ERIC ED540551: Solving the Unknown with Algebra: Poster/Teaching Guide for Pre-Algebra Students. Expect the Unexpected with Math[R]

2013·6.2 MB·English

by  ERIC

Lesson 1 Lesson 2 Worksheet Answer Key Lesson 1
continued Using Mathematical Analyzing Change/ Dear Teacher: Using Mathematical
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Incorporate the value of x back into the original equation to • W orksheet 2 (Content Connections: Percentages, Exponents) between a tree and its shadow. A one-foot 6. 1 0(1 + 1.0)6 = 640 bacteria per cubic .1 = .9 formula to solve for variables
show both sides being equal. • B onus Worksheet 2 (Content Connections: Conversion of shadow is cast by a three-foot tree, so a centimeter and 10(1 + 1.0)7 = 1,280 The probability of two green lights in a row Time: 30–40 minutes, depending on class review needs;
7. M ention that keeping an equation in balance applies to many Percentages to Decimals, Exponents) 30-foot shadow would be cast by a 90-foot bacteria per cubic centimeter, so the on Washington Road = .9 ˙ .9 = .81 or 81.0% additional time for worksheets
real-life situations. Show students the map and ask them to • T ake-Home Activity 2 (Content Connections: Percentages, tree. As a proportion this is shown as 30 ft./x sandwich has been in the locker between 3. The probability of four green lights in a row Materials:
explain how a map is a representation of the area it depicts. Exponents) feet = 1 foot /3 feet. Solving for x leaves x = six and seven weeks. on Washington Road = .9 ˙ .9 ˙ .9 ˙ .9 = .656 Getting Started: • B alance scale with weights (drawings on the board will also
90 feet. Therefore, 60 feet from the school Take-Home Activity 2: The Case of the or 65.6%
Answers should include that map distances are proportional suffice)
tmoe tahseu rreinagl dthiset adniscteasn tchee by ertewpereesne tnwt.o D meampo lnosctartaitoen bs ya nd D1.I R SehCoTIwO aN sSa: vings or investment ad to the class. Ask students to 2. b 1u8i0l dfeinegt. i sS neot tu ap saa pfero dpiostratinocne .6 ft./3 ft. = D1.e c Cauyrirnegn tC avarlue = 20,000(1 – .20)4 = 20,000 ˙ TBaakreg-aHino mHuen Atcetrivity 3: The Case of the Kid Poster/Teaching Guide for Pre-Algebra Students Tphreis- aplrgoegbrraam in iss tdruescitgionne din t oth seuspep aleremaesn: t your existing • Mcoaopr d(ein.ga.t,e ws,o arnldd, as tkaetye ,i nectclu.)d winigth s cloanlegitude/latitude and/or
using proportions to calculate the actual distance. For explain what the interest rate (e.g., 1.75%) means. Establish x ft./90 ft. Solving for x leaves x = 180 .84 = 20,000 ˙ .4096 = $8,192.00 1. Costs for each rental company: 1. solving for unknowns;
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appropriate assumption depending on the means of travel = $20. Work the calculation as necessary and generalize the For 32 servings: $5,242.88 = $2,949.12 The family should use Let’s Go because it’s world questions The materials are taught through this story line: arm. Ask why the scale is in balance (because each side has
(e.g., a driving speed of 55 mph or flying speed of 600 mph). formula by writing the following formula for simple interest: Butter: 1 2/3 cups Now Try This: the cheapest. Athena and Rick are two students frequently called the same weight). Ask what would happen if 5 grams were
Divide distance by rate to determine trip length. I = p . r . t, where I = interest, p = principal (amount Sugar: 1 cup 1. 30,000(1 – .20)3 = 30,000 ˙ .83 = 30,000 ˙ 2. t = $220w + .10n, where t = total cost, w = through powerful upon to use their powerful mathematical thinking added to both sides. Demonstrate, reiterating that the scale
9. W rite “Distance = Rate . Time” on the board and explain what deposited), r = rate (of interest), and t = time (in years). It Flour: 3 1/3 cups .512 = $15,360.00 number of weeks rented, n = number of miles mathematical to solve seemingly unsolvable mysteries, problems, is still balanced because the same amount was added to each
eoaf cah r etelartmio mnsehainps i.n I nthdeic raetael twhoartl ad ,f ojursmt ualsa a is m aa rpe pisr eas entation .mt hTieigm hCeDt) a.a lItsf om n baeetcu eurssisteayf rueylq, t ueoax slpshl aoPirwnin ttchoie pt hacllea + cs slIan tsthsea rtteh tsahtt e (t Phtroeitn iancli tvpeaarlelu s.e tR oaft e W A Colhmcioponpnuidnts:g: 4 4C rc 2eu/ap3ms c:u 6p 2s/3 tablespoons 32.. 4 205,,000000((11 –– ..2200))15 == 2450,,000000 ˙˙ ..885 = = $ 4200,,000000 ˙.0 0 Nowd rTirvye Tnhis: thinking! aanndd qthuee swtoiornkssh. eThetesy i pnosisdt ec aesneg fiaglees sotnu dae mntast hto b floolglo, w 2. s Aisdke w. hat would happen if we only operated on one side of the
representation of a real place. Show how it’s possible to .32768 = $13,107.20 Let’s Go formula is t = $220w + .10n along with the duo and solve problems by applying scale. What if 2 grams were taken off one side? Demonstrate
rate is in effect the entire term of the CD. (If a student asks, Chocolate: 10 2/3 squares
determine one of the terms if the other two are known, e.g., Worksheet 3: The Case of the Screeching Tires Develop a formula for Smooth Ride: mathematical skills and a logical, systematic approach. that the scale is out of balance because the action taken on
it may be necessary to explain that interest also comes into Now Try This: For 8 servings:
one could calculate rate if time and distance are known. 1. Tire Tracks 1 = 24 9 .375 = 225 = 15 mph t = $100w + .40n one side wasn’t repeated on the other. Return the 2 grams to
Wforrimteu tlha,e e f.ogr.m, ru =la d a/st da n=d r t. =t adn/dr. show how to manipulate the 3. ptA hlsaeky bs wtouhrdreoenwn ates rfi wpnahaynastc iitanhlte eiynr estthsitit nutokti wtohnoe ul ofildan nhasna pcmipaoeln nien iysf. tt iIhtnue tt bihoaenns.ek) cases, B S Fluuogtutarer:r: 5: 1 5//6/41 c c2uu pcpu p TTTiiirrreee TTTrrraaaccckkksss 324 === 222444 ˙˙˙ 62 3 47= . =5 1=45 479 6=0 =01 22= 4 m3 m0p hpm hp h Swoit nihnce ebr otshitnhec fteor itrphm eiusy lf abosor atthnw deo q swueeta elt khts.e ,m su ebqsutaitlu ttoe e2a fcohr Tesuhapcrehpe lle elmessseosnnotn efe dpa bltayun rase bst eoaan wcuhos r bwkasoshircke esphtre,e -aeantl dga enisbd ra talas coko en-cheopmtse; 3. b Aaskla wnhcea tt hweo sucladl he.appen if we tripled both sides. Demonstrate
10. Pt hoein mt aopu.t Dcoeoterrdminiantee ws hanetdh/eorr sltoundgeintutds ek/nloawti thuodwe mtoa erxkpinrgesss o an oinftfeerreedst a, itnwdoic-yaetea rt hCaDt. tBheec bauansek twhiilsl pisa ay cthasee i notfe sriemspt laet the W Alhmipopnidnsg: 1Cr 1e/a6m c:u 1p 2s/3 tablespoons 2. T Tihrree Ter oacf kths e5 fi=v e 2c4a r ˙˙s 1an5a0l =yz e3d, 6w0e0r e= s6p0e emdpinhg, 424200 +˙ .21 +0 n.1 =0 2n0 =0 1 +0 .04 ˙0 n2 + .40n atoc tailvliotwy. t Tehaec hmearst etroi aulsse a trhee dmes iing nwehda tweivtehr fl oerxdiebri lbiteys t tohpaetr athtieo sncsa, ldei visi dseti ltlh bea wlaenigcehdt .o Tnh eeanc, hto s iddeem boyn 3s,t srahtoew iinnvge rtshea t
location’s coordinates in x,y format (and longitude/latitude Coconut: 1 cup Subtract 200 and .10n from both sides: inverse operations “cancel each other out.”
end of each year. This results in a $20 payment at the end of some more than twice the speed limit. The supplements the scope and sequence of his/her pre-
format if desired). Make sure they can determine horizontal each year. Plug these numbers into the formula I = p . r . t to Chocolate: 2 2/3 squares principal should request that the town install 240 = .30n More Free Math Programs: algebra instruction and curriculum. The worksheets 4. I ndicate that equations work like the balancing scale, with
and vertical distance using coordinates. show how a total of $40 in interest will be paid. To show how Worksheet 2: A Case of Interest speed bumps. Divide both sides by .3 leaving n = 800 Visit www.actuarialfoundation.org/programs/for_teachers.shtml also include additional math content connections, the equal sign acting as the fulcrum. (Many students have
11. D istribute Worksheet 1. Read the introduction as a class 1. After two years students will have the $500 the common misconception that the equal sign means “the
much the depositor has in total, show how to add the interest Now Try This: highlighted in the materials section of each lesson.
and review the “Additional Clues” (key facts) before students they deposited plus interest calculated answer to the problem is” rather than “is the same as.”)
to the principal, i.e., $1,000 + $40 = $1,040. Tire Tracks 1 = 21 46 2/3 = 400 = 20 mph
complete the worksheet. Make sure students are able to using the formula I = p ˙ r ˙ t. Interest = ˙ The take-home activities further encourage students Before proceeding, make sure students understand this
determine each route’s distance using the coordinates and 4. A sk what would happen if the depositor kept the first year’s $500 ˙ .049 ˙ 2 = $49, so the students will Tire Tracks 2 = ˙ 2 64 6 2/3 = 1,600 = 40 mph to apply their math skills to real-world situations. distinction and the goal of keeping things in balance.
interest in the account to “grow.” Show how the depositor have $500 + $49 = $549 Tire Tracks 3 = 21 40 4 1/6 = 2,500 = 50 mph
the map’s scale. Review answers as a class. ˙ The classroom poster displays the important concept 5. W rite a simple equation on the board, e.g., x + 7 = 9. Indicate
would earn an extra 40¢ by using the $20 as principal for one 2–3. Using the growth formula y = a(1+r)n, after Two of the three cars exceeded the 20 mph DevelOPeD WITH
12. S tudents will build on these skills in Bonus Worksheet 1 of keeping equations in balance while solving for that the goal is, as with the scale, to keep the equation in
year (the second year of the CD’s term) at 2%. Explain that two years the students will have $500(1 speed limit.
and Take-Home Activity 1. If students require additional unknowns and reminds students, particularly visual balance. Indicate that we can solve for x if we isolate it on
this is an example of compound interest.
