Fractions - R evision

1. Fractions - Revision

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

3. 7 5 1 a) ___ ___ ___ squares? rhombuses? of a parallelogram? 6 4 3 b) c) 4 How to color:

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

5. 9 1 2 ___ ___ ___ 10 4 9 , , E.g. ... are _____ fractions. < 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. proper less They are ______ than 1.

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

7. 6 4 9 5 ___ ___ ___ ___ 2 4 4 3 , , , ... are ________ fractions. ≥ Improper fractions are fractions with the numerator greater than or equal to the denominator. improper E.g. greater than or equal to They are __________________ 1. 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!!

8. 6 4 9 5 ___ ___ ___ ___ 2 4 4 3 , , , ... are ________ fractions. ≥ Improper fractions are fractions with the numerator greater than or equal to the denominator. improper E.g. greater than or equal to They are __________________ 1. Any improper fraction can be changed into a mixed fraction or a natural number.

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

10. 6 4 5 4 9 6 ___ ___ ___ ___ ___ ___ 2 4 2 3 4 4 , , , ... are ________ fractions. ≥ = = Improper fractions are fractions with the numerator greater than or equal to the denominator. improper E.g. Which of these fractions can be changed into natural numbers ? Change them (look at the picture)! 1 3

11. 19 ___ 8 a) ___ 2 = 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): 3 8 Explanation: 19:8 equals 2 and remainder 3 Rewrite denominator! =

12. 19 42 68 ___ ___ ___ 9 8 7 c) a) b) ___ ___ 7 2 = = = 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): 3 8 5 9 6 Explanation: 42:7 equals 6 (no remainder) =

13. 19 42 36 68 2 ___ ___ ___ ___ ___ 8 9 9 7 4 b) a) d) c) e) ___ ___ 2 7 = = = = = 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? 3 8 We divided the numerator by the denominator. 5 9 Why? Because the fraction line alwaysmeans division! 6 Now let's change a fraction into a decimal number! How to do it? 9 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...

14. 19 19 ___ ___ 8 8 a) ___ 2 = = 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): 3 8 . 19 : 8 = 2 3 7 5 3 0 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 0 So, we changed the same fraction into both - mixed fraction and decimal number. 4 0 Let's change the number at task "a)" into a decimal number... How to do it? 0

15. 7 ___ 8 55 ___ = a) 6 Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 6 · 8 + 7 8 Rewrite denominator! =

16. 6 7 ___ ___ 8 7 24 55 16 69 8 ___ ___ ___ ___ ___ = = a) b) 9 6 = = = ... 1 2 3 Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 8 7 c) 8 = (When we divide numerator by denominator, the result must be 8 !) = = = = = = = ...

17. 6 7 ___ ___ 7 8 241 55 24 16 69 8 ____ ___ ___ ___ ___ ___ = = a) b) 6 9 = = = ... 1 2 3 Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: 8 7 c) 8 = Explanation: Rewrite the given number, but without decimal point... d) 2.41 = 100 Write the digit 1 and as many zeros as we have decimal digits in the given number... 2 decimal digits 2 zeros =

18. 3 7 6 ___ ___ ___ 7 8 7 2893 241 309 16 24 54 55 19 27 69 31 8 ____ ____ ____ ____ ___ ___ ___ ___ ___ ___ ___ ___ = = = b) a) h) 9 6 4 = = = ... = = ... 1 1 2 2 3 Now conversely! Let's revise how to change numbers from other forms into fractions... 2.) Change into fraction: f) 0.019 = 8 1000 g) 27 = 7 c) 8 = d) 2.41 = 7 100 i) 28.93 = e) 30.9 = 100 10

19. 241 ____ ___ 2 100 = Some decimal numbers can be changed into mixed fractions. Let's revise it... 3.) Change into mixed number: Recall: 2.41 can be changed not only into a mixed fraction, but into an improper fraction as well. Say that improper fraction... 41 a) 2.41 = 100 2 decimal digits 2 zeros

20. ____ 2 ____ ____ 30 15 45 ____ . 1000 Some decimal numbers can be changed into mixed fractions. Let's revise it... 3.) Change into mixed number: 41 a) 2.41 = 100 9 b) 30.9 = 10 7 c) 15.007 = 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, the solution would be

21. 10 5 ___ ___ 12 6 a) = What does it mean - "to reduce afraction" ? To reduce afraction means to divide both the numerator and denominator of the fraction by the samenumber. 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 2 We can reduceit by __. 6 So, we divide both the numerator and denominator by 2 and write the results… =

