1 Teaching the Lesson

Transcription

1 Gtting Startd Mathmatical Practics SMP1, SMP3, SMP4, SMP5, SMP6, SMP7, SMP8 Contnt Standards 5.NBT.4 Mntal Math and Rflxs Us your stablishd slat procdurs to dictat problms such as th following: Writ Circl th digit in th tnths plac. Put an X through th digit in th thousandths plac. 3.4O82X Writ Circl th digit in th hundrdths plac. Put an X through th digit in th tns plac. 91X7.20O3 Writ 6,000,387,420. Circl th digit in th millions plac. Put an X through th digit in th tn thousands plac. 6,000O,38X7,420 Math Mssag Solv th problm at th top of journal pag 360. Study Link 10 7 Follow-Up Hav studnts shar thir mystry graphs and th situations thy portray. Sid Primtr 10 cm 40 cm 1 Taching th Lsson 15 cm 60 cm 20 in. 80 in. 5 mi 20 mi Math Mssag Follow-Up (Math Journal 2, p. 360) ELL Sid Primtr _ Primtr Sid 10 cm 40 cm 40_ 10 = 4 15 cm 60 cm 60_ 15 = 4 20 in. 80 in. 80_ 20 = 4 5 mi 20 mi 20_ 5 = 4 A Problm from th National Assssmnt Th following problm was in th mathmatics sction of a 1975 national standardizd tst. A squar has a primtr of 12 inchs. What is th ara of th squar? 9 1. Your answr: in 2. Th tabl blow givs th national rsults for this problm. Algbraic Thinking Ask studnts to shar thir solutions. 9 squar inchs Hav studnts compar th prcnt of studnts in th class with th corrct answr with th rsults from th 1975 National Assssmnt of Educational Progrss. Th two most coon incorrct rsponss ar 144 squar inchs and 48 squar inchs. Ask studnts to brainstorm in small groups and thn complt Problms 2 and 3 on th journal pag. Whn studnts hav finishd, ask voluntrs to shar thir rsponss with th class. Ongoing Assssmnt: Rcognizing Studnt Achivmnt Journal Pag 360 Us journal pag 360 to assss studnts undrstandings of ara and primtr. Studnts ar making adquat progrss if thir xplanations dmonstrat accurat dscriptions of ara and primtr. [Masurmnt and Rfrnc Frams Goal 2] Answrs 13-Yar-Olds 17-Yar-Olds Young Adults Corrct answr 7% 28% 27% 144 sq inchs 12% 19% 25% 48 sq inchs 20% 10% 10% 24 sq inchs 6% 4% 2% 12 sq inchs 4% 3% 3% 6 sq inchs 4% 2% 1% 3 sq inchs 3% 2% 2% Othr incorrct answrs 16% 13% 10% No answr or I don t know 28% 19% 20% Explain why many studnts might hav givn th following answrs. Sampl answrs: squar inchs Studnts thought ach sid of th squar masurd 12 inchs and multiplid 12 º 12 to find th ara. Studnts thought ach sid of th squar masurd 12 inchs. Thy confusd ara with primtr and addd to find th ara squar inchs Math Journal 2, p. 360 Rviw with studnts that th primtr of a closd 2-dimnsional figur is th distanc around th figur. To support English languag larnrs, us an xampl and gsturs as you discuss th dfinition. Draw a squar on th board, labl its sid with a masurmnt, and ask for its primtr. Rpat for srvral squars. Collct th data in a tabl on th board. (S margin.) Explain that th rlationship btwn th primtr of a squar and th masur of a sid can b xprssd as a ratio. Add a third column labld _ Primtr to th tabl on th board. Ask Sid studnts to calculat th primtr-to-sid ratio for ach squar. (S margin.) 826 Unit 10 Using Data; Algbra Concpts and Skills

