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Fractions - Revision

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Fractions are numbers which mostly describe ______________. parts of a whole E.g. colored: ___512 of the rectangle not colored: ___712 of the rectangle a) colored: ___73 circles not colored: ___23 of a circle b) colored : not colored: __35 rectangles 2 ___25 of a rectangle Fractions can describe even more than one whole! E.g.

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How to color: a) ___56 of a parallelogram? b) ___74 squares? c) ___13 rhombuses? 4

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The parts of the fraction: __ab ? numerator numerator ? denominator denominator ? fraction line (or vinculum) fraction line (or vinculum) Denominator tells us into how many equal parts the whole Numerator tells us how many of those parts should be colored. Fraction line always means division. E.g. __84 = 8 : 4 = 2 We used these properties of numerator and denominator in the previous examples. = should be divided.

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Proper fractions are fractions with the numerator less than Improper fractions are fractions with the numerator greater than or equal to the denominator. E.g., ___14, ___29 ___ are _____ fractions. They are ______ than 1. less < proper the denominator.

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improper They are __________________ 1. greater than or equal to E.g., ___94, ___53 ___44... are ________ fractions., ___62 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!! Proper fractions are fractions with the numerator less than Improper fractions are fractions with the numerator greater than or equal to the denominator. the denominator.

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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. improper E.g., ___94, ___53 ___44... are ________ fractions., ___62 They are __________________ 1. greater than or equal to

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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. improper E.g., ___94, ___53 ___44... are ________ fractions., ___62 They are __________________ 1. greater than or equal to

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Which of these fractions can be changed into mixed fractions ? Change them (look at the picture)! ___94 = 2 ___14 ___53 = 1 ___23 Improper fractions are fractions with the numerator greater than or equal to the denominator. improper E.g., ___94, ___53 ___44... are ________ fractions., ___62

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Which of these fractions can be changed into natural numbers ? ___44 = 1 ___62 = 3 Improper fractions are fractions with the numerator greater than or equal to the denominator. improper E.g., ___94, ___53 ___44... are ________ fractions., ___62 Change them (look at the picture)!

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Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number (do what is possible) : ___198 = a) 3 8___2 Explanation: 19:8 equals 2 and remainder 3 Rewrite denominator! =

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Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number (do what is possible) : ___198 = a) 3 8___2 ___689 = b) 5 9___7 ___427 = c) 6 Explanation: 42:7 equals 6 (no remainder) =

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Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number (do what is possible) : ___198 = a) 3 8___2 ___689 = b) 5 9___7 ___427 = c) 6 ___364 = d) ___29 = e) This is proper fraction (numerator is less than denominator), so we can't change it into a mixed fraction or into natural number! 9 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? We should divide again, but now in writing... Let's revise it...

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Let's change the number at task "a)" into a decimal number... How to do it? ___198 = 19 : 8 = So, we changed the same fraction into both - mixed fraction and decimal number. Remember: When we change fraction into any another form, we always divide (because the fraction line means division)! Only when we change into decimal number, then we use long division. ___198 = a) ___2 3 8 Now, let's revise how to calculate it (without a picture)! 1.) Change into a mixed number or a natural number (do what is possible) :

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Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 855______78 = a) 6 6 · Rewritedenominator! =

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Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 855______78 = a) 6 769______67 = b) 9 c) 8 = 18___ =16___ 2= 24___ 3 =... (When we divide numerator by denominator, the result must be 8 !) ====== =...

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Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 855______78 = a) 6 769______67 = b) 9 c) 8 = 18___ =16___ 2= 24___ 3 =... d) 2.41 = ____ 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... 2 decimal digits 2 zeros Explanation: =

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Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 855______78 = a) 6 769______67 = b) 9 c) 8 = 18___ =16___ 2= 24___ 3 =... d) 2.41 = ____ e) 30.9 = 10309____ f) = ____ 731______37 = h) 4 g) 27 = 127___ =54___ 2 =... i) = ____

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Some decimal numbers can be changed into mixed fractions. Let's revise it... 3.) Change into mixed number: a) 2.41 = ____ ___ Recall: 2.41 can be changed not only into a mixed fraction, but into an improper fraction as well. Say that improper fraction... 2 decimal digits 2 zeros =

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Some decimal numbers can be changed into mixed fractions. Let's revise it... 3.) Change into mixed number: a) 2.41 = ____2 b) 30.9 = 10 9____30 c) = ____15 d) = 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, the solution would be ____.

