Area Formulas with Applications

Transcription

1 Formulas wit Applications Ojective To review and use formulas for perimeter, circumference, and area. epresentations etoolkit Algoritms Practice EM Facts Worksop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacer s Lesson Guide Teacing te Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Convert etween fractions, decimals, and percents. [Numer and Numeration Goal 5] Measure lengts to te nearest millimeter. [Measurement and Reference Frames Goal ] Use formulas to calculate te circumference of circles, as well as perimeter and area of figures. [Measurement and Reference Frames Goal ] Apply perimeter, circumference, and area formulas to solve for missing lengts and areas. [Measurement and Reference Frames Goal ] Key Activities Students use formulas to calculate area. Tey apply formulas for perimeter, circumference, and area to solve for missing lengts. Ongoing Assessment: Recognizing Student Acievement Use journal page 35. [Measurement and Reference Frames Goal ] Materials Mat Journal, pp Student Reference Book, pp. 5 8 and 377 Study Link 9 7 metric ruler calculator Using Nets to Find Surface Mat Journal, pp. 37A and 37B Students use drawings of nets to find te surface areas of 3-dimensional solids. Mat Boxes Mat Journal, p. 38 straigtedge Students practice and maintain skills troug Mat Box prolems. Study Link Mat Masters, p. 306 Students practice and maintain skills troug Study Link activities. Solving Perimeter Prolems Sir Cumference and te First Round Tale Students apply perimeter formulas to solve a prolem from a story. Using te Distriutive Property to Find Dimensions Mat Masters, pp. 306A and 306B Students use teir knowledge of te distriutive property to find te dimensions of ojects wit given areas. EXTRA PRACTICE Exploring s of Parallelograms and Triangles Mat Masters, pp scissors transparent tape Students derive area formulas for parallelograms and triangles. Informally Relating Circumference and Students give an informal derivation of te relationsip etween te circumference and area of a circle. ELL SUPPORT Relating Figures and Formulas Students illustrate relationsips etween geometric sapes and formulas. Advance Preparation For te optional Enricment activity in Part 3, otain a copy of te ook Sir Cumference and te First Round Tale y Cindy Neuscwander and Wayne Geean (Carlesridge, 00). Teacer s Reference Manual, Grades 6 pp. 9,, 88 Unit 9 More aout Variales, Formulas, and Graps

2 Getting Started Matematical Practices SMP, SMP, SMP, SMP5, SMP6, SMP8 Content Standards 6.RP.3a, 6.NS., 6.EE., 6.EE.c, 6.EE.9, 6.G., 6.G. Mental Mat and Reflexes Students convert from exponential notation to standard notation witout using a calculator. Suggestions: , , Mat Message Complete te prolems on journal page 35. Study Link 9 7 Follow-Up Discuss ow te spreadseet in Prolem was created. Ask: Wat formula is stored in cell B? = π A Wat formula is stored in cell C5? = π A5 A5 Compare te two graps in Prolem. Te first grap sows tat as te radius increases, te circumference also increases at a uniform rate. Te second grap sows tat area increases at a faster rate tan te radius. Use te second grap to solve Prolem 3. Find te point on te grap tat as a y-value of 3 on te vertical axis. From tat point, move straigt down to find te x-value on te orizontal axis. Te radius is etween _ and 3_ feet. Teacing te Lesson Mat Message Follow-Up WHOLE-CLASS DISCUSSION (Mat Journal, p. 35; Student Reference Book, pp. 5 8) Algeraic Tinking Review students strategies and solutions. Sow ow tey can draw te eigts in Prolems 5 and 6 outside te figures. Remind students tat tey must draw te eigts at rigt angles to te ases as sown elow. parallelogram trapezoid 3 cm If necessary, use Student Reference Book, pages 5 8 to review ow to calculate te areas of rectangles, parallelograms, and triangles. Ongoing Assessment: Recognizing Student Acievement Journal Page 35 Use journal page 35 to assess students ailities to apply area formulas. Students are making adequate progress if tey can calculate te areas in Prolems. Some students may e ale to apply te definition of eigt to draw a reasonaly correct eigt, measure it, and ten use it to calculate te area in Prolem. [Measurement and Reference Frames Goal ] Formulas Calculate te area of eac figure elow. A summary of useful area formulas appears on page 377 of te Student Reference Book. Measure dimensions to te nearest tent of a centimeter. Record te dimensions next to eac figure. You migt need to draw and measure or line segments on a figure. Round your answers to te nearest square centimeter... formula A = πr A = _ formula 0 cm cm circle formula formula rectangle parallelogram formula triangle trapezoid triangle A = l w A = _ 0 cm 6 cm Sample answer for A = A = 3 + _ formula: ( - 3) formula 3 cm 3 cm Mat Journal, p Try Tis 3 cm 35 Lesson 89

