Discovering the Better Buy

Transcription

1 Discvering the Better Buy Presented by: Cynthia Raff The Center fr Mathematics and Teaching, Inc. Califrnia Mathematics Cuncil Palm Springs, CA Nvember 6, 2015 Sessin #154 Evaluatin pll cde

2 TORTILLAS FLAT NROUND 3trtillasfr$0.60 WRAPITUP 4trtillasfr$0.76 What are sme ways we culd represent this infrmatin? 2

3 BAGELS MathLinks: Grade 6 SHMEAR NTHINGS 4bagelsfr$3.00 HOLEHYBREAD 5bagelsfr$4.00 Assume yu can buy any number f bagels at this rate. Cmplete the tables and answer the questins belw. 1. SHMEAR N THINGS # f cst bagels HOLE-Y BREAD # f bagels cst 2. Which shp has the better buy? Use entries in the tables abve t explain yur reasning. 3. Label and scale the grid. Graph the data using tw different clrs. Explain hw t use the graph abve t tell which bagel is the better buy. 4. Find the unit price fr bagels at bth shps. Use these numbers t explain which shp has the better buy. 5. Write an equatin t represent the cst (dependent variable), given the number f bagels (independent variable), fr: a. Shmear n Things: y = b. Hle-y Bread: y = 3

4 BAGELS MathLinks: Grade 7 SHMEAR NTHINGS 4bagelsfr$3.00 HOLEHYBREAD 5bagelsfr$ Cmplete the tables belw. Assume each shp will sell yu any number f bagels at the rates shwn abve. SHMEAR N THINGS HOLE-Y BREAD # f bagels cst in dllars # f bagels cst in dllars Tw variables (quantities that vary) are prprtinal if ne is a cnstant multiple f the ther. One methd f testing fr a prprtinal relatinship is t write and cmpare ratis. If the values f the ratis are all equal, then the quantities are prprtinal. 2. Write the values f sme ratis fr each shp in the table belw using the data tables abve. cst in dllars 3 6 # f bagels SHMEAR N THINGS HOLE-Y BREAD Simplify Unit price (in dllars/bagel) cst in dllars # f bagels Simplify Unit price Which shp has the better buy? Circle entries in yur table t justify yur answer and explain in wrds. 4

5 TESTING FOR PROPORTIONAL RELATIONSHIPS MathLinks: Grade 7 Each table belw shws a relatinship between quantities that vary. Write, simplify, and cmpare the values f the ratis (unit rates). Then state whether the pairs in each table represent a prprtinal relatinship between the quantities. 1. The number f meals Venya served t the hmeless and their cst. Mnday Tuesday Wednesday Thursday Friday # f meals cst $135 $180 $60 $300 $165 unit rate Prprtinal? 2. The number f bags f feathers Jaime used t make pillws. # f bags # f pillws unit rate Prprtinal? 3. The number f tables LaTnya rented fr a party and their cst. # f tables cst $20 $25 $30 $35 $40 unit rate Prprtinal? 4. The size f each rm in Satshi s huse and the number f plants in each rm. bedrm living rm den kitchen pati square feet # f plants unit rate Prprtinal? 5

6 TESTING FOR PROPORTIONAL RELATIONSHIPS (Cntinued) MathLinks: Grade 7 Graph the data frm each f the tables n the previus page. Be sure t label and scale the axes apprpriately. 5. The number f meals Venya served t the hmeless and their cst. 6. The number f bags f feathers Jaime used t make pillws (0,0) (0,0) 7. The number f tables LaTnya rented fr a party and their cst. 8. The size f each rm in Satshi s huse and the number f plants in each rm. (0,0) (0,0) 9. What d yu ntice abut graphs that represent prprtinal relatinships, cmpared t thse that d nt? 6

