9 4. CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association

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1 9. CISC - Curriculum & Instruction Steering Committee The Winning EQUATION A HIGH QUALITY MATHEMATICS PROFESSIONAL DEVELOPMENT PROGRAM FOR TEACHERS IN GRADES THROUGH ALGEBRA II STRAND: NUMBER SENSE: Rtionl Numbers MODULE TITLE: PRIMARY CONTENT MODULE V MODULE INTENTION: The intention of this module is to inform nd instruct prticipnts in the underlying mthemticl content in the re of rtionl numbers. THIS ENTIRE MODULE MUST BE COVERED IN-DEPTH. The presenttion of these Primry Content Modules is deprture from pst professionl development models. The content here, is presented for individul techer s depth of content in mthemtics. Presenttion to students would, in most cses, not ddress the generl cse or proof, but focus on presenttion with numericl exmples. In ddition to the underlying mthemticl content provided by this module, the fcilittor should use the clssroom connections provided within this binder nd referenced in the fcilittor s notes. TIME: 3 hours PARTICIPANT OUTCOMES: Demonstrte understnding of the frctions nd rtionl numbers. Demonstrte understnding of equivlent frctions nd how to reduce nd build frctions. Demonstrte how to dd, subtrct, multiply, nd divide rtionl numbers. Demonstrte how to use the bsic lgorithms for ddition nd subtrction of frctions tht my not hve common denomintors.

2 CISC - Curriculum & Instruction Steering Committee 2 PRIMARY CONTENT MODULE V NUMBER SENSE - Rtionl Numbers Fcilittor s Notes Prerequisite: Fmilirity with the rithmetic of the integers nd positive nd negtive whole numbers. The following re some helpful references tht give foundtion for rtionl numbers: T-1 Teching frctions in elementry school: A mnul for techers. (Mrch, 1998). Knowing nd Teching Elementry Mthemtics, by Liping M, Pre-Test Pretest: Give test on ddition, subtrction, multipliction, division, reducing frctions, nd solving some word problems. Questions on conceptul understnding re emphsized in post-test. T-2 1. Wht is Frction? This is not n esy question. The word frction comes from the Ltin word, frctio, which mens the ct of breking into pieces. Frctions rise nturlly in mesurement problems, especilly to express quntity less thn whole unit. Frctions indicte mounts or distnces in which bsic unit is subdivided into whole number of equl prts. T-3 T- & T-5 For exmple, if pie is divided into 8 pieces nd three of these re eten, then the frction, 3 expresses the frction of the pie tht ws 8 consumed. Use T- nd T-5 to illustrte the concept of frction further. There is no universlly ccepted definition of frction, but we will use this one: T-6 Definition 1: A frction is n ordered pir of integers nd b, with b not equl to zero, nd written s. The integer is clled the b numertor nd b is clled the denomintor. Some uthors require nd b to be whole numbers (so tht they re both positive).

3 CISC - Curriculum & Instruction Steering Committee 3 The words numertor nd denomintor re pproprite for frctions becuse the numertor enumertes (i.e., counts) while the denomintor specifies wht is being counted. T-7 T-8 T-8A Point out tht frctions cn be understood s points on number line. Mke n overhed slide with number line with some numbers bigger thn 1 nd less thn zero. Identify loction of some frctions on this number line. Include some frctions bigger thn 1. Tell prticipnts tht frctions with which re bigger thn one re clled improper frctions. Tell prticipnts tht number line is prticulrly convenient wy to visulize negtive frctions, like 3. Tell prticipnts tht it will be explined soon tht 3 = 3 = 3 nd tht similr sttements re true for ll frctions which hve negtive numertors or denomintors. Frctions my be thought of s loctions or points on number line nd therefore s numbers. So, it is nturl to develop rules to dd, subtrct, multiply, nd divide them. The set of frctions is lrger set of numbers thn the set of integers. Any integer m cn be regrded s frction by writing it s m 1. For exmple, 3 = 3 5, nd 5 = 1 1. Before ddition, subtrction, multipliction, nd division of frctions cn be defined, there is very importnt issue which must be ddressed. Tht is, when do two frctions represent the sme mount? When do they represent the sme point or number on number line? For exmple, 1 2 = 2 = 3 6 = = etc. So ll of these frctions re relly 8 the sme number. How does this work for frctions in generl? We will temporrily restrict our ttention to frctions with positive numertors nd denomintors. Wht is the esiest wy to tell when two frctions re relly the sme number? T-9 Use T-9 to explin tht frctions such s 1 2, 2, nd 8 sme mounts; s well s 1 2, 2 6, nd 3 9. represent the