2 This kind of comparison is calld a ratio comparison. Ask studnts why th ratio will always b 4. A squar consists of 4 sids of qual lngth. Thrfor, th primtr of a squar is 4 tims th lngth of on of its sids. Links to th Futur Studnts had xprinc with ratio comparisons in Grads 2 4. Ths xprincs will b continud in Unit 12 and will b xtndd in Grad 6. Rviwing th Manings of Diamtr and Radius (Studnt Rfrnc Book Glossary) ELL DISCUSSION Explain that this lsson is an xploration of a similar ratio for circls that compars th of th circl to its circumfrnc. Ask studnts to find th trm circumfrnc in th Studnt Rfrnc Book Glossary. St your board compass for a 6-inch radius and draw a circl on th board. Draw a, and labl it 1 foot. Ask studnts to find th trms and radius in th Studnt Rfrnc Book Glossary. Draw a radius in th circl on th board, and ask for its lngth. 6 inchs To support English languag larnrs, draw a circl and labl it with th trms circumfrnc,, and radius. Introducing th Circumfrnc Invstigation PROBLEM SOLVING DISCUSSION Ask studnts to stimat th circumfrnc of a 1-foot circl. Us th circl on th board and a 1-foot lngth of string to guid studnts to mak th following comparisons: Is th circumfrnc longr than th? Ys, that s clar from th figur. Is th circumfrnc longr than twic th? Ys, th top half of th circl is longr than 1, so th bottom half must also b longr than 1. Draw a squar with 1-foot sids around th circl. Ask: Which path is longr, th on around th squar or th on around th circl? Th path around th squar, which has a primtr of 4 ft, or 4 s, is longr. Ask studnts to giv a rang for th masur of th circumfrnc of th 1-foot circl. Th circumfrnc is mor than 2 s and lss than 4 s; th circumfrnc is about 3 tims th. Lsson

3 radius Ratio of Circumfrnc to Diamtr You ar going to xplor th rlationship btwn th circumfrnc and th of a circl. Answrs vary. 1. Using a mtric tap masur, carfully masur th circumfrnc and of a varity of round objcts. Masur to th narst millimtr (on-tnth of a cntimtr). 2. Rcord your data in th first thr columns of th tabl blow. 3. In th fourth column, writ th ratio of th circumfrnc to th as a fraction. 4. In th fifth column, writ th ratio as a dcimal. Us your calculator to comput th dcimal, and round your answr to two dcimal placs. i C r c u m f r n c Invstigating Circumfrnc (Math Journal 2, p. 361) PARTNER Discuss th journal pag. Studnts work in pairs or groups of thr. Each group should hav at last as many round objcts to masur as thr ar mmbrs in th group. Caution studnts that, to masur a circumfrnc, thy should pull th tap or string straight around th objct, not at an angl. Objct Ratio of Circumfrnc to Diamtr Circumfrnc Diamtr as a Fraction as a Dcimal (C) (d) C (from calculator) d Coff cup What is th mdian of th circumfrnc-to- ratios in th last column? 6. Th studnts in your class combind thir rsults in a stm-and-laf plot. Us that plot to find th class mdian valu for th ratio C d. Math Journal 2, p. 361 Corrct Incorrct Ratio of Circumfrnc to Diamtr Stms (1s and 0.1s) Lavs (0.01s) Corrct Incorrct Masuring th of ral objcts is difficult bcaus th cntr is usually not markd. Blow ar two mthods studnts can us: 1. Position th tap or string so it crosss th widst part of th objct. Th is th longst lin sgmnt, or chord, conncting two points on th circumfrnc. 2. Us a right-angld cornr of a book or pic of papr to find a. Position th right angl on th circl. Th points whr th sids of th angl intrsct th circl ar th ndpoints of a. (S margin.) Hav studnts complt th tabl on th journal pag. Answrs should b roundd to hundrdths. Making a Stm-and-Laf Plot (Math Journal 2, p. 361) Algbraic Thinking Ask studnts to complt Problm 5 on th journal pag. Copy th incomplt stm-and-laf plot onto th board or a transparncy. (S margin.) As studnts finish thir calculations, ach studnt finds th mdian C_ ratio for th objcts masurd. A d partnr or group voluntr ntrs this C_ ratio on th plot. d Whn studnts hav finishd, rwrit th numbrs in ach laf in