3 Perimeter, Circumference, and Solve eac prolem. Explain your answers.. Rectangle PERK as a perimeter of 0 feet. Lengt of side PE ft of rectangle PERK 96 ft Sample answer: P = l + w A = lw 0 = l + 6 A = 8 = l A = 96 = l. Te area of triangle BAC is 300 meters. Wat is te lengt of side AB? Lengt of side AB 0 m Sample answer: A = _ 300 = ( _ ) = 5 0 = 3. Te area of parallelogram LMNK is 7 square inces. Te lengt of side LX is 6 inces, and te lengt of side KY is 3 inces. Wat is te lengt of LY? Lengt of LY 8 in. Sample answer: A = 7 = (6 + 3) 7 = 9 8 = Mat Journal, p. 36 Adjusting te Activity Use a grapic organizer to demonstrate B C 5 m K L Y P K X N ELL te relationsips etween figures and area formulas. Follow te format and notation used in te Student Reference Book, page M E 8 ft R A Using Perimeter, Circumference, and Formulas (Mat Journal, pp. 36 and 37) PROBLEM SOLVING Algeraic Tinking Assign te prolems on journal pages 36 and 37. Te numers are relatively easy in all prolems, so expect tat some students may use trial-and-error strategies. Wen most students ave finised, ring te class togeter to sare solutions. Many students are likely to quote or write te relevant formula, sustitute te given values in te formula, and ten solve for te missing lengt or area. Students may or may not reduce te prolem to an equation. For example, in Prolem 3: Student A: Te area of te parallelogram is ase times eigt. Te ase is 6 + 3, or 9 inces. Because 9 times 8 is 7, te eigt must e 8 inces. Student B: I used te formula A =. I know te area is 7 square inces, and I figured out te ase is 9 inces. If I sustitute tese numers in te formula, I get 7 = 9. So, te eigt must e 8 inces. In Prolem 6, some students migt figure out tat te pats are te same lengt witout actually calculating te circumferences. Te circumference of te large circle is 0π, and te circumference of eac smaller circle is 0π, or 0π togeter. AUDITORY KINESTHETIC TACTILE VISUAL Perimeter, Circumference, and cont.. Te area of triangle ACE is square yards. Wat is te area of rectangle BCDE? of rectangle BCDE 0 yd Sample answer: A = _ = _ 7 8 = 7 = For Prolems 5 and 6, use 3. for π. 5. To te nearest percent, aout wat percent of te area of te square is covered y te area of te circle? Answer 0% 6. Wic pat is longer: once around te figure 8 from A to B to C to B and ack to A or once around te large circle? C B 3 yd A 7 yd A = A = (3 + 7) A = 0 Sample answer: Square: A = s A = 0 = 00 Circle: A = πr A = π(5( ) = / 00 = 0.965; ( )% = 9.65% A B C Tey are te same distance around. Sample answer: Large circle: C = πd; C = π(0 ft); C = 5.6 ft Smaller circle: C = πd; C = π 0 ft = 6.8 ft Figure 8: 6.8 ft = 5.6 ft 0 in. 0 in. 0 ft 0 ft 8 D E Surface Surface area is te total area of all te surfaces of a 3-dimensional oject. In Prolems and, a 3-dimensional solid is represented y a net. Use te net and te following formulas to answer te questions. of a rectangle: A = of a triangle: A = _ ( ) A is te area of te rectangle. A is te area of te triangle. is te lengt of its ase. is te lengt of its ase. is te eigt of te rectangle. is te eigt of te triangle.. Te net elow represents a rigt triangular prism. 6 in. in. in. in. a. Te area of eac triangular face is in... Te area of te larger rectangular face is in.. c. Te area of eac of te smaller rectangular faces is 5 in.. d. Te surface area of te triangular prism is 60 in.. 3 Mat Journal, p. 37 Mat Journal, p. 37A 830 Unit 9 More aout Variales, Formulas, and Graps

