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Fractions - Revision

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**Fractions are numbers which mostly describe ______________. **

parts of a whole E.g. ___ 5 12 of the rectangle colored: ___ 7 12 of the rectangle not colored: Fractions can describe even more than one whole! E.g. a) ___ 7 3 circles colored: ___ 2 3 of a circle not colored: b) __ 3 5 rectangles 2 colored : ___ 2 5 of a rectangle not colored:

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**How to color: a) of a parallelogram? squares? b) rhombuses? c) 4 5 ___**

6 of a parallelogram? b) ___ 7 4 squares? c) ___ 1 3 rhombuses? 4

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**a __ b 8 __ 4 The parts of the fraction: ? numerator ?**

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 4 = 8 : 4 = 2

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**< 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. E.g. , ___ 1 4 2 9 10 ... are _____ fractions. proper < They are ______ than 1. less

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**≥ 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 4 5 3 ... are ________ fractions. 6 2 E.g. improper ≥ They are __________________ 1. greater than or equal to 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!!

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**≥ Improper fractions are fractions with the numerator greater**

than or equal to the denominator. , ___ 9 4 5 3 ... are ________ fractions. 6 2 E.g. improper ≥ They are __________________ 1. greater than or equal to 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!!

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**≥ Improper fractions are fractions with the numerator greater**

than or equal to the denominator. , ___ 9 4 5 3 ... are ________ fractions. 6 2 E.g. improper ≥ They are __________________ 1. greater than or equal to Any improper fraction can be changed into a mixed fraction or a natural number.

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**≥ Improper fractions are fractions with the numerator greater**

than or equal to the denominator. , ___ 9 4 5 3 ... are ________ fractions. 6 2 E.g. improper ≥ Which of these fractions can be changed into mixed fractions ? Change them (look at the picture)! ___ 9 4 = 2 ___ 1 4 ___ 5 3 = 1 ___ 2 3

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**≥ Improper fractions are fractions with the numerator greater**

than or equal to the denominator. , ___ 9 4 5 3 ... are ________ fractions. 6 2 E.g. improper ≥ Which of these fractions can be changed into natural numbers ? Change them (look at the picture)! ___ 4 = ___ 6 2 = 1 3

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**= Now, let's revise how to calculate it (without a picture)! a) 2**

1.) Change into a mixed number or a natural number (do what is possible): ___ 19 8 = a) ___ 2 3 8 Explanation: 19:8 equals 2 and remainder 3 Rewrite denominator! =

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**= = = 6 Now, let's revise how to calculate it (without a picture)! a)**

1.) Change into a mixed number or a natural number (do what is possible): ___ 19 8 = a) ___ 2 3 8 ___ 68 9 = b) ___ 7 5 9 ___ 42 7 = 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): How did we calculate in all these tasks? ___ 19 8 = a) ___ 2 3 8 We divided the numerator by the denominator. ___ 68 9 = b) ___ 7 5 9 Why? ___ 42 7 = c) 6 Because the fraction line always means division! ___ 36 4 = d) Now let's change a fraction into a decimal number! How to do it? 9 ___ 2 9 = e) This is proper fraction (numerator is less than denominator), so we can't change it into a mixed fraction or into natural number! We should divide again, but now in writing... Let's revise it...

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**. = = Now, let's revise how to calculate it (without a picture)! a) 2**

1.) Change into a mixed number or a natural number (do what is possible): ___ 19 8 = a) ___ 2 3 8 ___ 19 8 = . 19 : 8 = 2 3 7 5 3 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. 6 So, we changed the same fraction into both - mixed fraction and decimal number. Let's change the number at task "a)" into a decimal number... How to do it? 4

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**= Now conversely! Let's revise how to change numbers**

from other forms into fractions... 2.) Change into fraction: 6 · 8 + 7 ___ 7 8 = a) 6 55 ___ 8 Rewrite denominator! =