support, review worksheets as a class. learners, of this fundamental pre-algebra concept. one side of the equation through manipulations that keep
5. Indicate that there is a formula that can be used to calculate
the equation in balance. Depending on class support needed,
how money grows with compound interest: y = a(1+r)n where
Lesson 1 Lesson 2 Worksheet Answer Key Lesson 1
continued Using Mathematical Analyzing Change/ Dear Teacher: Using Mathematical
skoethheqneoeu b pwasoiut nthibohgotn rwst aiha dctenoet dies os .q–on Wu7l,vM a rlueiettn iafeooodv n–rien d 7irxng tuue hbxsn eai=dln l9 eas2g rno. t/ tcnRhheP eete bph s7ryeec aa ootrtaline gkwp sthii nhmttoge hse iltrodtehateftehpt. e ihssCraioood morme,nne aope anfls-e cstdthtt eeieo p n OSiaannntBtuddejd reiceenoCnsvmTtteI sipsvs wtoe ianiuSslgln: G i b;dt ec pria neaoltrbcetulawreleia ntstostte; it hsdaoipe ms npaaptlvyilifen nyt h gwde h aDt ecay Formula WD1.io sr R˙Rtka oo1snuu,hc0tteeee0e 0ABt f==1e :8e1 Tt,00)h, 00e00 C 0aft sf.te (. 5o( 4fu tnuhinteist Ds+ o +3u 5ub nutfintuistl s= +8 1u nuintist 6Nfno0eole0lw+od/ .wT 10tri.o4yn1 8rg0Ta)h 2e2iis5 qsoe :u=r: Sa $6att50,ui o40adn9 e== .nt 1 oa$t5s( 5fi.1 4w gS.40iuol.5lr 2 ten)2h 2eo,is eu6 dit0s ht0 too h= wse ao bm(lve1uet.ct1 teh0hr 2et dh5 ee)a,y l . BT1r.oa nn Em(u2lsmis pnm o WuSirtntoteruaerstkte aeitsots hr nloaeiugte thltei tg3: 1h5: t-T˙ m2h . e5i˙n Cp.u5artos epeb rr aouobbnfi a tltibhitmiyel i eaTt tya+ rla id(gt2y hl itg 1h)t + Grades 6–8 ® Part 1 of 2 WAspsteloraglelcnv-eodiabnmalggrrea edf ,obts o rraa a u Snn nsodekk lwdinvl elomissnw iiaggnntncs ht,le uhpaddern oti dnoUg gm rnha uaekmnslnpi inaop slguwtig ulfannodt reeiwnmdngi tutw selh iaqpt shur,aa NcttiCoiTcnMes . O•B Sawj entiutC ehdTq IetuvhnaeettsS ei ow:q niul law ulo nsridkgMesn rl sisokteaerdn vaid nes tcghal alset /Proportions
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m2bxoa tfnhroi psmiud lbeaosti tbohyn s,4i fid kreesset, p alesd atdhviineng eg q8 1u t2ao =t bi o4onxt.h i Sn sh ibdoaewlsa ,hn tochewe, n dle isvauivdbiintnrgag cx t =in 3g. • Scatulcduelantto crasl)culators (for Worksheet 2; can also be done without P1.la n Tuopti cnaa gplcruolpaotret tiohne shheoigwhint go ft ah em raetluarteio tnrsehei,p s et ctthhueab nilco a cc kemenortn iamtth el.etears,t sfiov teh wee seaknsd wwhicichh w ias sm ino r e EpRxroopabrdea isbssi e1lid t–y a tosh fae a dp gerroceibemanab llii,lg i1thy0t 0o o%fn a = Wr e1ad.s hSliiogn,hg ttth ooenr 1 – STihnec eArcetluya,rial Foundation Scholastic Inc. mthaep v’as rsicaabllee sa nind tchoeo frodrimnautleas d t o= cra .l ctu; alantde mdiastnainpucela; tied ethneti fy
Incorporate the value of x back into the original equation to • W orksheet 2 (Content Connections: Percentages, Exponents) between a tree and its shadow. A one-foot 6. 1 0(1 + 1.0)6 = 640 bacteria per cubic .1 = .9 formula to solve for variables
show both sides being equal. • B onus Worksheet 2 (Content Connections: Conversion of shadow is cast by a three-foot tree, so a centimeter and 10(1 + 1.0)7 = 1,280 The probability of two green lights in a row Time: 30–40 minutes, depending on class review needs;
7. M ention that keeping an equation in balance applies to many Percentages to Decimals, Exponents) 30-foot shadow would be cast by a 90-foot bacteria per cubic centimeter, so the on Washington Road = .9 ˙ .9 = .81 or 81.0% additional time for worksheets
real-life situations. Show students the map and ask them to • T ake-Home Activity 2 (Content Connections: Percentages, tree. As a proportion this is shown as 30 ft./x sandwich has been in the locker between 3. The probability of four green lights in a row Materials:
explain how a map is a representation of the area it depicts. Exponents) feet = 1 foot /3 feet. Solving for x leaves x = six and seven weeks. on Washington Road = .9 ˙ .9 ˙ .9 ˙ .9 = .656 Getting Started: • B alance scale with weights (drawings on the board will also
90 feet. Therefore, 60 feet from the school Take-Home Activity 2: The Case of the or 65.6%
Answers should include that map distances are proportional suffice)
tmoe tahseu rreinagl dthiset adniscteasn tchee by ertewpereesne tnwt.o D meampo lnosctartaitoen bs ya nd D1.I R SehCoTIwO aN sSa: vings or investment ad to the class. Ask students to 2. b 1u8i0l dfeinegt. i sS neot tu ap saa pfero dpiostratinocne .6 ft./3 ft. = D1.e c Cauyrirnegn tC avarlue = 20,000(1 – .20)4 = 20,000 ˙ TBaakreg-aHino mHuen Atcetrivity 3: The Case of the Kid Poster/Teaching Guide for Pre-Algebra Students Tphreis- aplrgoegbrraam in iss tdruescitgionne din t oth seuspep aleremaesn: t your existing • Mcoaopr d(ein.ga.t,e ws,o arnldd, as tkaetye ,i nectclu.)d winigth s cloanlegitude/latitude and/or
using proportions to calculate the actual distance. For explain what the interest rate (e.g., 1.75%) means. Establish x ft./90 ft. Solving for x leaves x = 180 .84 = 20,000 ˙ .4096 = $8,192.00 1. Costs for each rental company: 1. solving for unknowns;
eshxaomw phloew, i 1f 1 in in./c2h0 =0 2m0i0. =m 2il.e5s i ann./dx y moui. more 5a0su0r medil e2s.5. inches, tfimhnoaantn etchyi eaw li iintnhtse ttrhietseut ti rinoastntei tt roue ttpihoreens .de enptos stihteo rr ainte r eotfu prany fmore dnet pboys tihtien g N4se5ot-wtfio nTorgyt -u Ttpha lials :bp urToihlpdeoi lnretgnio gcnat.hn Iobnfe ta dh seihst eacrdamoseiwn, eo1df/ a3b y = x/45. 2. $ Uye2sai0nr,sg0 =0t h0$ e2˙ d0 .e,20c60a20y1 (f41o4 r– m= . u2$l05a),,26 t4=h2 e$. 28v0a8,l.u0 eS0 i0an˙fct.ee8 r6t hs=ie x $SLem5t5o’s5o .Gt0ho0 :R $id2e2:0 $ ˙1 20 0+ ˙.1 20 +˙ .14,01 5˙0 1 =,1 45400 = + 2 10105 + = FAollilgonws t hwei atdhv NenCtTuMre sS tandards ClaPsossrtoeorm 2. mbaalnanipcuel;a ting equations while keeping them in •• W BC ooonnrnkuseshc Wteieootrn k1ss :(h MCeoeenta ts1eu narnet mdCo Tenannkte)e-cHtioomnse: AOcrdtievriteyd 1 P (aCirosn)tent
8. Utt rhsaaivnte gtlh bdeei stttiwmaeneec ndese t phcaeeln ctdwusloa o tpneo dhi,no atwss. kf S ahtsout wdoe nlnoetn isgs s tihrtao wvuoelduli lnidng dt.a iMckaeat kteeo an 2. Uwsitihn ga t2h%e einxtaemrepslte roaft ea, $ s1h,o0w0 0th CeD c daelcpuolsaittieodn f $o1r ,o0n0e0 y .e .a0r2 . 1 STPaoroklvpei-onHrgto ifmoonre sx A lecativveist yx 1=: 1 T5h efe Ceat.se of Sweet ctahaderd’ siat mvioaonluuaenl ttw woaofs t y$he8ea, rd1se9 wc2lo.i0nu0eld ai nbft eve ar$ lf8uo,eu1 ir9n 2y te.h0ae0rs –, U4Ch6ne0ca l=ep $ TWe6hd6de0ey.0’lss0:: $.6300 0˙ ˙1 ,21 5=0 $ =6 0$06.9000.00 oasf Rthiceky asnodlv Ae trheeanl-a Inside 34.. u wsoirnkgin fogr wmiuthla psr toop soorltvioen rse,a rla-wdiocrallds ,c hanaldle enxgpeosn; eanntds. D1.I RS ehCoTIwO sNtSu:dents a balancing scale without weights on either
appropriate assumption depending on the means of travel = $20. Work the calculation as necessary and generalize the For 32 servings: $5,242.88 = $2,949.12 The family should use Let’s Go because it’s world questions The materials are taught through this story line: arm. Ask why the scale is in balance (because each side has
(e.g., a driving speed of 55 mph or flying speed of 600 mph). formula by writing the following formula for simple interest: Butter: 1 2/3 cups Now Try This: the cheapest. Athena and Rick are two students frequently called the same weight). Ask what would happen if 5 grams were
Divide distance by rate to determine trip length. I = p . r . t, where I = interest, p = principal (amount Sugar: 1 cup 1. 30,000(1 – .20)3 = 30,000 ˙ .83 = 30,000 ˙ 2. t = $220w + .10n, where t = total cost, w = through powerful upon to use their powerful mathematical thinking added to both sides. Demonstrate, reiterating that the scale
9. W rite “Distance = Rate . Time” on the board and explain what deposited), r = rate (of interest), and t = time (in years). It Flour: 3 1/3 cups .512 = $15,360.00 number of weeks rented, n = number of miles mathematical to solve seemingly unsolvable mysteries, problems, is still balanced because the same amount was added to each
eoaf cah r etelartmio mnsehainps i.n I nthdeic raetael twhoartl ad ,f ojursmt ualsa a is m aa rpe pisr eas entation .mt hTieigm hCeDt) a.a lItsf om n baeetcu eurssisteayf rueylq, t ueoax slpshl aoPirwnin ttchoie pt hacllea + cs slIan tsthsea rtteh tsahtt e (t Phtroeitn iancli tvpeaarlelu s.e tR oaft e W A Colhmcioponpnuidnts:g: 4 4C rc 2eu/ap3ms c:u 6p 2s/3 tablespoons 32.. 4 205,,000000((11 –– ..2200))15 == 2450,,000000 ˙˙ ..885 = = $ 4200,,000000 ˙.0 0 Nowd rTirvye Tnhis: thinking! aanndd qthuee swtoiornkssh. eThetesy i pnosisdt ec aesneg fiaglees sotnu dae mntast hto b floolglo, w 2. s Aisdke w. hat would happen if we only operated on one side of the
representation of a real place. Show how it’s possible to .32768 = $13,107.20 Let’s Go formula is t = $220w + .10n along with the duo and solve problems by applying scale. What if 2 grams were taken off one side? Demonstrate
rate is in effect the entire term of the CD. (If a student asks, Chocolate: 10 2/3 squares
determine one of the terms if the other two are known, e.g., Worksheet 3: The Case of the Screeching Tires Develop a formula for Smooth Ride: mathematical skills and a logical, systematic approach. that the scale is out of balance because the action taken on
it may be necessary to explain that interest also comes into Now Try This: For 8 servings:
one could calculate rate if time and distance are known. 1. Tire Tracks 1 = 24 9 .375 = 225 = 15 mph t = $100w + .40n one side wasn’t repeated on the other. Return the 2 grams to
Wforrimteu tlha,e e f.ogr.m, ru =la d a/st da n=d r t. =t adn/dr. show how to manipulate the 3. ptA hlsaeky bs wtouhrdreoenwn ates rfi wpnahaynastc iitanhlte eiynr estthsitit nutokti wtohnoe ul ofildan nhasna pcmipaoeln nien iysf. tt iIhtnue tt bihoaenns.ek) cases, B S Fluuogtutarer:r: 5: 1 5//6/41 c c2uu pcpu p TTTiiirrreee TTTrrraaaccckkksss 324 === 222444 ˙˙˙ 62 3 47= . =5 1=45 479 6=0 =01 22= 4 m3 m0p hpm hp h Swoit nihnce ebr otshitnhec fteor itrphm eiusy lf abosor atthnw deo q swueeta elt khts.e ,m su ebqsutaitlu ttoe e2a fcohr Tesuhapcrehpe lle elmessseosnnotn efe dpa bltayun rase bst eoaan wcuhos r bwkasoshircke esphtre,e -aeantl dga enisbd ra talas coko en-cheopmtse; 3. b Aaskla wnhcea tt hweo sucladl he.appen if we tripled both sides. Demonstrate
10. Pt hoein mt aopu.t Dcoeoterrdminiantee ws hanetdh/eorr sltoundgeintutds ek/nloawti thuodwe mtoa erxkpinrgesss o an oinftfeerreedst a, itnwdoic-yaetea rt hCaDt. tBheec bauansek twhiilsl pisa ay cthasee i notfe sriemspt laet the W Alhmipopnidnsg: 1Cr 1e/a6m c:u 1p 2s/3 tablespoons 2. T Tihrree Ter oacf kths e5 fi=v e 2c4a r ˙˙s 1an5a0l =yz e3d, 6w0e0r e= s6p0e emdpinhg, 424200 +˙ .21 +0 n.1 =0 2n0 =0 1 +0 .04 ˙0 n2 + .40n atoc tailvliotwy. t Tehaec hmearst etroi aulsse a trhee dmes iing nwehda tweivtehr fl oerxdiebri lbiteys t tohpaetr athtieo sncsa, ldei visi dseti ltlh bea wlaenigcehdt .o Tnh eeanc, hto s iddeem boyn 3s,t srahtoew iinnvge rtshea t
location’s coordinates in x,y format (and longitude/latitude Coconut: 1 cup Subtract 200 and .10n from both sides: inverse operations “cancel each other out.”