ordr from last to gratst so finding th mdian will b asir. Working togthr, find th class mdian ratio. Expct it to b gratr than 3 and probably within 0.1 of π (3.14). Studnts rcord th class mdian on Problm 6 of th journal pag. Stms (1s and 0.1s) Lavs (0.01s) Discussing Pi (Studnt Rfrnc Book, p. 112) Rmind studnts of th _ primtr ratio for squars from th Math sid Mssag Follow-Up. In th circumfrnc invstigation, studnts find that th circumfrnc ratio also appars to b constant, with a valu of about 3.1 or 3.2. Th xact valu of this ratio is an irrational numbr namd for a lttr of th Grk alphabt. 828 Unit 10 Using Data; Algbra Concpts and Skills

4 Ask whthr anyon knows th nam of this Grk lttr. Th lttr π, pronouncd pi Ask studnts to find th trms rational numbr and irrational numbr in th Studnt Rfrnc Book Glossary. Rad Studnt Rfrnc Book, pag 112 as a class. Mak sur studnts undrstand that th digits in pi (π), to th right of th dcimal point, continu forvr without any discrnibl pattrn. Thus, whn calculating with π to find th circumfrnc of a circl, w us an approximation of its xact valu. Thrfor, th rsulting circumfrnc is not an xact valu. W can us th symbol, maning approximatly qual to, to dnot this. Writ and labl th symbol on th board or Class Data Pad. Guid studnts to find and us th π ky on thir calculators as in th Chck Your Undrstanding Problms on Studnt Rfrnc Book, pag 112. Not that ntring th ky squnc π Æ 5 on th TI-15 will display 5, or 5π. On th Casio fx-55, it will display a dcimal rsult. To s th answr as a dcimal on th TI-15, prss F D. Rmind studnts to round an appropriat numbr of dcimal placs. Convrting Clsius to Fahrnhit In th U.S. customary systm, tmpratur is masurd in dgrs Fahrnhit ( F). Th mtric systm masurs tmpratur in dgrs Clsius ( C). Watr frzs at 0 C, or 32 F. 1. What tmpratur is shown on th thrmomtr at th right? 18 C, or 0 F Th following formula convrts tmpraturs from dgrs Clsius to dgrs Fahrnhit, whr F stands for th numbr of dgrs Fahrnhit and C stands for th numbr of dgrs Clsius: Formula: F (1.8 º C ) 32 A rul of thumb givs a rough stimat of th convrsion. Rul of thumb: Doubl th numbr of dgrs Clsius and add th Fahrnhit frzing tmpratur. F (2 º C) Convrt Clsius tmpraturs to Fahrnhit using th formula and th rul of thumb. Compar th rsults. C F (formula) F (rul of thumb) Ar th rsults from using th rul of thumb accurat in most situations? Explain. Sampl answr: Ys. A diffrnc of about 6 dgrs dos not mattr in non-scintific situations. 4. Nam on situation whr you would us th formula rathr than th rul of thumb to convrt to dgrs Fahrnhit. Explain. Sampl answr: If you wr taking your body tmpratur, you would want as accurat a rading as possibl. 5. Normal body tmpratur is 37 C or F. 6. Watr boils at 100 C or 212 F Math Journal 2, p. 362 Dgrs Clsius ( C) Ç10 Ç20 Ç30 Dgrs Fahrnhit ( F) Ç10 Ç20 2 Ongoing Larning & Practic Convrting Tmpraturs (Math Journal 2, p. 362) Studnts larn a formula and a rul of thumb for convrting btwn Fahrnhit and Clsius dgrs. Math Boxs 10 8 (Math Journal 2, p. 363) Mixd Practic Math Boxs in this lsson ar paird with Math Boxs in Lsson Th skill in Problm 3 prviws Unit 11 contnt. Study Link 10 8 (Math Mastrs, p. 312) NOTE For practic with unit convrsions of mtric and customary units of lngth, s Math Boxs 1. Blow ar th bowling scors from th Pick s family runion bowling party a. Mak a stm-and-laf plot for th data. Idntify th landmarks. b. What is th maximum scor? 168 c. What is th mod for th scors? d. What is th mdian scor? Stms Lavs (100s and 10s) (1s) Hom Connction Studnts practic solving problms using th circumfrnc formula. Thy find th circumfrnc or th. Rmind studnts to bring xampls of ratios that thy can add to th Rats and Ratios Musum. 2. Solv a. b c d , f Complt th following quivalnts a. 1 cup ouncs b. 1 pint ouncs c. 1 quart ouncs d. 4 quarts ouncs. 1 gallon ouncs Math Journal 2, p. 363 Lsson

