Mathematics Instructional Cycle Guide

Transcription

1 Mathematics Instructinal Cycle Guide Cncept (7.RP.2) Rsemary Burdick, 2014 Cnnecticut Dream Team teacher 0

2 CT CORE STANDARDS This Instructinal Cycle Guide relates t the fllwing Standards fr Mathematical Cntent in the CT Cre Standards fr Mathematics: Rati and Prprtin 7.RP.1 Cmpute unit rates assciated with ratis f fractins, including ratis f lengths, areas and ther quantities measured in like r different units. 7.RP.2 Recgnize and represent prprtinal relatinships between quantities, fractinal quantities, by testing fr equivalent ratis in a table r graphing n a crdinate plane. This Instructinal Cycle Guide als relates t the fllwing Standards fr Mathematical Practice in the CT Cre Standards fr Mathematics: Insert the relevant Standard(s) fr Mathematical Practice here. MP.4: Mdel with mathematics MP.7: Lk fr and make use f structure. MP.8: Lk fr and express regularity in repeated reasning. WHAT IS INCLUDED IN THIS DOCUMENT? A Mathematical Checkpint t elicit evidence f student understanding and identify student understandings and misunderstandings (p. 21) A student respnse guide with examples f student wrk t supprt the analysis and interpretatin f student wrk n the Mathematical Checkpint (p.3) A fllw-up lessn plan designed t use the evidence frm the student wrk and address the student understandings and misunderstandings revealed (p.7)) Supprting lessn materials (p.20-25) Precursry research and review f standard 7.RP.1 / 7.RP.2 and assessment items that illustrate the standard (p. 26) HOW TO USE THIS DOCUMENT 1) Befre the lessn, administer the (Which Cylinder?) Mathematical Checkpint individually t students t elicit evidence f student understanding. 2) Analyze and interpret the student wrk using the Student Respnse Guide 3) Use the next steps r fllw-up lessn plan t supprt planning and implementatin f instructin t address student understandings and misunderstandings revealed by the Mathematical Checkpint 4) Make instructinal decisins based n the checks fr understanding embedded in the fllw-up lessn plan MATERIALS REQUIRED Large pster paper fr students t wrk n and then hang arund the rm. Needed fr students t cmplete their Gallery Walk If pssible, a smart bard TIME NEEDED Insert Checkpint Name: Which Cntainer? Insert time needed: 20 minutes Fllw-Up Lessn Plan: Insert time needed: 2-3 Class perids Timings are nly apprximate. Exact timings will depend n the length f the instructinal blck and needs f the students in the class. 1

3 Questin(s) (See attached dcument n page 18) Tw liquid strage cntainers f the same size are being filled. Liquid enters cntainer A at a rate f 2/3 galln per 1/4 minute. Step 1: Elicit evidence f student understanding Mathematical Checkpint Liquid purs int cntainer B at a rate f 3/5 gallns per 1/6 minute. Cntainer A: Cntainer B: CT Cre Standard: Target questin addressed by this checkpint: Purpse 7.RP.1 Cmpute unit rates assciated with ratis f fractins, including ratis f lengths, areas and ther quantities measured in like r different units. 7.RP.2 Recgnize and represent prprtinal relatinships between quantities. (Fractinal quantities) by testing fr equivalent ratis in a table r graphing n a crdinate plane. Hw d students apprach a Prprtinal cntext invlving Fractins, when the unit rate is given as a cmplex fractin? T what extent d they? Draw a mdel that crrectly represents the divisin f time and the amunt f cntainer filled. Cnnect their mdel t the situatin and hw it helped t determine the slutin. Use the mdel t make cnnectins t creating a table f equivalent ratis and t writing an equatin. Slve the equatin crrectly. Determine which cntainer is being filled faster. Justify yur answer. Supprt yur reasning by using evidence frm the mdels abve. 2

4 Step 2: Analyze and Interpret Student Wrk Student Respnse Guide Gt It Develping Getting Started What will a respnse include frm a student wh has demnstrated cnceptual understanding and mastery? What will a respnse include frm a student wh demnstrated sme understanding and pssibly sme misunderstandings r undevelped understanding? What will a respnse include frm a student wh demnstrated minimal understanding and pssibly miscnceptins?. 3

