Fractions – Basics

FRACTIONS – basics Fractions arise to express PART of a UNIT 1 What part of an HOUR is thirty minutes? (The UNIT here is HOUR.) Fifteen minutes? ( ( ( tw elve minutes? ( 2 What fraction of the children are happy? ; ; ; ; (The UNIT here is the entire GROUP or set of children.) In this example, w e may also perceive the fraction as a ratio. One person might say half or four-eighths the children are happy. Another might say four (are happy), of eight altogether, or one of every tw o. 3 It’ s 12 miles to Nana’ s house. We have driven 9 miles. What part of the drive have w e covered? (The UNIT here is the distance to Nana’ s house.) We compare 9 to 12. 4 Four children must divide three pies among them. How much of a PIE does each one get? rrr (The UNIT here is a PIE.) Most generally, a fraction is used to express part of a w hole. The denominator tells us into how many equal parts the w hole w as divided. The numerator tells us how many of those parts w e have. 3 ² numerator 4 ² denominator ) ËDefinition: The set of rational numbers, denoted Q, is { a/ b| a 0 Z and b 0 Z and b ' 0} . The numerator of a/ b is defined to be a, the denominator of a/ b is b. The concept of equivalent fractions is encoded in the ËFUNDA M ENT A L LA W OF a) = a@c )) b b@c if c ' 0, FRA CTIONS : In grade school this is called RENAMING fractions. For example, 1/2 = (1@4)/(2@4) = 4/8, illustrated at right: 1/2 is the simplest or reduced form of the fraction. A rational fraction is said to be in SIM PLEST or reduced form w hen the numerator and denominator are relatively prime. (Special case: if the denominator is 1, w e usually omit w riting the denominator.) 1 2 )) = A f t er w e def ine m ult iplicat ion ! ËReduce: 72 27 = 960 420 = 8@9 3@9 = 8 3 @ 9 9 = 8 3 @ 1 FRA CTION STRIPS : 2 4 4 8 )) = )) = COMPARING FRACTIONS: Are 3 and 4 7 7 equivalent? 3 4 and 2 ? 3 72 and 480? 27 210 130 and 42? 195 63 ËHow we know that a/b and c/d are equivalent: Reduce, express with common denominator. What about w hen they are NOT equivalent? How do w e put them in order? (1) use common denominator (then compare numerators) (2) compare to easily recognized fractions (3) if the numerators are the same, compare denominators. (Larger denominatorY smaller fraction!) (Would you rather have ½ or c of $1million?) Illustrating fractions w ith area models. To illustrate one-half, it is important that the UNIT be divided into tw o EQUA L parts. These are good illustrations of one-half: These are not: In different discussions of fractions the unit may vary, but one common unit should be used throughout a particular discussion. Diagrams used to illustrate addition, subtraction, multiplication and division, as w ell as fraction equivalence and comparison should follow suit. For instance, to show equivalence, use ONE unit and subdivide: 1 1 1 '2 = 2'4 '2 = 3'6 '3 = 2'6 Rectangles use: do NOT use: aren’ t“ glamorous” , but they’ re easy to = divide tw o w