Appendix A: Mathematics Unit 16
Delaware Mdel Unit Gallery Template This unit has been created as an exemplary mdel fr teachers in (re)design f curse curricula. An exemplary mdel unit has undergne a rigrus peer review and jurying prcess t ensure alignment t selected Delaware Cntent Standards. Unit Title: Cmparing Numbers Designed by: Michelle Hawley, Innvative Schls; adapted frm Cnnected Mathematics 3: Cmparing Bits and Pieces Cntent Area: Mathematics Grade Level(s): Grade 6 Summary f Unit In this unit, students will develp number sense arund fractins, decimals, ratis and rates. Using manipulatives and, later, visual mdels and symbls, students will wrk t develp flexibility when wrking with different representatins, and will develp skills fr slving real-wrld prblems as they becme fluent with using the representatins interchangeably. Of particular fcus in this unit are the cncepts f equivalence, fractins, and ratis. Stage 1 Desired Results What students will knw, d, and understand Delaware Cntent Standards CC.6.RP.1 Understand the cncept f a rati and use rati language t describe a rati relatinship between tw quantities. Fr example, The rati f wings t beaks in the bird huse at the z was 2:1, because fr every 2 wings there was 1 beak. Fr every vte candidate A received, Candidate C received nearly three vtes. CC.6.RP.2 Understand the cncept f a unit rate a/b assciated with a rati a:b with b nt 0, and use rate language in the cntext f a ratin relatinship. CC.6.RP.3 Use rati and rate reasning t slve real-wrld and mathematical prblems, e.g., by reasning abut tables f equivalent ratis, tape diagrams, duble number line diagrams, r equatins. CC.6.NS.4 Find the greatest cmmn factr f tw whle numbers less than r equal t 100 and the least cmmn multiple f tw whle numbers less than r equal t 12. Use the distributive prperty t express a sum t tw whle number 1-100 with a cmmn factr as a multiple f a sum f tw whle number with n cmmn factr. Fr example, express 36 + 8 as 4(9+2). CC.6.NS.6 Understand a ratinal number as a pint n the number line. Extend number line diagrams and crdinate axes familiar frm previus grades t represent pints n the line and in the plane with negative number crdinates. 1 17
MP.1 Make sense f prblems and persevere in slving them MP.2 Reasn abstractly and quantitatively MP.3 Cnstruct viable arguments and critique the reasning f thers MP.4 Mdel with mathematics MP.6 Attend t precisin MP.7 Lk fr and make use f structure MP.8 Lk fr and express regularity in repeated reasning Big Idea(s) Flexibility with understanding and cmparing values Multiple representatins can be used equivalently, and we shuld use the representatin that best helps t slve the given prblem While fractins and ratis may lk the same, they are nt necessarily Using mdels and pictures can help us understand and slve prblems Fractins, ratis and rates all help us slve prblems Unit Enduring Understanding(s) There are multiple ways t represent numbers. We can cnvert frm ne frm t anther in rder t slve prblems (r t slve prblems mre efficiently). Equivalence implies that tw representatins have the same value; they may r may nt lk similar. There are an infinite number f equivalent fractin and rati statements available. Scaling is used t find equivalent fractins and ratis; they are multiplicative, nt additive. Unit Essential Questins(s) Hw many ways can we cmpare numbers? Hw d rati statements help us understand and make cmparisns? Hw can fractin strips and ther mdels help us understand patterns fr finding equivalent fractins? Hw can fractin strips and ther mdels help us slve prblems? What is equivalence and what des it mean fr fractins and ratis? Knwledge and Skills Students will knw There are multiple ways t represent numbers There are infinite equivalent fractins pssible Vcabulary: Fractin Numeratr Denminatr 2 18
Equivalent fractins Number line Rati Rate Unit rate Students will be able t Mdel fractins, decimals, ratis and rates Make cmparisns between representatins Use number lines t help with making cmparisns Find equivalent fractins Demnstrate understanding f the Fundamental Law f Fractins Use ratis t cmpare quantities Distinguish between fractins representing numbers and ratis representing cmparisns Distinguish between a difference, which is an additive cmparisn, and a rati, which is a multiplicative cmparisn Apply a variety f scaling strategies t slve prblems using ratis and unit rates Stage 2 Assessment Evidence Evidence that will be cllected t determine whether r nt Desired Results are achieved Suggested Perfrmance/Transfer Task(s) Adapted frm: http://schls.nyc.gv/nr/rdnlyres/da04b2e8-94ce-4de4-b902- CEE331D651FB/0/NYCDOEG6MathRatiReasning_Final.pdf Mark was mixing blue paint and yellw paint in the rati f 2:3 t make green paint. He wants t make 45 liters f green paint. He began t make a table t help him think abut the prblem, but is unsure f what t d next. Liters f Blue Paint Liters f Yellw Paint Liters f Green Paint 2 3 3 19