4 CISC - Curriculum & Instruction Steering Committee T-10 The Jumbo Inch project tht follows cn be used with T-8 to further reinforce this ide. Model the mking of the jumbo inch using T-8, s if it were nrrow strip of pper, to record the steps s you go through the process on the overhed. After modeling for prticipnts, they will do their own jumbo inch using H-10. Mrk the left side of the strip with 0. Mrk the right side of the strip with 1. Fold the strip into hlves. Ask, How mny equl pieces do we 1 hve? Wht frction should we put on the fold? 2 Drw line on the fold bout one-third of the distnce cross. Mrk the line with 1 2 nd put 2 over the 1. 2 Refold the strip into hlves nd fold it into hlves gin. Fold the strip into hlves. Ask, How mny equl pieces do we 1 hve? Wht frction should we put on the folds?, 3 Drw lines on the new folds nd lbel them with 1 nd 3. Wht should we put on the middle fold? 1 2 nd under the 2 2. Continue the procedure through the 16ths. 2 Write 2 under the H-10 T-11 T-12 Discuss wht would hppen if you continued the process. Tlk bout wht would hppen if you hd more thn one inch. Drw mrk between 2 nd 3 nd sk how it would be lbeled. How would we write 2 3 in sixteenths? Hve prticipnts mke their own Jumbo Inch using H-10. This worksheet cn be kept nd used for other lessons throughout the workshop. T-11 represents wht the jumbo inch will look like when the project is finished. Indicte the use of frction strips or pper folding here with some frction of the pper shded to represent frction.

5 CISC - Curriculum & Instruction Steering Committee 5 T-13 T-13A T-1 By incresing the number of equl pieces of whole by multiple of the denomintor of frction, one cn see, using physicl models tht the number of smller pieces which re shded increses by the sme fctor. In symbolic form, frction like 2 2 k cn be rewritten s for ny 3 3 k whole number k. For exmple, 2 3 = 6 = 6 = etc. All of these frctions 9 represent the sme number. Give more exmples if needed. T-15 It follows tht the numertor nd denomintor of frction cn be divided, or multiplied, by the sme common fctor to get nother frction tht represents the sme point on the number line or number. For exmple, 6 9 = = 2 3 In mny cses, two frctions cn be seen to be equl in this wy; tht is, by multiplying or dividing both the numertor nd denomintor of one frction by the sme whole number to get the other frction. But this does not lwys work for whole number vlues of k.. T-16 Here s n exmple: Consider for exmple 3 6 nd 8. Ech of these is equl to 1, but there 2 is no whole number multiple of 3 tht gives nd no whole number multiple of 6 tht gives 8. With lrger numbers involved, it cn be hrder to tell whether two frctions re equl. For exmple, = becuse they cn both be reduced to 13, but it is not t ll obvious tht 17 they re equl by inspection. Is there simple wy to tell when two frctions re the sme number? YES, it is clled cross multipliction.

6 CISC - Curriculum & Instruction Steering Committee 6 Two frctions b nd c d re equl, i.e., b = c d d = bc. if nd only if (iff) For exmple, 3 6 = 8 becuse 3 8 = 6. T-17 T-18 And, = Why does this lwys work? becuse = Answer: To determine whether b = c, find common denomintor d for ech frction. One tht is sure to work, even without knowing wht, b, c, nd d re is the number bd. bd is common denomintor for b nd c d. As you go through this exmple: uncover the sttements to revel the numbers exmple next to the vribles on the trnsprency. This should clrify the concepts presented. Multiplying top nd bottom of b by d nd top nd bottom of c by b the two frctions re chnged s d follows: b = d bd c d = bc bd Since the two frctions on the right hve the sme denomintor, they re equl exctly when their numertors re the sme, i.e., when d = bc. This leds to the following importnt definition: T-19 T-19A Definition 2: Two frctions b nd c re equl, tht is, they represent d the sme number, if nd only if d = bc. Notice from the rgument leding to Definition 2 tht cross multipliction provides simple wy to tell whether one frction is lrger thn nother.