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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!! 4.) Reduce these fractions to non-reducible fractions: ___1012 = a) We can reduce it by __ ___56 = When we reduce a fraction to the non-reducible fraction, then we get this fraction in the lowest possible terms. So, we divide both the numerator and denominator by 2 and write the results…

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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. They represent equal parts!! 4.) Reduce these fractions to non-reducible fractions: ___2430 = b) We can reduce it by __ ___45 = When we reduce a fraction, does the reduced fraction represent a bigger or a smaller part than the original fraction? When we reduce a fraction to the non-reducible fraction, then we get this fraction in the lowest possible terms.

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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. They represent equal parts!! 4.) Reduce these fractions to non-reducible fractions: ___4263 = c) We can reduce it by __ ___69 Now we can reduce again, by __ ___23 = = = When we reduce a fraction, does the reduced fraction represent a bigger or a smaller part than the original fraction? When we reduce a fraction to the non-reducible fraction, then we get this fraction in the lowest possible terms.

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Now let's revise other properties of fractions... a) One whole equals ____ fourths. four 1 = ___44 = b) One whole equals _____ twelfths. twelve 1 = ___1212 = c) Two wholes equal ___ thirds. six 2 = ___63 = 5.) Complete these sentences:

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d) 4 days = week ___47 Now let's revise other properties of fractions... 5.) Complete: Explanation: First, let's recall that a week has ___ days. 7 So we can conclude that each day is of the week. __17 Now, we should think in this way: If 1 day is of the week, __17 then 2 days are of the week, __27 3 days are of the week, __37 and 4 days are of the week. __47

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d) 4 days = week ___47 Now let's revise other properties of fractions... 5.) Complete: We can explain it in another way: Again, let's recall that a week has ___ days. 7 Monday Tuesday Wednesday Thursday Friday Saturday Sunday week: So, let's consider it together with the rectangle divided into 7 equal parts! rectangle: 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... ? __47 of the week

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d) 4 days = week ___47 Now let's revise other properties of fractions... 5.) Complete: 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.

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d) 4 days = week ___47 e) 20 min. = hour ___2060 = hour ___13 We can reduce it by __ Now let's revise other properties of fractions... 5.) Complete: f) 5 months = year ___ __13 min.?20 hour ? January February March April May June Julyyear: August September Octobar November December __512 of the year ?

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6.) There are 7 tasks in the Ivan's homework. Ivan solved 2 of them. What portion of the homework did Ivan solve? Ivan solved of his homework. ___27 What portion of the homework does he still have to solve? He should solve of his homework yet. ___57 Now let's revise other properties of fractions... 1st task 2nd task 3rd task 4th task 5th task 6th task 7th task homework: ? __27 of the homework ?? ____5577

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Now let's revise other properties of fractions... If children ate (seven tenths) of a cake, which If children ate (seven tenths) of a cake, which fraction of the cake remained? fraction of the cake remained? ___710 7.) ___310 (three tenths) of the cake remained. ___310 of the cake ___710

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8.) Climber Dario climbed of his path in one hour, ___614 another of the path in the next one hour, and finally of his path in the third hour. Did he climb his whole path? ___ 4 14 ___ 3 14 ___1314 ___414 ___614 + ___314 += No, he didn't climb his whole path. There remained of his path. ___114 Now let's revise other properties of fractions...

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9.) Seven friends gathered some money and bought 3 chocolates ofequal size. They want to divide the chocolates equally. 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? How much chocolate will each of them get? 3 : 7 = ___37 Each friend will get of a chocolate. ___37 Explanation with pictures: When they divide the first chocolate into 7 equal parts, each friend will get __17 of the chocolate. When they divide the second chocolate into 7 equal parts, each friend will get __17 of the chocolate. When they divide the third chocolate into 7 equal parts, each friend will get __17 of the chocolate. So, after all divisions each friend will have __37 of a chocolate.

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10.) a) 12 chocolates should be divided among 5 friends. How many chocolates will each of them get? How many chocolates will each of them get? 12 : 5 = ___125 = 2 5___2 Each friend will get 2 chocolates. ___25 Explanation with pictures: Each friend gets __15 of the 11th chocolate. Each friend gets __15 of the 12th chocolate. So, after all divisions each friend will have __25 chocolates! 2

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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.

