Pearson's onramp to Algebra

Transcription

1 4 Repreenting Proportial Relatihip GOAL To ue proportial relatihip repreented in a table, a ormula, and a graph. Looking Bak In thi unit, tudent have worked with ratio table (Le 4 and ), unit ratio (Le 3), and the tant o proportiality (Le ). Student have alo had prior experiene with plotting imple graph to how pair o value (Unit, Le 6). Voabulary From Previou Le A ratio i a relatihip in whih or every x unit o e quantity there are y unit o another quantity. Unit ratio are ratio written a ome number to. A rate i a ratio involving two quantitie meaured in dierent unit. A unit prie i a unit rate that give the prie o e item. In a proportial relatihip, e quantity y i a tant multiple o the other quantity x. The tant multiple i alled the tant o proportiality. The tant o proportiality i equal to the ratio x. y Mathemati Learning Progrei Mathemati o the Le Key Point Student ue the tant o proportiality to write a ormula indiating the proportial relatihip between two quantitie. Key Point Student repreent proportial relatihip with ratio table and graph. Student plot the value o quantitie whoe ratio are alway the ame (e.g., ot to the number o an item, eet to inhe, irumerene to diameter o a irle). Comm Core State Standard Looking Ahead The ability to reognize and repreent proportial relatihip i eential or writing and graphing linear equati. Ctent Standard 7.RP.2 Reognize and repreent proportial relatihip between quantitie. 7.RP.2.b Identiy the tant o proportiality (unit rate) in table, graph, equati, diagram, and verbal deripti o proportial relatihip. Mathematial Pratie MP Make ene o problem and perevere in olving them. MP2 Rea abtratly and quantitatively. MP3 Ctrut viable argument and ritique the reaing o other. Pear' Ramp to Algebra LESSON 4 REPRESENTING PROPORTIONAL RELATIONSHIPS 93 ALGS3_TE_U4_L4.indd 93

2 4 Opening Launh Clik PLAY the irt reen o the Preentati Sreen etext or an engaging animati to tart the le. Thi animati how a enario about the ot o tomatoe per pound, whih relate to both the text and the ept o the le. Suggeted Time: about minute SCREEN A br a Diplay SCREEN A. Tell the tudent that Lia knew lb o tomatoe ot $.8. Thi i a ratio table he made. Ak tudent to reall the term unit prie, unit ratio, and tant o proportiality. Cot o Tomatoe Al Cot (dollar) $.8 per pound..8 :. Unit ratio o ot : weight am Ctant o proportiality or the relatihip o ot.8 and weight o tomatoe i. UNIT 4, LESSON 4 SCREEN A ALGS3_SCR_U4_L4.indd 2/6/2 7:48 AM ' INSTRUCTION R QQ What do you notie about the unit prie, the unit ratio, and the tant o proportiality? [All three o thee are very imilar and tain.8 or.8, whih are equivalent. The unit prie and the unit ratio are both ratio ompared to, but the unit prie tain unit and the unit ratio doe not. The tant o proportiality i not a ratio.] DIFFERENTIATED Unit prie o tomatoe i p QQ What i the tant o proportiality or thi relatihip? Say how you know. [.8; beaue w =.8 in every olumn o the ratio table.] Weight (pound) w to QQ What i the unit ratio o ot to weight o tomatoe? Say how you know. [.8 ; beaue a unit ratio expree a ratio a ome number to. In thi ae, the ratio w i alway equivalent to.8, regardle o whih olumn you look at.] ge QQ What i the unit prie or tomatoe? Say how you know. [$.8 per pound; beaue the unit prie tell the prie per eah unit (pound in thi ae).] ar Englih Language Learner Deine eah o the term and have tudent relate eah deiniti to the text o buying tomatoe. SCREEN B Pe Diplay SCREEN B. Ak tudent to provide the miing value or and 2 pound o tomatoe. Add inormati to the table a tudent anwer the ollowing queti. QQ How would you ue the tant o proportiality to ind the ot o lb o tomatoe? [Multiply.8 by.] Weight (pound) w Cot (dollar) QQ What i the ot o lb o tomatoe? 2 lb? [$8; $6] QQ How would you ind the ot o w lb o tomatoe? [Multiply the tant o proportiality (.8) by the number o pound o tomatoe:.8w.] w 4..8 Formula: = QQ How an you tate the ormula or the ot o tomatoe uing variable? [ =.8w] UNIT 4, LESSON 4 94 ~.8.8 ~ 2.8w w SCREEN B RATIO AND PROPORTIONALITY ALGS3_SCR_U4_L4.indd 2 ALGS3_TE_U4_L4.indd 94 9/6/2 : PM