7 CISC - Curriculum & Instruction Steering Committee 7 Proposition: b < c d if nd only if d < bc. For exmple, to decide whether 3 5 is lrger thn, we need to 7 compre 3 7 to 5. Since 5 < < 3 5. H-19 T-20 T-21 T-22 Use H-19 to prctice deciding when two frctions re equl nd if not, which is lrger. Cll ttention of the prticipnts to Definition 2, gin. Point out tht we hve been ssuming tht, b, c, nd d re ll positive. Tell them even so, this definition includes the cses where some or ll of, b, c, nd d re negtive. 3 Use T-20 to explin tht = 3 by cross multipliction nd, lso, by multiplying the top nd bottom of one frction by 1 to get the other frction. Do the sme for 3 3 nd to show they re equl. Tell prticipnts you will show them why 3 is the sme s 3 in the next subsection. This justifies the plcement of these frctions on the number line. Definition 3: A rtionl number is ny number which cn be expressed in the form of b where nd b re integers; b does not equl zero. In other words rtionl numbers re the numbers represented by frctions. So, frctions nd rtionl numbers re essentilly the sme thing. But s we hve seen bove, more thn one frction represents the sme rtionl number, e.g., 1 2 = 2 = 3 6 = etc. T-22A Sets of numbers. This shows the hierrchy of number systems developed so fr. The Arithmetic of Rtionl Numbers. 2. Addition nd Subtrction of Frctions T-23 Cse 1 Two frctions which re to be dded or subtrcted hve the sme denomintor.

8 CISC - Curriculum & Instruction Steering Committee = = = = 3 5 T-2 In generl: Cse 2 b + c b = + c b b c b = c b Two frctions which re to be dded or subtrcted hve the different denomintors. This should be defined using the following formul. After explining the vrible process, then go through with the numericl representtion for prticipnt understnding. b + c (d + cb) 2 = d bd 3 + ( ) = b + c d = = b d d + c d b b = = d bd + cb db = = T-2A Therefore: (d + cb) bd b + c d = (d + cb) bd ( ) = 5 15 Which simplifies to: = or It is importnt to give this s definition of frction ddition becuse it contributes to n understnding of lgebr. This formul works in ll cses, even when the frctions hve the sme denomintor. This importnt formul will be used repetedly in lgebr. Explin tht common denomintors less thn bd my be used, including the Lest Common Multiple (LCM) of b nd d nd this is covered in Number Sense: Module III. Use pproprite overheds lredy submitted.

9 CISC - Curriculum & Instruction Steering Committee 9 T-25 Use the formul for Cse 1 to justify the erlier sttement tht 3 = 3. This follows from: = becuse = 0 = 0 = 0 Therefore 3 = 3. T-26 Use T-26 to demonstrte how mixed numbers cn be converted to improper frctions using frction ddition. One definition tht needs to be cler is: = Using this informtion, we cn convert 5 2 to n improper frction = = = 17 3 Notice the shortcut, = 17 becuse = 17 (Multiply the 3 whole number times the denomintor nd then dd the numertor). H-26 H-26 sks prticipnts why these two wys of converting mixed number to n improper frction re relly the sme. Now, how do we go the other wy? T-27 How would you know to how to write 17 3 = = 5 2 3?

10 CISC - Curriculum & Instruction Steering Committee 10 Long division for 17 3 will lso do this conversion. More exmples should be presented s needed by prticipnts. 3. Multipliction of Frctions The principl im of this section is to motivte the formul for multiplying two frctions, i.e. to justify b c d = c bd T-28 Definition : The product of b nd c for ny, b, b, nd d (with d denomintors not zero) is given by b c d = c bd. The gol of this section is to explin why this is good definition on more intuitive level. Cse 1: n integer times unit frction. 1 5 = ( ) = 5 = 5 So, 1 5 = 5 T-29 Since = 1 this eqution cn be rewritten s =. This is lso 5 the result tht Definition gives. Do few more exmples, depending on how the prticipnts rect nd how they did on the pre-test.