end of each year. This results in a $20 payment at the end of some more than twice the speed limit. The supplements the scope and sequence of his/her pre-
format if desired). Make sure they can determine horizontal each year. Plug these numbers into the formula I = p . r . t to Chocolate: 2 2/3 squares principal should request that the town install 240 = .30n More Free Math Programs: algebra instruction and curriculum. The worksheets 4. I ndicate that equations work like the balancing scale, with
and vertical distance using coordinates. show how a total of $40 in interest will be paid. To show how Worksheet 2: A Case of Interest speed bumps. Divide both sides by .3 leaving n = 800 Visit www.actuarialfoundation.org/programs/for_teachers.shtml also include additional math content connections, the equal sign acting as the fulcrum. (Many students have
11. D istribute Worksheet 1. Read the introduction as a class 1. After two years students will have the $500 the common misconception that the equal sign means “the
much the depositor has in total, show how to add the interest Now Try This: highlighted in the materials section of each lesson.
and review the “Additional Clues” (key facts) before students they deposited plus interest calculated answer to the problem is” rather than “is the same as.”)
to the principal, i.e., $1,000 + $40 = $1,040. Tire Tracks 1 = 21 46 2/3 = 400 = 20 mph
complete the worksheet. Make sure students are able to using the formula I = p ˙ r ˙ t. Interest = ˙ The take-home activities further encourage students Before proceeding, make sure students understand this
determine each route’s distance using the coordinates and 4. A sk what would happen if the depositor kept the first year’s $500 ˙ .049 ˙ 2 = $49, so the students will Tire Tracks 2 = ˙ 2 64 6 2/3 = 1,600 = 40 mph to apply their math skills to real-world situations. distinction and the goal of keeping things in balance.
interest in the account to “grow.” Show how the depositor have $500 + $49 = $549 Tire Tracks 3 = 21 40 4 1/6 = 2,500 = 50 mph
the map’s scale. Review answers as a class. ˙ The classroom poster displays the important concept 5. W rite a simple equation on the board, e.g., x + 7 = 9. Indicate
would earn an extra 40¢ by using the $20 as principal for one 2–3. Using the growth formula y = a(1+r)n, after Two of the three cars exceeded the 20 mph DevelOPeD WITH
12. S tudents will build on these skills in Bonus Worksheet 1 of keeping equations in balance while solving for that the goal is, as with the scale, to keep the equation in
year (the second year of the CD’s term) at 2%. Explain that two years the students will have $500(1 speed limit.
and Take-Home Activity 1. If students require additional unknowns and reminds students, particularly visual balance. Indicate that we can solve for x if we isolate it on
this is an example of compound interest.
support, review worksheets as a class. learners, of this fundamental pre-algebra concept. one side of the equation through manipulations that keep
5. Indicate that there is a formula that can be used to calculate
the equation in balance. Depending on class support needed,
how money grows with compound interest: y = a(1+r)n where
Lesson 1 Lesson 2 Worksheet Answer Key Lesson 1
continued Using Mathematical Analyzing Change/ Dear Teacher: Using Mathematical
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Incorporate the value of x back into the original equation to • W orksheet 2 (Content Connections: Percentages, Exponents) between a tree and its shadow. A one-foot 6. 1 0(1 + 1.0)6 = 640 bacteria per cubic .1 = .9 formula to solve for variables
show both sides being equal. • B onus Worksheet 2 (Content Connections: Conversion of shadow is cast by a three-foot tree, so a centimeter and 10(1 + 1.0)7 = 1,280 The probability of two green lights in a row Time: 30–40 minutes, depending on class review needs;
7. M ention that keeping an equation in balance applies to many Percentages to Decimals, Exponents) 30-foot shadow would be cast by a 90-foot bacteria per cubic centimeter, so the on Washington Road = .9 ˙ .9 = .81 or 81.0% additional time for worksheets
real-life situations. Show students the map and ask them to • T ake-Home Activity 2 (Content Connections: Percentages, tree. As a proportion this is shown as 30 ft./x sandwich has been in the locker between 3. The probability of four green lights in a row Materials:
explain how a map is a representation of the area it depicts. Exponents) feet = 1 foot /3 feet. Solving for x leaves x = six and seven weeks. on Washington Road = .9 ˙ .9 ˙ .9 ˙ .9 = .656 Getting Started: • B alance scale with weights (drawings on the board will also
90 feet. Therefore, 60 feet from the school Take-Home Activity 2: The Case of the or 65.6%
Answers should include that map distances are proportional suffice)
tmoe tahseu rreinagl dthiset adniscteasn tchee by ertewpereesne tnwt.o D meampo lnosctartaitoen bs ya nd D1.I R SehCoTIwO aN sSa: vings or investment ad to the class. Ask students to 2. b 1u8i0l dfeinegt. i sS neot tu ap saa pfero dpiostratinocne .6 ft./3 ft. = D1.e c Cauyrirnegn tC avarlue = 20,000(1 – .20)4 = 20,000 ˙ TBaakreg-aHino mHuen Atcetrivity 3: The Case of the Kid Poster/Teaching Guide for Pre-Algebra Students Tphreis- aplrgoegbrraam in iss tdruescitgionne din t oth seuspep aleremaesn: t your existing • Mcoaopr d(ein.ga.t,e ws,o arnldd, as tkaetye ,i nectclu.)d winigth s cloanlegitude/latitude and/or
using proportions to calculate the actual distance. For explain what the interest rate (e.g., 1.75%) means. Establish x ft./90 ft. Solving for x leaves x = 180 .84 = 20,000 ˙ .4096 = $8,192.00 1. Costs for each rental company: 1. solving for unknowns;
eshxaomw phloew, i 1f 1 in in./c2h0 =0 2m0i0. =m 2il.e5s i ann./dx y moui. more 5a0su0r medil e2s.5. inches, tfimhnoaantn etchyi eaw li iintnhtse ttrhietseut ti rinoastntei tt roue ttpihoreens .de enptos stihteo rr ainte r eotfu prany fmore dnet pboys tihtien g N4se5ot-wtfio nTorgyt -u Ttpha lials :bp urToihlpdeoi lnretgnio gcnat.hn Iobnfe ta dh seihst eacrdamoseiwn, eo1df/ a3b y = x/45. 2. $ Uye2sai0nr,sg0 =0t h0$ e2˙ d0 .e,20c60a20y1 (f41o4 r– m= . u2$l05a),,26 t4=h2 e$. 28v0a8,l.u0 eS0 i0an˙fct.ee8 r6t hs=ie x $SLem5t5o’s5o .Gt0ho0 :R $id2e2:0 $ ˙1 20 0+ ˙.1 20 +˙ .14,01 5˙0 1 =,1 45400 = + 2 10105 + = FAollilgonws t hwei atdhv NenCtTuMre sS tandards ClaPsossrtoeorm 2. mbaalnanipcuel;a ting equations while keeping them in •• W BC ooonnrnkuseshc Wteieootrn k1ss :(h MCeoeenta ts1eu narnet mdCo Tenannkte)e-cHtioomnse: AOcrdtievriteyd 1 P (aCirosn)tent
8. Utt rhsaaivnte gtlh bdeei stttiwmaeneec ndese t phcaeeln ctdwusloa o tpneo dhi,no atwss. kf S ahtsout wdoe nlnoetn isgs s tihrtao wvuoelduli lnidng dt.a iMckaeat kteeo an 2. Uwsitihn ga t2h%e einxtaemrepslte roaft ea, $ s1h,o0w0 0th CeD c daelcpuolsaittieodn f $o1r ,o0n0e0 y .e .a0r2 . 1 STPaoroklvpei-onHrgto ifmoonre sx A lecativveist yx 1=: 1 T5h efe Ceat.se of Sweet ctahaderd’ siat mvioaonluuaenl ttw woaofs t y$he8ea, rd1se9 wc2lo.i0nu0eld ai nbft eve ar$ lf8uo,eu1 ir9n 2y te.h0ae0rs –, U4Ch6ne0ca l=ep $ TWe6hd6de0ey.0’lss0:: $.6300 0˙ ˙1 ,21 5=0 $ =6 0$06.9000.00 oasf Rthiceky asnodlv Ae trheeanl-a Inside 34.. u wsoirnkgin fogr wmiuthla psr toop soorltvioen rse,a rla-wdiocrallds ,c hanaldle enxgpeosn; eanntds. D1.I RS ehCoTIwO sNtSu:dents a balancing scale without weights on either
appropriate assumption depending on the means of travel = $20. Work the calculation as necessary and generalize the For 32 servings: $5,242.88 = $2,949.12 The family should use Let’s Go because it’s world questions The materials are taught through this story line: arm. Ask why the scale is in balance (because each side has
(e.g., a driving speed of 55 mph or flying speed of 600 mph). formula by writing the following formula for simple interest: Butter: 1 2/3 cups Now Try This: the cheapest. Athena and Rick are two students frequently called the same weight). Ask what would happen if 5 grams were
Divide distance by rate to determine trip length. I = p . r . t, where I = interest, p = principal (amount Sugar: 1 cup 1. 30,000(1 – .20)3 = 30,000 ˙ .83 = 30,000 ˙ 2. t = $220w + .10n, where t = total cost, w = through powerful upon to use their powerful mathematical thinking added to both sides. Demonstrate, reiterating that the scale
9. W rite “Distance = Rate . Time” on the board and explain what deposited), r = rate (of interest), and t = time (in years). It Flour: 3 1/3 cups .512 = $15,360.00 number of weeks rented, n = number of miles mathematical to solve seemingly unsolvable mysteries, problems, is still balanced because the same amount was added to each