a. Explain hw t cntinue t place values int the table. b. Write an explanatin in wrds t Mark abut hw he can use find ut hw many liters f blue paint and yellw paint he will need t make 45 liters f paint. c. What fractin f his mixture must be blue paint? Draw a mdel and explain hw yu knw this is crrect. Frmative Assessment Questins Hw many liters f green paint shuld there be in the first rw? Hw many liters f each clr wuld yu need t mix t make 10 liters f green paint? Will there be a time when yu ll add mre blue than yellw t mix the green paint? What happens if yu dn t fllw the rati when yu mix the paint? Differentiatin Opprtunities Interventin: Allw students t wrk in partners r in grups. Allw students t use manipulatives t mdel the prblem. Extensin: Have students create their wn rati prblem t share with thers. Rubric(s) 4 All f the fllwing must be present: Student recgnizes that the green paint is calculated by finding the ttal blue + yellw, and that additinal rws f data can be cmpleted using prprtinal reasning. (Student des nt need t refer t it as prprtinal reasning. ) Student prvides a strng explanatin abut filling in the table values. Student finds that Mark will need 18 liters f blue and 27 liters f yellw paint t create 45 liters f green paint. Student prvides a strng explanatin fr hw t determine the number f liters f blue and yellw paint. Student crrectly identifies that blue is 2/5 f the ttal mixture. Student prvides a sund explanatin fr their reasning. 3 Based upn the cmpnents f a 4, 1-2 f the explanatins may be inadequate r basic. Mathematical respnses must be accurate. Student recgnizes that the green paint is calculated by finding the ttal blue + yellw, and that additinal rws f data can be cmpleted using prprtinal reasning. (Student des nt need t refer t it as prprtinal reasning. ) Student prvides a strng explanatin abut filling in the table 4 20
values. Student finds that Mark will need 18 liters f blue and 27 liters f yellw paint t create 45 liters f green paint. Student prvides a strng explanatin fr hw t determine the number f liters f blue and yellw paint. Student crrectly identifies that blue is 2/5 f the ttal mixture. Student prvides a sund explanatin fr their reasning. 2 Student incrrectly respnds t ne f the three mathematical prtins, thse crrect mathematical respnses have reasnable explanatins. Student recgnizes that the green paint is calculated by finding the ttal blue + yellw, and that additinal rws f data can be cmpleted using prprtinal reasning. (Student des nt need t refer t it as prprtinal reasning. ) Student prvides a strng explanatin abut filling in the table values. Student finds that Mark will need 18 liters f blue and 27 liters f yellw paint t create 45 liters f green paint. Student prvides a strng explanatin fr hw t determine the number f liters f blue and yellw paint. Student crrectly identifies that blue is 2/5 f the ttal mixture. Student prvides a sund explanatin fr their reasning. 1 Student might answer ne f the mathematical parts crrectly, and is able t prvide an explanatin. Or Student prvides reasnable explanatins but des nt give any mathematical respnses. 0 Nne f the parts is answered crrectly Other Evidence Frmative Assessments: Daily warm ups Observatins Class wrk Unit prblems Math jurnals ACE questins (hmewrk assignments) Exit Tickets Class discussins Grup discussins Prtflis Summative Assessments: Quizzes and tests Perfrmance Task 5 21
Student Self-Assessment and Reflectin Math jurnals Self-crrectins Class and grup discussins Stage 3 Learning Plan (Design learning activities t align with Stage 1 and Stage 2 expectatins) Key learning events needed t achieve unit gals Lessn 1.1: Fundraising Fcus Questin: What are tw ways t cmpare a $500 fundraising gal t a $200 fundraising gal? Launch the lessn by activating prir knwledge abut students experiences with fundraising, and ideas abut cmparing numbers. Discuss the idea f a fundraising gal t make sure a cmmn language is develped. Tell the stry f the fundraising campaign given in Prblem 1.1. Make sure student understand their rle in the prblem: their gal is t evaluate each given claim, and be prepared t discuss whether each is true r nt. Allw students time t wrk cllabratively n parts A and B, circulating t grups. Use frmative assessment questins t make sure grups have sund strategies fr evaluating the claims. Lk fr grups wh have nvel