YACK IN THE BOX. Getting Ready. The Activity On Their Own (Part 1) What You ll Need. Overview

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1 YACK IN THE BOX Factions Equivalence Witin equations Gettin Rea What You ll Nee Cuisenaie Ros, set e ai Small shoe boxes o tissue boxes -centimete i ae, ae 20 Activit Maste, ae 02 Ovevie Stuents use a combination of to Cuisenaie Ros to fom a lone o sement. Assumin that this o eesents one hole unit, stuents fom aition an subtaction sentences involvin the factional lenths of the othe os. In this activit, stuents have the ootunit to: investiate factional ats of the hole ceate smbolic equations fin factional sums an iffeences eesent factions ith equivalent exessions Othe Sue Souce activities that exloe these an elate concets ae: Plaoun Equiment, ae 34 Foein Cuenc Exchane, ae 39 The Activit On Thei On (Pat ) Chen is ceatin a ne ame calle Yac in the Box, an he nees assistance in finin the values of the lain ieces. Ho can he etemine the factional values of his ame ieces? Wo ith a atne. A ac is a ne o hose lenth consists of ello Cuisenaie Ro an blac Cuisenaie Ro attache en to en. Usin Cuisenaie Ros, mae all one-colo combinations that ill match the lenth of the ac. Assume that the lenth of the ac eesents one hole unit. Fo each of the one-colo combinations, fin the factional at of a sinle o in elation to the hole ac. Reco the colo of each o an its factional value. No fin the factional values of each of the emainin os ith the ac eesentin hole unit. A ou finins to the ata alea collecte an aane ou ata in inceasin oe of value. Loo fo attens an elationshis in the ata. Be ea to exlain ou finins. ETA/Cuisenaie YACK IN THE BOX Numbe Gaes

2 Thinin an Shain Invite stuents to shae an exlain thei finins. Ceate a chat namin each o colo an its factional at of the hole. Use omts lie these to omote class iscussion: Which one-colo o combinations matche the lenth of the ac? Ho i ou etemine the factional at that each of these colo os eesente in elation to the hole? Ho i ou etemine the factional values of the emainin os? Ho i ou etemine the oe of the factions? What attens o elationshis i ou iscove? On Thei On (Pat 2) What if... Chen is ea fo ou to la a ame of Yac in the Box? Can ou etemine factional elationshis amon the os to hel ou in the ame? Wo in ais. Place a set of Cuisenaie Ros in a small box. Decie ho ill o fist. Plae A anoml selects 3 os fom the box. On a sheet of ae, Plae A ites to aition sentences about the 3 os. One equation shoul elate the colos of the os an the othe equation shoul elate thei factional values. Factional values ae to be exesse in tems of acs. Fo examle, + + = e; /6 ac + /4 ac + /3 ac = 3 /4 ac. Then Plae A selects 2 of the 3 os, an ites to subtaction sentences about them. One equation shoul elate the colos of the os an the othe equation shoul elate thei factional values. As befoe, factional values shoul be exesse in tems of acs. Fo examle, = ; /3 ac /4 ac = /2 ac. Plae B checs Plae A s equation sheet. If all statements ae coect, Plae A eans oint. If a mistae(s) is foun, Plae B can mae the coection(s), an then he o she eceives the oint. Afte etunin the 3 os to the box, Plae B selects 3 os fom the box, an he o she eeats the activit. Pla continues b altenatel ain os, itin the sets of equations, an checin esults. The fist lae to ean 8 oints is the inne. Be ea to iscuss the esults of ou ame. 30 Numbe Gaes 7 8 ETA/Cuisenaie

3 Thinin an Shain Invite ais to iscuss the esults of thei ame. Encouae stuents to justif thei equations an coections. Use omts lie these to omote class iscussion: Wee some aition statements hae to eneate than othes? Wee some easie? Which ones an h? Ho i ou select hich to os to use in the subtaction activit? What as ifficult about itin the subtaction equations? What as eas? What mistaes, if an, o ou thin ou coul have avoie? Wite a lette to Chen exlainin the ocesses use hen ain factions an subtactin factions. Inclue an exlanation about the sills ou use hile lain Yac in the Box. Teache Tal Whee s the Mathematics? If ou combine the ello an the blac Cuisenaie Ros, ou ill et a ne o calle a ac, hose lenth equals the sum of the lenths of its to comonents. When stuents seach fo onecolo o combinations that ill match the lenth of the ac, the soon iscove that onl the hite, e, liht een, ule, an a een os ill o. B countin ho man of each o ae neee to mae one-colo acs, stuents can etemine the factional at of the hole eesente b each o. { ac } 2 = ac = 2 ac 6 = ac = 6 ac 4 = ac = 4 ac 3 = ac = 3 ac 2 = ac = 2 ac ETA/Cuisenaie YACK IN THE BOX Numbe Gaes 7 8 3