ays! To model the addition of ½+ ½ do not use: ½+ ¼ do not use: ½+ a use: ½+ ¼ use: + aNot sam e unit ! _ aSam e unit ! _ NSU! Measurement models are very similar to area models, but are not necessarily 2-dimensional. Set models are rarely used for comparisons or illustrating arithmetic operations. (Usually “ find bof N” ) jjjj jjjj jjjj jjjj jjjj jjjj Many fractions do not exist in W, the set of w hole numbers. They belong to the set of “ rational numbers” , w hich includes the w hole numbers, and all these new ratios. To save space, w e w ill refer to the set of fractions as “ Q” . Getting ready for addition & subtraction of fractions: We start w ith common denominators: 1 ) 3 + 1 3 ) 2 ) 5 – 1 5 ) 1 ) 4 + 2 4 ) 1 ) + 2 1 2 ) We step up to denominators that are “ compatible” , as one divides the other: 1 ) 2 + 1 4 ) 3 ) 4 – 1 2 ) 1 ) 2 – 1 6 1 ) 5 + ) 5 ) 6 – ) ) Until w e are ready for the general case: 1 ) 2 + 1 3 ) 1 ) 2 – 1 3 ) 2 3 Then w e w ill go beyond that, to: 1 ) 6 + 1 9 ) 1 ) 6 – 1 9 ) 4 9 1 ) + 12 1 ) 16 1 ) 3 + 1 3 ) ADDITION AND SUBTRACTION OF RATIONAL NUMBERS ËDef'n ADDITION on Q: a b D D a+ b )) + )) = D b D D E.g. a!b D 1 2 if a,b,c ,d,D0 W 2@2 + 1@3 3@2 c b d ad bc bd bd )) + )) = ad + bc )))))) bd & aD> bD or ad> bc & D ' 0 & d ' 0 a c b d )) ! )) = numbers a/b & c/d = a )) + )) = in general, for rational ))))) 2 + 3 '0 & d'0 numbers a/b & c/d ËDef'n SUBTRA CTION on Q: a D in general, for rational ))))) )) ! )) = & if a,b,c ,d,D0W ad bc bd bd )) ! )) 4 2 /3 = 4 + 2 = = 2+ 5 = 2 + 5 1 1 ad ! bc )))))) bd 12 + 2 ¹Dealing w ith 3 “ mixed numbers” 3 Notice if the rational addends are integers, w e obt ain the accustomed results. = 2 @1 + 5 @1 1 @1 = Also notice if b = d, the above general addition formula yields: a c ad + dc d(a + c) a+ c ...as w e E.g. 2 3 )) + )) = ))))))) = ))))))) = ))))) ) + ) d d d@d d@d d expect. 7 7 = ))))))) Although the given formula for + " w orks" , it leads to unnecessarily large numbers plus extra w ork to reduce! The preferred met hod is to w rite equivalent fractions using the least common denominator, D — (LCD ) ... and then appeal to the intuitively clear " a/ D + b/ D = (a + b)/ D " . In short, rewrite the fractions with the LCD , then add numerators. (Note the LCD is the actually the Least Common Multiple of the denominators.) E.g. 4 @3 5 12 )) + _ 4 @4 7 16 )) = )) + )) )) = _ Sketch a model to illustrate the entire process of adding 1/2 + 2/5: . . . . . . . . . . . . 1 ) 2 . . . . . . . . . . . . + . . . . . . 2 ) 5 . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 )) + )) = 10 10 . . . . . . . . . . . . . . . . . . 