3 Opening tinued Saolding or Sue DIFFERENTIATED INSTRUCTION You an ue the Online Math Tool: Data & Graph to review the eential part o a graph, uing the data in Example. Have tudent upply appropriate name or the horiztal and vertial axe and or the graph. Step Supply the olumn label and put the data in the table. Then pre Selet Graph. Step 2 Step 3 Step 4 p Ak tudent whether the graph how a y proportial relatihip. [Ye, beaue the ratio x i tant.] Pre the Line butt and plae the line over the point. Pe ar ' R am The graph how the point determined by the pair o data in the table. Add a title. to Al ge br a Selet Satter Plot. Put in x-max and y-max value. Then pre OK. LESSON 4 ALGS3_TE_U4_L4.indd 9 REPRESENTING PROPORTIONAL RELATIONSHIPS 9

4 Student Page or Reerene 4 Repreenting Proportial Relatihip GOAL To ue proportial relatihip repreented in a table, a ormula, and a graph. When the ratio o two quantitie i tant, the quantitie are proportial. You an repreent a proportial relatihip a a ratio table. Example Lia knew pound (lb) o tomatoe ot $.8. She made thi table. Weight (lb) w Cot ($) The unit ratio o ot to weight i.8 :. The unit prie i $.8 per pound. To ind the tant ratio o ot to weight, divide any value o by it orrepding value w. w =.8 =.6 2 = = = 4. The tant o proportiality i.8 or.8. Example 2 Suppoe you want to know the ot o any number o pound o tomatoe w? * w Weight (lb) w w Cot ($) w * w The ratio table how that the ot i.8w, whih i the produt o the tant o proportiality and the weight w. LESSON 4 REPRESENTING PROPORTIONAL RELATIONSHIPS When Time I Short Aign 4 and 7 8. In thi Work Time, tudent will ee the number a a multiplier in a table, a the tant o proportiality, a a number in a ormula, and a the growth between point the vertial axi o a graph. Through thi experiene, they net the multiple repreentati or a ituati. A you irulate between pair or group o tudent, plan whih tudent you wih to have preent during the Cloing and look or the ollowing to diu A problem that i giving many tudent trouble. An unuual method or olving a problem. A eptual miundertanding that need to be diued.. Work Time Number o Flag Number o Star 2 Example 3 You an ue the tant o proportiality to write a ormula to how the proportial relatihip between ot and weight. Thi ormula an then be ued to olve or. tant o proportiality =.8w total ot o tomatoe weight o tomatoe What i the ot o 2 lb o tomatoe? =.8w =.8 2 z Subtitute 2 or w. = 6 z Simpliy. The ot o 2 lb o tomatoe i $6. Example 4 You an alo ue the ormula =.8w to olve or w. How many pound o tomatoe an you buy or $? =.8w COMMENT =.8w z Subtitute or. Invere Equati w =.8 For ab = Ex: 4 3 = 2 z Rewrite a an invere equati. b w = 62. z Simpliy. = a 2 3 = 4 a = b 2 For $, you an buy 62. lb o tomatoe. 4 = 3 You an how the relatihip o ot and weight o tomatoe a graph. The graph below how the data rom the table in Example. Cot o Tomatoe w Weight (lb) 2 Cot ($) Student Editi (page 3) Work Time RATIO AND PROPORTIONALITY Dwayne aunt work or a ompany that manuature and ell lag. Dwayne aunt prepare all o the tar. Sometime an order i or a large quantity o lag. Dwayne aunt want to determine a rule o he know how many tar to prepare.. Dwayne aunt begin by tarting a ratio table howing the unit ratio or the number o tar to the number o lag. Find three more orrepding pair o value or the two quantitie (number o lag and number o tar) that he ould inlude in thi table. Complete the table. Pear' Ramp to Algebra Number o Flag Number o Star Coah Student may hooe other value or the number o lag than thoe hown above. I you ee an error, ak tudent to ind the ratio o tar to lag or eah pair o data. 96 RATIO AND PROPORTIONALITY ALGS3_TE_U4_L4.indd 96

