 3 times the adius squaed. The aea of the pond and fountain is about ft. The aea of the fountain is about ft. The aea of the pond is about ft. The aea of the duck pond is appoximately 56.6 ft. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC The diamete of a basketball hoop is 18 inches. What is the lagest cicumfeence of a basketball that will fit though the hoop? Show you wok inches 3 A cicula window fame has a diamete of inches. The glass window inside the fame has a adius of 10 inches. What is the aea of the window fame not including the glass? squae inches CHECKLIST READ PLAN SOLVE CHECK CHECKLIST READ PLAN SOLVE CHECK 15 Lesson Suppot PoblemSolving Pactice Have students use the Checklist as they complete each step. Pompt: Can you tell me what you need to find? Pompt: What fomula will you use? Pompt: Ae thee othe calculations you need to make? Exploe Student Thinking MP1 Invite students to explain how they found the aea o cicumfeence. Have patnes compae thei wok on a poblem and discuss thei esults. COMMON ERRORS Students may stop at finding the aea o cicumfeence of one component of the poblem but fail to answe the question asked. Have students undeline the value they need to find and pedict the units thei answe should have. Assess Use the table below to obseve whethe students accuately find the aea o cicumfeence of a cicle, and to addess any difficulties as needed befoe the quiz. When all students ae eady, assign the Lesson 15 Quiz. 1 3 Obsevation Eos in finding aea o cicumfeence ae fequent o iegula; geneal confusion about finding aea o cicumfeence. Obsevation Makes occasional eos in finding aea o cicumfeence; some undestanding of finding aea o cicumfeence. Obsevation Wites, models, and solves poblems involving cicles accuately. Action Have students daw cicles on gid pape and count squae units to check aea calculations. Have them estimate cicumfeence by using 3 fo p befoe using the fomula. Action Povide additional pactice poblems fo finding aea and cicumfeence. Encouage students to model aea and cicumfeence. Action Assign the Lesson 15 Quiz. 014 Tiumph Leaning, LLC Solving Poblems with Cicles 11
10 LESSON S uppo tcoa c h, Ma t he ma t i c s, S t ude nte di t i on, G a de7 15 Solving Poblems with Cicles Radius, Diamete, and Cicumfeence of a Cicle Plug In Evey cicle has a cente, a adius (), a diamete (d), and a cicumfeence (C). The length of the diamete, d, of a cicle is two times the length of the adius,. The cicumfeence is the peimete of a cicle. Multiply pi ( ) by the diamete, d, to find the cicumfeence. can appoximated by If d 5 6 cm, then C cm. d C d C d If 5 3 cm, then d cm. I see! I use when I find the cicumfeence because 3.14 is the appoximate value of pi. Wods to adius a segment fom the cente of a cicle to any point on it, epesented by Know CU S S DIS diamete a segment acoss a cicle though its cente, epesented by d cicumfeence the distance aound a cicle, epesented by C pi the atio of the cicumfeence of a cicle to its diamete, C4d C 5 d o 3.14 How can you find the cicumfeence of a cicle if you know only the adius? A You can use multiplication to find the diamete of a cicle Find the diamete of the cicle. Wite the fomula fo the diamete of a cicle. Plug in the length of the adius in the fomula and multiply. Wite the length of the diamete. d5 d5 cm 3 5 The diamete measues cm cm. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC The length of the adius is half the length of the diamete! d. So 5 d 4, o _ L e ss o n 1 5 CC13_Mth_G7_SE_PDF.indd /05/13 7:06 PM
11 15 Solving Poblems with Cicles B You can use the adius to appoximate the cicumfeence of a cicle. Find the cicumfeence of the cicle. 1 Wite the fomula fo the cicumfeence using the diamete. Descibe the elationship between diamete and adius. 3 Wite a cicumfeence fomula using. 4 Plug in 3.14 fo and the length of the adius in the cicumfeence fomula. 5 Multiply. Wite the appoximate cicumfeence. PRACTICE C 5 d The length of the diamete is times the length of the adius. d 5 C 5 d o C 3 3 C 3 C m The cicumfeence is appoximately metes. Find the length of the adius o diamete when given the othe measue. I emembe! The symbol means is appoximately equal to. Rose measues the adius of he bicycle wheel to be 14 inches. She odes a new tie fo the wheel by finding its cicumfeence. Howeve, she calculates its value using 3 fo p instead of What should she expect when he tie aives? 4.5 m Cicle Fomulas can be found on p. 47. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 1 If d 5 8, then 5. If 5 15, then d 5. Find the cicumfeence of the cicle. Use 3.14 fo p. Show you wok ft d 35 in. 7 m 145