5 Student Respnse Example What will a respnse include frm a student wh demnstrated minimal understanding and pssibly miscnceptins? Getting Started Indicatrs What pssible indicatrs may be included in a student respnse wh has demnstrated minimal understanding f the standard? Lack f understanding between the relatinship f the time and gallns flwing int the tw tanks. Lack f a rati table mdeling the relatinship between time and gallns. Lack f understanding a unit rate and hw that is cnnected t this situatin. The student calculated 1 1 and f a minute int secnds 4 6 incrrectly. What strategies, and representatins will r will nt be used? What understandings r prcedural fluency des the student respnse reveal? N rati table t shw the prgressin f time and gallns in each cntainer. The student tried t calculate the unit rate by using percentages. He shwed 3% f the tank filled per secnd. That cannt be determined, as the amunt f water each tank hlds was nt given. What undevelped understandings, miscnceptins, and cmmn mistakes may be revealed in the student respnse t this item? Student made n cnnect between the amunt f time and number f gallns flwing int the tanks. Student culd nt recgnize a unit rate. Student did nt make the cnnectin between tw equal representatins f a rati table. There was nt clear understanding that a cmparisn had t be made cmparing the rates t select which cntainer wuld be filler faster. 4

6 In the Mment Questins/Prmpts What questins culd yu ask, r feedback culd yu prvide in the mment t develp student understanding, create disequilibrium, r advance student thinking? What relatinship is represented in this prblem Lking at Cntainer A hw much water is flwing int the cntainer and in what time perid. Let s break dwn the time perids and see hw much water will be in the tank at each time: 1 4 hur hw much water A secnd 1 hur hw much water 4 After a third 1 hur hw much water 4 Clsing the Lp (Interventins/Extensins) LZ vide lessn links that may help develp cnceptual understanding and prcedural skill needed I als use with the student a visual mdel shwing the relatinship. This wuld be fund at: Here students will visually see the cnnectin and relatinship between numbers. After the student wrked thrugh the guided lessns here, I wuld use Thinking blcks t mdel this prblem With this students I wuld draw the bar mdel t have them fill in with time and then the gallns that entered the tank after each 1 4 hur. I wuld ask the student hw many 1 hurs are in 1 whle hur. 4 Then I wuld ask fr them t mark ff that time n a Bar mdel and see if they can determine the amunt f water in the tank. 1 4 hur 1 4 hur 1 4 hur 1 4 hur If a student cntinues t have a hard time with the cncept I wuld then mve t mdel using half hurs r ONLY whle numbers t see if the student can then see the relatinship between the numbers. 5

7 A duble number line culd als be used t help the students the relatinship between time and gallns. ¼ hur gallns 6

8 Student Respnse Example What will a respnse include frm a student wh demnstrated sme understanding and pssibly sme misunderstandings r undevelped understanding? Develping Indicatrs What pssible indicatrs may be included in a student respnse wh has demnstrated sme understanding f the standard? Student shwed n rati table fr the relatinship between time and gallns. Student did nt write the unit rate fr each tank. N equatin was given t shw the relatinship f this situatin. What strategies, and representatins will r will nt be used? What understandings r prcedural fluency des the student respnse reveal? Student did understand that the filling f the cntainers was in a linear relatin they shw the water-level mving in a straight line. What undevelped understandings, miscnceptins, and cmmn mistakes may be revealed in the student respnse t this item? Fr Cntain A the student shwed a dubling f time increments. He was nt cnsistent with his measurements. The cntainer that wuld fill the fastest was nt indicated. Student did nt shw recgnitin f unit rate. 7

9 In the Mment Questins/Prmpts What questins culd yu ask, r feedback culd yu prvide in the mment t develp student understanding, create disequilibrium, r advance student thinking? Yur idea f drawing water, was an awesme idea. Can yu talk t me abut yur mdel? What is it shwing and hw d yu feel it mdels the situatin given in the questin? Clsing the Lp (Interventins/Extensins) LZ vide lessn links that may help develp cnceptual understanding and prcedural skill needed Can yu explain yur increments f time fr cntainer A? Why did yu nt use the same idea fr cntainer B? D yu think yu culd have used anther type f mdel t help yu shw hw much water was ging int the tank fr each time perid? 8

