NUMBER AND NUMBER SENSE MMW class math.pptx

computation

1. NUMBER AND NUMBER SENSE

2. LEARNING OBJECTIVES: At the end of the 3-hour session, you should be able to: perform operations involving fractions, change fractions to decimals, and vice versa and solve problems about ratio and proportion.

3. Numerator = # on top Denominator = # on the bottom (down below) The Denominator How many equal parts are described? How many equal parts are in all (total)?

4. Parts of a Whole Whole = 1 complete thing Different from a hole in the ground

5. 1 WHOLE  fraction The top number and the bottom are the same! 4 5 8 55 86 4 5 8 55 86

6. Parts of a Whole 4 6

7. Parts of a Whole What’s the fraction we should write for each?

8. Write a fraction for the part of each flag that is red. Italy Indonesia Taiwan Austria Germany Spain

9. Parts of a Set What fraction of this set are bananas? What fraction of this set are oranges? What fraction of this set are apples?

10. Proper Fraction Proper fraction: fraction with a smaller number on top (numerator) than the bottom (denominator)

11. Improper Fraction Improper Fraction: fraction with the top number (numerator) larger than the bottom number (denominator).

12. Whole Number Whole number: regular whole numbers without fractions.

13. Mixed Number Mixed Number: A whole number WITH a fraction. Whole Number Fraction

14. Complete These Statements: A mixed number is the sum of a _____________ and a ________________. The __________________ is the number on the bottom of a fraction, and the _________________ is the number on the top of a fraction.

15. HOW CAN WE CHANGE an IMPROPER FRACTION TO MIXED NUMBER?

16. To change an improper number to mixed number Divide the numerator by the denominator. After the division, the quotient will be the whole number, the remainder becomes the new numerator and copy the same denominator.  

17. HOW CAN WE CHANGE MIXED NUMBER TO IMPROPER FRACTION?

18. To change a mixed number to improper fraction Multiply the whole number to the denominator Add the product to the numerator then copy the same denominator.  

19. A. B.  

20. similar fractions dissimilar fractions Fractions whose denominators are the same. Fractions whose denominators are different/ not the same.

21. Operations involving fractions

22. Adding and subtracting similar fractions To add/ subtract similar fractions, simply add/ subtract their numerator and copy the denominators.

23. = = = = = =  

24. Adding and subtracting DISsimilar fractions To add/ subtract dissimilar fractions, change first the fractions to similar fractions by finding the LCD and express them into their equivalent fractions.

25. + We need a common denominator to add these fractions.

26. 7, 14, 21, 28, 35… 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 Count by 2's Count by 7's + We need a common denominator to add these fractions.

27. 7, 14, 21, 28, 35… 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 Count by 2's Count by 7's The first number IN COMMON that appears on both lists becomes the common denominator

28. + = x 2 x 2 = X 7 x 7 7 6 7 + 6 = 13 Add the numerators 13 Make equivalent fractions.

29. + We need a common denominator to add these fractions. 5, 10, 15, 20, 25, 30, 35, 40, 45 7, 14, 21, 28, 35, 42, 49, 56, 63 Count by 7's Count by 5's

30. = x 7 x 7 = X 5 x 5 15 7 + 15 + 7 = 22 Add the numerators 22 Make equivalent fractions.

31. We need a common denominator to subtract these fractions. 7, 14, 21, 28, 35, 42, 49, 56, 63 8, 16, 24, 32, 40, 48, 56, 64, 72 Count by 7's Count by 8's

32. = x 7 x 7 = x 8 x 8 32 21 Subtract the fractions. 32 - 21 = 11 11 Make equivalent fractions.

33. + We need a common denominator to add these fractions. 3, 6, 9, 12, 15, 18, 21, 24, 27 5, 10, 15, 20, 25, 30, 35, 40, 45 Count by 3's Count by 5's

34. X 3 = x 5 x 5 = x 3 9 10 + 9 + 10 = 19 Add the numerators 19

35. We need a common denominator to subtract these fractions. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33 11, 22, 33, 44, 55, 66, 77… Count by 3's Count by 11's

36. = x 11 x 11 = X 3 x 3 22 15 Subtract the fractions. 22 - 15 = 7 7 Make equivalent fractions.

37. + = x 3 x 3 = x 2 x 2 19 10 + 9 = 19 Add the numerators. Make equivalent fractions. 9 10

38. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 - We need a common denominator to subtract a fraction from another. 11, 22, 33, 44, 55, 66, 77… Count by 11's Count by 5's

39. = x 5 x 5 = x 11 x 11 4 15 - 11 = 4 Subtract. Make equivalent fractions. - 15 11

40. - We need a common denominator to subtract one fraction from another. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 … 10, 20, 30… Count by 2's Count by 10's

41. = x 1 x 1 = x 5 x 5 2 5 – 3 = 2 Subtract. Make equivalent fractions. - 5 3

42. Try These A F E B C D

43. Multipying fractions If the fractions are proper fractions, simply multiply the numerator to the numerator, then denominator to the denominator.