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**= = Now conversely! Let's revise how to change numbers**

from other forms into fractions... 2.) Change into fraction: ___ 7 8 = a) 6 55 ___ 8 ___ 6 7 = b) 9 69 ___ 7 1 8 ___ = 16 ___ 24 ___ c) 8 = 2 = 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: ___ 7 8 = a) 6 55 ___ 8 ___ 6 7 = b) 9 69 ___ 7 1 8 ___ = 16 ___ 24 ___ c) 8 = 2 = 3 = ... Explanation: 241 ____ d) = Rewrite the given number, but without decimal point... 100 Write the digit 1 and as many zeros as we have decimal digits in the given number... 2 decimal digits 2 zeros =

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**= = = Now conversely! Let's revise how to change numbers**

from other forms into fractions... 2.) Change into fraction: ___ 7 8 = a) 6 55 ___ 19 ____ f) = 8 1000 ___ 6 7 = b) 9 69 ___ 1 27 ___ = 54 ___ g) 27 = 2 = ... 7 1 8 ___ = 16 ___ 24 ___ c) 8 = 2 = 3 = ... ___ 3 7 = h) 4 31 ___ 241 ____ d) = 7 100 309 ____ 2893 ____ e) = i) = 10 100

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**Some decimal numbers can be changed into mixed fractions. **

Let's revise it... 3.) Change into mixed number: ____ 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... a) = 100 2 decimal digits 2 zeros = 100 241 ___

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**Some decimal numbers can be changed into mixed fractions. **

Let's revise it... 3.) Change into mixed number: ____ 2 41 a) = 100 ____ 30 9 b) = 10 ____ 15 7 c) = 1000 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 1000 45 ____ .

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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!! 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: 5 ___ 10 12 = a) ___ 5 6 We can reduce it by __. 2 6 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. 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: 4 ___ 24 30 = b) ___ 4 5 We can reduce it by __. 6 5 =

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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!! 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: 6 2 ___ 42 63 = c) ___ 6 9 ___ 2 3 = Now we can reduce again, by __. We can reduce it by __. 7 3 9 3 =

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**Now let's revise other properties of fractions...**

5.) Complete these sentences: ___ 4 a) One whole equals ____ fourths. four 1 = = ___ 12 b) One whole equals _____ twelfths. twelve 1 = = ___ 6 3 c) Two wholes equal ___ thirds. six 2 = =

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**Now let's revise other properties of fractions...**

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

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**Now let's revise other properties of fractions...**

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

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**Now let's revise other properties of fractions...**

5.) Complete: ___ 4 7 d) 4 days = week 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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**Now let's revise other properties of fractions...**

5.) Complete: ___ 4 7 d) 4 days = week ? 20 min. 1 ___ 20 60 ___ 1 3 12 3 9 6 __ 1 3 ? hour e) 20 min. = hour = hour We can reduce it by __. 20 3 ___ 5 12 f) 5 months = year January February March April May June July year: August September Octobar November December __ 5 12 ? of the year

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**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? Ivan solved of his homework. ___ 2 7 What portion of the homework does he still have to solve? He should solve of his homework yet. ___ 5 7 homework: 1st task __ 2 7 ? of the homework 2nd task 3rd task __ 5 7 4th task of the homework ? 5th task 6th task 7th task

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**Now let's revise other properties of fractions...**

If children ate (seven tenths) of a cake, which fraction of the cake remained? ___ 7 10 7.) ___ 3 10 (three tenths) of the cake remained. ___ 3 10 of the cake ___ 7 10 of the cake

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**Now let's revise other properties of fractions...**

8.) Climber Dario climbed of his path in one hour, ___ 6 14 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 3 ___ 4 14 6 + 3 = ___ 13 14 No, he didn't climb his whole path. ___ 1 14 There remained of his path.