4 To etemine hat factional at of the hole the emainin ello, blac, bon, blue, an oane os ae, stuents nee to etemine that the factional value of the hite o is 2 of the ac o. A iven numbe of these hite os can be attache en to en to match the lenth of each of the othe os. B matchin the lenths, stuents can fin the folloin infomation: ello = 5 2 ac blac = 7 2 ac n e bon = 8 2 o 2 3 ac blue = 9 2 o 3 4 ac o oane = 0 2 o 5 6 ac B comain the bon o to the ule o, stuents ma notice that 2 ule os ae neee to equal the lenth of the bon o. In othe os, the bon o also eesents 2( 3), o 2 3 ac. Comain the blue o to the liht een o, stuents notice that 3 liht een os ae neee to equal the lenth of the blue o. Thus, the blue o also eesents 3( 4), o 3 4 ac. Lieise, the oane o is the same lenth as 5 e os an must also eesent the faction 5( 6), o 5 6 ac. Befoe the stat to la Yac in the Box, in Pat 2, stuents miht fin it convenient to a a iaam simila to the one belo, in hich inceasin lenths ae matche u ith thei factional values. o e n 2 2 o o o o o o o 3 4 ac o Plain Yac in the Box ives stuents the ootunit to o ith concete moels of factions an then tanslate these moels into smbolic equations. The oblems eneate in the ame ovie actice in ain an subtactin factions. The esults of these aithmetic oblems ma involve oe factions, hole numbes, imoe factions, o mixe numbes. Usin os to visualize the subtaction oeation, in hich the shote o is taen aa fom the lone o, ill hel stuents avoi the oblem of subtactin the lae faction fom the smalle faction an eneatin a neative numbe. 32 Numbe Gaes 7 8 ETA/Cuisenaie

5 Man stuents ill lace the thee os an fom the box en to en an comae thei total lenth to each of the os in the ac famil to fin one that matches. If a o is foun, its factional value eesents the sum of the thee os. If the colos of the thee os ae the same, stuents can fin thei sum b ain thei numeatos an eein the same enominato. Fo examle, hen thee e os ae ae toethe, the esultin sum of becomes 3 6. If the colos of the thee os ae not the same, hoeve, the 3 os must be exchane fo equivalent lenth os all usin the same colo. Fo examle, hen e, liht een, an ule os ae an fom the box, the e o can be thouht of as 2 hites ( 2 2), the liht een o as 3 hites ( 3 2), an the ule o as 4 hites ( 4 2). Once the colos ae the same, stuents can a the factions to et 9 hites ( 9 2). Some stuents ma then choose to ite the equations usin the factional name fo each o exesse in loest tems, as shon belo = = 9 2 o 3 4 Othe stuents ho ae oin ith the same e, liht een, an ule os ma continue to seach until the fin the minimum numbe of one-colo os that ill match the lenth of these thee os. The sum can then be itten in tems of liht een os = = 9 2 Man iffeent numbe sentences o equations can be itten to exess the esults of ain the e, liht een, an ule os toethe. Talin about ho these sentences iffe, but ae eall equivalent, ill hel stuents see the ole of equivalent factions in fomin sums. Stuents ma also eneate iffeent equations b usin the Commutative Poet of Aition to eaane the oe of the factions bein ae. Once stuents have selecte the to os to be subtacte, the can visuall eesent thei iffeence b lain the to os sie b sie. In the case hee the to os ae ientical, thei iffeence ill equal zeo. If the to os ae of iffeent lenths, the istance fom the en of the shote o to the en of the lone o eesents the iffeence beteen the os. The examle belo shos the oblem of subtactin the liht een o fom the ule o an the aoiate smbolic an numeic equations. o = iffeence smbolic 3 ac 4 ac = 4 2 ac 3 2 ac = 2 ac numeic 3 o o Usin lettes to ite equations base on the colos of Cuisenaie Ros seves as an intouction to late o ith vaiables an alebaic equations. ETA/Cuisenaie YACK IN THE BOX Numbe Gaes