10 Student Respnse Example What will a respnse include frm a student wh has demnstrated cnceptual understanding and mastery? Gt it Indicatrs What indicatrs must be included in an exemplar student respnse Students hw understd this questin wuld have shwn: A clear understanding f rati table Given the crrect equivalent fractinal ratis. Use f divisin with cmplex fractins. Understanding f unit rate and hw it is used in an equatin. What strategies, and representatins will r will nt be used? What understandings r prcedural fluency des the student respnse reveal? The student has the ability t: Describe and identify cmplex fractins Recgnize the difference between a unit rate and a rati Recgnize that a unite rate can be fractinal Recgnize that tw equivalent ratis represent a prprtin. Recgnize the cnnectins between the equivalent ratis and the values in the table. Recgnize the unit rate as the cnstant f prprtinality Use the cnstant f prprtinality in an equatin. What undevelped understandings, miscnceptins, and cmmn mistakes may be revealed in the student respnse t this item? A student may nt make a cnnectin between the cnstant f prprtinality and the unit rate, r hw that is used in an equatin. 9

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12 In the Mment Questins/Prmpts What questins culd yu ask, r feedback culd yu prvide in the mment t extend r push student understanding, create disequilibrium, r advance student thinking? I wuld ask that student hw they felt the prblem wuld change if I tld them each cntainer held 35 gallns r gallns. Hw d yu think a picture f this rati table wuld lk? If I wanted t graph it what d yu think I shuld d? Clsing the Lp (Interventins/Extensins) LZ vide lessn links that may extend and deepen student understanding and prcedural fluency

13 Steps 3 and 4: Act n Evidence frm Student Wrk and Adjust Instructin Lessn Objective: Cntent Standard(s): Targeted Practice Standard : Mdeling prprtinal relatinships and unit rate with fractinal units. 7.RP.2 Recgnize and represent prprtinal relatinships between quantities, (Fractinal quantities) by testing fr equivalent ratis in a table r graphing n a crdinate plane. MP.4: Mdel with mathematics. MP.7: Lk fr and make use f structure. MP.8: Lk fr and express regularity in repeated reasning. Mathematical Gals Students will mve frm the Bar Mdel apprach (6 th grade) t understanding a prprtinal representatin with fractinal units and learn hw t the use a rati table t find unit rate. Success Criteria Students need t understand, recgnize and utilize prprtinal relatinship in tables, graphs and equatins. This lessn is t help the students mve frm the Bar Mdel intrduced in 6 th grade t the rati table t the determining the unit rate. Launch (Prbe and Build Backgrund Knwledge) Prir Knwledge: Students shuld understand the cncept f rati and use precise language and symbls t describe a rati relatinship. Students shuld als understand unit rate and use rati and rate reasning t slve mathematical prblems Purpse: Asses and Activate Prir Knwledge: Students will cmpute unit rate assciated t fractinal quantities, which measure different units t find which water tank will be filled the fastest. The student s D Nw wuld be: Withut slving cmpare the tw prblems belw. Use the questins prvided t help yu cmpare them. 1) Oliver is training fr a marathn. In practice he runs 15 kilmeters in 70 minutes. Hw far will Oliver run in 1 minute? 2) On the snwiest day in the past 5 years, it snwed 7 feet in 27 1 hurs. Hw much did it snw per hur? 2 Hw are these tw prblems the same? Hw are they different? What can we learn frm ding prblem #1 that will help us slve prblem #2. Instructinal Task Purpse: A brief descriptin f the mathematics and/r the mathematical practices the task is intended t engage students in and what students will be ding. Students will refine their skills frm the 6 th grade curriculum, transferring them t prblem slving t find unit rates when given fractin/decimal quantities. a) Students will demnstrate the slutins t finding unit rates when given tw quantities including whle numbers, fractins, and decimals. 12