eoaf cah r etelartmio mnsehainps i.n I nthdeic raetael twhoartl ad ,f ojursmt ualsa a is m aa rpe pisr eas entation .mt hTieigm hCeDt) a.a lItsf om n baeetcu eurssisteayf rueylq, t ueoax slpshl aoPirwnin ttchoie pt hacllea + cs slIan tsthsea rtteh tsahtt e (t Phtroeitn iancli tvpeaarlelu s.e tR oaft e W A Colhmcioponpnuidnts:g: 4 4C rc 2eu/ap3ms c:u 6p 2s/3 tablespoons 32.. 4 205,,000000((11 –– ..2200))15 == 2450,,000000 ˙˙ ..885 = = $ 4200,,000000 ˙.0 0 Nowd rTirvye Tnhis: thinking! aanndd qthuee swtoiornkssh. eThetesy i pnosisdt ec aesneg fiaglees sotnu dae mntast hto b floolglo, w 2. s Aisdke w. hat would happen if we only operated on one side of the
representation of a real place. Show how it’s possible to .32768 = $13,107.20 Let’s Go formula is t = $220w + .10n along with the duo and solve problems by applying scale. What if 2 grams were taken off one side? Demonstrate
rate is in effect the entire term of the CD. (If a student asks, Chocolate: 10 2/3 squares
determine one of the terms if the other two are known, e.g., Worksheet 3: The Case of the Screeching Tires Develop a formula for Smooth Ride: mathematical skills and a logical, systematic approach. that the scale is out of balance because the action taken on
it may be necessary to explain that interest also comes into Now Try This: For 8 servings:
one could calculate rate if time and distance are known. 1. Tire Tracks 1 = 24 9 .375 = 225 = 15 mph t = $100w + .40n one side wasn’t repeated on the other. Return the 2 grams to
Wforrimteu tlha,e e f.ogr.m, ru =la d a/st da n=d r t. =t adn/dr. show how to manipulate the 3. ptA hlsaeky bs wtouhrdreoenwn ates rfi wpnahaynastc iitanhlte eiynr estthsitit nutokti wtohnoe ul ofildan nhasna pcmipaoeln nien iysf. tt iIhtnue tt bihoaenns.ek) cases, B S Fluuogtutarer:r: 5: 1 5//6/41 c c2uu pcpu p TTTiiirrreee TTTrrraaaccckkksss 324 === 222444 ˙˙˙ 62 3 47= . =5 1=45 479 6=0 =01 22= 4 m3 m0p hpm hp h Swoit nihnce ebr otshitnhec fteor itrphm eiusy lf abosor atthnw deo q swueeta elt khts.e ,m su ebqsutaitlu ttoe e2a fcohr Tesuhapcrehpe lle elmessseosnnotn efe dpa bltayun rase bst eoaan wcuhos r bwkasoshircke esphtre,e -aeantl dga enisbd ra talas coko en-cheopmtse; 3. b Aaskla wnhcea tt hweo sucladl he.appen if we tripled both sides. Demonstrate
10. Pt hoein mt aopu.t Dcoeoterrdminiantee ws hanetdh/eorr sltoundgeintutds ek/nloawti thuodwe mtoa erxkpinrgesss o an oinftfeerreedst a, itnwdoic-yaetea rt hCaDt. tBheec bauansek twhiilsl pisa ay cthasee i notfe sriemspt laet the W Alhmipopnidnsg: 1Cr 1e/a6m c:u 1p 2s/3 tablespoons 2. T Tihrree Ter oacf kths e5 fi=v e 2c4a r ˙˙s 1an5a0l =yz e3d, 6w0e0r e= s6p0e emdpinhg, 424200 +˙ .21 +0 n.1 =0 2n0 =0 1 +0 .04 ˙0 n2 + .40n atoc tailvliotwy. t Tehaec hmearst etroi aulsse a trhee dmes iing nwehda tweivtehr fl oerxdiebri lbiteys t tohpaetr athtieo sncsa, ldei visi dseti ltlh bea wlaenigcehdt .o Tnh eeanc, hto s iddeem boyn 3s,t srahtoew iinnvge rtshea t
location’s coordinates in x,y format (and longitude/latitude Coconut: 1 cup Subtract 200 and .10n from both sides: inverse operations “cancel each other out.”
end of each year. This results in a $20 payment at the end of some more than twice the speed limit. The supplements the scope and sequence of his/her pre-
format if desired). Make sure they can determine horizontal each year. Plug these numbers into the formula I = p . r . t to Chocolate: 2 2/3 squares principal should request that the town install 240 = .30n More Free Math Programs: algebra instruction and curriculum. The worksheets 4. I ndicate that equations work like the balancing scale, with
and vertical distance using coordinates. show how a total of $40 in interest will be paid. To show how Worksheet 2: A Case of Interest speed bumps. Divide both sides by .3 leaving n = 800 Visit www.actuarialfoundation.org/programs/for_teachers.shtml also include additional math content connections, the equal sign acting as the fulcrum. (Many students have
11. D istribute Worksheet 1. Read the introduction as a class 1. After two years students will have the $500 the common misconception that the equal sign means “the
much the depositor has in total, show how to add the interest Now Try This: highlighted in the materials section of each lesson.
and review the “Additional Clues” (key facts) before students they deposited plus interest calculated answer to the problem is” rather than “is the same as.”)
to the principal, i.e., $1,000 + $40 = $1,040. Tire Tracks 1 = 21 46 2/3 = 400 = 20 mph
complete the worksheet. Make sure students are able to using the formula I = p ˙ r ˙ t. Interest = ˙ The take-home activities further encourage students Before proceeding, make sure students understand this
determine each route’s distance using the coordinates and 4. A sk what would happen if the depositor kept the first year’s $500 ˙ .049 ˙ 2 = $49, so the students will Tire Tracks 2 = ˙ 2 64 6 2/3 = 1,600 = 40 mph to apply their math skills to real-world situations. distinction and the goal of keeping things in balance.
interest in the account to “grow.” Show how the depositor have $500 + $49 = $549 Tire Tracks 3 = 21 40 4 1/6 = 2,500 = 50 mph
the map’s scale. Review answers as a class. ˙ The classroom poster displays the important concept 5. W rite a simple equation on the board, e.g., x + 7 = 9. Indicate
would earn an extra 40¢ by using the $20 as principal for one 2–3. Using the growth formula y = a(1+r)n, after Two of the three cars exceeded the 20 mph DevelOPeD WITH
12. S tudents will build on these skills in Bonus Worksheet 1 of keeping equations in balance while solving for that the goal is, as with the scale, to keep the equation in
year (the second year of the CD’s term) at 2%. Explain that two years the students will have $500(1 speed limit.
and Take-Home Activity 1. If students require additional unknowns and reminds students, particularly visual balance. Indicate that we can solve for x if we isolate it on
this is an example of compound interest.
support, review worksheets as a class. learners, of this fundamental pre-algebra concept. one side of the equation through manipulations that keep
5. Indicate that there is a formula that can be used to calculate
the equation in balance. Depending on class support needed,
how money grows with compound interest: y = a(1+r)n where
Lesson 2 6. A sk how we can find out what s equals. Indicate that we can WorKshEEt #1 WorKshEEt #2 WorKshEEt #3
continued analyzing Change/ take the square root of both sides. Referring to the scale
Growth and decay Formula example, remind students that what we do to one side of an
equation we must do to the other. Provide other examples as The Case of the Doubtful Distance A Case of Interest The Case of the Screeching Tires
y = ending value, a = starting amount (in this case principal needed using perfect squares for the area.
or amount invested), r = interest rate, and n = the number of 7. I ntroduce an example where the area is not a perfect square,
time periods. The amount of interest can be determined by for example, A = 29. Using a calculator, show that 29 is
subtracting the starting principal amount from the ending approximately equal to 5.385. Solving the Unknown with Algebra—a new blog by There are three “Looks like we have another case!” shouted Trhaeis e6dth $ g5r0a0d earts “SCREECH—help!” was the subject line tHo ogwe tw pilrl owoef bweit ahboluet I think we have
principal amount. Demonstrate with calculations from the 8. To show how squares are inverse operations of square 8th-graders Rick and Athena to solve questions different routes. Rick as he scanned his e-mail. Athena and their bake sale. Will they need I researched of an e-mail Rick and Athena received a radar gun? our answer!
I think the
two-year CD. roots, ask what 2 i9s equal to. Break down the problem for and mysteries by using math—has received its formula d = r . t Rick have received an anxious query from the to have a second compound interest. from their principal, who suspects that
bake sale? This should help us
6. D istribute Worksheet 2 and classroom calculators. Read the students as needed, showing step-by-step how 92 = 81 and first e-mail request! is key here. president of the 6th-grade class. The middle solve the case! drivers are exceeding the 15 mph speed
introduction with the class and review the key facts before 81 = 9. To generalize, this means that 2 =x x. Provide other
students complete the worksheet. Review answers as a class. examples with perfect squares as needed. “We’ve been asked to be advisors for the annual school has a proud tradition which involves limit along the road leading to the school.
