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Fractions - Revision parts of a whole Fractions are numbers which mostly describe ______________. E.g. colored: 5 ___ of the rectangle 12 7 ___ of the rectangle not colored: 12 Fractions can describe even more than one whole! E.g. a) colored: 7 ___ circles 3 2 not colored: ___ of a circle 3 b) colored : 3 5 2 __ rectangles 2 not colored: ___ of a rectangle 5 How to color: a) 5 ___ of a parallelogram? 6 b) 7 ___ squares? 4 c) 1 ___ rhombuses? 4 3 The parts of the fraction: ?numerator a __ b ?fraction line (or vinculum) ?denominator Denominator tells us into how many equal parts the whole should be divided. Numerator tells us how many of those parts should be colored. We used these properties of numerator and denominator in the previous examples. Fraction line always means division. E.g. 8 __ = 8 : 4 = 2 4 = Proper fractions are fractions with the numerator less than the denominator. Improper fractions are fractions with the numerator greater than or equal to the denominator. 1 , ___ E.g. 4 2 , ___ 9 ___ ... are proper _____ fractions. 9 10 < less than 1. They are ______ Proper fractions are fractions with the numerator less than the denominator. Improper fractions are fractions with the numerator greater than or equal to the denominator. 9 , ___ E.g. 4 5 , ___ 4 , ___ 6 ___ improper fractions. ... are ________ 3 4 2 ≥ greater than or equal to 1. They are __________________ Which of the fractions above are equal to 1? How can we recognize fractions which are equal to 1? The numerator is equal to the denominator!! Improper fractions are fractions with the numerator greater than or equal to the denominator. 9 , ___ E.g. 4 5 , ___ 4 , ___ 6 ___ improper fractions. ... are ________ 3 4 2 ≥ greater than or equal to 1. They are __________________ Which of the fractions above are greater than 1? How can we recognize fractions which are greater than 1? The numerator is greater than the denominator!! Improper fractions are fractions with the numerator greater than or equal to the denominator. 9 , ___ E.g. 4 5 , ___ 4 , ___ 6 ___ improper fractions. ... are ________ 3 4 2 ≥ greater than or equal to 1. They are __________________ Any improper fraction can be changed into a mixed fraction or a natural number. Improper fractions are fractions with the numerator greater than or equal to the denominator. 9 , ___ E.g. 4 5 , ___ 4 , ___ 6 ___ improper fractions. ... are ________ 3 4 2 ≥ Which of these fractions can be changed into mixed fractions ? Change them (look at the picture)! 9 1 ___ = 2 ___ 4 4 2 5 ___ ___ = 1 3 3 Improper fractions are fractions with the numerator greater than or equal to the denominator. 9 , ___ E.g. 4 5 , ___ 4 , ___ 6 ___ improper fractions. ... are ________ 3 4 2 ≥ Which of these fractions can be changed into natural numbers ? Change them (look at the picture)! 4 ___ = 1 4 6 ___ = 3 2 Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number a) 19 3 ___ = 2 ___ 8 8 Explanation: 19:8 equals 2 and remainder 3 Rewrite denominator! = (do what is possible): Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number a) 19 3 ___ = 2 ___ 8 8 b) 68 5 ___ = 7 ___ 9 9 c) 42 ___ = 7 6 Explanation: 42:7 equals 6 (no remainder) = (do what is possible): Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number a) b) 19 3 ___ = 2 ___ 8 8 68 5 ___ = 7 ___ 9 9 c) 42 ___ = 7 6 d) 36 ___ = 4 9 e) 2 ___ = 9 (do what is possible): How did we calculate in all these tasks? We divided the numerator by the denominator. Why? Because the fraction line always means division! Now let's change a fraction into a decimal number! How to do it? This is proper fraction (numerator is less than denominator), We should divide again, so we can't change it into mixedin writing... buta now fraction or into natural number! Let's revise it... Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number a) (do what is possible): 19 3 ___ = 2 ___ 8 8 19 ___ = 19 : 8 = 2 .3 7 5 8 30 Remember: 60 So, we changed same fraction 0 Let's change the 4number at task "a)" intochange athe decimal number... When we fraction into both - mixed fraction How to do it? into any another form,and 0 decimal number. we always divide (because the fraction line means division)! Only when we change into decimal number, then we use long division. Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 6·8+7 a) 7 55 ___ ___ = 6 8 8 Rewrite denominator! = Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: a) 7 55 ___ ___ = 6 8 8 b) 6 69 ___ ___ = 9 7 7 c) 8 16 24 ___ ___ ___ 8 = = = = ... 1 2 3 (When we divide numerator by denominator, the result must be 8 !) = = = = = = = ... Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: a) 7 55 ___ ___ = 6 8 8 b) 6 69 ___ ___ = 9 7 7 c) 8 16 24 ___ ___ ___ 8 = = = = ... 1 2 3 d) 241 2.41 = ____ 100 2 decimal digits 2 zeros = Explanation: Rewrite the given number, but without decimal point... Write the digit 1 and as many zeros as we have decimal digits in the given number... Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: a) 7 55 ___ ___ = 6 8 8 f) 0.019 = 19 ____ 1000 b) 6 69 ___ ___ = 9 7 7 g) 27 54 27 = ___ = ___ = ... 1 2 c) 8 16 24 ___ ___ ___ 8 = = = = ... 1 2 3 d) 241 2.41 = ____ 100 h) 3 ___ = 4 7 e) 309 30.9 = ____ 10 i) 2893 28.93 = ____ 100 31 ___ 7 Some decimal numbers can be changed into mixed fractions. Let's revise it... 3.) Change into mixed number: a) 41 2.41 = 2 ____ 100 2 decimal digits 2 zeros = Recall: 2.41 can be changed not only into a mixed fraction, but into an improper fraction as well. Say that improper fraction... 241 ___ 100 Some decimal numbers can be changed into mixed fractions. Let's revise it... 3.) Change into mixed number: a) 41 2.41 = 2 ____ 100 b) 9 30.9 = 30 ____ 10 c) 7 15.007 = 15 ____ 1000 d) 0.045 = This decimal number can't be changed into a mixed fraction because it has got zero wholes. We can only change it into a fraction. If we should change it into fraction, 45 the solution would be ____ . 1000 What does it mean - "to reduce a fraction" ? To reduce a fraction means to divide both the numerator and denominator of the fraction by the same number. When we reduce a fraction, does the reduced fraction represent a bigger or a smaller part than the original fraction? They represent equal parts!! When we reduce a fraction to the non-reducible fraction, then we get this fraction in the lowest possible terms. 4.) Reduce these fractions to non-reducible fractions: a) 5 10 5 ___ = ___ 12 6 6 = 2 We can reduce it by __. So, we divide both the numerator and denominator by 2 and write the results… What does it mean - "to reduce a fraction" ? To reduce a fraction means to divide both the numerator and denominator of the fraction by the same number. When we reduce a fraction, does the reduced fraction represent a bigger or a smaller part than the original fraction? They represent equal parts!! When we reduce a fraction to the non-reducible fraction, then we get this fraction in the lowest possible terms. 