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11.) Little Ana ate strawberries. How many strawberries did she eat actually? did she eat actually? ___ : 2 = = ___122 Little Ana ate 6 strawberries. 6 Explanation with pictures: strawberries ___122 = 6 strawberries

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12.) There are 48 apples in the box. ___ 3 8 of the box are red apples, ___ 5 12 are green apples and the rest of the apples are yellow. a) How many apples are of which color? ___38 of 48 is 18red: ___512 20green: ( we calculated 48:8·3 ) = 38,yellow: = 10 There are 18 red, 20 green and 10 yellow apples in that box. b) What portion of the box do yellow apples form? ___ We can reduce it by __. 2 = ___524 Yellow apples form (five twenty- fourths) of the box. ___524

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13.) Complete these expressions: a) of 15 is ___25 6 We calculated: 15 : 5 · 2· 2· 2· 2=6

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How to explain this calculation? we divide it into 5 equal parts, and then color 2 parts. Recall: If we want to color of some figure, then ___25 Here we do just about the same thing! number 15 into 5 equal parts, and then take 2 parts. of 15 can be calculated so that we divide ___25 15 : 5 · 2 =6 a) of 15 is ___25 13.) Complete these expressions: 6

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b) of 72 is ___49 32 c) of 10 is ___34 Here we can't divide 10:4, so we must calculate in some another way! The word of means multiplication! ___34 · We can reduce it by __. 2 ___152 = 7___1 2 7 ___12 a) of 15 is ___25 13.) Complete these expressions: Are we allowed to calculate in that way in the a and b tasks? Yes, we are!!! =E.g. ___25 · We can reduce it by __. 5 ___61 = 6 = Both procedures give equal results!!! 6

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b) of 72 is ___49 32 c) of 10 is ___34 7 ___12 a) of 15 is ___25 13.) Complete these expressions: How can we imagine the problem in part c? 6 We have 10 pieces of something, e.g. 10 pears... The denominator 4 tells us that we should divide this "bulk" into 4 equal parts... 1st2nd3rd4th The numerator 3 tells us to take 3 of these 4 parts... How many pears do we have in these three parts? __12 7 We got the same result !!!

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d) of is ___1213 ___3944 ___1213 · ___3944 = 3 11 We can reduce it by __ ___911 ___911 a) of 15 is ___25 13.) Complete these expressions: b) of 72 is ___49 32 c) of 10 is ___34 7 ___12 6

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14.) Complete: a) is ______ than 1 ___25 less, by ___35. Explanation with pictures: of the rectangle is less than 1 rectangle, of the rectangle is less than 1 rectangle, ___25 < by the uncolored part, and this part is ___35 of the rectangle.

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14.) Complete: a) is ______ than 1 ___25 less, by ___35. b) is _______ than 1 ___1915 greater, by ___415. Explanation with pictures: rectangles is greater than 1 rectangle, rectangles is greater than 1 rectangle, ___1915 > by the part determined by second rectangle, and it is ___415 of the rectangle.

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15.) In each inequality below, which number is the greater? a) ___89 ___59 > > 14.) Complete: a) is ______ than 1 ___25 less, by ___35. b) is _______ than 1 ___1915 greater, by ___415.

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15.) In each inequality below, which number is the greater? a) ___89 ___59 > < b) ___710 ___15 24 < 14.) Complete: a) is ______ than 1 ___25 less, by ___35. b) is _______ than 1 ___1915 greater, by ___415.

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15.) In each inequality below, which number is the greater? a) ___89 ___59 > < b) ___710 ___15 24 < c) ___ < 14.) Complete: a) is ______ than 1 ___25 less, by ___35. b) is _______ than 1 ___1915 greater, by ___415.

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15.) In each inequality below, which number is the greater? a) ___89 ___59 > < b) ___710 ___15 24 < c) ___ d) ___35 ___15 66 > e) ___83 ___52 We multiply through diagonals... ·16 ·15 > > 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) is ______ than 1 ___25 less, by ___35. b) is _______ than 1 ___1915 greater, by ___415.