12 POWER UP Aea of a Cicle The cicumfeence of a cicle is elated to the aea of a cicle. b This octagon is divided into 8 equivalent tiangles. Its shape is close to, but not exactly, a cicle. If cut apat, the 8 tiangles can be aanged as a paallelogam. If half of the fist tiangle is cut off and moved to the end, the paallelogam becomes a ectangle. The bases of fou tiangles, o about half the cicumfeence of the cicle, make up the length of the ectangle. We can appoximate the length: l 1 (p) 5 p. The width of the ectangle is equal to the height, h, of the tiangle. The height, h, is about the same as the adius,, of the cicle, so we can appoximate the width: w. The fomula fo the aea of a ectangle is A 5 lw. If we substitute p fo l and fo w, the esult is the fomula fo the aea of a cicle. A 5 p 3, o p. I see! The aea of a cicle is p, o a little moe than thee times the adius squaed! h l w Wods to Know aea of a cicle the total numbe of squae units that fit inside the cicle; The fomula fo the aea of a cicle is A 5 p. A Juanita calculates the cicumfeence and aea fo a cicle with adius cm. She finds they ae both equal to 4p. She asks how the distance aound a cicle can be the same as the aea inside a cicle. What would you tell he? You can use the adius to appoximate the aea of a cicle. Find the aea of the cicle. 1 Wite the fomula fo the aea of a cicle. Plug in 3.14 fo p and the length of the adius in the fomula. 3 Multiply. Wite the appoximate aea. A 5 A 3 A 3 3 A in. The aea is appoximately squae inches. 5 in. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 146 Lesson 15
13 B You can use diamete to find the appoximate aea of a cicle. Find the aea of the cicle. 1 Find the adius of the cicle. 15 Solving Poblems with Cicles Always check whethe the poblem o diagam lists the diamete o the adius. Wite the fomula fo the aea of the cicle. 3 Plug in 3.14 fo p and the length of the adius in the fomula. 4 Multiply. Wite the appoximate aea. d 8 cm The adius is half the diamete, so 5 cm. A 5 A 3 A 3 3 A cm The aea is appoximately centimetes. squae Rob made an eo when he solved fo the aea of the cicle shown. What can you tell him about his wok? What is the coect answe? A 5 p 3 3 Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC PRACTICE A d 6 cm A cm Find the appoximate aea of the cicle. Use 3.14 fo. Show you wok. 1 1 mi 7 mm 3 4 d 5 m d 13 ft Cicle Fomulas can be found on p
14 READY TO GO Solving Poblems with Cicles You can find the cicumfeence and aea of a cicle using fomulas. Find the cicumfeence and aea of a cicula building with diamete of 16 yads. Use the fomula fo the cicumfeence of a cicle. Substitute the value fo d. C 5 pd C 5 p 3 16 C C 50.4 yd The cicumfeence of the cicula building is appoximately 50.4 yads. I emembe! I can use 3.14 fo. And I use the symbol to find an appoximate value. Use the fomula fo the aea of a cicle. Substitute the value fo. A 5 p A A A A yd The aea of the cicula building is appoximately squae yads. I have to fist find the adius. Since the diamete is 16, the adius is Tina made an eo when solving fo the aea of the cicle shown. Identify he eo, and wite the coect answe. How fa off was he answe? d 11 mm A 5 p11 A A A mm lesson link Plug In POWER UP GO! The diamete o the adius can be used to find the cicumfeence of a cicle. d C 5 pd o p The adius can be used to find the aea of a cicle. 3 ft A I get it! I can use what I leaned about cicles to solve poblems involving cicumfeence and aea. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 148 Lesson 15