4.) Reduce these fractions to non-reducible fractions: b) 4 24 4 ___ = ___ 30 5 5 = 6 We can reduce it by __. What does it mean - "to reduce a fraction" ? To reduce a fraction means to divide both the numerator and denominator of the fraction by the same number. When we reduce a fraction, does the reduced fraction represent a bigger or a smaller part than the original fraction? They represent equal parts!! When we reduce a fraction to the non-reducible fraction, then we get this fraction in the lowest possible terms. 4.) Reduce these fractions to non-reducible fractions: c) 42 ___ 63 6 = 9 2 6 ___ 9 = 3 = = 2 ___ 7 by __. 3 Now We can we can reduce reduce it byagain, __. 3 Now let's revise other properties of fractions... 5.) Complete these sentences: four fourths. a) One whole equals ____ 4 ___ 4 1 = = twelve twelfths. b) One whole equals _____ 1 = 12 ___ 12 = six thirds. c) Two wholes equal ___ 2 = = 6 ___ 3 Now let's revise other properties of fractions... 5.) Complete: 4 d) 4 days = ___ week 7 Explanation: 7 days. First, let's recall that a week has ___ 1 of the week. So we can conclude that each day is __ 7 Now, we should think in this way: 1 __ of the week, 7 2 __ then 2 days are of the week, 7 3 __ 3 days are of the week, 7 4 __ and 4 days are of the week. 7 If 1 day is Now let's revise other properties of fractions... 5.) Complete: 4 d) 4 days = ___ week 7 We can explain it in another way: 7 days. Again, let's recall that a week has ___ So, let's consider it together with the rectangle divided into 7 equal parts! week: Monday Tuesday Wednesday Thursday Friday Saturday Sunday rectangle: 4? of the week __ 7 Left picture: We are interested in which part of the week consist of 4 days... Right picture: Which part of the rectangle consist of 4 parts... Now let's revise other properties of fractions... 5.) Complete: 4 d) 4 days = ___ week 7 Shortly: Rewrite the given number into numerator. In the denominator write the total number of days in a week ! As we already know, the denominator is the total number of equal parts, and the numerator describes the number of parts we are interested in. Now let's revise other properties of fractions... 5.) Complete: 4 d) 4 days = ___ week 7 20 ? 1 12 1 __ 20 1 ___ ___ ? 20 3 e) 20 min. = hour = hour We can reduce 9 it by3 __. 60 3 3 5 f) 5 months = ___ year 12 min. hour 6 year: January February March April May June July August September Octobar November December 5 __ ? 12 of the year Now let's revise other properties of fractions... 6.) There are 7 tasks in the Ivan's homework. Ivan solved 2 of them. What portion of the homework did Ivan solve? 2 Ivan solved ___ of his homework. 7 What portion of the homework does he still have to solve? He should solve 5 ___ of his homework yet. 7 homework: 5 __ ? of the homework 7 1st task 2nd task 3rd task 4th task 5th task 6th task 7th task 2 __ ? of the homework 7 Now let's revise other properties of fractions... 7 7.) If children ate ___ (seven tenths) of a cake, which 10 fraction of the cake remained? 3 ___ (three tenths) of the cake remained. 10 3 ___ of the cake 10 7 ___ of the cake 10 Now let's revise other properties of fractions... 6 ___ 8.) Climber Dario climbed of his path in one hour, 14 4 another ___ of the path in the next one hour, 14 3 and finally ___ of his path in the third hour. 14 Did he climb his whole path? 6 4 3 ___ + ___ + ___ = 14 14 14 13 ___ 14 No, he didn't climb his whole path. There remained 1 ___ of his path. 14 9.) Seven friends gathered some money and bought 3 chocolates of equal size. They want to divide the chocolates equally. How much chocolate will each of them get? 3 3 : 7 = ___ 7 3 Each friend will get ___ of a chocolate. 