2.A2.F.BF.A.1 Write a Function That Describes a Relationship Between Twoquantities

CoinCarpet

StandardsAddressed

1.8.G.C.7 Know and understand the formulas for the volumes of cones, cylinders, and spheres, and use themtosolve real-world and mathematicalproblems

2.A2.F.BF.A.1 Write a function that describes a relationship between twoquantities.

3.Standards for Mathematical Practice, especially: 1. make sense of problems and persevere in solving them;4.model with mathematics; and 5. use appropriate toolsstrategically.

Materials Needed

Eachstudentgroupshouldhaveapenny(preferablyadozenormore).Eachstudent(orgroup)shouldhaveacopyof the studentworksheet.

Before conducting theactivity

1.Measure your classroom and determine its area in squareinches.

2.Handoutarulertoeachstudentorstudentgroup.(Thelastpagehasseveralifyouwouldliketoprintthem—but be sure to tell the printer to print the pdf “actual size” instead ofscaling.)

Conducting theActivity

1.Review,ifnecessarytheratiosofthesidesof30◦–60◦–90◦triangles(e.g.,standardB.G.SRT.B.3)andtheuseofproportions (standards 7.RP.A.2 and7.RP.A.3).

2.Divide the class up into small groups and have them work the first three problems. After most havefinished,

pause

anddiscusstheanswers—especiallyhoweachgroupworkedtoimprovetheiraccuracy.Weoftenskip

over how to obtain accurate measurements inmathematics.

3.Pauseagain

after the next three problems. Show the class why the area of a hexagon circumscribing the

coinshouldbeusedinsteadandhavetheclassworkouttheareaofthishexagon(seetheteacher’sguide).

4.Now let the class finish the final three problems and discuss theiranswers.

Variations

1.Ratherthanusetheroom,considercarpetingthewholeschool(oreachstudent’shouse).Estimatingthefloorarea would be an interesting exercise in thesecases!

2.Use different shapes for tiling the floor: dollar bills, or regular pentagons. (The class could explore whyusingregular 5-gons or 7-gons are much harder than 3, 4, 6, or8-dons.)

3.What is the cheapest coin in the world to carpet the floor with? The cheapest paperbill?

Source

Thisactivityislooselybasedon“CoinCarpet”byDanMeyer(from

accessedJune2016,oneofhis“ThreeActMathTasks”

Chris K. Caldwell, UT Martin,CC BY-SA3.0

Coin Carpeting—StudentWorksheetName:

Date:

1.Your teacher will give you a coin. Write what it is herehere:. Use this choice of coin to answerallof the followingquestions.

2.Using just a ruler determine the diameter of yourcoinand itsradius. Whatdidyou do to improve your accuracy?

3.Whatistheareaofthefaceofyourcoin?Itsvolume?Did you record thesewithcorrectunits?

4.Use a ruler to measure the dimensions of this rectangle. What is itsarea?

5.Howmanyofyourcoincanyou fitintherectangle(laythecoinsflatandtrytominimizethespacebetweenthecoins)?What would this number of coinscost?

6.Basedonyourareacomputationinquestions3and4,whatisthemaximumnumberofyourcoinsthatyoushouldbeabletofitintherectangle? Doesthisanswermatchyouranswerto5?Explainwhyorwhynot.

7.Estimate the area of your classroom’sfloor.How far off might you be (as apercentage)?

8.How many of your coins would it take to carpet the entirefloor?How muchwould

thatcost?How far off might you be (as a percentage)?Round youranswersappropriately (and be able to defend yourchoice)!

9.Which of the following coins would be the cheapest to carpet your classroom floor with: pennies, nickels, dimes,orquarters?Whichwouldcostthemost?Usethebackofthispagetocarefullyexplainwhyyouransweriscorrect.

CoinCarpeting—Teacher’sGuide

1.Yourteacherwillgiveyoucoin.Writewhatitisherehere:penny.Usethischoiceofcointoanswerallofthefollowingquestions.

2.Using just a ruler determine the diameter of your coin (0.750 in or 19.05 mm; see error is divided by 10. Another would be to measure the circumference and divide byπ.)

3.What is the area of the face of your coin? (0.3752π ≈ 0.442 sq in) Its volume? (0.3752π(0.0598in) ≈ 0.0264 cuin)Didyourecordthesewithcorrectunits?(Itwouldbereasonabletoalsoaskyourstudentsthesurfaceareaandthen discuss the accuracy of their answers. The three significant digits I used here is probably more than theywillachieve,butitisimportantforstudentsto have aroughhandleonthesizeoftheirerrors(sotheerrorexpectedintheanswer).

4.Usearulertomeasurethedimensionsofthisrectangle.Whatisitsarea?(Therectangleis3”by6”,so18sqininarea.)