22. 24 4 ___ ___ 30 5 b) = What does it mean - "to reduce afraction" ? To reduce afraction means to divide both the numerator and denominator of the fraction by the samenumber. 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 6 We can reduce it by __. 5 =

23. 42 2 6 ___ ___ ___ 63 3 9 c) = = = What does it mean - "to reduce afraction" ? To reduce afraction means to divide both the numerator and denominator of the fraction by the samenumber. 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 = 7 3 Now we can reduce again, by __. We can reduce it by __. 9 3

24. 12 6 4 ___ ___ ___ 12 3 4 = = = Now let's revise other properties of fractions... 5.) Complete these sentences: four a) One whole equals ____ fourths. 1 = twelve b) One whole equals _____ twelfths. 1 = six c) Two wholes equal ___ thirds. 2 =

25. 4 ___ 7 3 1 2 1 4 __ __ __ __ __ 7 7 7 7 7 If 1 day is of the week, Now let's revise other properties of fractions... 5.) Complete: d) 4 days = week Explanation: 7 First, let's recall that a week has ___ days. So we can conclude that each day is of the week. Now, we should think in this way: then 2 days are of the week, 3 days are of the week, and 4 days are of the week.

26. 4 ___ 7 4 __ 7 Now let's revise other properties of fractions... 5.) Complete: d) 4 days = week We can explain it in another way: 7 Again, let's recall that a week has ___ days. So, let's consider it together with the rectangle divided into 7 equal parts! week: rectangle: Monday Tuesday ? 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...

27. 4 ___ 7 Now let's revise other properties of fractions... 5.) Complete: 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.

28. 20 5 4 1 ___ ___ ___ ___ 60 12 7 3 year: 5 1 __ __ 12 3 January February 12 March 9 3 April May 6 June July August September Octobar November December Now let's revise other properties of fractions... 5.) Complete: d) 4 days = week ? 20 min. 1 ? hour e) 20 min. = hour = hour 20 We can reduce it by __. 3 f) 5 months = year ? of the year

29. 5 2 ___ ___ 7 7 Ivan solved of his homework. 5 2 __ __ 7 7 He should solve of his homework yet. 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? What portion of the homework does he still have to solve? homework: 1st task ? of the homework 2nd task 3rd task 4th task ? of the homework 5th task 6th task 7th task

30. 7.) 7 3 3 7 ___ ___ ___ ___ 10 10 10 10 If children ate (seven tenths) of a cake, which fraction of the cake remained? (three tenths) of the cake remained. of the cake of the cake Now let's revise other properties of fractions...

31. 8.) Climber Dario climbed of his path in one hour, another of the path in the next one hour, 13 1 6 3 6 4 4 3 ___ ___ ___ ___ ___ ___ ___ ___ 14 14 14 14 14 14 14 14 and finally of his path in the third hour. Did he climb his whole path? + + = Now let's revise other properties of fractions... No, he didn't climb his whole path. There remained of his path.

32. 3 3 ___ ___ 7 7 Each friend will get of a chocolate. of the chocolate. 3 1 1 1 __ __ __ __ 7 7 7 7 of the chocolate. of the chocolate. of a chocolate. 9.) Seven friends gathered some money and bought 3 chocolatesofequal size. They want to divide the chocolates equally. How much chocolate will each of them get? 3 : 7 = Explanation with pictures: When they divide the first chocolate into 7 equal parts, each friend will get When they divide the second chocolate into 7 equal parts, each friend will get When they divide the third chocolate into 7 equal parts, each friend will get So, after all divisions each friend will have

33. 12 2 ___ ___ 5 5 ___ 2 of the 11th chocolate. of the 12th chocolate. 2 1 1 __ __ __ 5 5 5 Each friend will get 2 chocolates. chocolates! 2 10.) a) 12 chocolates should be divided among 5 friends. How many chocolates will each of them get? 2 12 : 5 = = 5 Explanation with pictures: Each friend gets Each friend gets So, after all divisions each friend will have

34. 10.) b) What about dividing 12 chocolates among 3 friends? 12 : 3 = 4 Each friend will get 4 chocolates. Explanation with pictures: 4 chocolates. After division each friend will have

35. 11.) Little Ana ate strawberries. How many strawberries did she eat actually? 12 12 12 ___ ___ ___ 2 2 2 = strawberries 12 : 2 = 6 Little Ana ate 6 strawberries. Explanation with pictures: = 6 strawberries