7 Explanation with pictures: When they divide the first chocolate into 7 equal parts, 1 each friend will get __ of the chocolate. 7 When they divide the second chocolate into 7 equal parts, 1 each friend will get __ of the chocolate. 7 When they divide the third chocolate into 7 equal parts, 1 each friend will get __ of the chocolate. 7 3 So, after all divisions each friend will have __ of a chocolate. 7 10.) a) 12 chocolates should be divided among 5 friends. How many chocolates will each of them get? 2 12 12 : 5 = ___ = 2 ___ 5 5 2 Each friend will get 2 ___ chocolates. 5 Explanation with pictures: Each friend gets 1 __ of the 11th chocolate. 5 Each friend gets 1 __ of the 12th chocolate. 5 So, after all divisions each friend will have 2 2 __ chocolates! 5 10.) b) What about dividing 12 chocolates among 3 friends? 12 : 3 = 4 Each friend will get 4 chocolates. Explanation with pictures: After division each friend will have 4 chocolates. 12 11.) Little Ana ate ___ strawberries. How many strawberries 2 did she eat actually? 12 ___ = 12 : 2 = 6 2 Little Ana ate 6 strawberries. Explanation with pictures: 12 ___ strawberries 2 = 6 strawberries 12.) There are 48 apples in the box. 3 5 ___ of the box are red apples, ___ are green apples 8 12 and the rest of the apples are yellow. a) How many apples are of which color? red: green: yellow: 3 ___ of 48 is 8 5 ___ of 48 is 12 18 + 20 = 38, 18 ( we calculated 48:8·3 ) 20 48 - 38 = 10 There are 18 red, 20 green and 10 yellow apples in that box. b) What portion of the box do yellow apples form? 5 5 ___ 10 5 Yellow apples form (five twenty___ 2 24 = ___ We can reduce it by __. fourths) of the box. 48 24 24 13.) Complete these expressions: a) 2 ___ of 15 is 6 5 We calculated: 15 : 5 · 2 = 6 13.) Complete these expressions: a) 2 ___ of 15 is 6 5 How to explain this calculation? 2 Recall: If we want to color ___ of some figure, then 5 we divide it into 5 equal parts, and then color 2 parts. Here we do just about the same thing! 2 ___ of 15 can be calculated so that we divide 5 number 15 into 5 equal parts, and then take 2 parts. 15 : 5 · 2 = 6 13.) Complete these expressions: a) 2 ___ of 15 is 6 5 b) 4 ___ of 72 is 9 32 c) 3 ___ of 10 is 4 1 7 ___ 2 E.g. 2 ___ · 15 51 3 6 = ___ = 6 1 5 Both procedures We can reduce it by __. give equal results!!! Here we can't so1 we must calculate in 3 word 15 10:4, The of5 divide means multiplication! ___ ___ ___ = 10 = · 7 some another way! 42 2 2 2 in that way in the can reduce it by __. Are weWe allowed to calculate a and b tasks? Yes, we are!!! 13.) Complete these expressions: a) 2 ___ of 15 is 6 5 b) 4 ___ of 72 is 9 32 c) 3 ___ of 10 is 4 1 7 ___ 2 We got the same result !!! The denominator tells us should divide this "bulk" We 10 imagine pieces of something, 10c?pears... The Howhave numerator can we 3 4tells the us problem tothat takewe in 3e.g. of part these 4 parts... into 4 equal parts... 1st 1 2 2nd 3 4 3rd 5 6 7 4th 8 How many pears do we have in these three parts? 9 10 7 1 __ 2 13.) Complete these expressions: a) 2 ___ of 15 is 6 5 b) 4 ___ of 72 is 9 32 c) 3 ___ of 10 is 4 1 7 ___ 2 d) 12 39 ___ of ___ is 13 44 12 ___ 13 3 1 · 9 ___ 11 3 9 39 ___ = ___ 11 44 11 4 We We can can reduce reduce it it by by __. 13 __. 