4 Ongoing Learning & Practice Surface continued. Te net elow represents a square pyramid. Te triangular faces are all congruent. Using Nets to Find Surface (Mat Journal, pp. 37A and 37B) Have students look at journal page 37A. Tell tem tat a net is a -dimensional representation of a 3-dimensional oject tat sows wat te oject would look like if it were unfolded. Explain tat since a net includes all of a 3-dimensional solid s faces, te area of a net is equal to te surface area of te 3-dimensional solid it represents. Because te nets are -dimensional polygons, students can decompose te polygons into triangles and rectangles to find te areas. Students use tis information to complete journal pages 37A and 37B. 6.3 cm 5.5 cm 6.3 cm 6.3 cm 6.3 cm a. Te area of eac triangular face is cm.. Te area of te square face is cm. c. Te surface area of te square pyramid is cm. 3. Roxanne cut out te sape elow from a piece of cardoard. Se will fold it to make a rigt rectangular prism. Se plans to decorate it y completely covering te outside wit paper. How muc paper does Roxanne need to decorate te prism? 0 in. 6 in. 8 in Mat Boxes (Mat Journal, p. 38) Roxanne needs 376 in. of paper to decorate te prism. Mat Journal, p. 37B Mixed Practice Mat Boxes in tis lesson are paired wit Mat Boxes in Lesson 9-6. Te skills in Prolems and 5 preview Unit 0 content. Study Link (Mat Masters, p. 306) Home Connection Students solve area prolems, some of wic involve variales and extraneous information. Study Link Master Name STUDY LINK Prolems Calculate te area of eac figure in Prolems 6. Rememer to include te unit in eac answer.. parallelogram 7 in. 6 in.. triangle 8 in.. rectangle ft 5. triangle 30 in. 3. parallelogram 0 cm 6. trapezoid cm in..5 ft 08 cm 9 cm 5 7 Mat Boxes. Use te distriutive property to write a numer model for te prolem. Ten solve. Niesa ougt 3 sets of screwdrivers at $9.9 eac and 3 oxes of screws at $.98 eac. Wat was te total cost of Niesa s purcase? Numer model x = 3( ) Solution $ Simplify eac expression y comining like terms. a. 3m + p + 6m + 9. k + 7n - 9n - k c. 3_ w - (- 5 ) + 5_ w d. -6x - 6y - 6x - 7y 9m + p + 9 -k - n w + _ 5 -x -3y 5 3 mm 7 mm 0 mm 0 ft.9 ft ft 30 m 0 m 30 m m 6 m 3. Seven out of 9 cards are faceup. If 56 cards are faceup, ow many cards are tere altogeter?. Draw te line(s) of symmetry for eac figure elow. 5.5 mm 55 ft 696 m 8 m Try Tis In Prolems 7 and 8, all dimensions are given as variales. Write a true statement in terms of te variales to express te area of eac figure. Example: a d c _ Practice c a a º c º d a (n + m) y x m 9. x 5.3 = x = = -3w w = 0. Mat Masters, p. 306 y n 7 cards 5. Identify weter te preimage () and image () are related y a translation, a reflection, or a rotation. Write your answer on te line elow. rotation Mat Journal, p y x 80 8 Lesson 83

5 Name Teacing Master Finding Dimensions Pete is considering uying a tent for a ackpacking trip. Te package says tat te ase of te enclosed part of te tent is rectangular and as an area of 5 square feet, and tat tere is a rectangular awning at te front of te tent tat covers an area of 0 square feet. From te picture, Pete can tell tat te main tent and awning ave te same widt. He draws tis picture to illustrate te information on te lael. Main tent 5 sq ft m Awning 0 sq ft Pete laeled is picture wit te variale m to represent te lengt of te main tent, te variale a to represent te lengt of te awning, and te variale w to represent te widt. He wants to know te value of eac of tese dimensions. Use wat you know aout area models and te distriutive property to elp you find te dimensions of te tent. Assume te dimensions are in wole numers of feet. Explain your tinking. Sample answer: From te area formula for a rectangle, I know tat te total area of te main tent and awning togeter is (m + a) w. From te picture, I know tat te total area is To write tis in te form (m + a) w, I can use te distriutive property. First, I need to find a common factor of 5 and 0. Te only common factor is = = (9 + ) 5. Tis sows tat m = 9, a =, and w = 5. Mat Masters, p. 306A a w 3 Differentiation Options SMALL-GROUP Solving Perimeter Prolems 5 30 Min Literature Link To furter explore circles, students read te ook Sir Cumference and te First Round Tale, y Cindy Neuscwander and Wayne Geean (Carlesridge, 00), a fantasy aout ow te knigts of Camelot decided to use a round tale. After reading te story, students calculate te various perimeters for te first tree configurations mentioned. Using previous configurations and applying teir knowledge of te scale factor ( cm : ft), students sould conclude tat te lengt etween two parallel sides of te exagon is 0 feet and tat te sides are aout feet long. Tey will need to estimate te circumference of te final circular tale. A good estimate is etween 6 and 7 ft, assuming Lady Di is aout 5 _ feet tall. Radius (r) Name Circumference (C) Teacing Master Finding Dimensions continued Use te distriutive property to solve te prolems. Assume all dimensions are in wole-numer units. On te line elow eac drawing, sow ow you used te distriutive property to find te dimensions.. Te We site for a otel says it as a pool wit a 9-square-meter deep end and a 63-square-meter sallow end. Te diagram elow sows te sape of te pool. Sallow end Deep end Sample answer: = 7 (9 + 7) a. How wide is te pool? 7 m. How long is te sallow end? 9 m c. How long is te deep end? 7 m. A diagram of te stage in a scool auditorium is sown elow. Te area of te part eind te curtain, or te main stage, is 70 square feet. Te area of te section in front of te curtain, or te apron, is 5 square feet. Main stage Apron Sample answer: = 7 (0 + 3) a. How far does te apron extend in front of te curtain? 3 ft. How far does te main stage extend eind te curtain? 0 ft c. How wide is te stage? 7 ft (A) A_ C 8 in cm ft m Using te Distriutive Property to Find Dimensions (Mat Masters, pp. 306A and 306B) 5 30 Min To furter explore area and area formulas, students use information aout te areas of rectangles and teir knowledge of te distriutive property to find te dimensions of rectangles. If students ave troule getting started, you migt ave tem write tis form of te distriutive property on teir papers: a c + c = (a + ) c. Remind tem ow tey used te distriutive property to factor sums in Lesson 9-. Informally Relating Circumference and 5 30 Min To explore te relationsip etween area and circumference of a circle, draw te tale in te margin on te oard wit only te Radius (r) column filled in. Students complete te oter columns and descrie te patterns tey oserve in te tale. Help students notice tat te ratio of a circle s area to its circumference is alf te measure of its radius. Round all values to te nearest wole numer. Mat Masters, p. 306B 83 Unit 9 More aout Variales, Formulas, and Graps