appraches t evaluatin. When finished, discuss each claim, and have a few grups share their wn cmparisns. Have grups wrk n part C. Use frmative assessment questins t make sure grups have sund strategies fr answering the questin. Summarize the lessn. The gal is t help students make sense f multiplicative cmparisns. Students shuld realize that cmparing by subtracting gives different results than using part:whle fractins r ratis. Begin discussing the cncept f fr every t help lay the grundwrk fr ratis in future lessns, cmparing these types f statements t fractin statements. During the summary, encurage students t self-reflect upn their wrk and make ntes/changes in their ntes. Lessn 1.2: Fundraising Thermmeters Fcus Questin: Hw des a fr every statement shw a rati cmparisn? Launch the lessn by watching the animatin vide prvided with the text. Students shuld ntice that, while the fundraising gal thermmeters lk the same, the ttal amunt is different, implying that the fractinal parts are each wrth different amunts. Ensure that students understand the thermmeter mdel. Have students wrk individually, then with partners, t answer part A. Use frmative assessment questins t make sure grups have sund strategies fr determining the values at the unlabeled marks. Review these as a class. 6 22
Have grups wrk n parts B-D. Use frmative assessment questins t make sure grups have sund strategies fr answering the questins. Summarize the lessn. The primary gals are tw get students t develp strategies fr partitining, and t use partitining t generate/evaluate rati statements using the phrase fr every. Be sure t include ppular strategies in the discussin (e.g., the ne-tenth and ne-half strategies and hw they relate t each ther), and begin t lay the fundatin fr equivalence. As students review ratis, be sure t discuss cnventins (the first number in the rati describes the first quantity) as well. During the summary, encurage students t self-reflect upn their wrk and make ntes/changes in their ntes. Lessn 1.3: On the Line (tw days) Fcus Questin: When yu fld fractin strips, what relatinship d yu see emerge that shw hw the numeratr and denminatr change t make equivalent fractins? Launch the lessn by cnnecting t the 6 th grade fundraising gal prgress. The intrductin t this prblem shuld peak students interest t the new challenge. Make sure students understand that tdays prblem invlved develping a mdel fr measuring the fundraising prgress ver time. Discuss the term equivalent fractins. Allw partners t wrk n parts A and B. Use frmative assessment questins t make sure grups have sund strategies fr flding their paper and answering the questins. Discuss partners strategies fr flding the fractins. Spend time with thse in part B, as this thinking will help develp the cncept f equivalent fractins. Summarize fr parts A and B. Begin building the cncept f equivalent fractins by lking at cmmnalities amng the fractin strips, and the cncept f fractins as values n a number line. During the summary, encurage students t self-reflect upn their wrk and make ntes/changes in their ntes. Have grups wrk n parts C-E, making sure that each student creates his r her wn individual number line in the prcess. Use frmative assessment questins t make sure grups have sund strategies fr answering the questins. If time permits, have students create a clthesline number line as a class. Summarize the lessn. The fcus shuld be n explicit strategies fr finding equivalent fractins, as well as using number lines as a way t interpret fractinal values. During the summary, encurage students t self-reflect upn their wrk and make ntes/changes in their ntes. Lessn 1.4: Measuring Prgress Fcus Questin: Hw can fractin strips help yu t find part f a number? Launch the prblem by revisiting the fractin strip mdel. Students will be able t use these t measure prgress in this prblem. Make sure students understand they are t fcus n determining daily prgress fr just the 6 th graders. 7 23
Allw students t wrk in grups t answer questins A-E. Use frmative assessment questins t make sure grups have sund strategies fr answering the questins. Summarize the lessn. Students shuld be cmfrtable with part:whle relatinships and using partitining t represent these relatinships. The summary shuld als fcus upn the cncept f equivalence, especially n the relatinships rather than the quantities at hand. Ensure that students are able t cmmunicate the meanings f the numeratr and denminatr in a fractin, and are able t cmpare fractins with cmmn denminatrs. During the summary, encurage students t self-reflect upn their wrk and make ntes/changes in their ntes. Check fr understanding: If the sixth graders and seventh