7. S tudents will build on these skills in bonus Worksheet 2 9. Finally, provide an example of an equation involving taking 7th-grade field trip to the state park,” said each 6th-grade class raising money that The principal has urged the city to install
and Take-Home Activity 2. If students require additional the square root of both sides of an equation. Write x2 + 6 = Athena. “One of the homerooms is not sure of they will use at the end of their 8th-grade speed bumps, but wants to provide proof
support, review worksheets as a class. 15. Ask students to solve and explain. Show the solution
the best route to take to win the Champions year for a community service project. This that cars are speeding. Rick thinks a speed
on the board, first subtracting 6 from both sides, leaving x2
= 9. Indicate that the equation is still in balance and that it Cup race!” year, the 6th-graders held a bake sale which radar gun is the only way, but Athena
will stay in balance if we take the square root of both sides, raised $500. The class treasurer is concerned suggests they first inspect the scene. The
Let’s work
For the Champions Cup race, each homeroom
leaving x = 3. Remind students that whatever is done to one the math! because the class wants to build flower beds pair discover different sets of tire tracks
side of an equation must be done to the other. receives a map showing alternate routes to
Lesson 3
at the town’s senior center at a cost of $545. where cars had slammed on their brakes
Functions and 10. D istribute Worksheet 3 and calculators. Read the the cup. “I think we should consider the
Formulas/square roots introduction and review the key facts. Depending upon formula Distance = Rate . Time (d = r . t),” Athena researches savings opportunities for before a stop sign. “I think our answer is
student support needed, it might be necessary to compute the class and finds that: right here!” Athena declares.
said Rick. “Check out this map”:
the first vehicle’s speed as a class and/or review the
ObjECTIvES: calculator’s square root function. • First National Bank is offering a two-year CD with 4.9% simple interest.
Students will understand what a 11. Ask students to complete the worksheet. Review answers as a 5 • Second National Bank is offering a two-year CD with 4.8% compound interest (compounded yearly).
Athena’s first step is to assemble the key facts. She connects with police investigators
square root is and that squares and class. WorK thE math
square roots are inverse operations K EY: Do the students need to hold another fund-raiser? Rick thinks they might, but Athena has another idea and opens and discovers that the formula used to analyze tire tracks is s = 24 . d where s = speed
12. S tudents will further develop skills at working with equations
and can be used to manipulate in bonus Worksheet 3 and Take-Home Activity 3. If 4 = Route A = 1,000 feet her laptop to get to work. in miles per hour and d = length of the tire tracks in feet. She then takes measurements of different tire tracks.
equations as long as “whatever is 0 1
students require additional support, review worksheets as a = Route B Show your work—use separate paper as needed.
done to one side of the equation is
class.
done to the other.” 3 = Route C
Time: 20–30 minutes, depending on WorK thE math 1 For each set of measurements below, calculate each NoW trY this:
class review needs; additional time for worksheets ADDITIONAL CLUES: vehicle’s speed:
M•• ScW aattoulecrdrukiesalanhltsteo :ce rats l)3c u(lCaotnotres n(fto Cro Wnonrekcsthioenest: 3 S;q cuaanr ea lRsoo obtes )done without FTwfohowerr _ mwAtc.oeatruaceact rufhireaaelrr eFsia o.mslufhanottdumhan trldieoasnto iWuorencb.e ossri,tg ev/ iasptirt:o grams/ S21TART RaRRteoosuu otteef sABp ((erseowdua fgmohrp teyea)r:cr ha3 i,rn0o)0u:0 t2 ef, e0be0at0s pe feder eo htn op puerrre hvoiouurs years: S1how HFyoioruswrt w mNoaurktci—hou nwsaeill lsb etaphnaekr ac CltaeDs p?sa hpaerv ea si nn etewdoe dy.ears if it buys a H iyn =tI: a =r( ep1m .+ re r .m) nt b ((ecsorim mthppleoesu ienn dtfeo irnremtsetur)elasts): t#i1r:E 9 t.3ra7C5K fseet VEhiClE spEEd Astou p barmninaictlyispz daeil fp ifones raseninbotlt eth iserper e stecradhcionkog ld oatna
• B onus Worksheet 3 (Content Connections: Probability) n Th e Math Academy Series: Using Math in Route C (flat and dry): 6,000 feet per hour #2: 6 feet a road near her school:
• T ake-Home Activity 3 (Content Connections: Variables) 0
the Real World 0 1 2 3 4 5
Tire Tracks #1: 16 2/3 feet
DIRECTIONS:
n “E xpect the Unexpected With Math®” #3: 37.5 feet Tire Tracks #2: 66 2/3 feet
1. A sk the class to define what a square is. After students
NoW trY this:
mention that a square has four equal sides and four right Series: 2 How much will the class have in two years if it buys a Tire Tracks #3: 104 1/6 feet
WorK thE math
angles, draw a square on the board. Probability: Shake, Rattle, & Roll Second National bank CD?
#4: 24 feet The speed limit is 20 mph.
2. Label one of the sides “4 feet.” Ask how the area of the Show your work—use separate paper as needed. Assume next year’s 6th-grade
square can be determined (by multiplying 4 . 4 or, more Graphing: Bars, Lines, & Pies For each measurement above,
class needs $600 for its service
generally, squaring one of the sides) and write A = s2 where Converting decimals, fractions, and calculate each vehicle’s speed
A represents area and s represents the length of a side. If percents: Conversions Rock 1 How long is each route? 2 Using the formula d = r . t, how long project. If it can buy a two-year #5: 150 feet to determine whether or not
CD with a compound interest rate
necessary, explain how the superscript “2” is used to note should it take to complete each route? there is a speeding issue.
exponents. Perimeter, area, surface area, and Route A 3 Which CD is the better deal? Explain your thinking. Did of 5%, how much does it need to
3. Ask the class how to find the length of a side of a square if we volume: Setting the Stage with Geometry Route b Route A your calculations surprise you? raise at its fund-raiser?
know its area, for example when A = 25. Depending on the Route b 2 What should Rick and Athena report to the principal about
level of class support needed, indicate that the problem can Printable copies also available at: Route C speeding cars?
be solved by finding out what number times itself equals 25. www.scholastic.com/unexpectedmath Route C
4. I ntroduce the radical sign notation, e.g., that 5 = 25.
5. G o back to the formula for area, A = s2. Then write 25 = s2.
Lesson 2 6. A sk how we can find out what s equals. Indicate that we can WorKshEEt #1 WorKshEEt #2 WorKshEEt #3
continued analyzing Change/ take the square root of both sides. Referring to the scale
Growth and decay Formula example, remind students that what we do to one side of an
equation we must do to the other. Provide other examples as The Case of the Doubtful Distance A Case of Interest The Case of the Screeching Tires
y = ending value, a = starting amount (in this case principal needed using perfect squares for the area.
or amount invested), r = interest rate, and n = the number of 7. I ntroduce an example where the area is not a perfect square,
time periods. The amount of interest can be determined by for example, A = 29. Using a calculator, show that 29 is
subtracting the starting principal amount from the ending approximately equal to 5.385. Solving the Unknown with Algebra—a new blog by There are three “Looks like we have another case!” shouted Trhaeis e6dth $ g5r0a0d earts “SCREECH—help!” was the subject line tHo ogwe tw pilrl owoef bweit ahboluet I think we have
principal amount. Demonstrate with calculations from the 8. To show how squares are inverse operations of square 8th-graders Rick and Athena to solve questions different routes. Rick as he scanned his e-mail. Athena and their bake sale. Will they need I researched of an e-mail Rick and Athena received a radar gun? our answer!
I think the
two-year CD. roots, ask what 2 i9s equal to. Break down the problem for and mysteries by using math—has received its formula d = r . t Rick have received an anxious query from the to have a second compound interest. from their principal, who suspects that
bake sale? This should help us
6. D istribute Worksheet 2 and classroom calculators. Read the students as needed, showing step-by-step how 92 = 81 and first e-mail request! is key here. president of the 6th-grade class. The middle solve the case! drivers are exceeding the 15 mph speed
introduction with the class and review the key facts before 81 = 9. To generalize, this means that 2 =x x. Provide other
students complete the worksheet. Review answers as a class. examples with perfect squares as needed. “We’ve been asked to be advisors for the annual school has a proud tradition which involves limit along the road leading to the school.
7. S tudents will build on these skills in bonus Worksheet 2 9. Finally, provide an example of an equation involving taking 7th-grade field trip to the state park,” said each 6th-grade class raising money that The principal has urged the city to install
and Take-Home Activity 2. If students require additional the square root of both sides of an equation. Write x2 + 6 = Athena. “One of the homerooms is not sure of they will use at the end of their 8th-grade speed bumps, but wants to provide proof
support, review worksheets as a class. 15. Ask students to solve and explain. Show the solution
the best route to take to win the Champions year for a community service project. This that cars are speeding. Rick thinks a speed
on the board, first subtracting 6 from both sides, leaving x2
= 9. Indicate that the equation is still in balance and that it Cup race!” year, the 6th-graders held a bake sale which radar gun is the only way, but Athena
will stay in balance if we take the square root of both sides, raised $500. The class treasurer is concerned suggests they first inspect the scene. The
Let’s work
For the Champions Cup race, each homeroom
leaving x = 3. Remind students that whatever is done to one the math! because the class wants to build flower beds pair discover different sets of tire tracks
side of an equation must be done to the other. receives a map showing alternate routes to
Lesson 3
at the town’s senior center at a cost of $545. where cars had slammed on their brakes
Functions and 10. D istribute Worksheet 3 and calculators. Read the the cup. “I think we should consider the
Formulas/square roots introduction and review the key facts. Depending upon formula Distance = Rate . Time (d = r . t),” Athena researches savings opportunities for before a stop sign. “I think our answer is
student support needed, it might be necessary to compute the class and finds that: right here!” Athena declares.
said Rick. “Check out this map”:
the first vehicle’s speed as a class and/or review the
ObjECTIvES: calculator’s square root function. • First National Bank is offering a two-year CD with 4.9% simple interest.
Students will understand what a 11. Ask students to complete the worksheet. Review answers as a 5 • Second National Bank is offering a two-year CD with 4.8% compound interest (compounded yearly).
