Working+With+Algebra+Tiles+(Project)

7/29/2019 Working+With+Algebra+Tiles+(Project)

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Working With Algebra Tiles(Grade 9 projects)

The members of the group are: -

R A A (ID; 739)

A M (ID; 1737)

F A (ID; 778)

Section: 53Math Teacher: Mrs. Inass Ibrahim


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Algebra tiles

Introduction:

Studentswillinvestigatethefactorizationofx

2

+bx+candax

2

+bx+c.Itisthe intent of the project to develop the students insight into the

trinomialfactorizationprocess.Herex 2+bx+c,wherea,b,andcdenote

constantvaluesandthextermsignifiestheunknownvariable,often

canbefactoredintotheproductoftwoterms.Studentswillusealgebra

tilestoidentifythebinomialfactors.Thissetsthestageforthestudents

to later learn how to solve polynomial equations and explore

relationshipsbetweenthetrinomialx2+bx+canditsfactoredform(x

+m)(x+n).

Learning Objectives:

1)Factortrinomialsoftheformx 2+bx+cintotwobinomialfactors.

2)Identifytherelationshipsthatexistbetweenb,c,m,andnwhenx 2+

bx+cisfactoredas(x+m)(x+n).

3)Generalizetheprocessforfactoringtrinomialsoftheform:x 2-bx+c,

x2-bx-cwherebandcarepositiveintegers.


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Task 1

Navigate the web to search about the definition of the word "factor

and the use of algebra tiles to factor trinomials.

(Factoring)

Isthe process ofbreakingdown algebraic equationandexpression into factors,

which when multiplied together give the original. For example, the number 15

factorsintoprimesas35,andthepolynomialx24factorsas(x2)(x+2).Inall

cases,aproductofsimplerobjectsisobtained.

(Usesofalgebratiles)

Weusealgebratilestoidentifythebinomialfactorsandwewilllaterlearnhowtosolvepolynomialequationsandexplorerelationshipbetweenthetrinomialandit

factoredform.

Access the Algebra Tiles (Appendix I) and work on the problems

below.

Exampleof

factoringtrinomials

byusingAlgebra

Tilesfromthesite

http://courses.w

ccnet.edu/~rwh

atcher/VAT/Fact

oring/Factoring/

Level One.html


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Continue Task 1

Model the following quadratic trinomials using Algebra Tiles (Appendix

I), and thenwrite the binomial factors to the right.

Polynomials: X2+bx+c Factors:(x + m)(x + n)

X2+2X+1 (X+1)(X+1)

X2+3X+2 (X+2)(X+1)

X2+7X+6 (X+6)(X+1)

X2+5X+6 (X+3)(X+2)

X2+5X+4 (X+4)(X+1)

2X2+3X+1 (X+1)(2X+1)

2X2+7X+3 (X+3)(2X+1)

3X2+7X+2 (X+2)(3X+1)

4X2+8X+3 (2X+1)(2X+3)


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Task 2

Design and present solid colored model on one of the above examples.


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Continue Task 2

Generalize the process for factoring trinomials of the form: x2 + bx + c

where b and c are positive integers.

Tofactoratrinomialintheform x2 + bx + c wefindtwointegers,m

andp,withasumofbandaproductofc.Thenwritex2+bx+cas

(x+m)(x+p).

x2+bx+c=(x+m)(x+p)whenm+p=bandmp=c

e.g.x2+6x+8=(x+2)(x+4),because2+4=6and(2)(4)=8.

By relating factoring a quadratic trinomial to an area model.

Notethattilesshouldnotbeplacedontopofothertiles,addingthatthe

piecesmustbeusedtocreatearectangle.Andthatswhythedimensions

oftherectanglearethefactorsofthetrinomial.Wecancreatetwo

rectanglesthatlookdifferentbutareactuallycongruentbecause(x+

1)(x+2)isthesamefactorizationas(x+2)(x+1)byvirtueofthe

commutativepropertyofmultiplication.Sowemustarrangeallthetiles

toformarectanglewiththeareax2+5x+6.

((Youshouldberemindedthatyouareworkingwithan

areamodelandthatbyplacingadditionalxtiles,theyareaddingareatothex2tile.))

@Whatarethedimensionsoftherectangle?

Answer:Thedimensionsare(x+3)by(x+2).

Multiplythedimensionsyoufoundfortherectangletoprove

thatx2+5x+6istheareaoftherectangle.

Answer:(x+3)(x+2)=x2+3x+2x+6 =x2+5+6

@Howdothedimensionsoftherectanglerelatetothenumbers5and

6?

Answer:2+3=5and23=6.


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Task 3

Generalize the process for factoring trinomials of the form: x2- bx + c ,

x2

- bx - c where b and c are positive integers.

Whencispositive,itsfactorshavethesamesigns.Bothofthefactors

arepositiveornegativebaseuponthesighofb.Ifbispositive,the

factorsarepositive.Ifbisnegative,thefactorsarenegative.Whencis

negative,itsfactorsareoppositesigns.Todeterminewhichfactorin

positiveandwhichisnegative,lookatthesignofb.Thefactorwiththe

greaterabsolutevaluehasthesamesighasb.

Provide six examples on the above two forms.( 3 examples on each

form).

ProcessForFactoringTrinomials.

X2-bx+c

=(x-m)(x-p)

Example:-

1)4x213x+10

(4x-5)(x-2)

2)3g27g+2

(3g-1)(g-2)

3)2t211t+15

(2t-5)(t-3)

X2-bx-c

=(x-m)(x+p)

Example:-

1)2x23x-9

(2x+3)(x-3)

2)2x2x-1

(2x+1)(x-1)

3)3x211x+20

(3x+4)(x-5)


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In conclusion

Manypeoplebelievethatalgebratilesarenewinmathematics

instruction,however,algebratileshavebeenusedinclassroomssincethemid1980s.Sincethen,algebratileshavegainedpopularityin

teachingcirclesandareregularlypresentedinconjunctionwith

traditionalmethodsinmosttextbooks.Unlikeacalculatororananswer

key,whichdoesthethinkingforyou,algebratilesareatoolthatcan

guidethelearnertowardsagreaterunderstandingoftheconcepts

involvedwithaparticularskill.

Studiesshowthatover80%ofstudentsarevisuallearnersratherthan

auditorylearners.Therefore,itmakessensethatthemajorityof

studentsbenefitfromavisual,concreterepresentationofabstract

algebraicconcepts.Algebratilesdojustthis?theyallowstudentsto

matchaconcepttoatileversustryingtoimagineeverythingintheir

minds.

SoasastudentIbelievethattheAlgebraTilesaresousefuland

especiallyforwhomareweakincalculations.AndintheendIwould

liketothankmymathteacher,Ms.InassIbrahimforgivingmethe

opportunityindoingaprojectinmyfavoritelesson.SothanksAlot!

Best Whishes,

Al- Ain Female campus

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Working+With+Algebra+Tiles+(Project)

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