7 TORTILLAS MathLinks: Grade 7 FLAT N ROUND 3 trtillas fr $0.60 WRAP IT UP 4 trtillas fr $0.76 Cmplete the tables belw. Assume each shp will sell any number f trtillas at the rates shwn. FLAT N ROUND # f trtillas cst WRAP IT UP # f trtillas cst 3. Label and scale the grid. Graph the data using tw different clrs. 1. Find the unit price at each trtilla stre. 4. Identify the y-crdinate when x = 1. FLAT N ROUND (1, ) 2. Write equatins t relate the number f trtillas t the cst. FLAT N ROUND y = WRAP IT UP (1, ) 5. Explain the meaning f these crdinate pairs in this cntext. WRAP IT UP y = The equatins abve are in the frm y = mx and represent prprtinal relatinships. Fr each respective equatin, y is prprtinal t x. The number m is called the cnstant f prprtinality. 6. Hw are the crdinates fr the rdered pairs in prblem 5 related t the equatin in the frm y = mx? 7

8 1. Cmpletethetables.Assumeaprprtinal relatinshipbetweenthenumberfbagels andthecst. SHMEAR N THINGS # f cst bagels SHMEAR NTHINGS 4bagelsfr$3.00 HOLE-Y BREAD # f bagels BAGELS MathLinks: Grade 8 cst Whichshphasthebetterbuy?Useentries inthetablestexplainyurreasning. 3. Writeequatinstrelatethenumberf bagelstcst. SHMEAR NTHINGSy= HOLEHYBREADy= Thelinearfunctinsyuwrteabvearebth inthefrmy=mx.thisiscalledadirect prprtinequatinbecauseyisdirectly prprtinalt(isamultiplef)x. 4. Hwisthedirectprprtinequatin differentfrmthelinearfunctin y=mx+b? 5. Title,label,andscalethegrid.Graphthe datausingtwdifferentclrs. HOLEHYBREAD 5bagelsfr$ Explainwhichgraphillustratesaslwer riseinprice. 7. Identifythecrdinateswhenx=1 SHMEAR NTHINGS(1, ) HOLEHYBREAD(1, ) 8.WhatdtheseyHcrdinatesrepresentin thecntextftheprblem? 8

9 TORTILLAS MathLinks: Grade 8 1. Cmpletethetables.Assumeaprprtinal relatinshipbetweenthenumberf trtillasandcst. FLAT N # f trtilla s ROUND FLAT NROUND 3trtillasfr$0.60 cst WRAP IT UP # f trtilla s cst 5. Title,label,andscalethegrid.Graphthe datausingtwdifferentclrs. WRAPITUP 4trtillasfr$ Whichshphasthebetterbuy?Useentries inthetablestexplainyurreasning. 3. Writeequatinstrelatethenumberf trtillastcst. FLAT NROUNDy= WRAPITUPy= 4. Identifythecrdinateswhenx=1 FLAT NROUND(1, ) WRAPITUP(1, ) Hwarethesecrdinatesrelatedtthe unitratefrnetrtilla? 6. Explainwhichgraphillustratesaslwer riseinprice. Inthelinearfunctiny=mx+b, brepresentstheyhintercept. 7. WritecrdinatesfrtheyHinterceptsfr eachfunctin. FLAT NROUND(0, ) WRAPITUP(0, ) WhatdtheseyHcrdinatesrepresentin thecntextftheprblem? 9

10 PITA BREAD MathLinks: Grade 8 PAPA SPITA EATHAPITA 6pitasfr$ 5pitasfr$ 1. ThegraphfrEATHAPITAisprvidedbelw.HwdyuknwbylkingatthegraphfrEATHA PITAthatitdepictsaprprtinalrelatinship? 2. CmpletethetablesbelwandgraphfrPAPA SPITA.ApartialtablefrPAPA SPITAisprvided. Usetablesandgraphstcmpletethepricinginfrmatinabve.Assumeeachshpwillsellany numberfpitasattheratesshwn. $5 PAPA SPITA #f pitas cst EATHAPITA #f pitas cst EATHAPITA 3. Findtheunitpricesatbthpitastres. 4. Writeequatinstrelatethenumberf pitastthecst. PAPA SPITAy= EATHAPITAy= 5. Verifythattherelatinshipbetweenthenumber fpitasandthepricepaidat EATHAPITAisprprtinalbycreatingratis andcheckingfrequivalentfractins. 6. DesthegraphfrPAPA SPITAgthrughthe rigin?whatdesthatmeaninthiscntext? 7. Whichgraphillustratesaslwerriseinprice? Hwdyuknw? $