Athena’s first step is to assemble the key facts. She connects with police investigators
square root is and that squares and class. WorK thE math
square roots are inverse operations K EY: Do the students need to hold another fund-raiser? Rick thinks they might, but Athena has another idea and opens and discovers that the formula used to analyze tire tracks is s = 24 . d where s = speed
12. S tudents will further develop skills at working with equations
and can be used to manipulate in bonus Worksheet 3 and Take-Home Activity 3. If 4 = Route A = 1,000 feet her laptop to get to work. in miles per hour and d = length of the tire tracks in feet. She then takes measurements of different tire tracks.
equations as long as “whatever is 0 1
students require additional support, review worksheets as a = Route B Show your work—use separate paper as needed.
done to one side of the equation is
class.
done to the other.” 3 = Route C
Time: 20–30 minutes, depending on WorK thE math 1 For each set of measurements below, calculate each NoW trY this:
class review needs; additional time for worksheets ADDITIONAL CLUES: vehicle’s speed:
M•• ScW aattoulecrdrukiesalanhltsteo :ce rats l)3c u(lCaotnotres n(fto Cro Wnonrekcsthioenest: 3 S;q cuaanr ea lRsoo obtes )done without FTwfohowerr _ mwAtc.oeatruaceact rufhireaaelrr eFsia o.mslufhanottdumhan trldieoasnto iWuorencb.e ossri,tg ev/ iasptirt:o grams/ S21TART RaRRteoosuu otteef sABp ((erseowdua fgmohrp teyea)r:cr ha3 i,rn0o)0u:0 t2 ef, e0be0at0s pe feder eo htn op puerrre hvoiouurs years: S1how HFyoioruswrt w mNoaurktci—hou nwsaeill lsb etaphnaekr ac CltaeDs p?sa hpaerv ea si nn etewdoe dy.ears if it buys a H iyn =tI: a =r( ep1m .+ re r .m) nt b ((ecsorim mthppleoesu ienn dtfeo irnremtsetur)elasts): t#i1r:E 9 t.3ra7C5K fseet VEhiClE spEEd Astou p barmninaictlyispz daeil fp ifones raseninbotlt eth iserper e stecradhcionkog ld oatna
• B onus Worksheet 3 (Content Connections: Probability) n Th e Math Academy Series: Using Math in Route C (flat and dry): 6,000 feet per hour #2: 6 feet a road near her school:
• T ake-Home Activity 3 (Content Connections: Variables) 0
the Real World 0 1 2 3 4 5
Tire Tracks #1: 16 2/3 feet
DIRECTIONS:
n “E xpect the Unexpected With Math®” #3: 37.5 feet Tire Tracks #2: 66 2/3 feet
1. A sk the class to define what a square is. After students
NoW trY this:
mention that a square has four equal sides and four right Series: 2 How much will the class have in two years if it buys a Tire Tracks #3: 104 1/6 feet
WorK thE math
angles, draw a square on the board. Probability: Shake, Rattle, & Roll Second National bank CD?
#4: 24 feet The speed limit is 20 mph.
2. Label one of the sides “4 feet.” Ask how the area of the Show your work—use separate paper as needed. Assume next year’s 6th-grade
square can be determined (by multiplying 4 . 4 or, more Graphing: Bars, Lines, & Pies For each measurement above,
class needs $600 for its service
generally, squaring one of the sides) and write A = s2 where Converting decimals, fractions, and calculate each vehicle’s speed
A represents area and s represents the length of a side. If percents: Conversions Rock 1 How long is each route? 2 Using the formula d = r . t, how long project. If it can buy a two-year #5: 150 feet to determine whether or not
CD with a compound interest rate
necessary, explain how the superscript “2” is used to note should it take to complete each route? there is a speeding issue.
exponents. Perimeter, area, surface area, and Route A 3 Which CD is the better deal? Explain your thinking. Did of 5%, how much does it need to
3. Ask the class how to find the length of a side of a square if we volume: Setting the Stage with Geometry Route b Route A your calculations surprise you? raise at its fund-raiser?
know its area, for example when A = 25. Depending on the Route b 2 What should Rick and Athena report to the principal about
level of class support needed, indicate that the problem can Printable copies also available at: Route C speeding cars?
be solved by finding out what number times itself equals 25. www.scholastic.com/unexpectedmath Route C
4. I ntroduce the radical sign notation, e.g., that 5 = 25.
5. G o back to the formula for area, A = s2. Then write 25 = s2.
Lesson 2 6. A sk how we can find out what s equals. Indicate that we can WorKshEEt #1 WorKshEEt #2 WorKshEEt #3
continued analyzing Change/ take the square root of both sides. Referring to the scale
Growth and decay Formula example, remind students that what we do to one side of an
equation we must do to the other. Provide other examples as The Case of the Doubtful Distance A Case of Interest The Case of the Screeching Tires
y = ending value, a = starting amount (in this case principal needed using perfect squares for the area.
or amount invested), r = interest rate, and n = the number of 7. I ntroduce an example where the area is not a perfect square,
time periods. The amount of interest can be determined by for example, A = 29. Using a calculator, show that 29 is
subtracting the starting principal amount from the ending approximately equal to 5.385. Solving the Unknown with Algebra—a new blog by There are three “Looks like we have another case!” shouted Trhaeis e6dth $ g5r0a0d earts “SCREECH—help!” was the subject line tHo ogwe tw pilrl owoef bweit ahboluet I think we have
principal amount. Demonstrate with calculations from the 8. To show how squares are inverse operations of square 8th-graders Rick and Athena to solve questions different routes. Rick as he scanned his e-mail. Athena and their bake sale. Will they need I researched of an e-mail Rick and Athena received a radar gun? our answer!
I think the
two-year CD. roots, ask what 2 i9s equal to. Break down the problem for and mysteries by using math—has received its formula d = r . t Rick have received an anxious query from the to have a second compound interest. from their principal, who suspects that
bake sale? This should help us
6. D istribute Worksheet 2 and classroom calculators. Read the students as needed, showing step-by-step how 92 = 81 and first e-mail request! is key here. president of the 6th-grade class. The middle solve the case! drivers are exceeding the 15 mph speed
introduction with the class and review the key facts before 81 = 9. To generalize, this means that 2 =x x. Provide other
students complete the worksheet. Review answers as a class. examples with perfect squares as needed. “We’ve been asked to be advisors for the annual school has a proud tradition which involves limit along the road leading to the school.
7. S tudents will build on these skills in bonus Worksheet 2 9. Finally, provide an example of an equation involving taking 7th-grade field trip to the state park,” said each 6th-grade class raising money that The principal has urged the city to install
and Take-Home Activity 2. If students require additional the square root of both sides of an equation. Write x2 + 6 = Athena. “One of the homerooms is not sure of they will use at the end of their 8th-grade speed bumps, but wants to provide proof
support, review worksheets as a class. 15. Ask students to solve and explain. Show the solution
the best route to take to win the Champions year for a community service project. This that cars are speeding. Rick thinks a speed
on the board, first subtracting 6 from both sides, leaving x2
= 9. Indicate that the equation is still in balance and that it Cup race!” year, the 6th-graders held a bake sale which radar gun is the only way, but Athena
will stay in balance if we take the square root of both sides, raised $500. The class treasurer is concerned suggests they first inspect the scene. The
Let’s work
For the Champions Cup race, each homeroom
leaving x = 3. Remind students that whatever is done to one the math! because the class wants to build flower beds pair discover different sets of tire tracks
side of an equation must be done to the other. receives a map showing alternate routes to
Lesson 3
at the town’s senior center at a cost of $545. where cars had slammed on their brakes
Functions and 10. D istribute Worksheet 3 and calculators. Read the the cup. “I think we should consider the
Formulas/square roots introduction and review the key facts. Depending upon formula Distance = Rate . Time (d = r . t),” Athena researches savings opportunities for before a stop sign. “I think our answer is
student support needed, it might be necessary to compute the class and finds that: right here!” Athena declares.
said Rick. “Check out this map”:
the first vehicle’s speed as a class and/or review the
ObjECTIvES: calculator’s square root function. • First National Bank is offering a two-year CD with 4.9% simple interest.
Students will understand what a 11. Ask students to complete the worksheet. Review answers as a 5 • Second National Bank is offering a two-year CD with 4.8% compound interest (compounded yearly).
Athena’s first step is to assemble the key facts. She connects with police investigators
square root is and that squares and class. WorK thE math
square roots are inverse operations K EY: Do the students need to hold another fund-raiser? Rick thinks they might, but Athena has another idea and opens and discovers that the formula used to analyze tire tracks is s = 24 . d where s = speed
12. S tudents will further develop skills at working with equations
and can be used to manipulate in bonus Worksheet 3 and Take-Home Activity 3. If 4 = Route A = 1,000 feet her laptop to get to work. in miles per hour and d = length of the tire tracks in feet. She then takes measurements of different tire tracks.
equations as long as “whatever is 0 1
students require additional support, review worksheets as a = Route B Show your work—use separate paper as needed.
done to one side of the equation is
class.
done to the other.” 3 = Route C
Time: 20–30 minutes, depending on WorK thE math 1 For each set of measurements below, calculate each NoW trY this:
class review needs; additional time for worksheets ADDITIONAL CLUES: vehicle’s speed:
M•• ScW aattoulecrdrukiesalanhltsteo :ce rats l)3c u(lCaotnotres n(fto Cro Wnonrekcsthioenest: 3 S;q cuaanr ea lRsoo obtes )done without FTwfohowerr _ mwAtc.oeatruaceact rufhireaaelrr eFsia o.mslufhanottdumhan trldieoasnto iWuorencb.e ossri,tg ev/ iasptirt:o grams/ S21TART RaRRteoosuu otteef sABp ((erseowdua fgmohrp teyea)r:cr ha3 i,rn0o)0u:0 t2 ef, e0be0at0s pe feder eo htn op puerrre hvoiouurs years: S1how HFyoioruswrt w mNoaurktci—hou nwsaeill lsb etaphnaekr ac CltaeDs p?sa hpaerv ea si nn etewdoe dy.ears if it buys a H iyn =tI: a =r( ep1m .+ re r .m) nt b ((ecsorim mthppleoesu ienn dtfeo irnremtsetur)elasts): t#i1r:E 9 t.3ra7C5K fseet VEhiClE spEEd Astou p barmninaictlyispz daeil fp ifones raseninbotlt eth iserper e stecradhcionkog ld oatna
• B onus Worksheet 3 (Content Connections: Probability) n Th e Math Academy Series: Using Math in Route C (flat and dry): 6,000 feet per hour #2: 6 feet a road near her school:
• T ake-Home Activity 3 (Content Connections: Variables) 0
the Real World 0 1 2 3 4 5
Tire Tracks #1: 16 2/3 feet
DIRECTIONS:
n “E xpect the Unexpected With Math®” #3: 37.5 feet Tire Tracks #2: 66 2/3 feet
1. A sk the class to define what a square is. After students
NoW trY this:
mention that a square has four equal sides and four right Series: 2 How much will the class have in two years if it buys a Tire Tracks #3: 104 1/6 feet
WorK thE math
angles, draw a square on the board. Probability: Shake, Rattle, & Roll Second National bank CD?
#4: 24 feet The speed limit is 20 mph.