5 Dwayne aunt want to determine a rule o he know how many tar to prepare.. Dwayne aunt begin by tarting a ratio table howing the unit ratio or the number o tar to the number o lag. Find three more orrepding pair o value or the two quantitie (number o lag and number o tar) that he ould inlude in thi table. Complete the table. Work Time tinued Number o Flag number o tar 2. ; by inding number o lag Student and 4) 2. Ue your ratio table to ind the tant o proportiality in the relatihip between the two quantitie. Explain how you ound it. Coah Student may write the tant o proportiality a or. Both are orret, but or number o tar to number o lag will be more ueul. For tudent who anwer, ak i they an write another tant o proportiality that relate the number o tar to the number o lag. 3. For lag, the number o tar i Number o(page Star 3 Editi 3. Ue the ratio table to alulate the number o tar Dwayne aunt will need in order to make lag. 4. Let tand or the number o lag and tand or the number o tar. Ue the tant o proportiality to write a ormula that repreent the relatihip between and ( = ). or.. Ue your ormula to ind the number o tar Dwayne aunt need in order to make 26 U.S. lag. 6. Dwayne ound the number o lag hi aunt ould make with 3 tar. I Dwayne orret? Why or why not? Coah Student ometime have trouble undertanding that the number o tar an be repreented by uing the variable or the number o lag. Uing omething in your laroom, like boxe o paper lip, may make the ituati more tatile or tudent. Suppoe e box ha paper lip. Then two boxe ha. I there are b number o boxe, then the total number o paper lip i time b, or b. 3 = = 3 br a = 7 My aunt an make 7 lag. Work Time tinued HINT Think about what interval are appropriate between tik mark the vertial axi. ge 7. Sketh the graph o the relatihip between the number o tar and the number o lag. 4. = Al Wrap It Up REPRESENTING PROPORTIONAL RELATIONSHIPS LESSON 4 to p R 9. Look at your anwer or Problem. You ould have olved thi problem by uing a ratio table. Would you ue a graph? Say why or why not. 4 ALGS3_SE_U4_L4.indd 3 Number o Star.,3 tar; i there are 26 lag, then = 26. Uing the ormula, = 26 =,3. Dwayne aunt would need,3 tar. 7. am Coah To get tudent to think about the ormula they have written, ak them how to deribe it in their own word. Ak what operati i hidden inide the exprei. I they have written the ormula a = or =, ak them to ue their ormula to ind the number o tar in lag and probe the reaablene o their anwer. 8. Explain the role o the tant o proportiality in the ormula that you ound in Problem tar to tripe the urrent U.S. lag 3. tar thenumber Fort Sumter Flag to tar the urrent o Flag U.S. lag Coah graphflag: o the relatihip between the For thethe Fort Sumter number oa ormula tar and the number lagohould beo 4. Write or the number o taro in term the number lag (in the and orm o = ). aled. Enourage tudent appropriately evenly to label the aaxe oorthat Number o Flag. a. Write ormula the number o tripe in termio the the number horiztal axi. Thi i obeaue o lag (in the orm P = ). they have written the ormula o that quantity othe tar depend up the o b. Sketh a graph that how relatihip between the number and the lag. numbertripe o lag, number =.oat thi point, it i aeptable or tudent totant eitheronet thewhen point or not. I 6. What i the proportiality omparing the number o tripearie, the Fort Flag toto thedeend number otheir lag? hoie. dierene aksumter tudent ' ar Coah I a tudent i having trouble, reer him or her to Example 4. Pe The lag at the right i the Fort Sumter Flag. Write eah o the ollowing ratio a implet whole number ratio. 6. No; 7 lag an be made rom 3 tar. Dwayne did not impliy 3 orretly. 8. Anwer will vary. Sample anwer: The tant o proportiality in the ormula i a tant multiplier. To ind the number o tar needed or any number o lag, you multiply by. 9. Anwer will vary. Sample anwer: Problem would not be olved uing a hand-drawn graph tarting at the origin, beaue the number o lag, 26, i beyd a ale eaily ued the horiztal axi. (Student may want to ue a graph i they were uing a graphing alulator or omputer.) /6/2 2:9 PM Online Math Tool, Data & Graph Online Cept Book, ee page 3 39 Homework 2. tar to tripe the Fort Sumter Flag Coah Reer tudent having trouble to Example 3. Wrap It Up 3 3 Work Skill Time tinued Calulate exprei. between the HINT Think about what interval 7. Sketh the the value graphooeah the relatihip appropriate between tik mark and the Student Editi 4)number o lag 8.. (- 78.4), 2 are 7. number 6.66 (-o 2) tar, 2(page (-the 3) vertial axi. 9. (- 2) 3,. (- ), (- 3) (- 6) Wrap It Up 8. Explain the role o the tant o proportiality in the ormula that you ound in Problem 4. 4 RATIO AND PROPORTIONALITY 9. Look at your anwer or Problem. You ould have olved thi problem by uing a ratio table. Would you ue a graph? Say why or why not. ALGS3_SE_U4_L4.indd 4 Homework /6/2 2:9 PM Online Math Tool, Data & Graph Online Cept Book, ee page 3 39 The lag at the right i the Fort Sumter Flag. Write eah o the ollowing ratio a implet whole number ratio.. tar to tripe the Fort Sumter Flag 2. tar to tripe the urrent U.S. lag LESSON 4 REPRESENTING PROPORTIONAL RELATIONSHIPS 3. tar the Fort Sumter Flag to tar the urrent U.S. lag 97 For the Fort Sumter Flag: 4. Write a ormula or the number o tar in term o the number o lag (in the orm o = ). ALGS3_TE_U4_L4.indd 97. a. Write a ormula or the number o tripe in term o the number o lag (in the orm o P = ).

