Benchmark 6: Grade 6 Expressions and Equations Performance Task
6.EE.9
The Task:How Much Should That Specialty Pizza Cost?
Part1
Usewhatyouknowaboutindependent anddependent variablesalongwiththefollowinginformation toanswerthequestions.
1. What arethetwoquantities thatwillchangewhendeciding whichpizzatoorder?
______
2. a)Which quantity represents theindependent variable? ______
b) How do you know this is the independent variable? ______
______
3. a)Which quantityrepresentsthedependentvariable? ______
b) How do you know this is the dependent variable? ______
______
4. Writea wordequationtorepresent thecostofasmallpizzainterms ofthenumber oftoppings ordered. Explain how you came up with this equation.
5. Substitute variablesintotheequation youwroteforproblem 4 to write a mathematical equation.
6. Usethesamevariables youchoseinproblem 5towriteanequationtorepresentthecostofa mediumpizzainterms ofthenumber oftoppings ordered.
7. Usethesamevariables asinproblem 5towriteanequationtorepresentthecostofalargepizza intermsofthenumberoftoppings ordered.
Part2
Thepizzashopalsooffersthefollowing specialtypizzas.Eachpizzaincludes tomatosauce, mozzarellacheese,athin crust,andadditional toppingsaslisted.
• ILoveMeat
pepperoni, ham, ground beef,bacon, sausage,chicken
Small:$12.00 Medium: $16.25 Large:$19.50
• ILoveVegetables
greenpeppers, tomatoes, onions, olives,mushrooms, spinach, jalapenos, broccoli, garlic
Small:$9.50 Medium: $13.75 Large:$17.00
• ILoveCheese
ricotta cheese,Parmesan cheese,Asiagocheese
Small:$8.25 Medium: $11.00 Large:$13.25
• ILoveSimplicity
pepperoni
Small:$7.00 Medium: $9.50 Large:$11.50
• ILoveEverything
pepperoni, ham,ground beef,bacon, sausage,chicken,greenpeppers, tomatoes, onions, olives
Small:$15.50 Medium: $19.00 Large:$22.75
8. Anadinthepizzashopsaysyoucan"Savebyordering ourspecialtypizzas!"Butsincethename of theshopisCheatza Pizza,youwonder ifordering aspecialtypizzaactuallycostsmorethan creating yourownpizzawiththesamenumber of toppings. Youwanttoknowifcustomers are beingcheated.
Usetheequations youcreated inPart1tofindoutwhether itischeaper topurchase the specialtypizzaortocreateyourownforeachspecialtypizzacombination.
Make a table that compares the prices of the Specialty Pizzas to the prices of “create your own pizza”.
(Differentiate here by allowing students to create their own table; give them the first section; or give them the whole blank table)
Specialty pizza: I Love MeatSmall / Medium / Large
Specialty price
“Create your own” price
OR
Specialty pizza: I Love MeatSmall / Medium / Large
Specialty price
“Create your own” price
Specialty pizza: I Love Vegetables
Small / Medium / Large
Specialty price
“Create your own” price
Specialty pizza: I Love Cheese
Small / Medium / Large
Specialty price
“Create your own” price
Specialty pizza: I Love Simplicity
Small / Medium / Large
Specialty price
“Create your own” price
Specialty pizza: I Love Everything
Small / Medium / Large
Specialty price
“Create your own” price
9. Isitevercheapertocreateyourownpizza? Justifyyourresponse foreachspecialtypizzabyusingyourequations.
Answers
1. thecostofthepizzaandthenumber oftoppings
2. a)thenumberoftoppings b) explanation required
3 a)thecostofthepizza b) explanation required
4. costofpizza=$6.00+number oftoppings •$0.75
5. y=6+0.75x
6. y=8+x
7. y=11+1.25x
8. Completedtable:
Specialty pizza: I Love MeatSmall / Medium / Large
Specialty price / $12.00 / $16.25 / $19.50
“Create your own” price / $10.50 / $14.00 / $18.50
Specialty pizza: I Love Vegetables
Small / Medium / Large
Specialty price / $9.50 / $13.75 / $17.00
“Create your own” price / $12.75 / $17.00 / $22.25
Specialty pizza: I Love Cheese
Small / Medium / Large
Specialty price / $8.25 / $11.00 / $13.25
“Create your own” price / $8.25 / $11.00 / $14.75
Specialty pizza: I Love Simplicity
Small / Medium / Large
Specialty price / $7.00 / $9.50 / $11.50
“Create your own” price / $6.75 / $9.00 / $12.25
Specialty pizza: I Love Everything
Small / Medium / Large
Specialty price / $15.50 / $19.00 / $22.75
“Create your own” price / $13.50 / $18.00 / $23.50
9. Students should show evidence of having used their equations to arrive at the following conclusions for each type of pizza:
I LoveMeat:It'scheaper tocreateyourownforallthreesizes.