5.Howmanyofyourcoincanyoufitintherectangle?(Laythecoinsflatandminimizetheareabetweenthecoins.)Ifyou filled the rectangle with

yourcoin,whatwouldthetotalcostbe?Firstmakesurethestudent’scarpet(tile)correctly(asabove).Usingcoinsinarectangulararrayismuchmoredifficultvisually—itisveryhardtokeepthelinesstraight.Justgoogle“pennycarpet”forahundredexamples.Next,theanswerdependsonhowyoucount–specificallywhatyoudowiththecoinsattheedges.Itisusuallytosayifyouneedhalfapennyforaspot(e.g.,thetoprowofcoinsabove),tocountitashalf a penny. Done this way, the answer should be 36 or 37 (need parts of a coin to fill the right and left edges).Ifyou count each fraction of a coin as a whole coin, then you might get about 48. Note that the differencebetweenthese methods decreases as the are gets larger (the number of coins on the edge grows linearly, but the numbertocarpetgrowsasthesquareofaside,soifwemadetherectangle100timesaslarge,thepercentagedifferencebetweenthese methods of estimating would decrease by a factor of100).

6.Basedonyourareacomputationinquestions3and4,whatisthemaximumnumberofyourcoinsthatshouldyoube able to fit in the box? Does this answer match your answer to 5? Explain why or whynot.

(The goal is to get the student to just divide the rectangular area by the coin’s area. This will give a value thatisslightlytoolarge(40.7)becausethecoinsdonotcover100%oftherectangle.However,iftheycalculatetheareaof

thehexagon–theyshouldgetananswerthatisveryclose(36.9).Considerleadingyourclasstodothisusingoneofthe three following methods (or another f yourown.

One approach: notice that the triangle shown is the classic 30–60–90 degree trianglesohassidesproportionalto1,2and√3.Thestudentsshouldusethistofinditssides

must be the radius r, r√

3 and its hypotenuse 2r√

3.This means that its areais

11√

2(base)(height)=2(r)(r/

3). Multiply this by 12 (because the hexagon is made of12

of these right triangles) to get 2√3r2 ≈ 3.464r2 ≈ 1.1027πr2. So about 10.27% of the

floorisuncoveredbetweenthecircularcoins(nomatterwhatsizetheyare.)

Anotherapproachwouldbetodissectthehexagonandconvert

ittoarectanglewhoseheightisthediameterofthecoin2rand

√√√

whose base isr/

3 + 2r/ 3=

3r (again from the30–60–90

⇒triangle). This approach could be used by students who donot

know the 30–60–90 triangle because they could use a rulertoestimate the base (hence thearea).

Finally, in the unlikely case that your students know trigonometry, it is easy to show the area of a regular n-gon

circumscribed on a circle of radius ris.

tan(n−2)180◦

7.Estimate the area of your classroom’s floor? (Depends on the room.) How far off might yoube?

(Dependsontheshapeoftheroom.Ifitwasarectangleandtheywereoff1%inwidthand1%inlength,theycouldbeoff2%inarea(or1.01×1.01−1=0.0201);buttheshapemayintroducemanymoreerrors.Havethemcometo

a reasonable conclusion. Trying to be within 10% iscommon.)

8.Howmanyofyourcoinswouldittaketocarpettheentirefloor(trytominimizethespacebetweenthecoins)?Howmuch would that cost? How far off might you be (as a percentage)? Round your answers appropriately (be abletodefend yourchoice)!

(A close estimate of the priceis

area ofroom

area of coin’s hexagon × value ofcoin.

Justusethevalueyoucalculated....Ifthereerrorisestimatedat10%,itwouldmakesensetoroundtotwosignificant digits.)

9.Which of the following coins would be the cheapest to caret your classroom floor with: pennies, nickels, dimes,orquarters. Which would be the most expensive? Carefully explain why your answer iscorrect.

(Asufficientanswer:Apennycoversadimeandadimecosts10timesasmush,soitwouldcostateast10timesasmuch to use dimes. Four penny cover a nickel and six pennies cover each of the other coins, so it would becheaperto use pennies than any othercoin.

A better answer: A close estimate of the priceis

area ofroom

area of coin’s hexagon × value ofcoin.

Foreachofthecoins,thefloor’sareaisthesame.Alsotheareaofthehexagonisaconstanttimestheradiusofthecoin, so the cost is proportionalto

cost factor=Below are the cost factors for variouscoins.

value ofcoin

coin’sdiameter

penny / nickel / dime / quarter / half / dollar
value(cents) / 1 / 5 / 10 / 25 / 50 / 100
diameter(in) / 0.750 / 0.835 / 0.705 / 0.955 / 1.205 / 1.043
costfactor / 1.778 / 7.171 / 20.12 / 27.41 / 34.42 / 91.92
ratio to pennycost / 1.000 / 4.033 / 11.32 / 15.42 / 19.36 / 51.70

Thistableshowswhyitiseasytofindpennycarpetsonline,andtherearealsonickelcarpets,butfewusingtheothercoins.)

Ruler’s(shouldyourclassneedthem.)Measuringaroomwithjustaseveninchrulerisaninterestingtask(unlesstheflooristiled),theclassshoulddiscussstrategiesandcompareanswers.Tellyourpdfviewerto“printactualsize”(notscale to fit) to make sure these rulers arecorrect.

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