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15.) In each inequality below, which number is the greater? a) ___89 ___59 > < b) ___710 ___15 24 < c) ___ d) ___35 ___15 66 > e) ___83 ___52 > > 14.) Complete: a) is ______ than 1 ___25 less, by ___35. b) is _______ than 1 ___1915 greater, by ___415.

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16.) Complete: a) If the numerator increases, then the fraction _____________. increases as well E.g. ___13 ___23 ___33 ___43 ___53 If we are looking from the left to the right, the numerators ________. increase Colored parts, that is the fractions _____________. increase as well

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b) If the denominator increases, then the fraction _________. decreases E.g. ___11 ___12 ___13 ___14 ___15 If we are looking from the left to the right, the denominators ________. increase Colored parts, that is, the fractions of the whole _______. decrease 16.) Complete: a) If the numerator increases, then the fraction _____________. increases as well

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16.) Complete: a) If the numerator increases, then the fraction _____________. b) If the denominator increases, then the fraction _________. decreases increases as well

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17.) Figure out: ___56 + a) ___49 = ________ = +8 ___2318 = 1___5 18 ________ 27 6 = -1 ___266 = 4___2 6 ___12 - b) ___16 = 4 ___92 - ___16 = = We can reduce it by __ ___13 =

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 Explanation with pictures:

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 2___ 5 - b) ___35 = 1 Explanation with pictures:

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 2___ 5 - b) ___35 = 1 75 ___ 9 - c) ___29 = 6 Explanation with pictures:

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 2___ 5 - b) ___35 = 1 75 ___ 9 - c) ___29 = 6 ___12 + d) = ___ 2 Explanation with pictures:

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 2___ 5 - b) ___35 = 1 75 ___ 9 - c) ___29 = 6 ___12 + d) = ___ 2 ___211 - e) = 6 62___ 11 Explanation with pictures:

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 2___ 5 - b) ___35 = 1 75 ___ 9 - c) ___29 = 6 ___12 + d) = ___ 2 ___211 - e) = 6 62___ 11 ___13 - f) ___13 = 8 8 Explanation with pictures:

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18.) Try to figure these out mentally! 53 ___ 7 + a) ___57 = 3 2___ 5 - b) ___35 = 1 75 ___ 9 - c) ___29 = 6 ___12 + d) = ___ 2 ___211 - e) = 6 62___ 11 ___13 - f) ___13 = g) ___78 = ___ 8 Explanation with pictures:

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19.) Figure out: ___2749 · a) ___219 = 3 1 We can reduce it by __ ___97 = 1___2 7 ___29 · b) ___5419 = 4 ___389 · ___5419 = ___121 = 12 · c) ___16 = 9 2 ___91 · ___ = ___392 = 19___1 2

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20.) Figure out: : a) ___127 = 8 ___81 · ___ We can reduce it by __. 4 = ___143 = 4___2 3 : b) ___78 = 3 ___18 ___18 : ___318 = ___18 · ___ = ___131

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Let's recall the number line Place the following numbers onto the number line: a) 2 ___56 between __ and __ 23

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01234 a) 2 ___56 marked part of the number line should be divided into ____________ 6 equal parts Place the following numbers onto the number line: Let's recall the number line..

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01234 a) 2 ___56 we should count __ parts of the whole from the left 5 2 __56 Place the following numbers onto the number line: Let's recall the number line..

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01234 a) 2 ___56 2 __56 b) ___72 = 3___1 2 between __ and __ 34, exactly in the ______ middle 3 __12 __72 Place the following numbers onto the number line: Let's recall the number line..

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01234 a) 2 ___56 2 __56 b) ___72 = 3___1 2 3 __12 __72 c) ___23 it is not possible to change it into a mixed number, there are no wholes, so this number lies between __ and __ 01 Place the following numbers onto the number line: Let's recall the number line..

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01234 a) 2 ___56 2 __56 b) ___72 = 3___1 2 3 __12 __72 c) ___23 marked part of the number line should be divided into ____________ 3 equal parts Place the following numbers onto the number line: Let's recall the number line..

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01234 a) 2 ___56 2 __56 b) ___72 = 3___1 2 3 __12 __72 c) ___23 we count __ parts of the whole from the left 2 __ 2 3 Place the following numbers onto the number line: Let's recall the number line..

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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...

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Author of presentation: Antonija Horvatek Croatia, October 2008.

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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).

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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

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