6 Cloing In the Cloing, it i important that tudent link all the dierent repreentati together and that they undertand how they are related to the undamental idea o tant o proportiality. Problem 2, 4, 7, and 8 are partiularly important or making thee neti. Summary Queti QQ How i a proportial relatihip repreented in a table? [The ratio o orrepding value i equivalent.] in a ormula? [The tant o proportiality relate the two varying quantitie.] in a graph? [For any point (a, b), the ratio b a i the ame.] Have tudent put their anwer in text, like the tomato problem and the lag problem. Colidating Cept A ept map will help tudent make the neti between a deripti, a table, a ormula, and a graph. Work with a group o tudent to develop the ollowing a a poter. Pot and diu with all tudent. Deripti There are tar in a lag. The unit ratio i. The unit rate i tar per lag. The tant o proportiality i. Formula number o tar = tant o proportiality number o lag Table Graph Number o Flag Number o Star 2 Pear' Ramp to Algebra Number o Star Number o Flag DIFFERENTIATED INSTRUCTION 98 RATIO AND PROPORTIONALITY ALGS3_TE_U4_L4.indd 98

7 Homework. The ratio o tar to tripe the Fort Sumter lag i The ratio o tar to tripe the urrent U.S. lag i The ratio o tar the Fort Sumter lag to tar the urrent U.S. lag i = 33. a. P = 3 b. S 6. 3 Skill Student do omputati involving both divii and multipliati with poitive and negative number. Extra Pratie page 2 22 Name Cla Date 4-4 Extra Pratie Repreenting Proportial Relatihip Dwayne aunt work or a ompany that manuature and ell lag. Dwayne aunt i preparing all o the tripe or an order o a large quantiy o lag. She want to determine a rule o he know how many tripe to prepare. She think about the relatihip between the number o tripe (S) he will need in order to make a ertain number o U.S. lag ().. Name the quantitie that are proportial to eah other. the number o tripe and the number o lag 2. Write the unit ratio or the number o tripe to the number o lag. (The U.S. lag ha 3 tripe.) 3 : 3. Expre your anwer to Problem 2 a a rate. 3 tripe per lag 4. What i the tant o proportiality in thi relatihip? 3 Number o Stripe Number o Flag Student Editi (page 4) Homework The lag at the right i the Fort Sumter Flag. Write eah o the ollowing ratio a implet whole number ratio.. tar to tripe the Fort Sumter Flag 2. tar to tripe the urrent U.S. lag 3. tar the Fort Sumter Flag to tar the urrent U.S. lag For the Fort Sumter Flag: 4. Write a ormula or the number o tar in term o the number o lag (in the orm o = ).. a. Write a ormula or the number o tripe in term o the number o lag (in the orm o P = ). b. Sketh a graph that how the relatihip between the number o tripe and the number o lag. 6. What i the tant o proportiality when omparing the number o tripe the Fort Sumter Flag to the number o lag? Skill Calulate the value o eah exprei (-2), 2 8. (-78.4), 2 (-3) 9. (-2) 3,. (-), (-3) (-6) Name Cla Date Extra Pratie tinued 6. Sketh a graph o the number o tripe and the number o lag. Put the number o tripe the vertial axi and mark the number o lag rom to the horiztal axi. Number o Stripe S Number o Flag Pear' Ramp to Algebra 7. I Dwayne aunt need to make lag, how many tripe doe he need? Show how you alulated your anwer. 6 tripe; S = 3 = 3 ~ = 6 8. How many lag an Dwayne aunt make i he ha prepared 8 tripe? Show how you alulated your anwer. 4 lag; 8 = 3, o = 8 3 = 4 Online Math Tool, Data & Graph Online Cept Book, ee page Write a ormula or the number o tripe S in term o the number o lag. S = 3 LESSON 4 REPRESENTING PROPORTIONAL RELATIONSHIPS 99 ALGS3_TE_U4_L4.indd 99

8 Pear' Ramp to Algebra RATIO AND PROPORTIONALITY ALGS3_TE_U4_L4.indd