2. Label one of the sides “4 feet.” Ask how the area of the Show your work—use separate paper as needed. Assume next year’s 6th-grade
square can be determined (by multiplying 4 . 4 or, more Graphing: Bars, Lines, & Pies For each measurement above,
class needs $600 for its service
generally, squaring one of the sides) and write A = s2 where Converting decimals, fractions, and calculate each vehicle’s speed
A represents area and s represents the length of a side. If percents: Conversions Rock 1 How long is each route? 2 Using the formula d = r . t, how long project. If it can buy a two-year #5: 150 feet to determine whether or not
CD with a compound interest rate
necessary, explain how the superscript “2” is used to note should it take to complete each route? there is a speeding issue.
exponents. Perimeter, area, surface area, and Route A 3 Which CD is the better deal? Explain your thinking. Did of 5%, how much does it need to
3. Ask the class how to find the length of a side of a square if we volume: Setting the Stage with Geometry Route b Route A your calculations surprise you? raise at its fund-raiser?
know its area, for example when A = 25. Depending on the Route b 2 What should Rick and Athena report to the principal about
level of class support needed, indicate that the problem can Printable copies also available at: Route C speeding cars?
be solved by finding out what number times itself equals 25. www.scholastic.com/unexpectedmath Route C
4. I ntroduce the radical sign notation, e.g., that 5 = 25.
5. G o back to the formula for area, A = s2. Then write 25 = s2.
Lesson 2 6. A sk how we can find out what s equals. Indicate that we can WorKshEEt #1 WorKshEEt #2 WorKshEEt #3
continued analyzing Change/ take the square root of both sides. Referring to the scale
Growth and decay Formula example, remind students that what we do to one side of an
equation we must do to the other. Provide other examples as The Case of the Doubtful Distance A Case of Interest The Case of the Screeching Tires
y = ending value, a = starting amount (in this case principal needed using perfect squares for the area.
or amount invested), r = interest rate, and n = the number of 7. I ntroduce an example where the area is not a perfect square,
time periods. The amount of interest can be determined by for example, A = 29. Using a calculator, show that 29 is
subtracting the starting principal amount from the ending approximately equal to 5.385. Solving the Unknown with Algebra—a new blog by There are three “Looks like we have another case!” shouted Trhaeis e6dth $ g5r0a0d earts “SCREECH—help!” was the subject line tHo ogwe tw pilrl owoef bweit ahboluet I think we have
principal amount. Demonstrate with calculations from the 8. To show how squares are inverse operations of square 8th-graders Rick and Athena to solve questions different routes. Rick as he scanned his e-mail. Athena and their bake sale. Will they need I researched of an e-mail Rick and Athena received a radar gun? our answer!
I think the
two-year CD. roots, ask what 2 i9s equal to. Break down the problem for and mysteries by using math—has received its formula d = r . t Rick have received an anxious query from the to have a second compound interest. from their principal, who suspects that
bake sale? This should help us
6. D istribute Worksheet 2 and classroom calculators. Read the students as needed, showing step-by-step how 92 = 81 and first e-mail request! is key here. president of the 6th-grade class. The middle solve the case! drivers are exceeding the 15 mph speed
introduction with the class and review the key facts before 81 = 9. To generalize, this means that 2 =x x. Provide other
students complete the worksheet. Review answers as a class. examples with perfect squares as needed. “We’ve been asked to be advisors for the annual school has a proud tradition which involves limit along the road leading to the school.
7. S tudents will build on these skills in bonus Worksheet 2 9. Finally, provide an example of an equation involving taking 7th-grade field trip to the state park,” said each 6th-grade class raising money that The principal has urged the city to install
and Take-Home Activity 2. If students require additional the square root of both sides of an equation. Write x2 + 6 = Athena. “One of the homerooms is not sure of they will use at the end of their 8th-grade speed bumps, but wants to provide proof
support, review worksheets as a class. 15. Ask students to solve and explain. Show the solution
the best route to take to win the Champions year for a community service project. This that cars are speeding. Rick thinks a speed
on the board, first subtracting 6 from both sides, leaving x2
= 9. Indicate that the equation is still in balance and that it Cup race!” year, the 6th-graders held a bake sale which radar gun is the only way, but Athena
will stay in balance if we take the square root of both sides, raised $500. The class treasurer is concerned suggests they first inspect the scene. The
Let’s work
For the Champions Cup race, each homeroom
leaving x = 3. Remind students that whatever is done to one the math! because the class wants to build flower beds pair discover different sets of tire tracks
side of an equation must be done to the other. receives a map showing alternate routes to
Lesson 3
at the town’s senior center at a cost of $545. where cars had slammed on their brakes
Functions and 10. D istribute Worksheet 3 and calculators. Read the the cup. “I think we should consider the
Formulas/square roots introduction and review the key facts. Depending upon formula Distance = Rate . Time (d = r . t),” Athena researches savings opportunities for before a stop sign. “I think our answer is
student support needed, it might be necessary to compute the class and finds that: right here!” Athena declares.
said Rick. “Check out this map”:
the first vehicle’s speed as a class and/or review the
ObjECTIvES: calculator’s square root function. • First National Bank is offering a two-year CD with 4.9% simple interest.
Students will understand what a 11. Ask students to complete the worksheet. Review answers as a 5 • Second National Bank is offering a two-year CD with 4.8% compound interest (compounded yearly).
Athena’s first step is to assemble the key facts. She connects with police investigators
square root is and that squares and class. WorK thE math
square roots are inverse operations K EY: Do the students need to hold another fund-raiser? Rick thinks they might, but Athena has another idea and opens and discovers that the formula used to analyze tire tracks is s = 24 . d where s = speed
12. S tudents will further develop skills at working with equations
and can be used to manipulate in bonus Worksheet 3 and Take-Home Activity 3. If 4 = Route A = 1,000 feet her laptop to get to work. in miles per hour and d = length of the tire tracks in feet. She then takes measurements of different tire tracks.
equations as long as “whatever is 0 1
students require additional support, review worksheets as a = Route B Show your work—use separate paper as needed.
done to one side of the equation is
class.
done to the other.” 3 = Route C
Time: 20–30 minutes, depending on WorK thE math 1 For each set of measurements below, calculate each NoW trY this:
class review needs; additional time for worksheets ADDITIONAL CLUES: vehicle’s speed:
M•• ScW aattoulecrdrukiesalanhltsteo :ce rats l)3c u(lCaotnotres n(fto Cro Wnonrekcsthioenest: 3 S;q cuaanr ea lRsoo obtes )done without FTwfohowerr _ mwAtc.oeatruaceact rufhireaaelrr eFsia o.mslufhanottdumhan trldieoasnto iWuorencb.e ossri,tg ev/ iasptirt:o grams/ S21TART RaRRteoosuu otteef sABp ((erseowdua fgmohrp teyea)r:cr ha3 i,rn0o)0u:0 t2 ef, e0be0at0s pe feder eo htn op puerrre hvoiouurs years: S1how HFyoioruswrt w mNoaurktci—hou nwsaeill lsb etaphnaekr ac CltaeDs p?sa hpaerv ea si nn etewdoe dy.ears if it buys a H iyn =tI: a =r( ep1m .+ re r .m) nt b ((ecsorim mthppleoesu ienn dtfeo irnremtsetur)elasts): t#i1r:E 9 t.3ra7C5K fseet VEhiClE spEEd Astou p barmninaictlyispz daeil fp ifones raseninbotlt eth iserper e stecradhcionkog ld oatna
• B onus Worksheet 3 (Content Connections: Probability) n Th e Math Academy Series: Using Math in Route C (flat and dry): 6,000 feet per hour #2: 6 feet a road near her school:
• T ake-Home Activity 3 (Content Connections: Variables) 0
the Real World 0 1 2 3 4 5
Tire Tracks #1: 16 2/3 feet
DIRECTIONS:
n “E xpect the Unexpected With Math®” #3: 37.5 feet Tire Tracks #2: 66 2/3 feet
1. A sk the class to define what a square is. After students
NoW trY this:
mention that a square has four equal sides and four right Series: 2 How much will the class have in two years if it buys a Tire Tracks #3: 104 1/6 feet
WorK thE math
angles, draw a square on the board. Probability: Shake, Rattle, & Roll Second National bank CD?
#4: 24 feet The speed limit is 20 mph.
2. Label one of the sides “4 feet.” Ask how the area of the Show your work—use separate paper as needed. Assume next year’s 6th-grade
square can be determined (by multiplying 4 . 4 or, more Graphing: Bars, Lines, & Pies For each measurement above,
class needs $600 for its service
generally, squaring one of the sides) and write A = s2 where Converting decimals, fractions, and calculate each vehicle’s speed
A represents area and s represents the length of a side. If percents: Conversions Rock 1 How long is each route? 2 Using the formula d = r . t, how long project. If it can buy a two-year #5: 150 feet to determine whether or not
CD with a compound interest rate
necessary, explain how the superscript “2” is used to note should it take to complete each route? there is a speeding issue.
exponents. Perimeter, area, surface area, and Route A 3 Which CD is the better deal? Explain your thinking. Did of 5%, how much does it need to
3. Ask the class how to find the length of a side of a square if we volume: Setting the Stage with Geometry Route b Route A your calculations surprise you? raise at its fund-raiser?
know its area, for example when A = 25. Depending on the Route b 2 What should Rick and Athena report to the principal about
level of class support needed, indicate that the problem can Printable copies also available at: Route C speeding cars?
be solved by finding out what number times itself equals 25. www.scholastic.com/unexpectedmath Route C