11 PIZZA MathLinks: Grade 8 DOORHTOHDOORPIZZA PIZZAPALACE $ perpizza $ perpizza $5frdelivery 1. Cmpletethetablesandgraphs.ThegraphfrPIZZAPALACEisprvided.Usetablesandgraphst cmpletethepricinginfrmatinabve. DOORHTOHDOOR (includedelivery) #f cst pizzas $45 $0 PIZZAPALACE #f pizzas cst PIZZAPALACE 2. Usingdatafrmthetable,explain/shwwhich relatinshipisprprtinal. 3. Dbthgraphsrepresentprprtinal relatinships?whyrwhynt? 4. Usingthedatafrmthetables,determinewhich shpffersthebetterbuy.explainyur reasning. 5. Identifythecrdinatesfthepintwherethe twgraphsintersect. 6. Whatdesthispintmeanwithinthecntextf theprblem? 11

12 MATHEMATICAL CONTENT STANDARDS ASelectinfrmGrades638thatmaybediscussedduringthispresentatin 6.RP.1 6.RP.2 6.RP.3a 6.RP.3b 7.RP.A.2a 7.RP.A.2b 7.RP.A.2c 7.RP.A.2d 8.EE.5 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. Understand the cncept f a unit rate a/b assciated with a rati a:b with b 0, and use rate language in the cntext f a rati relatinship. Fr example, This recipe has a rati f 3 cups f flur t 4 cups f sugar, s there is 3/4 cup f flur fr each cup f sugar. We paid $75 fr 15 hamburgers, which is a rate f $5 per hamburger. 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: Make tables f equivalent ratis relating quantities with whle number measurements, find missing values in the tables, and plt the pairs f values n the crdinate plane. Use tables t cmpare ratis. 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: Slve unit rate prblems including thse invlving unit pricing and cnstant speed. Fr example, if it tk 7 hurs t mw 4 lawns, then at that rate, hw many lawns culd be mwed in 35 hurs? At what rate were lawns being mwed? Recgnize and represent prprtinal relatinships between quantities. Decide whether tw quantities are in a prprtinal relatinship, e.g., by testing fr equivalent ratis in a table r graphing n a crdinate plane and bserving whether the graph is a straight line thrugh the rigin Identify the cnstant f prprtinality (unit rate) in tables, graphs, equatins, diagrams, and verbal descriptins f prprtinal relatinships. Represent prprtinal relatinships by equatins. Fr example, if ttal cst t is prprtinal t the number n f items purchased at a cnstant price p, the relatinship between the ttal cst and the number f items can be expressed as t = pn Explain what a pint (x, y) n the graph f a prprtinal relatinship means in terms f the situatin, with special attentin t the pints (0, 0) and (1, r) where r is the unit rate. Graph prprtinal relatinships, interpreting the unit rate as the slpe f the graph. Cmpare tw different prprtinal relatinships represented in different ways. Fr example, cmpare a distance-time graph t a distance-time equatin t determine which f tw mving bjects has greater speed. 12