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**How much chocolate will each of them get?**

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 7 3 : 7 = Each friend will get of a chocolate. ___ 3 7 Explanation with pictures: When they divide the first chocolate into 7 equal parts, each friend will get __ 1 7 of the chocolate. When they divide the second chocolate into 7 equal parts, each friend will get __ 1 7 of the chocolate. When they divide the third chocolate into 7 equal parts, each friend will get __ 1 7 of the chocolate. __ 3 7 of a chocolate. So, after all divisions each friend will have

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**10.) a) 12 chocolates should be divided among 5 friends. **

How many chocolates will each of them get? ___ 12 5 ___ 2 2 12 : 5 = = 5 Each friend will get chocolates. ___ 2 5 Explanation with pictures: Each friend gets __ 1 5 of the 11th chocolate. Each friend gets __ 1 5 of the 12th chocolate. __ 2 5 chocolates! So, after all divisions each friend will have

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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? ___ 12 2 = ___ 12 2 12 : 2 = 6 Little Ana ate 6 strawberries. Explanation with pictures: strawberries ___ 12 2 = 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? ___ 3 8 red: of 48 is 18 ( we calculated 48:8·3 ) ___ 5 12 green: of 48 is 20 yellow: = 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 Yellow apples form (five twenty- fourths) of the box. ___ 5 24 ___ 10 48 ___ 5 24 = We can reduce it by __. 2 24

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**13.) Complete these expressions:**

a) of 15 is ___ 2 5 6 We calculated: 15 : 5 · 2 = 6

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**13.) Complete these expressions:**

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

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**· · 13.) Complete these expressions: a) of 15 is 6 6 15 = =**

___ 2 5 ___ 2 5 3 ___ 6 1 6 6 E.g. 15 = = 1 b) of 72 is ___ 4 9 32 Both procedures give equal results!!! We can reduce it by __. 5 c) of 10 is ___ 3 4 7 ___ 1 2 Here we can't divide 10:4, so we must calculate in some another way! The word of means ___ 3 4 5 ___ 15 2 multiplication! 7 ___ 1 10 = = 2 2 Are we allowed to calculate in that way in the a and b tasks? We can reduce it by __. 2 Yes, we are!!!

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**13.) Complete these expressions:**

a) of 15 is ___ 2 5 6 b) of 72 is ___ 4 9 32 c) of 10 is ___ 3 4 7 ___ 1 2 We got the same result !!! The denominator 4 tells us that we should divide this "bulk" into 4 equal parts... We have 10 pieces of something, e.g. 10 pears... The numerator 3 tells us to take 3 of these 4 parts... How can we imagine the problem in part c? 1st 2nd 3rd 4th 1 2 3 4 5 6 7 8 9 10 __ 1 2 7 How many pears do we have in these three parts?

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**· 13.) Complete these expressions: a) of 15 is 6 b) of 72 is 32**

___ 2 5 6 b) of 72 is ___ 4 9 32 c) of 10 is ___ 3 4 7 ___ 1 2 d) of is ___ 12 13 39 44 ___ 9 11 3 3 ___ 12 13 ___ 39 44 ___ 9 = 11 1 11 We can reduce it by __. 4 We can reduce it by __. 13

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**< 14.) Complete: a) is ______ than 1 . less , by 2 ___ 3 ___ 5 5**

Explanation with pictures: of the rectangle is less than 1 rectangle, ___ 2 5 < ___ 3 5 of the rectangle. by the uncolored part, and this part is

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**> 14.) Complete: a) is ______ than 1 . less , by**

2 5 ___ 3 5 . less , by b) is _______ than 1 ___ 19 15 ___ 4 15 . greater , by Explanation with pictures: rectangles is greater than 1 rectangle, ___ 19 15 > by the part determined by second rectangle, and it is ___ 4 15 of the rectangle.