44. Multiplying Fractions × = Multiply the numerators. Multiply the denominators. Cancel down.   2 × 3 6 3 × 4 12 = =  

45. × = × =   × = × =  

46. Multiplying mixed numbers. 2 × Change the mixed number to an improper fraction. Multiply as before. Cancel down and change to mixed number if necessary.   = ×   =   =  

47. 1 × = 2 × = × 1 = 1 × 1 =   × = = 1   × = = =2  

48. Multiplying fractions and whole numbers 12 × =   Whole numbers have a denominator of 1. Multiply numerators and denominators. Cancel down and change to a mixed number if necessary. ×   =   10  

49. × 6 = × 12 = 5 × =  

50. Dividing mixed numbers. 2 ÷ Change the mixed number to an improper fraction. Divide as before. Cancel down and change to a mixed number if necessary.   = ÷   = × =   =  

51. 1   2   3  

52. Dividing fractions and whole numbers 8 ÷ =   Whole numbers have a denominator of 1. Turn dividing fraction upside down and multiply numerators and denominators. Cancel down and change to a mixed number if necessary. =      

53. = = = = = =    

54. Fractions, Decimals, and Percents

55. Warm Up Write each fraction in the simplest form. Course 2 2 Fractions, Decimals, and Percents 1. 4 10 8 50 2. 3. 75 100 4. 12 36 5. 6. 18 48 2 5 3 4 1 3 3 5 3 8 4 25 9 15

56. Learn to write equivalent fractions, decimals, and percents . Course 2 2 Fractions, Decimals, and Percents

57. Course 2 2 Fractions, Decimals, and Percents The word percent means “per hundred.” So 40% means “40 out of 100.” Reading Math A percent is the ratio of a number to 100. The symbol % is used to indicate that a number is a percent. For example, 40% is the ratio 40 to 100, or . Percents can be written as fractions or decimals 40 100

58. Write 88% as a fraction in simplest form. Additional Example 1: Writing Percents as Fractions Course 2 2 Fractions, Decimals, and Percents 88% = 88 100 Write the percent as a fraction with a denominator of 100. = 22 25 Simplify.

59. Write 45% as a fraction in simplest form. Try This : Example 1 Course 2 2 Fractions, Decimals, and Percents 45% = 45 100 Write the percent as a fraction with a denominator of 100. = 9 20 Simplify.

60. Additional Example 2: Writing Percents as Decimals Course 2 6-1 Fractions, Decimals, and Percents Write 51% as a decimal. 51% = 51 100 = 0.51 Write the percent as a fraction with a denominator of 100. Write the fraction as a decimal.

61. Try This : Example 2 Course 2 6-1 Fractions, Decimals, and Percents Write 67% as a decimal. 67% = 67 100 = 0.67 Write the percent as a fraction with a denominator of 100. Write the fraction as a decimal.

62. Course 2 2 Fractions, Decimals, and Percent Notice that both 43% and 0.43 mean “43 hundredths.” Another way to write a percent as a decimal is to delete the percent sign and move the decimal point two places to the left. 43 . % = . 43

63. Write each decimal as a percent. Additional Example 3A & 3B: Writing Decimals as Percents Course 2 2 Fractions, Decimals, and Percents A. 0.08 = 8% B. 0.7 = 70% Write the decimal as a fraction. Write the fraction as a percent. Write the decimal as a fraction. Write an equivalent fraction with a denominator of 100. Write the fraction as a percent. 0.08 = 8 100 0.7 = 7 10 70 100 =

64. Write each decimal as a percent. Try This : Example 3A & 3B Course 2 2 Fractions, Decimals, and Percents A. 0.01 = 1% B. 0.1 = 10% Write the decimal as a fraction. Write the fraction as a percent. Write the decimal as a fraction. Write an equivalent fraction with a denominator of 100. Write the fraction as a percent. 0.01 = 1 100 0.1 = 1 10 10 100 =

65. Course 2 2 Fractions, Decimals, and Percent Both 0.07 and 7% mean “7 hundredths.” you can write a decimal as a percent by moving the decimal point two places to the right and adding the percent sign. . 07 = 7 . 0%

66. Course 2 2 Fractions, Decimals and Percents Write each fraction as a percent. A. 3 5 3 5 = 3 · 20 5 · 20 = 60 100 = 60% Write an equivalent fraction with a denominator of 100. Write the fraction as a percent. The method shown in example 4A works only if the given denominator is a factor or multiple of 100. Helpful Hint Additional Example 4A: Writing Fractions as Decimals

67. Write each fraction as a percent. Additional Example 4B: Writing Fractions as Decimals Course 2 2 Fractions, Decimals, and Percents B. 7 40 7 40 = 7 ÷ 40 = 0.175 = 17.5% Use division to write a fraction as a decimal. Write the decimal as a percent. The method shown in 4B works for any denominator. Helpful Hint

68. Course 2 6-1 Fractions, Decimals and Percents Write each fraction as a percent. A. 3 4 3 4 = 3 · 25 4 · 25 = 75 100 = 75% Write an equivalent fraction with a denominator of 100. Write the fraction as a percent. Try This : Example 4A

69. Write each fraction as a percent. Try This : Example 4B Course 2 2 Fractions, Decimals, and Percents B. 9 60 9 60 = 9 ÷ 60 = 0.15 = 15% Use division to write a fraction as a decimal. Write the decimal as a percent.