11 CISC - Curriculum & Instruction Steering Committee 11 We cn look t this result in different wy. 1 5 = 1 nd we cn 5 think of this s 1 5 of. In other words, this is 5. T-30 In the context of frctions, it is useful, especilly in the context of word problems to think of the multipliction symbol s representing the word of. So, once gin, 1 5 = 1 of = divided by 5. 5 Since 1 5 = 5, the frction 5 my be thought of s divided by 5. This is conceptul brekthrough. Frctions my thus be understood s the nswers to division problems tht hve no nswer if you only hd whole numbers or integers. With frctions, division problems, no longer need to hve reminders. T-31 Cse 2: The product of two unit frctions, e.g., Use n re model to explin this. Use unit squre (one by one) nd divide one side into equl subdivision nd the other side into 3 equl subdivisions. These generte subrectngles with dimensions 1 by 1 3 which prtition the unit squre. Therefore the re of one of these subrectngles is 1 1. There re 3 = 12 of these subrectngles 3 which cover the unit squre nd they ll hve the sme re. Therefore the re of one of them is the re of the 1 x 1 squre divided by 12, or Therefore = 1 nd this is consistent with Definition. 12 T-32 Do other exmples like this one. Use the unit squre. The time spent on prctice depends on the rections of prticipnts nd the results of the pre-test. Cse 3: The generl cse. Using the ssocitive nd commuttive properties of multipliction (which we tke s xioms).

12 CISC - Curriculum & Instruction Steering Committee 12 b c d =( 1 b ) ( 1 d c) = ( 1 b 1 d ) c =( c) ( 1 b 1 d ) = c bd where both Cse 1 nd Cse 2 hve been used. H-32 Hve prticipnts do H-32 to prctice multipliction of frctions using Cse 3.. Division of Frctions The im of this subsection is to explin nd motivte the formul for division of frctions given by: T-33 Definition 5: For ny frctions b nd c d, b c d = b d c. Do some exmples using this formul. The time spent on exmples depends on the understnding of prticipnts. The immedite gol is just to be ble to understnd wht the definition sys. E.g., = = 9 28 T-3 Why is Definition 5 tken s the definition of division of frctions? There re severl wys to understnd this rule for division of frctions, including: recognizing division in reltion to multipliction, first finding common denomintors of the two frctions. Division my be understood in the following wy: A B = C mens the sme s C B= A.

13 CISC - Curriculum & Instruction Steering Committee 13 The numbers A, B, nd C my represent ny numbers including frctions. The bove sttement my be tken s generl definition of division. For exmple, 12 = 3 becuse 3 = 12. Now try this with frctions. b c d = e f mens the sme s e f c d = b T-35 Now solve for e f by multiplying both sides by d c : e f c d d c = b d c e f c d d c = b d c e f cd cd = b d c e f 1 = b d c e f = b d c Since e f = b c d, this mens, b c d = b d c verified. nd the definition is T-36 Use T-36 to work through this definition with = T = mens how mny one-fourths re in 1 whole?

14 CISC - Curriculum & Instruction Steering Committee 1 Ech subrectngle is one-fourth of the unit squre. There re of the one-qurter units in the unit squre. So: 1 1 = 1 1 = T-38 Common denomintors pproch. Strt with n exmple is nlogous to 6 pples divided by 2 7 pples. The nswer in either cse is 3. Think of s 6 7 groups of 1 7 of something divided by 2 groups of 1 7 of tht thing. T-39 T-0 Illustrtion Dividing frctions with the sme denomintor cn be done by just dividing the numertors. T-0 presents nother method for division of frctions. To do the generl cse, b c, first find common denomintor for d both of these frctions. A common denomintor is bd. Rewrite b = d c nd bd d = bc bd Then b c d = d bd bc bd Since these lst two frctions hve the sme denomintor, we cn just divide the numertors. So the nswer is d divided by bc or d bc. Putting this ltogether sys tht b c d = b d, nd this is exctly c Definition 5. T-1 T-2 This might be clled the "frctions of frctions" pproch.