14 b) Students find unit rates within real wrld situatins using whle and nn-whle number quantities. (Examples: finding the rate f speed distance/time, cst per unit). Students can be given scenaris where given different distances and times, they cmpare rates/speeds. Faced with tw retail items students can determine the unit price and reasn which is the better deal. Engage (Setting Up the Task) Using the Bike prblem, abve n page 13, ask the students t wrk in grups and discuss hw they will shw the distance vs. the time rati. Check fr prir knwledge and use f the Bar Mdel t develp a picture f the time distance relatinship. Students will pst their wrk n large paper arund the rm and share these ideas. A Gallery Walk will then be cmpleted by the ther students wh will cmment n the wrk and write any questins they may have. Students will then share their wrk and answer any f the questins psted n their paper. Explre (Slving the Task) Present the Cntainer Prblem. After a brief intrductin t the prblem, have the students wrk in grups and be prepared t place their wrk n large paper t hang up arund the rm. What questins will yu ask as students wrk n the task t elicit evidence f their understanding and supprt mathematical cnnectins? Fcusing Questins Prbing Questins Advancing Questins What infrmatin d yu knw that will help yu with this questin? What type f mdel will yu be using t shw that infrmatin? What is a unit rate? What decisin helped yu determine what mdel t use? Hw des yur mdel relate t divisin/multiplicatin f fractins? Hw des yur mdel relate back t each cntainer? Is there any ther way yu culd have answered this questin? Hw wuld we change this questin t determine which cntainer wuld be filled first? 13

15 Hw can we find a unit rate? What cnnectin is there t the unit rate and yur mdel? What cnnectin can yu make between the unit rate and the equatin? Hw des yur mdel shw the slutin t the questin? Have yur checked yur slutin? What are sme anticipated student respnses r slutin paths? Hw d we knw which will fill faster we d nt knw hw much water each cntainer will hld? Give students sme directin t the type f mdel that can use: Bx mdel / Bar mdel / Duble line graph / rati table. If students are having a huge prblem wrking with the fractins, ask them t make the prblem simpler and wrk with whle numbers first. Then mve t fractins that might be easier fr them t wrk with. Elabrate (Discuss Task and Related Mathematical Cncepts) Hw will yu facilitate the sharing f student wrk and discussin t supprt students in making mathematical cnnectins? Students wrk will be dne n large paper and hung arund the classrm. Students will then cmplete a Gallery Walk. They must give ne psitive feedback. They ask a questin abut the wrk? Did yu think abut using a duble line chart instead f mdel? Why did yu set up yur rati table as yu did? Each grup will have an pprtunity t share their wrk and answer any questin(s) place n their paper. Call the class back tgether t discuss their bservatins. Refer back t the Questin presented befre this lessns Hw can we mdel prprtinal relatinships when ur unit rates are fractins? As students share their answer t this questin their respnses will be written n the white bard. Finally cme t a class agreement n a respnse. Checking fr Understanding Purpse: Pse the fllwing as an exit card t elicit evidence f students understanding f mdeling prprtinal relatinship using fractinal unit rates. (See Attached n page 20) Exit Ticket: Name 2 cncepts yu learned tday. What questin(s) d yu still have with tday s lessn? Can yu help Ms. Albert s students? Ms. Albert decided t make lemnade t serve t her math club students. The directins said t mix 2 scps t pwered drink mix t ½ galln f water t make each pitcher f lemnade. One f Ms. Albert students tells her she will need t add 8 scps f pwered mix with 2 gallns f water s she can make 4 pitchers f lemnade. Hw can yu use the cncept f tday s lessn- rati tables - t tell whether this student is crrect? 14

16 Cmmn Misunderstanding Purpse: T address cmmn misunderstanding students ften have abut fractins and hw they relate t tday s lessn remind the students f the task used in the D Nw. 1) Oliver is training fr a marathn. In practice he runs 15 kilmeters in 70 minutes. Hw far will Oliver run in 1 minute? 2) On the snwiest day in the past 5 years, it snwed 7 feet in 27 1 hurs. Hw much did it snw per hur? 2 Hw are these tw prblems the same? Hw are they different? What can we learn frm ding prblem #1 that will help us slve prblem #2. Depending n their cncern: Remind students f the cnnectin between ratis and unit rates. Hw are rates and rati alike? Hw are they different If the student has nt develped fr themselves a persnal knwledge / understanding f the cnceptual mdels used in divisin f fractins. Explre ways that it can be dne. What des unit rate mean? Can yu think f ways yu have heard it used in yur daily life? In questin ne hw did yu find the unit rate? D yu think that prcess wuld change in the secnd questin? If the students are having a prblem creating the rati table, ask the fllwing: D yu remember hw we fund equivalent fractins? Can yu describe that prcess t me? Hw can we use that prcess t find equivalent ratis? Help the students t make a cnnectin with the unit rate and hw it prgress thrughut the rati table As yu mved frm ¼ minute t ½ a minute what did yu ntice abut what yu were ding t the time? Therefre what must yu d with the amunt f water flwing int the tanks? D yu think there is a cnnectin between yur unit rate and the rest f the table? Hw might this unit rate help me slve this prblem? Hw can this unit rate be used in an equatin? Checking fr Understanding Purpse: A brief descriptin f what questins r prmpts yu will use t elicit evidence f student understanding and the strategy yu will use t elicit the evidence at the end f the lessn. Using the same basic questin f which cntainer is filling faster ask the fllwing. Students wuld still have time t wrk with their partner they wuld d a THINK PAIR SHARE with the rest f the class. What wuld happen if I tld yu Cntainer A wuld hld 50 gallns f water and Cntainer B wuld hld 65 gallns f water? Wuld that change yur answer t the questin, which cntainer is filling faster? If bth f the Cntainers wuld hld 75 gallns f water which cntainer wuld fill first and why? Can yu explain the difference between filling faster and which is filled first? What infrmatin d yu have nw that wuld determine this? 15