36. 12.) There are 48 apples in the box. of the box are red apples, are green apples 10 5 3 5 3 5 5 ___ ___ ___ ___ ___ ___ ___ 12 24 48 24 12 8 8 and the rest of the apples are yellow. Yellow apples form (five twenty- fourths) of the box. a) How many apples are of which color? red: 18 of 48 is ( we calculated 48:8·3 ) green: 20 of 48 is yellow: 38, 10 18 + 20 = 48 - 38 = There are 18 red, 20 green and 10 yellow apples in that box. b) What portion of the box do yellow apples form? 5 = 2 We can reduce it by __. 24

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

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

39. E.g. b) of 72 is a) of 15 is c) of 10 is 15 2 2 1 6 3 4 3 ___ ___ ___ ___ ___ ___ ___ ___ 5 1 4 9 4 2 5 2 7 ___ 7 6 13.) Complete these expressions: 3 · 6 = 15 = 1 Both procedures give equal results!!! 5 We can reduce it by __. 32 Here we can't divide 10:4, so we must calculate in some another way! 5 The word of means multiplication! 1 · = 10 = 2 2 Are we allowed to calculate in that way in the a and b tasks? 2 We can reduce it by __. Yes, we are!!!

40. c) of 10 is b) of 72 is a) of 15 is 1 3 4 2 ___ ___ ___ ___ 5 9 4 2 7 1 __ 2 7 13.) Complete these expressions: 6 32 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 weimagine the problem in part c? 1st 2nd 3rd 4th 1 2 3 4 5 6 7 8 9 10 How many pears do we have in these three parts?

41. c) of 10 is b) of 72 is a) of 15 is 12 39 39 12 4 2 9 1 3 ___ ___ ___ ___ ___ ___ ___ ___ ___ 13 13 44 44 11 4 5 2 9 7 d) of is 9 ___ 13.) Complete these expressions: 6 32 3 3 · = 11 1 11 4 We can reduce it by __. We can reduce it by __. 13

42. a) is ______ than 1 . of the rectangle is less than 1 rectangle, of the rectangle. 3 2 3 2 ___ ___ ___ ___ 5 5 5 5 14.) Complete: less , by Explanation with pictures: < by the uncolored part, and this part is

43. b) is _______ than 1 a) is ______ than 1 . . rectangles is greater than 1 rectangle, of the rectangle. 19 3 2 4 19 4 ___ ___ ___ ___ ___ ___ 15 15 5 5 15 15 14.) Complete: less , by greater , by Explanation with pictures: > by the part determined by second rectangle, and it is

44. a) is ______ than 1 b) is _______ than 1 . . 19 3 8 4 2 5 ___ ___ ___ ___ ___ ___ 15 15 9 9 5 5 a) 14.) Complete: less , by greater , by 15.) In each inequality below, which number is the greater? > >

45. b) is _______ than 1 a) is ______ than 1 . . 19 5 2 3 4 8 1 7 ___ ___ ___ ___ ___ ___ ___ ___ 10 15 15 5 9 5 9 5 a) b) 2 4 14.) Complete: less , by greater , by 15.) In each inequality below, which number is the greater? > < <

46. a) is ______ than 1 b) is _______ than 1 . . 19 4 5 3 8 1 3 2 7 ___ ___ ___ ___ ___ ___ ___ ___ ___ 15 10 11 15 5 9 9 5 5 a) b) 2 4 c) 4 5 14.) Complete: less , by greater , by 15.) In each inequality below, which number is the greater? > < < <

47. a) is ______ than 1 b) is _______ than 1 . . 19 8 4 7 5 3 1 3 1 2 8 5 3 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 15 10 11 15 5 5 2 3 5 9 5 9 5 · · e) a) d) b) 2 6 4 6 c) 4 5 14.) Complete: less , by greater , by 15.) In each inequality below, which number is the greater? > > > < 15 16 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. > >

48. a) is ______ than 1 b) is _______ than 1 . . 19 5 8 4 3 1 7 3 2 3 1 5 8 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 10 11 15 15 5 2 5 3 5 5 9 5 9 e) a) d) b) 6 2 6 4 c) 4 5 14.) Complete: less , by greater , by 15.) In each inequality below, which number is the greater? > > < > < >

49. 3 5 1 2 4 ___ ___ ___ ___ ___ 3 3 3 3 3 E.g. 16.) Complete: increases as well a) If the numerator increases, then the fraction _____________. If we are looking from the left to the right, the numerators ________. increase Colored parts, that is the fractions _____________. increase as well

50. 1 1 1 1 1 ___ ___ ___ ___ ___ 2 4 5 3 1 E.g. 16.) Complete: increases as well a) If the numerator increases, then the fraction _____________. decreases b) If the denominator increases, then the fraction _________. If we are looking from the left to the right, the denominators ________. increase decrease Colored parts, that is, the fractions of the whole _______.