5 Study Link Mastr Nam STUDY LINK 10 8 Finding Circumfrncs Th formula for th circumfrnc of a circl is: Circumfrnc π º, or C π º d Diffrntiation Options Us th π ky on your calculator to solv ths problms. If your calculator dosn t hav a π ky, ntr 3.14 ach tim you nd π. Find th circumfrnc of ach circl blow. Show answrs to th narst tnth. 1. a. b. 7" " 6.4 cm Circumfrnc inchs Circumfrnc cntimtrs 2. Th whls on Will s bicycl hav a of about 27 inchs, including th tir. a. What is th circumfrnc of th tir? About 85 inchs b. About how far will Will s bicycl travl if th whls go around xactly onc? About 85 inchs 3. Sofia masurd th circumfrnc of hr bicycl tir. Sh found it was 66 inchs. What is th of th tir? About 21 inchs 27" 40.2 READINESS Rading about Circumfrnc SMALL-GROUP 5 15 Min Litratur Link To xplor th rlationship btwn circumfrnc and, rad and discuss th following book with th group. Highlight th masurmnt tchniqus and th mathmatical concpts. Th Librarian Who Masurd th Earth Suary: Th Grk mathmatician Eratosthns (c B.C.E.) calculatd th circumfrnc of Earth to within 200 mils of th modrn valu. This book rcounts how Eratosthns mad his calculation and provids multipl opportunitis for discussion. Math Mastrs, p. 312 Nam Can Your Body Pass through an Indx Card? 1. Fold a 5-inch-by-8-inch indx card in th middl, along th 5-inch width of th card. 2. Hold th halvs of th foldd card togthr. Cut th card as shown by th lins in Diagram A. Som cuts start at th fold and go almost to th dg of th card. Othr cuts start at th dg and go almost to th fold. B sur th first and last cuts start at th fold. Cuts should altrnat btwn starting at th fold and starting at th dg. 3. Opn th card. Cut along th fold from X to Y as shown in Diagram B. B carful not to cut to th dgs of th card. X Y Diagram A X Y Diagram B 4. Pull th card apart carfully. You ll hav a papr ring. Is it larg nough for your body to pass through? Try This Taching Mastr Us anothr 5-inch-by-8-inch indx card. Can you cut out a ring that has a primtr twic th primtr of th ring you just mad? Explain how you would do it. Sampl answr: Mak twic as many cuts. ENRICHMENT PARTNER Passing Your Body through an Indx Card (Math Mastrs, p. 313) Min To furthr xplor studnts undrstanding of circumfrnc, hav thm rconfigur a rctangular pic of papr into a circl. Explain to studnts that thy ar going to figur out a way to pass thir ntir body through a 5-inch-by-8-inch pic of papr. Th only tool thy may us is a pair of scissors. Hav partnrs com up with a plan and implmnt it. Expct that most studnts will nd to follow th dirctions on Math Mastrs, pag 313. Th circumfrnc of th ring is approximatly qual to th sum of th lngths of all th cuts that ar mad. ELL SUPPORT SMALL-GROUP Building a Math Word Bank (Diffrntiation Handbook, p. 143) 5 15 Min To provid languag support for ratios, hav studnts us th Word Bank Tmplat found on Diffrntiation Handbook, pag 143. Ask studnts to writ th trms rat, ratio, and ratio comparison; to draw picturs rlating to ths trms; and to writ othr rlatd words. S th Diffrntiation Handbook for mor information. Planning Ahad Stor th cylindrical objcts you collctd for us again in Lsson Math Mastrs, p Unit 10 Using Data; Algbra Concpts and Skills