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 3 . ___ 5 Explanation with pictures: 2 ___ of the rectangle is less than 1 rectangle, 5 < 3 of the rectangle. by the uncolored part, and this part is ___ 5 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 b) 19 ___ greater than 1 , by is _______ 15 3 . ___ 5 4 . ___ 15 Explanation with pictures: 19 ___ rectangles is greater than 1 rectangle, 15 > by the part determined by second rectangle, and it is 4 of the rectangle. ___ 15 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 b) 19 ___ greater than 1 , by is _______ 15 3 . ___ 5 4 . ___ 15 15.) In each inequality below, which number is the greater? a) 8 ___ 9 > > 5 ___ 9 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 b) 19 ___ greater than 1 , by is _______ 15 3 . ___ 5 4 . ___ 15 15.) In each inequality below, which number is the greater? a) b) 8 5 ___ ___ > 9 9 7 1 ___ ___ < 2 4 10 5 < 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 b) 19 ___ greater than 1 , by is _______ 15 3 . ___ 5 4 . ___ 15 15.) In each inequality below, which number is the greater? a) b) c) 8 5 ___ ___ > 9 9 7 1 ___ ___ < 2 4 10 5 3 ___ 4 11 < < 5 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 b) 19 ___ greater than 1 , by is _______ 15 3 . ___ 5 4 . ___ 15 15.) In each inequality below, which number is the greater? a) b) c) 8 5 ___ ___ > 9 9 7 1 ___ ___ < 2 4 10 5 3 ___ 4 11 3 d) 6 ___ 5 < > > 5 1 ___ 6 5 e) 8 5 ___ ·>· ___ 3 2 16 > 15 We multiply through diagonals... Instead of cross-multiplying, we can find the common denominator and then compare numerators... If we would take the common denominator 3·2, that is 6, then we would get numerators 16 and 15. Multiplication through diagonals is shortcut for that. 14.) Complete: a) 2 ___ less is ______ than 1 , by 5 b) 19 ___ greater than 1 , by is _______ 15 3 . ___ 5 4 . ___ 15 15.) In each inequality below, which number is the greater? a) b) c) 8 5 ___ ___ > 9 9 7 1 ___ ___ < 2 4 10 5 3 ___ 4 11 3 d) 6 ___ 5 < 5 > 1 ___ 6 5 e) 8 ___ 3 > > 5 ___ 2 16.) Complete: increases as well a) If the numerator increases, then the fraction _____________. E.g. 1 ___ 3 2 ___ 3 3 ___ 3 4 ___ 3 5 ___ 3 If we are looking from the left to the right, increase the numerators ________. Colored parts, that is the fractions _____________. increase as well 16.) Complete: increases as well a) If the numerator increases, then the fraction _____________. decreases b) If the denominator increases, then the fraction _________. E.g. 1 ___ 1 1 ___ 2 1 ___ 3 1 ___ 4 1 ___ 5 If we are looking from the left to the right, increase the denominators ________. Colored parts, that is, the fractions of the whole decrease _______. 16.) Complete: increases as well a) If the numerator increases, then the fraction _____________. decreases b) If the denominator increases, then the fraction _________. 17.) Figure out: a) 5 4 15 + 8 23 5 ___ ___ ________ ___ ___ + = = = 1 6 9 18 18 18 b) 4 1 1 9 1 27 - 1 26 ___ - ___ = ___ - ___ = ________ = ___ = 2 6 2 6 6 6 2 ___ = 4 6 1 3 1 = 4 ___ 3 2 We can reduce it by __. 