17 Clsure Purpse: Befre students are given the exit card prvide an pprtunity fr them t answer the Essential questin fr themselves in the fllwing I can statements: With n prblem!! I still have a little cncern. Need help I can draw a mdel r use a rati table t help me determine equivalent fractinal ratis? I can use the unit rate t help me determine the relatinship in my mdel? I can use the unit rate in an equatin. I can justify my answer thrugh using examples frm my mdel? Were yu are creative prblem slver tday? Explain hw. 16

18 Extensin Task Purpse: Prvide an extensin task fr thse students wh are ready t deepen their understanding f prprtinal relatinships and fractinal rates. A lcal stre makes a special citrus salad dressing t be sld in their stres and used n their salads. Here is a listing f ingredients f the mixture is live il f the mixture is balsamic vinegar f the mixture is range juice f the mixture is lemn juice The stre wants t make sure that very batch is kept in the same prprtins. Olive Oil Single Batch Duble Batch Triple Batch Quadruple Batch A batch that is X size f the riginal Balsamic vinegar Orange Juice Lemn juice Des the prcess change nw since yu are making a mre extensive rati table? Hw are yu finding the amunt f ingredients fr the Batches? Hw d yu use yur unit rate in this prcess? Suppse the number f Batches I wanted t make were nt n yur rati table? Explain hw yu wuld determine hw much f each ingredient I wuld use t make that batch? Use yur ideas listed abut, and determine hw much f each ingredient I wuld need t make 12 batches. Were yur ideas in line with what yu did? If s explain hw? 17

19 Name: Date: Bike Prblem Barry wants t enter a lcal bike race. He begins his training by biking 8 1/2 miles every 1/2 hur. What are the tw units be cmpared in this prblem? Create a mdel t determine hw far Barry will bike in 2 1/2 hurs. If this rate cntinues, hw lng will Barry have t ride, t reach his gal f 51 miles? Using the mdel yu created, write an equatin t mdel this situatin fr any number f miles. 18

20 Name: Date: Which Cntainer? Tw liquid strage cntainers f the same size are being filled. Liquid enters cntainer A at a rate f 2 3 galln per 1 4 minute. Liquid purs int cntainer B at a rate f 3 5 galln per 1 6 minute. Cntainer A: Cntainer B: Determine which cntainer is being filled faster. Justify yur answer. Supprt yur reasning by using evidence frm the mdels abve. 19

21 Name: Date: Special Salad Dressing A lcal stre makes a special citrus salad dressing t be sld in their stres and used n their salads. Here is a listing f ingredients f the mixture is live il f the mixture is balsamic vinegar f the mixture is range juice f the mixture is lemn juice The stre wants t make sure that very batch is kept in the same prprtins. Olive Oil Single Batch Duble Batch Triple Batch Quadruple Batch Batch that is x time greater. Balsamic vinegar Orange Juice Lemn juice Des the prcess change nw since yu are making a mre extensive rati table? Hw are yu finding the amunt f ingredients fr the Batches? Hw d yu use yur unit rate in this prcess? Suppse the number f Batches I wanted t make were nt n yur rati table? Explain hw yu wuld determine hw much f each ingredient I wuld use t make that batch? Use yur ideas listed abut, and determine hw much f each ingredient I wuld need t make 12 batches. Were yur ideas in line with what yu did? If s explain hw? 20