6 EXTRA PRACTICE Exploring s of Parallelograms and Triangles (Mat Masters, pp ) 5 30 Min To provide extra practice wit formulas, ave students transform parallelograms into rectangles to derive te area formula for parallelograms and comine two identical triangles to derive te area formula for triangles. 9 8 Teacing Master Name s of Parallelograms. Do not cut out te sapes on tis page. Instead cut out Parallelogram A on Mat Masters, page 309 and follow te directions tere. Parallelogram A Tape your rectangle in te space elow. ase cm lengt cm eigt 3 cm widt 3 cm of parallelogram cm of rectangle cm. Do te same wit Parallelogram B on Mat Masters, page 309. Parallelogram B Tape your rectangle in te space elow. Sample answer: ELL SUPPORT SMALL-GROUP Relating Figures and Formulas 5 5 Min To provide language support, use a grapic organizer suc as te one elow to present te area formulas in tis lesson. 6 ase cm lengt cm eigt 5 cm widt 5 cm of parallelogram 30 cm of rectangle 30 cm 3. Write a formula for te area of a parallelogram. A Mat Masters, p Sample answer: Square Rectangle Parallelogram s Triangles Formulas Circles d d ) r Name 9 8 Teacing Master s of Triangles. Do not cut out te triangle elow. Instead cut out Triangles C and D from Mat Masters, page 309 and follow te directions tere. Triangle C Tape your parallelogram in te space elow. D C ase cm lengt cm eigt cm widt cm of triangle 8 cm of parallelogram 6 cm. Do te same wit Triangles E and F. Triangle E 6 Tape your parallelogram in te space elow. ase cm lengt cm eigt cm widt cm of triangle 6 cm of parallelogram cm 3. Write a formula for te area of a triangle. A E 6 F Mat Masters, p. 308 Lesson 833

7 Name Finding Dimensions Pete is considering uying a tent for a ackpacking trip. Te package says tat te ase of te enclosed part of te tent is rectangular and as an area of 5 square feet, and tat tere is a rectangular awning at te front of te tent tat covers an area of 0 square feet. From te picture, Pete can tell tat te main tent and awning ave te same widt. He draws tis picture to illustrate te information on te lael. Main tent 5 sq ft Awning 0 sq ft w m a Pete laeled is picture wit te variale m to represent te lengt of te main tent, te variale a to represent te lengt of te awning, and te variale w to represent te widt. He wants to know te value of eac of tese dimensions. Use wat you know aout area models and te distriutive property to elp you find te dimensions of te tent. Assume te dimensions are in wole numers of feet. Explain your tinking. Copyrigt Wrigt Group/McGraw-Hill 306A

8 Name Finding Dimensions continued Use te distriutive property to solve te prolems. Assume all dimensions are in wole-numer units. On te line elow eac drawing, sow ow you used te distriutive property to find te dimensions.. Te We site for a otel says it as a pool wit a 9-square-meter deep end and a 63-square-meter sallow end. Te diagram elow sows te sape of te pool. Sallow end Deep end a. How wide is te pool?. How long is te sallow end? c. How long is te deep end?. A diagram of te stage in a scool auditorium is sown elow. Te area of te part eind te curtain, or te main stage, is 70 square feet. Te area of te section in front of te curtain, or te apron, is 5 square feet. Main stage Apron a. How far does te apron extend in front of te curtain? Copyrigt Wrigt Group/McGraw-Hill. How far does te main stage extend eind te curtain? c. How wide is te stage? 306B