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 Explanation with pictures: 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 b) 3 1 - ___ = 5 2 ___ 5 Explanation with pictures: 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 b) 3 1 - ___ = 5 2 ___ 5 c) 2 7 ___ ___ = 5 6 9 9 Explanation with pictures: 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 b) 3 1 - ___ = 5 2 ___ 5 c) 2 7 ___ ___ = 5 6 d) 9 1 ___ + 5 4 2 9 1 = 9 ___ 2 Explanation with pictures: 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 b) 3 1 - ___ = 5 2 ___ 5 c) 2 7 ___ ___ = 5 6 9 d) 1 ___ + 5 4 2 e) 6 2 ___ - 6 11 9 1 = 9 ___ 2 2 = ___ 11 Explanation with pictures: 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 b) 3 1 - ___ = 5 2 ___ 5 c) 2 7 ___ ___ = 5 6 9 9 d) 1 ___ + 5 4 2 e) 6 f) 1 1 ___ ___ = 8 3 3 2 ___ - 6 11 1 = 9 ___ 2 2 = ___ 11 8 Explanation with pictures: 18.) Try to figure these out mentally! a) 5 ___ = 3 + 7 5 ___ 3 7 b) 3 1 - ___ = 5 2 ___ 5 c) 2 7 ___ ___ = 5 6 9 9 d) 1 ___ + 5 4 2 e) 6 f) 1 1 ___ ___ = 8 3 3 g) 7 6 - 2 ___ = 8 2 ___ - 6 11 1 = 9 ___ 2 2 = ___ 11 Explanation with pictures: 8 1 3 ___ 8 19.) Figure out: a) b) c) 3 3 27 21 9 2 ___ ___ ___ ___ · = = 1 49 7 9 1 7 7 9 We can reduce it by2__. 7 reduce 2We can___ 54 ___ · = 4 9 19 6 38 54 12 ___ ___ ___ = = 12 · 9 19 1 1 1 19 We can reduce it by __. 9 can 1 9 3We ___ 13 reduce 39it by __. 1 ___ ___ ___ ___ = = = 19 · 9 · 2 6 1 6 2 2 2 3 We can reduce it by __. 20.) Figure out: a) 12 ___ = 8 : 7 8 2 ___ 7 14 2 ___ ___ ___ = 4 · = 1 12 3 3 3 4 We can reduce it by __. b) 1 1 7 1 31 1 ___ ___ 8 1 ___ ___ ___ ___ ___ : 3 = : = · = 8 8 8 8 8 31 31 1 8 We can reduce it by __. Let's recall the “number line”.. 0 1 2 3 Place the following numbers onto the number line: a) 5 ___ 2 6 2 and __ 3 between __ 4 Let's recall the “number line”.. 0 1 2 3 Place the following numbers onto the number line: a) 5 ___ 2 6 marked part of the number line 6 equal parts should be divided into ____________ 4 Let's recall the “number line”.. 5 2 __ 6 0 1 2 3 4 Place the following numbers onto the number line: a) 5 ___ 2 6 5 parts of the whole from the left we should count __ Let's recall the “number line”.. 5 2 __ 6 0 1 2 1 3 __ 2 3 Place the following numbers onto the number line: a) 5 ___ 2 6 b) 7 1 ___ = 3 ___ 2 2 middle 3 and __ 4 , exactly in the ______ between __ 7 __ 2 4 Let's recall the “number line”.. 5 2 __ 6 0 1 2 1 3 __ 2 3 7 __ 2 4 Place the following numbers onto the number line: a) 5 ___ 2 6 b) 7 1 ___ = 3 ___ 2 2 c) 2 ___ 3 it is not possible to change it into a mixed number, there are no wholes, 0 and __ 1 so this number lies between __ Let's recall the “number line”.. 5 2 __ 6 0 1 2 1 3 __ 2 3 Place the following numbers onto the number line: a) 5 ___ 2 6 b) 7 1 ___ = 3 ___ 2 2 c) 2 ___ 3 marked part of the number line 3 equal parts should be divided into ____________ 7 __ 2 4 Let's recall the “number line”.. 5 2 __ 6 0 2 __ 3 1 2 1 3 __ 2 3 7 __ 2 Place the following numbers onto the number line: a) 5 ___ 2 6 b) 7 1 ___ = 3 ___ 2 2 c) 2 ___ 3 2 parts of the whole from the left we count __ 4 We shall continue this revision in writing... Now we should be able to solve several more complex tasks with more ‘fraction calculation’ operations. Open your notebooks... Author of presentation: Antonija Horvatek Croatia , October 2008. With thanks to: GSC for support, great suggestions and preliminary help with the translation into English and Rex Boggs for support and help with the translation into fluent U.S. idiom (a.k.a. ‘American’). You are welcome to use this presentation in your teaching. Additionally, you can change some parts of it if used solely for teaching. However, if you want to use it in public lectures, workshops, in writing books, articles, on CDs or any public forum or for any commercial purpose, please ask for specific permission from the author. Antonija Horvatek http://public.carnet.hr/~ahorvate [email protected]