9 )) 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ËProperties of + on Q: Closure: Given any rationals a/b and c/d, their sum, (ad + bc)/bd 0 Q. (Sinc e: a,d,b,c 0 Z guarant ees ad+ bc and bd are bot h 0 Z by closure of m ult iplicat ion and addit ion on Z; and b ' 0 & d ' 0 guarant ees bd ' 0 . Commutativity: a/b + c/d = (ad + bc)/bd The only dif f erence bet w een t hese is in t he num erat ors. c/d + a/b = (cb + da)/bd But t hey are equal, by com m ut at iv it y of m ult iplicat ion and addit ion on t he set W [or Z] of w hole-numbers[or int egers] . Theref ore, w e m ight say com m ut at iv it y of addit ion on Q is " inherit ed" f rom t he com m ut at iv it y of addit ion and m ult iplicat ion on W [or Z ]. Associativity: Identity: Inverses: A re sim ilarly " inherit ed" f rom t he set of W , since, af t er rew rit ing as equiv alent f rac t ions w it h LCD ' s, w e sim ply add t he w hole num bers in t he num erat ors. ËProperties of ! on Q: Mixed numbers. Properties are the same as for W. Common error. Counting up. Compensation. 7 = 6+ 1 = 6+ 8 / 8 Mixed numbers: 7 3 /8 ! 2 3 /4 Alternative: 7 3 /8 ! 2 6/8 6 !2 4 The traditional method: Subtract the fraction from the fraction and the w hole number from the w hole. Exchange a unit for the appropriate fraction if needed. 11 /8 6 /8 5 /8 Use ‘ improper’ fractions. 59 7 3 / 8 ! 2 3 / 4 = (7+ 3 / 8 ) ! (2 + 3 / 4 ) = /8 ! 1 1 /4 = 5 9 /8 ! 2 2/8 = 3 7/8 = 4 5 /8 Plus: avoids mixed numbers at the point of subtracting. Minus: Look at those FUN big numbers, and the extra conversion steps at the beginning & end. Alternative: “ Count up” + 1 /4 * 2 3 /4 + 4 * * 3 4 5 So w e see the difference is 4 + Alternative: “ compensation” 1 /4 + 8 + 3 6 /8 = /8 * 7 3/8 7 4 5 /8 (Based in part on some perceptions from the above picture): + 2/ + 2/ 7 3 /8 8 ! 2 6/8 8 = shifts both #s up the # line the same amount. 7 5 /8 ! 3 = 45/8 8Motivation: It’ s so much easier to subtract 3! Addition compensation: !1 / 6 + 1 / 6 3 1 /3 + 2 5 /6 = 3 2 /6 + 2 5 /6 = (Boost one, drop the other so the total is unchanged.) 3 1/6 + 3 Common error: 2 ) 3 How silly is t hat ? > ½ 3 ) 5 + + > ½ = 5 ) 8 = < 1 ???? 1 ) 2 + 1 ) 3 = 2 ) 5 1 ) 2 + 1 ) 2 = 2 ) 4 This is all BAD SCIENCE!!! ???? Word problems Three-fifths of the children in Ms Jones’ class participate on school teams. If 8 children in Ms Jones’ class do not participate on school teams, how many children are in Ms. Jones’ class? ²Fifths partitions the class int o 5 equal-size groups. ÆÉÉÉÉÉÉÉÉÈÉÉÉÉÉÉÉÉÉÇ ÆÉÉÉÉÈÉÉÉÉÉÇ Since the 8 Boys make up 2 of those groups, we see that each Girls Boys – 8 group must contain 8/2 = 4 children. So there are 5@4 = 20 children altogether in Ms. Jones’ class. Mabel is w rapping gifts for the teachers. Each gift needs 2½ feet of ribbon. If there are 20 teachers getting gifts, how much ribbon does Mabel need? If she bought a spool w ith 10 yards of ribbon does she have enough? Etc 0 1 2 3 4 5 6 7 8 9 10 Introduction to multiplication: Whole number × Fraction via REPEA TED 1 1 1 2 EG 2× ) = ) + )= ) 3 3 3 3 Fraction × Whole 1 EG ) × 2 = 3 A DDITION : 1 3 × ) = 4 number via THE W ORD “ OF” : 1 ) of 2 = ? 