15 15 Solving Poblems with Cicles wok togethe You can estimate the cicumfeence and aea. Then check you estimate. The adius is 9 cm. The cicumfeence is a little moe than 3 times the diamete, o twice the adius. So the cicumfeence will be about , o 54 cm. I can estimate the cicumfeence and aea to see if my answes ae easonable. 1 cm The aea is a little moe than 3 times the adius squaed. So the aea will be about , o 43 cm. C 5 p A 5 p C 5 p9 A 5 p9 C A A You can use the cicumfeence of a cicle to solve poblems. C 56.5 cm A cm Find the length of fencing needed to enclose the playgound. Use 3.14 fo p. Cicle Fomulas can be found on p Estimate the cicumfeence. Use the fomula to find the cicumfeence of the playgound. 3 Wite the length of fencing needed. 50 yd Playgound Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC The diamete is little moe than yd. The cicumfeence is a times the diamete. So the cicumfeence will be about 3, o yd. C 5 C 3 C yd The length of fencing needed to enclose the playgound is appoximately yads. Look at the fomulas fo the cicumfeence and aea of a cicle. How is the fomula fo cicumfeence elated to the fomula fo aea? 149
16 Ready to Go pactice Use the fomulas to find the cicumfeence and aea of the cicle. Use 3.14 fo p. d 0 km REMEMBER means to multiply a numbe by itself. 1 C 5 pd A 5 p C 3 A 3 10 C km A 3 3 A km 11 mi C 5 p A 5 p C 3 3 A 3 C 3 A 3 3 C mi A mi Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 150 Lesson 15
17 15 Solving Poblems with Cicles Estimate fist. Then solve. Use 3.14 fo p. 3 Ali needs a pool cove fo he cicula pool. What is the cicumfeence of the pool? 1 ft 4 Tom is planning a cicula flowe gaden fo the yad. What aea of the yad will be planted with flowes? 6 ft Solve. Use 3.14 fo p. 5 A spinkle spays wate within a 5mete adius as it spins in a cicle. What is the appoximate aea of lawn that is wateed by the spinkle? Sketch a diagam to make sue you undestand what the poblem is descibing. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 6 A hockey puck is 7.6 centimetes in diamete. What is the cicumfeence of the puck? Make Sense of Poblems Kate keeps a cicula photogaph in a cicula pictue fame. The photogaph has a diamete of 7 cm. The fame has a diamete of 15 cm. Descibe the steps Kate can take to find the total distance aound the outside and the inside of the pictue fame. What is the total distance aound the outside and the inside of the pictue fame? Making a diagam can help you sot though the infomation given in a poblem! 151
18 Ready to Go Poblem solving READ PLAN DUCKS AMUCK A fountain, 3 ft in diamete, sits in the cente of a cicula duck pond. The pond measues 3 ft fom the fountain to its edge. What is the aea of the duck pond? What is the poblem asking you to find? the of the duck pond 3 f t 3 f t How can you solve this poblem? Find the between the aeas of the pond and the fountain. What fomula can you use? SOLVE 3 3 CHECK Find the diamete of the d ft pond and fountain. Find the adius of the pond and fountain. Find the aea of the pond and fountain. Find the adius of the fountain. Find the aea of the fountain. Subtact to find the aea of the duck pond. 5 d ft A 3 ft 5 d ft A 3 Estimate each solution to see if you answes ae easonable. The aea is a little moe than 3 times the adius squaed. The aea of the pond and fountain is about 3 3 The aea of the fountain is about 3 3 ft A 5 5 ft 5 ft. 5 ft. The aea of the pond is about 5 ft. The aea of the duck pond is appoximately ft. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 15 Lesson 15
19 15 Solving Poblems with Cicles pactice Use the poblemsolving steps to help you. Use 3.14 fo p. Be sue to include the units in you answe! 1 Bad is eplacing the topsoil in the community gaden. If the aea he is eplacing is in the shape of a half cicle with a diamete of 30 dm, how many squae decimetes of the gaden will he eplace? CHECKLIST Read Plan Solve Check The diamete of a basketball hoop is 18 inches. What is the lagest cicumfeence of a basketball that will fit though the hoop? Show you wok. CHECKLIST Read Plan Solve Check Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 3 A cicula window fame has a diamete of inches. The glass window inside the fame has a adius of 10 inches. What is the aea of the window fame not including the glass? CHECKLIST Read Plan Solve Check 153
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