ILoveVegetables:It'scheapertoorderthespecialtypizzaforallthree sizes.
I Love Cheese: It costs the same to create your own small or medium as it would to order the specialty pizza, but the speciality large is cheaper than creating your own.
I LoveSimplicity:It'scheapertocreateyourownforthesmallandmedium,butnotthelarge.
I Love Everything: It’s cheaper to create your own for the small and medium but not the large
Task Overview: Background
Thistaskallowsstudents tothink about independent anddependent variables intherealworld. Students thenrepresent therelationship between thetwovariables usinganequation. Part 1guides thestudents throughtheprocedure forasmallpizza,thenasksthemtowriteanequation forthe medium andlargepizzasindependently. Part2asksstudents toconsiderthecostofspecialtypizzas inrelationtothe"createyourown"pizzas. Students must then determine ifcustomers whoorderspecialty pizzasarepaying morethan customers whocreatetheir own,andwhenitiscosteffectiveto order aspecialtypizzaversus creating theirown.
Thistaskalsoprovides practicewith:
• writing equations torepresent real-world situations
• comparing values
• calculating withdecimals/money
Implementation Suggestions
Students canworkindividually oringroups forthistask.Thistaskshouldfollowanintroduction or reviewofindependent anddependent variables.
Introduction
Askstudents tothinkabout dependent andindependent variablesintheir lives.Havestudentssuggestquantities that aredependent onother quantities. Examples mayinclude cellphonebillsand number oftextssent,orthefinalscoreofabaseball game andthetotal number ofhits.
Monitoring/FacilitatingtheTask
Askquestions andprompt studentthinking sothat they:
• Notice thattheresponses forquestions 1-7 useonlytwopossible answers: thecostofpizza andthenumber oftoppings.
• Realizethatthecostofthepizzachangesbased onthenumber oftoppings selected, andthat thisdiffersfrom thenumber oftoppings changing basedonthecostofthepizza.
• Selectthecorrect variable,xor y,torepresent their quantities.
• Justify theirsolutions inquestion 12usingtheequations theycreated.
DebriefingtheTask
Havestudents form smallgroupstodiscusstheirresults. Students should comparetheirequations, lookingforsimilarities and differences. Theyshoulddebate anyinconsistencies intheir solutions, usingtheir equations tojustify theanswertheybelievetobecorrect. Eachgroup shoulddecide on theirgroup solutionstoquestion 12andwritetheiranswersontheboard.
Asaclass,reviewthesolutions totheworksheet, leavingtheinformation ontheboardforthefinal discussion. Topicsfordiscussioninclude:
•howstudents determined whichquantitywasdependent andwhichwasindependent
• howstudents representedthedifferent information inanequation
Differentiation
Youmayconsider assigningonlyonesizeortypeof pizzatospecificindividuals inplaceof completing question 12forallsizesandtypes.Students mayalsowishtodesigntheirfavoritepizza anddetermine itscostiftheycreatedtheirownpizzawiththeavailabletoppings.
TechnologyConnection
Asanextstep,toextend thistasktothesecondpart ofthestandard ("analyze therelationship between dependent andindependent variables usinggraphs andtables..."),students canuse graphing software toproduce graphs and/or spreadsheet softwaretoproduce tables usingthegiven information andtheir answers.
ChoicesforStudents
Students maydecidetousethemenu fromalocalpizzeria tocreate thesamecomparisons.This information couldthen beusedtocreate a"BestPizzaDealatJoe'sPizza"pamphlet forcirculation in theclass,school,orcommunity.
MeaningfulContext
Independent anddependent variables areseeninmanydifferentcontexts asidefrom thoseusedin mathematics. Students canexploreindependent anddependent variablesinavarietyofcontexts.Forexample,ifglobalwarmingistheindependent variable, havestudents compose alistofpossibledependentvariablesthat mayberelatedtotheindependent variable. Havestudents research quantitative valuestorepresent eachvariable.
Recommended Resources
• Independent vs.Dependent Variables
Thisanimated videocoversthe-basics ofdependent andindependent variables by comparing playerperformance inasoccergame.
• InternetPizzaServerOrdering Area
Thissiteallowsyoutoselecttraditional aswellasunusual/impossiblepizzatoppings and" order" apizza.Theapplication thenreturnsyourpizzaalongwiththecostofthe pizza.Advanced studentsmayenjoythechallengeoftryingtofigureouttheequation usedtorepresentthecostofthepizza.
• OneThingDepends onAnother
Thismusicvideospoofofan'80spopsongprovidesashort reviewofindependent anddependent variables andincludes domain, range, andfunction information.