3 1 ) of 3 Fraction × Fraction 1 1 ) × ) = 2 2 Consider: 1 1 ) × ) 3 2 1 4 1 4 ) + ) + 1 = 4 3 4 ) ) two units W it h t his & a f ew m ore ex am ples w e can see t hat t his m ult iplicat ion is com m ut at iv e, just like in W 2 is ) 3 One Half W e see t he w hole # m ult iplies t he num erat or. Half of one-half is: E.g. ² Half of one-half 1 ) 3 × 3 ) 5 2 ) 3 × 3 ) 5 ) × c ) d W hic h lead t o our... ËDef'n. M ULTIPLICATION on Q: a b M ak e y our ow n M odels of m ult iplic at ion f or: Multiply: 15 24 )) )) = 8 25 3@5 @ 3@8 8 @ 5@5 ))))))) a@c b@d ))))) = 1 4 of 1 2 3 4 of 1 2 1 4 of 1 3 3 4 of 1 3 Since w e know all t he f act ors in t he num & den bef ore w e m ult iply , w e know w e can “ cancel” BEFORE w e m ult iply .... I pref er w hat ’ s show n. 3@3 5 )) = ac bd ))) = What about “ Mixed Numbers” (they should call these “ Mixed Up” ) 2 1 2 10 ) × 4 ) = ( 10 + ) ) × ( 4 + 3 2 3 1 2 ) ) @@@ = (FOIL aw ay) BAD– Common Error: multiply w holes, multiply fractions, add. “ Improper” fractions are NOT bad: 2 1 2 1 10 ) × 4 ) = ( 10+ ) ) × ( 4 + ) ) = 3 2 3 2 APP: 32 3 9 = 2 )) @ ) 32@9 ))) = 48 3@2 One fourth of the students at WHS are freshmen. Of these, 3 fourths are in a geometry class. What part of the WHS students are freshmen taking geometry? ËProperties of Multiplication on Q: (assume a/b. c/d, e/f are rationals) Closure: ac/bd 0 Q since ac and bd are in W [Z], and bd Commutativity: sem i-inherit ed f rom W [Z]: a/b @ c/ d = Associativity: dit t o: (a/b @ c/d) @ e/f = Identity: is st ill 1 (ak a 1 /1 ): a/b @ 1 = Inverses f or all non-zero rat ionals a/b! : If a ac/bd @ e/f = ac/bd = 0 st ill t akes all! ca/db = (ac)e/(bd)f = a/b @ 1 /1 = a@1 /b@1 = 0 @ w hat ev er = 0 Distributive property of multiplication over addition: st ill holds. c/d @ a/b a(ce)/b(df ) = a/b. ' 0 and b ' 0 , t hen a/b x b/a = 1 . (The m ult iplicat iv e inv erse of a/b, prov ided a Annihilator: ' 0 because b ' 0 and d ' 0 . ' 0 , is b/a .) et c! Notice that: Div ision by a w hole num ber seem s t o be m ult iplicat ion by t he reciprocal. ½ of 8 is the same as 8 ÷ 2 ¼ of 12 is 12÷ 4 and so on. Further, consider that: 1 ÷ ½ = 2, 1÷ a = 3, 1÷ ¼ = 4 2÷½ Div ision by 1 /n seem s t o be m ult iplicat ion by n, t he reciprocal. = 4, 3 ÷ ½ = 6, and so on. 6 Just as ËDef'n. DIV ISION on Q: IF c ' 0 (& b,d ' 0) a b ÷ 2 c d = = ÷ = 3 ad bc a b x ...since 2x3 ...since c x ad = d bc d c = 6 acd bcd d C? = 2 3 Other view s: c d = ÷ 3 1 ÷ = 4 5 a ÷ c b d a b 2 d 3 @ c = 15 4 ÷ 20 20 ad bd ÷ = c ÷) = x d ) ÷ c ) d since = 15 = 4 ad bc = bc bd = c d ) b ))) c ) = a b ) @ bd b )))) = c ) @ bd b 2 3 d c = a ) b (Then solve for x.) a@d Provided c,b,d ' 0 ))) c@b d d @ c ) @x d a = 2d 3c = @ ) 3@5 4@1 ...x must satisfy: a a ) b 2d 3c = 1 5 ÷4 a b i.e. ad is t he “ m agic num ber” bc t hat solv es c a ) E.g. = A mixed number example: 10 2 ÷ 1 3 3 5 = 32 ÷ 8 3 5 = 32 3 8 5 32 3 8 5 = x x 15 = 15 32 @ 5 8@3 = "Cancellation" (?) 