22 Name: Exit Ticket: Name 2 cncepts yu learned tday. What questin(s) d yu still have with tday s lessn? Can yu help Ms. Albert s students? Ms. Albert decided t make lemnade t serve t her math club students. The directins said t mix 2 scps t pwered drink mix t ½ galln f water t make each pitcher f lemnade. One f Ms. Albert students tells her she will need t add 8 scps f pwered mix with 2 gallns f water s she can make 4 pitchers f lemnade. Hw can yu use the cncept f tday s lessn - rati tables - t tell whether this student is crrect? Name: Exit Ticket: Name 2 cncepts yu learned tday. What questin(s) d yu still have with tday s lessn? Can yu help Ms. Albert s students? Ms. Albert decided t make lemnade t serve t her math club students. The directins said t mix 2 scps t pwered drink mix t ½ galln f water t make each pitcher f lemnade. One f Ms. Albert students tells her she will need t add 8 scps f pwered mix with 2 gallns f water s she can make 4 pitchers f lemnade. Hw can yu use the cncept f tday s lessn - rati tables - t tell whether this student is crrect? 21

23 Name: Self-Reflectin: This will be placed in yur prtfli, stapled t tday s cntainer questin. With n prblem! I still have a little cncern. Need help I can draw a mdel r use a rati table t help me determine equivalent fractinal ratis? I can use the unit rate t help me determine the relatinship in my mdel? I can use the unit rate in an equatin. I can justify my answer thrugh using examples frm my mdel? Explain hw t using a rati table helped yu with tday s questin? Were yu a creative prblem slver tday? Explain hw. 22

24 Scenari: Table: Equatin/Algebraic Mdel: 23

25 Cntent Standard(s): What standard was this item designed t assess? Rati and Prprtin 7.RP.1 Cmpute unit rates assciated with ratis f fractin, including ratis f length, areas and ther quantities measured in like r different units. (COMPLEX Fractins) 7.RP.2 Recgnize and represent prprtinal relatinships between quantities. (Fractinal quantities) by testing fr equivalent ratis in a table r graphing n a crdinate plane. Smarter Balanced Claim Claim 1: Cncepts and Prcedures Students can explain and apply mathematical cncepts and interpret and carry ut mathematical prcedure with precisin and fluency. Research and review f standard Standard(s) fr Mathematical Practice: What Standard(s) fr Mathematical Practice are implicit in this item r cntent standard? MP.2: Reasn abstractly MP.3: Cnstruct Viable arguments and critique the reasning f thers. MP.4: Mdel with mathematics. MP.7: Lk fr and make use f structure. Smarter Balanced Item Select a Smarter Balanced released item that addresses yur selected standard(s). If pssible, insert a screensht here. 24

26 CPR Pre-Requisites (Cnceptual Understanding, Prcedural Skills, and Representatins) Lk at the Prgressins dcuments, Learning Trajectries, LZ lessn library, unpacked standards dcuments frm states, NCTM Essential Understandings Series, NCTM articles, and ther prfessinal resurces. Yu ll find links t great resurces n yur PLC Platfrm. 7.RP.1 is extensins f the students wrk in sixth grade n divisin f fractins. In sixth grade students had t use the Bar Mdel, and Duble number lines t gain an understanding f divisin f fractins. 7.RP.1 is a cmbinatin f divisin f fractins mving int unit rate. 7.RP.2 students will begin t use this cncept in the creatin f rati tables anther mdel t aid them in understanding ratis. It is a full understanding f the unit rate that will then help the students mve n t the understanding f the equatin. Cnceptual Understanding and Knwledge What are the cnceptual understandings students must have in rder t achieve mastery f the standard Clear understanding f what a fractin is. Remainder is a prtin f the grup serving size the remainder is a piece f what we are lking fr. Cnnectin between rati and fractins and their differences. Clear distinctin between multiplying and dividing fractins. Example: 8 x 4 same as That a Cmplex fractin is anther way t represent a basic fractin divisin prblem is the same as