graders each cllected $250 ut f their gals f $300 and $450, what fractins f each f their gals have they cllected? Lessn 1.5: Cmparing Fundraising Gals Fcus Questin: What des it mean fr tw fractins t be equivalent? What des it mean fr tw ratis t be equivalent? Launch the lessn with the vide prvided with the text. Ensure that students understand the examples in the vide. Intrduce the Day 10 fundraising thermmeters, cmparing and cntrasting t previus representatins. Review the cln ntatin if necessary. Have students wrk in grups t cmplete parts A-E. Use frmative assessment questins t make sure grups have sund strategies fr answering the questins. If grups are struggling, it culd be useful t guide them tward creating a new set f fractin strips fr these thermmeters. Summarize the lessn. Fcus n strategies used t make sense f differentsized thermmeters, and hw part:whle relatinships are useful when lking at the mney raised by each grup. Reinfrce the beginnings f prprtinal reasning. During the summary, encurage students t selfreflect upn their wrk and make ntes/changes in their ntes. Have students cmplete the fur mathematical reflectin questins in their math jurnals. Resurces and Teaching Tips Student text Number lines, fractins strips, physical manipulatives Equivalent Fractins nline manipulatives: http://illuminatins.nctm.rg/activity.aspx?id=3510 Equivalent Fractins game: http://www.funbrain.cm/fract/ Khan Academy Fractins: https://www.khanacademy.rg/math/arithmetic/fractins/equivalent_fractin s/v/equivalent-fractins Khan Academy Ratis: https://www.khanacademy.rg/math/pre- algebra/rates-and-ratis/ratis_and_prprtins/v/intrductin-t-ratis-- new-hd-versin 8 24
Equivalent Ratis game: http://www.mathplaygrund.cm/asb_ratiblaster.html Differentiatin Students will wrk in hetergeneus cllabrative grupings t prvide additinal supprt Manipulatives will be available fr students t use as needed Hmewrk assignments will be differentiated based upn student needs Scafflded questining strategies will be used t elicit student thinking and cnstructin f knwledge Additinal practice prblems may be available fr students wh need additinal expsure t the cncepts Extensin prblems will be available fr students wh demnstrate understanding Design Principles fr Unit Develpment At least ne f the design principles belw is embedded within unit design. Internatinal Educatin - the ability t appreciate the richness f ur wn cultural heritage and that f ther cultures in t prvide crss-cultural cmmunicative cmpetence. Universal Design fr Learning - the ability t prvide multiple means f representatin, expressin and engagement t give learners varius ways t acquire and demnstrate knwledge. 21 st Century Learning the ability f t use skills, resurces, & tls t meet the demands f the glbal cmmunity and tmrrw s wrkplace. (1) Inquire, think critically, and gain knwledge, (2) Draw cnclusins make infrmed decisins, apply knwledge t new situatins, and create new knwledge, (3) Share knwledge and participate ethically and prductively as members f ur demcratic sciety, (4) Pursue persnal and aesthetic grwth.(aasl,2007) Universal Design fr Learning: Students will have pprtunities t build their understanding thrugh different ways f representing the relatinships in the prblems. Students will als cnvey their learning thrugh manipulatives, written and ral respnses, and will have pprtunities t cmplete extensin prblems. 21 st Century Learning: Student will use critical thinking skills while develping understanding f equivalence with fractins and ratis. They will wrk t develp their wn thinking, while questining and prviding feedback t thers. Students will wrk cllabratively t build and apply this understanding t multiple real-wrld prblems. 9 25
Technlgy Integratin The ability t respnsibly use apprpriate technlgy t cmmunicate, slve prblems, and access, manage, integrate, evaluate, and create infrmatin Students will learn hw t use calculatrs t slve prblems invlving equivalence Additinal nline resurces fr supprt: Equivalent Fractins nline manipulatives: http://illuminatins.nctm.rg/activity.aspx?id=3510 Equivalent Fractins game: http://www.funbrain.cm/fract/ Equivalent Ratis game: http://www.mathplaygrund.cm/asb_ratiblaster.html Cntent Cnnectins Cntent Standards integrated within instructinal strategies ELA cnnectin: Students will be speaking, listening and writing abut mathematics n a daily basis. Students will be expected t prvide respnses, defend their respnses, and have discussins abut mathematics. These learning experiences will als be reflected in their written respnses t prblems (including explanatins fr why their slutins wrk), as well as their math jurnal entries daily. 10 26