7 1 1) )) = 8 ÷ 1 24 15 8 )) @ 24 = 25 )) 15 @ 24 = 8 @ 25 )))))) 5@3@8@3 8@5@5 ))))))) Properties of division on Q: Almost closure: Closure, except for division by 0. = a/b ÷ c/d = ad/bc, a rational, provided c ' 0. The ability to divide any number by almost any number expands the problems for w hich w e can compute answ ers. This property goes hand in hand w ith the property of INVERSES for multiplication. 1 is the multiplicative identity. If w e ask, in multiplication, is there a w ay to “ get back to the identity, 1" using multiplication, the answ er is now yes, just multiply by the reciprocal. a ) b @ ? = 1 has an answ er for every fraction except 0. a But that, in turn means that 1 ÷ ) b has an answ er except w hen a = 0. Word problems put the arithmetic to use. How ever, you may also observe the truth of the “ rules” via the setting of a w ord problem. Take, for instance, this problem from the homew ork: After reading 186 pages, Jennifer had read 3/5 of her book. How many pages has the book? 186 ÂÅÅÅÅÅÅÄÅÅÅÅÅÅà Solution by scaling: 186 is 3 parts (of 5 parts). Each part is 186 ÷ 3 = 62. The w hole is 5 parts, or 5@62 = 310 pages. Look at the algebra: Let X = total pages So ÆÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÈÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÇ ? 3 ) of X = 186 5 3 X = 186 ÷ / 5 See process above!! a a # ÷ /b b 3 You w ill be asked to make up a w ord problem yielding (for example) 22 ÷ / 5 . The above observation indicates that a problem of the form: “ John used 3/5 of his fuel, or 22 gallons. How much fuel did he have to start?” w ill have that result. ) of X = # results in the division X = Also important: Notice that the problem (4a) Make a w ord problem for 135 ÷ 5 using measurement division. 135 athletes w ere assigned to practice in teams of 5. How many teams ? (4b) ...Ditto, but this time partitive division. 135 athletes w ere assigned equally to 5 service committees. How many on each committee? (4c) ...Measurement division 32 ÷ 3¾ It takes 3¾ minutes to grade 1 page of a test, how many pages can be graded in 32 minutes? 3 (4d) ...Partitive division for 72 ÷ / 5 3 After 72 hours grading, Mary found she w as / 5 done. How long to grade all the papers? Bar diagrams can be especially helpful w hen Fractions are involved. Consider solving this problem w ithout bar diagrams, w ithout algebra: (5e) 1 5 A pet shop sold / 4 of its puppies in the 1 st w eek, / 6 of the rest in the 2 nd w eek. The pet shop sold 28 puppies altogether during those tw o w eeks. How many puppies had they to start? Our text emphasizes using ONE bar diagram. That’ s fine for this problem, but w e show here the development of the diagram. b 1 st w eek 2 nd w eek:: ÂÄà ÂÅÅÅÅÅÅÅÅÅÄÅÅÅÅÅÅÅÅà Altogether: ÆÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÈÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÇ 28 28 ÷ 7 = 4 (each part) 8 @ 4 = 32 puppies to start. Using algebra: Let X = number of puppies to start 1 5 3 ( / 4 )X + ( / 6 ) ( / 4 )X = 7 ( /8) X = X = 28 28 8 28 ( / 7 ) = 32 There w ere 32 puppies to start. Fraction “Rules” Summary (Fundamental Law ): )ba ac (+ – Definition): a ) b Fraction as ÷ : a÷b = Multiplication : a c ) b @) d Division : a c ) b ÷ ) d = )) bc c ad bc )) b d ± )) bd ± ) d = ad ± b c ))))) b d = a ) b (b 0) (c 0) ac )) bd = = ad )) bc a The set of fractions (all numbers that can be w ritten as / b) has the follow ing properties: All the arithmetic rules for the Whole numbers apply. In addition: 1 For every member x of Q, except if x = 0, there is another member of Q, called / x , such that 1 x@) X = 1 1 ) X is called the reciprocal of x, and is also called the multiplicative inverse of x. 1 1 1 1 For instance, if x is a w hole number, 1, 2, 3, @@@ then ) X is ) 1 , ) 2 , ) 3 , @@@ (respectively). a 1 Other fractions such as ) b can be thought of as a @ () b ) w hen it suits. 2 What is the multiplicative inverse of ) 3 ? )32 @ = 1 See text for proofs of that these numbers satisfy the “ rules” stated in the summary above. Complex fractions: 2 ) @ )13  —‰ 3 + 3 9 ) ) ‰2 ÷ 2 ž ÷ 10 )) 27 a a ( b 2 ÷ c) + c ÷ ) 2 2 ) 9 2 + )) 27 ))))))))))) 2 2 ) – )) 9 27 1 3 ) – 2 4 )))))))) 2 1 ) + 1 3 2 3×9 5× 7 4×5 20 27 18 20 × 27 × 35 )) × )) ÷ )) = ))))))))) 21 5 35 21 × 5 × 18 3×7 2×9 20 27 18 2 @10 @3 @9 @5 @7 )) × )) ÷ )) = )))))))))) = 21 5 35 3 @7 @5 @2 @ 9 2 3 —‰ ) @ )31 3 9 10 2 ÷ ) 2  ž ÷ )) 27 + ‰ ) 2 ) 9 – + 3 ) 9  @ 27 )) 10 5 ) 9 @ 27 )) 10 Order of operat ions: parent heses f irst ! = Parent heses st ill f irst , but w e can get ready t o do t he division, by invert ing.... 3 ) Then m ult iplying 2 a a ( b 2 ÷ c) + c ÷ ) 2 W rit e t he divisions in f ract ion f orm . 2 b a( c ) + Ready t o m ult iply and add... ) 2 c @) a 2 a b )aa ) ) c) + 2 c )cc ) ) a) 22 Get a com m on denom inat or — ac. 2 a b + 2c )) )) a ) c)) 2 ) 2 + )) 9 27 ))))))))))) 2 2 ) – )) 9 27 & A dd = = = 2 2 ˆ) + ) ) 9 27  2 7 ))))))))))) 2 2 ˆ) – ) ) 9 27  2 7 6 + 2 ))))))) 6 – 2 The easy w ay t o clean t his up— m ult iply num erat or & denom inat or by LCM (9 ,2 7 ), w hich is 2 7 . Get : no more f ract ions w it hin f ract ion ! 2 Do t he sam e t hing here—m ult iply num erat or & denom inat or by 1 2 . 1 3 ) – 2 36+ 3 – 24 15 4 Get )))))))))) )))))))) = ))) 2 1 ) + 1 12 + 8 + 12 32 3 A lt ernat ely , com bine t he w hole num bers 1 1 3 ) – 2 1 + ) 4 4 )))))))) = ))))) = 2 2 1 ) + 1 2 + ) 3 3 f irst , t hen m ult iply by 1 2 : 12 + 3 )))))) = 24 + 8 15 ))) 32 This is problem # 5 in St 2 9 (§6 .6 p 1 6 5 ). Although not assigned, it illustrates the problem created by disorganized “ cancellation” . Here is Ky le’ s w ork, if deciphered correct ly: It ’ s hard t o see w hat ’ s been “ cancelled” and w hat ’ s lef t . 2 3×9 5× 7 4×5 20 27 18 20 × 27 × 35 )) × )) ÷ )) = ))))))))) 21 5 35 21 × 5 × 18 3×7 2×9 Here is t he sam e w ork, organized. A nt icipat ing reducing t he f ract ion, 2 0 w as f act ored int o 2 @1 0 , & 1 8 int o 2 @9 , so t he 2 s can be reduced. Et c. Try t he “ cancellat ion” yourself here: 20 27 18 )) × )) ÷ )) 21 5 35 2 @10 @3 @9 @5 @7 = )))))))))) 3 @7 @5 @2 @ 9 = 10