13 STANDARDS FOR MATHEMATICAL PRACTICE The Standards fr Mathematical Practice describe a variety f prcesses and prficiencies that educatrs seek t develp in mathematically prficient and fluent students acrss all grade levels. Many prcesses and prficiencies in these practice standards verlap, several may be used tgether n any given prblem r task, and rarely wuld we expect students t use them all at nce. We d expect that ver time students will use them frequently. In additin, sme will be used naturally within the cntext f slving particular prblems, and thers will nly ccur in an envirnment in which students are prvided ample pprtunities t cllabrate and discuss. One way t think abut the practices is in grupings (graphic frm CCSS-M authr, Bill McCallum). REASONING AND EXPLAINING HABITS OF MIND MP1 Make sense f prblems and persevere in slving them MP6 Attend t precisin MP2 Reasn abstractly and quantitatively MP3 Cnstruct viable arguments and critique the reasning f thers MODELING AND USING TOOLS MP4 Mdel with mathematics MP5 Use apprpriate tls strategically SEEING STRUCTURE AND GENERALIZING MP7 Lk fr and make use f structure MP8 Lk fr and make use f repeated reasning Frm CCSS-M: Students wh lack understanding f a tpic may rely n prcedures t heavily...in shrt, a lack f understanding effectively prevents a student frm engaging in the mathematical practices. In this respect, thse cntent standards which set an expectatin f understanding are ptential pints f intersectin between the Standards fr Mathematical Cntent and the Standards fr Mathematical Practice. Withut understanding, a student may rely n prcedures and may nt represent prblems cherently, justify cnclusins, apply the mathematics t ther situatins, use technlgy mindfully t wrk with the mathematics, explain the mathematics accurately t ther students, step back fr an interview, r deviate frm a knwn prcedure t find a shrtcut. 13

14 STANDARDS FOR MATHEMATICAL PRACTICE (Cntinued) MP1 Make sense f prblems and persevere in slving them Habits f Mind MP6 Attend t precisin Understand a prblem and lk fr entry pints Cnsider given infrmatin, cnstraints, and relatinships Cnsider simpler r analgus prblems Make cnjectures, mnitr prgress and alter their slutin curse as needed Explain crrespndences using multiple representatins Understand and analyze the appraches f thers Cntinually ask, Des this make sense? Cmmunicate mathematical ideas precisely Use clear definitins State meaning f symbls and use them prperly Attend t units f measures and labeling f axes Calculate accurately and give slutin with apprpriate degree f accuracy MP2 Reasn abstractly and quantitatively Reasning and Explaining MP3 Cnstruct viable arguments and critique the reasning f thers Attend t the meaning f quantities Decntextualize a prblem using symbls, and manipulate them as if they have a life f their wn Cntextualize manipulatins t create a cherent representatin f a prblem Use assumptins, definitins, and established results t create arguments (deductive reasning) Make and test cnjectures based n evidence (inductive reasning) Analyze situatins by breaking them int cases Use cunterexamples t disprve a statement Identify flaws in an argument Listen t r read t arguments and ask useful questins t clarify reasning 14

15 STANDARDS FOR MATHEMATICAL PRACTICE (Cntinued) MP4 Mdel with Mathematics Mdeling and Using Tls MP5 Use apprpriate tls strategically Apply mathematics t slve everyday prblems Make reasnable assumptins and apprximatins t simplify a situatin Identify imprtant quantities in a situatin Use multiple representatins t analyze relatinships and draw cnclusins Interpret results in the cntext f the situatin Imprve the mathematical apprach (mdel) if it has nt served its purpse Select useful tls such as paper and pencil, graph paper ruler, calculatr, cncrete mdel, spreadsheet, r statistical sftware t slve prblems Use cncrete mdels and technlgy tls t explre cncepts Recgnize limitatins f tls Identify and use relevant external resurces, such as the internet MP7 Lk fr and make use f structure Structure and Generalizing MP8 Lk fr and make use f repeated reasning Identify patterns and apply them t slve prblems Recgnize the structure f a symblic representatin and generalize it See cmplicated bjects as cmpsed f chunks f simpler bjects Identify repeated calculatins and patterns Generalize prcedures based n repeated patterns r calculatins Find shrtcuts based n repeated patterns r calculatins 15