15 CISC - Curriculum & Instruction Steering Committee 15 T-3 A specil note: Mny students through the yers complin bout not understnding frctions. They will often void problems involving frctions. Trditionl sequence of dding, subtrcting, multiplying, nd dividing nturl numbers, whole numbers, nd integers leds us to do the sme with rtionl numbers. However, when we get to Algebr, the order is often reversed when working with polynomils involving rtionl expressions. Techers might give students greter security when working with frctions by cpitlizing on their successes. Students tend to find reducing of frctions n esier tsk thn finding common denomintors when dding or subtrcting them. Some students my find greter success with frctions by multiplying nd dividing them first nd then getting to one of the most difficult concepts to lern; ddition nd subtrction of frctions with unlike denomintors. Some techers my wish to tech ddition nd subtrction of frctions with Cse 1 conditions, move to multipliction nd division of frctions s n extension of reducing or building frctions, nd then conclude with dding nd subtrcting frctions with unlike denomintors s in Cse 2 conditions when they re more experienced in working with frctions. T- nd T-A T-5 Post-Test Conceptul Problem. Before showing T-A hve prticipnts shre their problem solving strtegies. The generl understnding should be emphsized: C nd D re both less thn 1 nd positive so their product hs to be less thn either C or D nd positive. B is the only possibility. Optionl Word Problems Prticipnts should tke the Post-Test t the end of this module. The Post-Test hs more conceptul questions thn the Pre-Test. Problems should be discussed with prticipnts to cler up ny confusions. Stndrds for Rtionl Numbers Grde 3 Number Sense 3.0 Students understnd the reltionship between whole numbers, simple frctions, nd decimls. 3.1 Compre frctions represented by drwings or concrete mterils to show equivlency nd to dd nd subtrct simple frction in context. 3.2 Add nd subtrct simple frctions.

16 CISC - Curriculum & Instruction Steering Committee 16 Grde Number Sense 1.5 Explin different interprettions of frctions, for exmple, prts of whole, prts of set, nd division of whole numbers by whole numbers; explin equivlents of frctions. 1.7 Write the frctions represented by drwing of prts of figure; represent given frction by using drwings; nd relte frction to simple deciml on number line. Grde 5 Number Sense 1.5 Identify nd represent on number line decimls, frctions, mixed numbers, nd positive nd negtive integers. 2.0 Students perform clcultions nd solve problems involving ddition, subtrction, nd simple multipliction nd division of frctions nd decimls. 2.3 Solve simple problems, including ones rising in concrete solutions, involving the ddition nd subtrction of frctions nd mixed numbers (like nd unlike denomintors of 20 or less), nd express nswers in the simplest form. 2. Understnd the concept of multipliction nd division of frctions. 2.5 Complete nd perform simple multipliction nd division of frctions nd pply procedures to solving problems. Grde 6 Number Sense 2.0 Students clculte nd solve problems involving ddition, subtrction, multipliction, nd division. 2.1 Solve problems involving ddition, subtrction, multipliction, nd division of positive frctions nd explin why prticulr opertion ws used for given sitution. 2.2 Explin the mening of multipliction nd division of positive frctions nd perform the clcultions. 2. Determine the lest common multiple nd the gretest common divisor of whole numbers; use them to solve problems with frctions (e.g., to find common denomintor to dd two frctions or to find the reduced form for frction).

17 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers T-1 References: Teching Frctions in Elementry School: A Mnul for Techers. (Mrch, 1998). Knowing nd Teching Elementry Mthemtics, by Liping M,

18 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers Pre Test Pre-Test = = = = = = = = = 10. There re 25 children in the clss; 3 of the children in the clss re 5 boys. How mny girls re in the clss? 11. Hlf of the children in our school wtch television every night. Threefourths of those children wtch for more thn n hour. Wht frction of the totl children wtch for more thn n hour night?

19 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers Pre Test Key Pre-Test Key girls

20 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers T-2 WHAT IS A FRACTION? The word frction comes from the Ltin word, frctio which mens the ct of breking into pieces. Frctions rise nturlly in mesurement problems proper frctions express quntity less thn whole unit indicte mounts or distnces in which bsic unit is subdivided into whole number of equl prts

21 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers T-3 An Exmple A pie is divided into 8 pieces nd three of them re eten. The frction 3 8 expresses the reltive mount of the pie tht ws consumed.

22 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers T

23 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers T color in 3 8 of the circle color in 6 8 of the circle

24 PRIMARY CONTENT MODULE V NUMBER SENSE: Rtionl Numbers T-6 Definition 1: A frction is rtio of integers nd b, b 0 written s b is the numertor b is the denomintor the numertor; counts or enumertes the denomintor; specifies wht is being counted or gives the denomintion