4. I ntroduce the radical sign notation, e.g., that 5 = 25.
5. G o back to the formula for area, A = s2. Then write 25 = s2.
Lesson 1 Lesson 2 Worksheet Answer Key Lesson 1
continued Using Mathematical Analyzing Change/ Dear Teacher: Using Mathematical
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Incorporate the value of x back into the original equation to • W orksheet 2 (Content Connections: Percentages, Exponents) between a tree and its shadow. A one-foot 6. 1 0(1 + 1.0)6 = 640 bacteria per cubic .1 = .9 formula to solve for variables
show both sides being equal. • B onus Worksheet 2 (Content Connections: Conversion of shadow is cast by a three-foot tree, so a centimeter and 10(1 + 1.0)7 = 1,280 The probability of two green lights in a row Time: 30–40 minutes, depending on class review needs;
7. M ention that keeping an equation in balance applies to many Percentages to Decimals, Exponents) 30-foot shadow would be cast by a 90-foot bacteria per cubic centimeter, so the on Washington Road = .9 ˙ .9 = .81 or 81.0% additional time for worksheets
real-life situations. Show students the map and ask them to • T ake-Home Activity 2 (Content Connections: Percentages, tree. As a proportion this is shown as 30 ft./x sandwich has been in the locker between 3. The probability of four green lights in a row Materials:
explain how a map is a representation of the area it depicts. Exponents) feet = 1 foot /3 feet. Solving for x leaves x = six and seven weeks. on Washington Road = .9 ˙ .9 ˙ .9 ˙ .9 = .656 Getting Started: • B alance scale with weights (drawings on the board will also
90 feet. Therefore, 60 feet from the school Take-Home Activity 2: The Case of the or 65.6%
Answers should include that map distances are proportional suffice)
tmoe tahseu rreinagl dthiset adniscteasn tchee by ertewpereesne tnwt.o D meampo lnosctartaitoen bs ya nd D1.I R SehCoTIwO aN sSa: vings or investment ad to the class. Ask students to 2. b 1u8i0l dfeinegt. i sS neot tu ap saa pfero dpiostratinocne .6 ft./3 ft. = D1.e c Cauyrirnegn tC avarlue = 20,000(1 – .20)4 = 20,000 ˙ TBaakreg-aHino mHuen Atcetrivity 3: The Case of the Kid Poster/Teaching Guide for Pre-Algebra Students Tphreis- aplrgoegbrraam in iss tdruescitgionne din t oth seuspep aleremaesn: t your existing • Mcoaopr d(ein.ga.t,e ws,o arnldd, as tkaetye ,i nectclu.)d winigth s cloanlegitude/latitude and/or
using proportions to calculate the actual distance. For explain what the interest rate (e.g., 1.75%) means. Establish x ft./90 ft. Solving for x leaves x = 180 .84 = 20,000 ˙ .4096 = $8,192.00 1. Costs for each rental company: 1. solving for unknowns;
eshxaomw phloew, i 1f 1 in in./c2h0 =0 2m0i0. =m 2il.e5s i ann./dx y moui. more 5a0su0r medil e2s.5. inches, tfimhnoaantn etchyi eaw li iintnhtse ttrhietseut ti rinoastntei tt roue ttpihoreens .de enptos stihteo rr ainte r eotfu prany fmore dnet pboys tihtien g N4se5ot-wtfio nTorgyt -u Ttpha lials :bp urToihlpdeoi lnretgnio gcnat.hn Iobnfe ta dh seihst eacrdamoseiwn, eo1df/ a3b y = x/45. 2. $ Uye2sai0nr,sg0 =0t h0$ e2˙ d0 .e,20c60a20y1 (f41o4 r– m= . u2$l05a),,26 t4=h2 e$. 28v0a8,l.u0 eS0 i0an˙fct.ee8 r6t hs=ie x $SLem5t5o’s5o .Gt0ho0 :R $id2e2:0 $ ˙1 20 0+ ˙.1 20 +˙ .14,01 5˙0 1 =,1 45400 = + 2 10105 + = FAollilgonws t hwei atdhv NenCtTuMre sS tandards ClaPsossrtoeorm 2. mbaalnanipcuel;a ting equations while keeping them in •• W BC ooonnrnkuseshc Wteieootrn k1ss :(h MCeoeenta ts1eu narnet mdCo Tenannkte)e-cHtioomnse: AOcrdtievriteyd 1 P (aCirosn)tent
8. Utt rhsaaivnte gtlh bdeei stttiwmaeneec ndese t phcaeeln ctdwusloa o tpneo dhi,no atwss. kf S ahtsout wdoe nlnoetn isgs s tihrtao wvuoelduli lnidng dt.a iMckaeat kteeo an 2. Uwsitihn ga t2h%e einxtaemrepslte roaft ea, $ s1h,o0w0 0th CeD c daelcpuolsaittieodn f $o1r ,o0n0e0 y .e .a0r2 . 1 STPaoroklvpei-onHrgto ifmoonre sx A lecativveist yx 1=: 1 T5h efe Ceat.se of Sweet ctahaderd’ siat mvioaonluuaenl ttw woaofs t y$he8ea, rd1se9 wc2lo.i0nu0eld ai nbft eve ar$ lf8uo,eu1 ir9n 2y te.h0ae0rs –, U4Ch6ne0ca l=ep $ TWe6hd6de0ey.0’lss0:: $.6300 0˙ ˙1 ,21 5=0 $ =6 0$06.9000.00 oasf Rthiceky asnodlv Ae trheeanl-a Inside 34.. u wsoirnkgin fogr wmiuthla psr toop soorltvioen rse,a rla-wdiocrallds ,c hanaldle enxgpeosn; eanntds. D1.I RS ehCoTIwO sNtSu:dents a balancing scale without weights on either
appropriate assumption depending on the means of travel = $20. Work the calculation as necessary and generalize the For 32 servings: $5,242.88 = $2,949.12 The family should use Let’s Go because it’s world questions The materials are taught through this story line: arm. Ask why the scale is in balance (because each side has
(e.g., a driving speed of 55 mph or flying speed of 600 mph). formula by writing the following formula for simple interest: Butter: 1 2/3 cups Now Try This: the cheapest. Athena and Rick are two students frequently called the same weight). Ask what would happen if 5 grams were
Divide distance by rate to determine trip length. I = p . r . t, where I = interest, p = principal (amount Sugar: 1 cup 1. 30,000(1 – .20)3 = 30,000 ˙ .83 = 30,000 ˙ 2. t = $220w + .10n, where t = total cost, w = through powerful upon to use their powerful mathematical thinking added to both sides. Demonstrate, reiterating that the scale
9. W rite “Distance = Rate . Time” on the board and explain what deposited), r = rate (of interest), and t = time (in years). It Flour: 3 1/3 cups .512 = $15,360.00 number of weeks rented, n = number of miles mathematical to solve seemingly unsolvable mysteries, problems, is still balanced because the same amount was added to each
eoaf cah r etelartmio mnsehainps i.n I nthdeic raetael twhoartl ad ,f ojursmt ualsa a is m aa rpe pisr eas entation .mt hTieigm hCeDt) a.a lItsf om n baeetcu eurssisteayf rueylq, t ueoax slpshl aoPirwnin ttchoie pt hacllea + cs slIan tsthsea rtteh tsahtt e (t Phtroeitn iancli tvpeaarlelu s.e tR oaft e W A Colhmcioponpnuidnts:g: 4 4C rc 2eu/ap3ms c:u 6p 2s/3 tablespoons 32.. 4 205,,000000((11 –– ..2200))15 == 2450,,000000 ˙˙ ..885 = = $ 4200,,000000 ˙.0 0 Nowd rTirvye Tnhis: thinking! aanndd qthuee swtoiornkssh. eThetesy i pnosisdt ec aesneg fiaglees sotnu dae mntast hto b floolglo, w 2. s Aisdke w. hat would happen if we only operated on one side of the
representation of a real place. Show how it’s possible to .32768 = $13,107.20 Let’s Go formula is t = $220w + .10n along with the duo and solve problems by applying scale. What if 2 grams were taken off one side? Demonstrate
rate is in effect the entire term of the CD. (If a student asks, Chocolate: 10 2/3 squares
determine one of the terms if the other two are known, e.g., Worksheet 3: The Case of the Screeching Tires Develop a formula for Smooth Ride: mathematical skills and a logical, systematic approach. that the scale is out of balance because the action taken on
it may be necessary to explain that interest also comes into Now Try This: For 8 servings:
one could calculate rate if time and distance are known. 1. Tire Tracks 1 = 24 9 .375 = 225 = 15 mph t = $100w + .40n one side wasn’t repeated on the other. Return the 2 grams to
Wforrimteu tlha,e e f.ogr.m, ru =la d a/st da n=d r t. =t adn/dr. show how to manipulate the 3. ptA hlsaeky bs wtouhrdreoenwn ates rfi wpnahaynastc iitanhlte eiynr estthsitit nutokti wtohnoe ul ofildan nhasna pcmipaoeln nien iysf. tt iIhtnue tt bihoaenns.ek) cases, B S Fluuogtutarer:r: 5: 1 5//6/41 c c2uu pcpu p TTTiiirrreee TTTrrraaaccckkksss 324 === 222444 ˙˙˙ 62 3 47= . =5 1=45 479 6=0 =01 22= 4 m3 m0p hpm hp h Swoit nihnce ebr otshitnhec fteor itrphm eiusy lf abosor atthnw deo q swueeta elt khts.e ,m su ebqsutaitlu ttoe e2a fcohr Tesuhapcrehpe lle elmessseosnnotn efe dpa bltayun rase bst eoaan wcuhos r bwkasoshircke esphtre,e -aeantl dga enisbd ra talas coko en-cheopmtse; 3. b Aaskla wnhcea tt hweo sucladl he.appen if we tripled both sides. Demonstrate
10. Pt hoein mt aopu.t Dcoeoterrdminiantee ws hanetdh/eorr sltoundgeintutds ek/nloawti thuodwe mtoa erxkpinrgesss o an oinftfeerreedst a, itnwdoic-yaetea rt hCaDt. tBheec bauansek twhiilsl pisa ay cthasee i notfe sriemspt laet the W Alhmipopnidnsg: 1Cr 1e/a6m c:u 1p 2s/3 tablespoons 2. T Tihrree Ter oacf kths e5 fi=v e 2c4a r ˙˙s 1an5a0l =yz e3d, 6w0e0r e= s6p0e emdpinhg, 424200 +˙ .21 +0 n.1 =0 2n0 =0 1 +0 .04 ˙0 n2 + .40n atoc tailvliotwy. t Tehaec hmearst etroi aulsse a trhee dmes iing nwehda tweivtehr fl oerxdiebri lbiteys t tohpaetr athtieo sncsa, ldei visi dseti ltlh bea wlaenigcehdt .o Tnh eeanc, hto s iddeem boyn 3s,t srahtoew iinnvge rtshea t
location’s coordinates in x,y format (and longitude/latitude Coconut: 1 cup Subtract 200 and .10n from both sides: inverse operations “cancel each other out.”
end of each year. This results in a $20 payment at the end of some more than twice the speed limit. The supplements the scope and sequence of his/her pre-
format if desired). Make sure they can determine horizontal each year. Plug these numbers into the formula I = p . r . t to Chocolate: 2 2/3 squares principal should request that the town install 240 = .30n More Free Math Programs: algebra instruction and curriculum. The worksheets 4. I ndicate that equations work like the balancing scale, with
and vertical distance using coordinates. show how a total of $40 in interest will be paid. To show how Worksheet 2: A Case of Interest speed bumps. Divide both sides by .3 leaving n = 800 Visit www.actuarialfoundation.org/programs/for_teachers.shtml also include additional math content connections, the equal sign acting as the fulcrum. (Many students have
11. D istribute Worksheet 1. Read the introduction as a class 1. After two years students will have the $500 the common misconception that the equal sign means “the
much the depositor has in total, show how to add the interest Now Try This: highlighted in the materials section of each lesson.
and review the “Additional Clues” (key facts) before students they deposited plus interest calculated answer to the problem is” rather than “is the same as.”)
to the principal, i.e., $1,000 + $40 = $1,040. Tire Tracks 1 = 21 46 2/3 = 400 = 20 mph
complete the worksheet. Make sure students are able to using the formula I = p ˙ r ˙ t. Interest = ˙ The take-home activities further encourage students Before proceeding, make sure students understand this
determine each route’s distance using the coordinates and 4. A sk what would happen if the depositor kept the first year’s $500 ˙ .049 ˙ 2 = $49, so the students will Tire Tracks 2 = ˙ 2 64 6 2/3 = 1,600 = 40 mph to apply their math skills to real-world situations. distinction and the goal of keeping things in balance.
interest in the account to “grow.” Show how the depositor have $500 + $49 = $549 Tire Tracks 3 = 21 40 4 1/6 = 2,500 = 50 mph
the map’s scale. Review answers as a class. ˙ The classroom poster displays the important concept 5. W rite a simple equation on the board, e.g., x + 7 = 9. Indicate
would earn an extra 40¢ by using the $20 as principal for one 2–3. Using the growth formula y = a(1+r)n, after Two of the three cars exceeded the 20 mph DevelOPeD WITH
12. S tudents will build on these skills in Bonus Worksheet 1 of keeping equations in balance while solving for that the goal is, as with the scale, to keep the equation in
year (the second year of the CD’s term) at 2%. Explain that two years the students will have $500(1 speed limit.
and Take-Home Activity 1. If students require additional unknowns and reminds students, particularly visual balance. Indicate that we can solve for x if we isolate it on
this is an example of compound interest.
support, review worksheets as a class. learners, of this fundamental pre-algebra concept. one side of the equation through manipulations that keep
5. Indicate that there is a formula that can be used to calculate
the equation in balance. Depending on class support needed,
how money grows with compound interest: y = a(1+r)n where