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**> > 14.) Complete: a) is ______ than 1 . less , by**

2 5 ___ 3 5 . less , by b) is _______ than 1 ___ 19 15 ___ 4 15 . greater , by 15.) In each inequality below, which number is the greater? a) ___ 8 9 5 > >

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**> < < 14.) Complete: a) is ______ than 1 . less , by**

2 5 ___ 3 5 . less , by b) is _______ than 1 ___ 19 15 ___ 4 15 . greater , by 15.) In each inequality below, which number is the greater? a) ___ 8 9 5 > b) ___ 7 10 1 5 2 4 < <

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**> < < < 14.) Complete: a) is ______ than 1 . less , by**

2 5 ___ 3 5 . less , by b) is _______ than 1 ___ 19 15 ___ 4 15 . greater , by 15.) In each inequality below, which number is the greater? a) ___ 8 9 5 > b) ___ 7 10 1 5 2 4 < c) ___ 3 11 4 5 < <

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**> > > < < > > · · 14.) Complete:**

a) is ______ than 1 ___ 2 5 ___ 3 5 . less , by b) is _______ than 1 ___ 19 15 ___ 4 15 . greater , by 15.) In each inequality below, which number is the greater? a) ___ 8 9 5 e) ___ 8 3 5 2 > > b) ___ 7 10 1 5 2 4 > < 16 15 Instead of cross-multiplying, we can find the common denominator and then compare numerators... We multiply through diagonals... c) ___ 3 11 4 5 < 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. d) ___ 3 5 1 6 > >

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**> > < > < > 14.) Complete: a) is ______ than 1 .**

2 5 ___ 3 5 . less , by b) is _______ than 1 ___ 19 15 ___ 4 15 . greater , by 15.) In each inequality below, which number is the greater? a) ___ 8 9 5 e) ___ 8 3 5 2 > > b) ___ 7 10 1 5 2 4 < > c) ___ 3 11 4 5 < d) ___ 3 5 1 6 >

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16.) Complete: a) If the numerator increases, then the fraction _____________. increases as well E.g. ___ 1 3 2 4 5 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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16.) Complete: a) If the numerator increases, then the fraction _____________. increases as well b) If the denominator increases, then the fraction _________. decreases E.g. ___ 1 2 3 4 5 If we are looking from the left to the right, the denominators ________. increase Colored parts, that is, the fractions of the whole _______. decrease

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

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**+ = 1 = = - = - = = = 4 4 = = 17.) Figure out: a) b) 4 ___ 5 6 4 9**

________ 15 + 8 ___ 23 1 ___ 5 = = 18 18 18 ___ 1 2 - b) 6 = 4 ___ 9 ___ 1 6 ________ 27 - 1 ___ 26 - = = = 2 6 6 1 4 ___ 2 4 ___ 1 3 = = We can reduce it by __. 2 6 3

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**+ = 3 18.) Try to figure these out mentally! a) 3 ___ 5 7 5 ___ 7**

Explanation with pictures:

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**+ = 3 - = 18.) Try to figure these out mentally! a) 3 b) 1 ___ 5 7 5**

2 ___ 5 Explanation with pictures:

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**+ = 3 - = - = 5 18.) Try to figure these out mentally! a) 3 b) 1 c) 6**

___ 5 7 = 3 5 3 ___ 7 - b) ___ 3 5 = 1 2 ___ 5 - c) ___ 2 9 = 6 7 5 ___ 9 Explanation with pictures:

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**+ = 3 - = - = 5 + = 5 9 18.) Try to figure these out mentally! a) 3 b)**

___ 5 7 = 3 5 3 ___ 7 - b) ___ 3 5 = 1 2 ___ 5 - c) ___ 2 9 = 6 7 5 ___ 9 ___ 1 2 + d) = 4 5 1 9 ___ 2 Explanation with pictures:

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**+ = 3 - = - = 5 + = 5 9 - = 18.) Try to figure these out mentally! a)**

___ 5 7 = 3 5 3 ___ 7 - b) ___ 3 5 = 1 2 ___ 5 - c) ___ 2 9 = 6 7 5 ___ 9 ___ 1 2 + d) = 4 5 1 9 ___ 2 ___ 2 11 - e) = 6 2 ___ 11 Explanation with pictures:

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**+ = 3 - = - = 5 + = 5 9 - = - = 18.) Try to figure these out mentally!**

