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 Meat
Small / Medium / Large
Specialty price
“Create your own” price

OR

Specialty pizza: I Love Meat
Small / 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 Meat
Small / 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.