27 Prcedural Skills Ratis are ften expressed in fractinal ntatin, althugh ratis and fractins d nt have identical meaning. Ratis are ften used t make part-t-part cmparisns; fractins are nly a part-t-whle cmparisn. What are the pre-requisite prcedural skills and strategic cmpetencies students must have in rder t achieve mastery f the standard Multiplying Fractins Divisin f Fractins Understand the cncept f unit rate Recgnize the difference between unit rate and a rati Recgnize that a unit rate can be fractinal The ability t recgnize and represent the cnnectin between equivalent ratis and values in a table The ability t determine the unit rate as the cnstant f prprtinality The ability t use the cnstant f prprtinality in an equatin Representatinal What representatins shuld students be able t understand and use in rder t achieve mastery f the standard Use f Bar Mdel fr divisin f fractins Use f Number line t understand divisin f measurement. (Time and distance) Duble Number line fr cmplex fractins It is the use f these representatinal mdels that will help us lead students t the rati table and the understanding f unit rate. Equivalent rati table Equatin t represent the relatinship fund in the rati table. Scial knwledge What terms, definitins, and cnventins must students have knwledge f in rder t achieve mastery f the standard Numeratr / denminatr / part-t-whle rati / cmplex fractins. Distance can be represented n a number line. 26

28 Standards Prgressin *Lk at LearnZillin lessns and expert tutrials, the Prgressins dcuments, learning trajectries, and the Wiring Dcument t help yu with this sectin Grade(s) belw Target grade Grade(s) abve What previus grade level standards build up t the grade level standard this item assesses? Numbers and Operatins: Fractins 5.NF.3 Interpret a fractin as divisin f the number by the denminatr. 5.NF.4 a, b Extend and Apply previus understanding f multiplicatin t multiply a fractin r whle number by a fractin. 5.NF.7: Apply and extend previus understanding f divisin t divide unit fractins by whle numbers and whle numbers by a unit fractin. Number System: 6.NS.1: Interpret and cmpute qutients f fractins and slve wrd prblems invlving divisin f fractins by fractins. What ther grade level standards are cnnected t the standard this item assesses? Number System: 7.NS.2: Apply and extend previus understandings f multiplicatin and divisin and f fractins t multiply and divide ratinal numbers. 7.NS.2c: Apply prperties f peratins as strategies t multiply and divide ratinal numbers. 7.RP.2c: Represent prprtinal relatinships by equatins 7.RP. 3 Use prprtinal relatinships t slve multistep rati and percent prblems. Examples: simple interest, tax, markups and markdwns, gratuities and cmmissins, fees, percent increase and decrease, percent errr What subsequent grade level standards build ff f the grade level standard this item assesses? Number System: 8.NS.1: Knw that numbers that are ratinal are called irratinal. 8.NS.2 Use ratinal apprximatins f irratinal numbers t cmpare the size f irratinal numbers 8.F.1: Functins 27

29 Cmmn Miscnceptins/Radblcks What characteristics f this prblem may cnfuse students? Students lack a general cnceptual understanding f fractins Student lack a general cnceptual understanding f ratis and unit rates. Students have nt develped fr themselves a persnal knwledge / understanding f the cnceptual mdels used in divisin f fractins. Student might see the numeratr and denminatrs as whle numbers and nt as a unit by themselves. Students d nt understand a fractin represents a number between 0 and 1. Students will nt understand hw a fractin in itself can be a numeratr r a denminatr. Divisin f fractins Will divisin always make things smaller? Will multiplicatin always make things bigger? When will this be true? When will it NOT be true? Students have nt mastered the cncept f relatinships between quantities r hw equivalent ratis are built. What are the cmmn miscnceptins and undevelped understandings students ften have abut the cntent addressed by this item and the standard it addresses? Students may nt fully understand a fractin is less than 1. Fractins in this questin can easily add up t 1 minutes, which then becmes the unit rate. Additin f fractins with like denminatrs culd als be a miscnceptin fr students wh d nt fully understand they add nly adding numeratrs. 2 3 The relatinship between a cmplex fractin 1 and the basic algrithm f Fr sme students with a miscnceptin f divisin f fractins, it may becme necessary t take the example back t the beginning f whle numbers divided by 1 such as 8 1 =. If we divide a number by 1 what happens? Hw d yu think this will change if we divided 8 by a number less than ne? What ver generalizatins may students make frm previus learning leading them t make false cnnectins r cnclusins? That divisin will als make things smaller. That a rati is just anther way t name a fractin. Meaning f a fractin with a denminatr f 1 - Hw that relates t unit rate. That the unit rate fr a relatinship such as ¼:1/2 wuld be less than 1 since bth ratis are less than 1. 28