___ 5 7 = 3 5 3 ___ 7 - b) ___ 3 5 = 1 2 ___ 5 - c) ___ 2 9 = 6 7 5 ___ 9 ___ 1 2 + d) = 4 5 1 9 ___ 2 ___ 2 11 - e) = 6 2 ___ 11 ___ 1 3 - f) = 8 8 Explanation with pictures:

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**18.) Try to figure these out mentally!**

+ a) ___ 5 7 = 3 5 3 ___ 7 - b) ___ 3 5 = 1 2 ___ 5 - c) ___ 2 9 = 6 7 5 ___ 9 ___ 1 2 + d) = 4 5 1 9 ___ 2 ___ 2 11 - e) = 6 2 ___ 11 ___ 1 3 - f) = 8 Explanation with pictures: 8 - g) ___ 7 8 = 6 2 1 3 ___ 8

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**· = 1 = · = · = = 12 · = 19 · = = 19.) Figure out: a) b) 4 c) 9 2 ___**

3 3 ___ 27 49 a) 21 9 = ___ 9 1 ___ 2 = 7 7 7 1 We can reduce it by __. We can reduce it by __. 2 9 7 6 ___ 2 9 b) 54 19 = 4 ___ 38 ___ 54 19 ___ 12 = = 12 9 1 1 1 3 We can reduce it by __. We can reduce it by __. 19 9 c) ___ 1 6 = 9 2 ___ 9 1 ___ 13 ___ 39 19 ___ 1 = = 6 2 2 2 We can reduce it by __. 3

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**: = 4 · = = : = : = · = 20.) Figure out: a) 8 b) 3 ___ 12 7 ___ 8 1**

14 4 ___ 2 = = 3 3 3 We can reduce it by __. 4 : b) ___ 7 8 = 3 1 ___ 1 8 ___ 31 ___ 1 8 ___ 8 31 1 ___ 1 : = = 8 31 1 We can reduce it by __. 8

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**2 Let's recall the “number line”.. 1 2 3 4**

1 2 3 4 Place the following numbers onto the number line: a) 2 ___ 5 6 between __ and __ 2 3

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**2 Let's recall the “number line”.. 1 2 3 4**

1 2 3 4 Place the following numbers onto the number line: a) 2 ___ 5 6 marked part of the number line should be divided into ____________ 6 equal parts

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**2 Let's recall the “number line”.. 1 2 3 4**

__ 5 6 1 2 3 4 Place the following numbers onto the number line: a) 2 ___ 5 6 we should count __ parts of the whole from the left 5

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**2 3 = Let's recall the “number line”.. 1 2 3 4**

__ 1 2 2 __ 5 6 1 2 3 4 __ 7 2 Place the following numbers onto the number line: a) 2 ___ 5 6 b) ___ 7 2 3 ___ 1 = 2 between __ and __ 3 4 , exactly in the ______ middle

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**2 3 = Let's recall the “number line”.. 1 2 3 4**

__ 1 2 2 __ 5 6 1 2 3 4 __ 7 2 Place the following numbers onto the number line: a) 2 ___ 5 6 b) ___ 7 2 3 ___ 1 = 2 c) ___ 2 3 it is not possible to change it into a mixed number, there are no wholes, so this number lies between __ and __ 1

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**2 3 = Let's recall the “number line”.. 1 2 3 4**

__ 1 2 2 __ 5 6 1 2 3 4 __ 7 2 Place the following numbers onto the number line: a) 2 ___ 5 6 b) ___ 7 2 3 ___ 1 = 2 c) ___ 2 3 marked part of the number line should be divided into ____________ 3 equal parts

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**2 3 = Let's recall the “number line”.. 1 2 3 4**

__ 1 2 2 __ 5 6 1 2 3 4 __ 2 3 __ 7 2 Place the following numbers onto the number line: a) 2 ___ 5 6 b) ___ 7 2 3 ___ 1 = 2 c) ___ 2 3 we count __ parts of the whole from the left 2

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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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**GSC Rex Boggs With thanks to:**

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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