70. Insert Lesson Title Here Course 2 2 Fractions, Decimals, and Percents 1. Write 40% as a fraction. 2. Write 0.65 as a percent. 3. Write 72% as a decimal. 4. Write as a percent. 6 10 5. About 95% of all animals are insects. Express this percent as a fraction.

71. A A A A A A 3.1 Ratio 3 Ratio and proportion 3.3 Direct proportion 3. 2 Dividing in a given ratio 3.4 Inverse proportion

72. Ratio A ratio compares the sizes of parts or quantities to each other. For example, What is the ratio of red counters to blue counters? red : blue = 9 : 3 = 3 : 1 For every three red counters there is one blue counter .

73. Ratio A ratio compares the sizes of parts or quantities to each other. The ratio of blue counters to red counters is not the same as the ratio of red counters to blue counters. blue : red = 3 : 9 = 1 : 3 For every blue counter there are three red counters . For example, What is the ratio of blue counters to red counters?

74. What is the ratio of red counters to yellow counters to blue counters? Ratio red : yellow : blue = 12 : 4 : 8 = 3 : 1 : 2 For every three red counters there is one yellow counter and two blue counters .

75. Simplifying ratios Ratios can be simplified like fractions by dividing each part by the highest common factor. For example, 21 : 35 = 3 : 5 ÷ 7 ÷ 7 For a three-part ratio all three parts must be divided by the same number. For example, 6 : 12 : 9 = 2 : 4 : 3 ÷ 3 ÷ 3

76. Comparing ratios The ratio of boys to girls in class 9P is 4:5. The ratio of boys to girls in class 9G is 5:7. Which class has the higher proportion of girls? The ratio of boys to girls in 9P is 4 : 5 ÷ 4 ÷ 4 = 1 : 1.25 The ratio of boys to girls in 9G is 5 : 7 ÷ 5 ÷ 5 = 1 : 1.4 9G has a higher proportion of girls.

77. Finding the missing number in a ratio Suppose the picture is reduced in size so that its width is 7.5 cm. What is the height of the reduced picture? ? 7.5 cm We have established that the ratio of the height to the width is 3 : 5. The ratio of the height to the width must remain the same or the picture will be distorted. We must therefore find a ratio equivalent to 3 : 5 but with the second part equal to 7.5. 3 : 5 ? : 7.5

78. Finding the missing number in a ratio To find the missing number in the ratio we have to work out what we have multiplied 5 by to get 7.5: 3 : 5 ? : 7.5 To do this divide 7.5 by 5. 7.5 ÷ 5 = 1.5 The 5 is multiplied by 1.5 … × 1.5 × 1.5 … so the 3 must be multiplied by 1.5. 4.5 So when the width of the rectangle is 7.5 cm this height is 4.5 cm.

79. Finding the missing number in a ratio The ratio of boys to girls in year 10 of a particular school is 6 : 7. If there are 72 boys, how many girls are there? 6 : 7 72 : ? To do this divide 72 by 6. 72 ÷ 6 = 12 … so the 7 must be multiplied by 12. × 12 × 12 The 6 is multiplied by 12 … 84 Again we can work this out by finding the missing number in the ratio. If there are 72 boys there must be 84 girls.

80. A A A A 3.3 Direct proportion Contents 3.1 Ratio 3 Ratio and proportion 3. 2 Dividing in a given ratio 3.4 Inverse proportion

81. Direct proportion Two quantities are said to be in direct proportion if they increase and decrease at the same rate. That is, if the ratio between the two quantities is always the same. For example, the speed that a car travels is directly proportional to the distance it covers. If the car doubles its speed it will cover double the distance in the same time. If the car halves its speed it will cover half the distance in the same time. If the car is at rest it won’t cover any distance. That is, if its speed is zero the distance covered is zero.

82. Direct proportion problems 3 packets of crisps weigh 84 g. How much do 12 packets weigh? 3 packets weigh 84 g. × 4 12 packets weigh × 4 336 g. If we multiply the number of packets by four then we have to multiply the weight by four. If all the packets weigh the same then the ratio between the number of packets and the weight is constant.

83. Direct proportion problems 3 packets of crisps weigh 84 g. How much does 1 packet weigh? 3 packets weigh 84 g. ÷ 3 1 packet weighs ÷ 3 28 g. We divide the number of packets by three and divide the weight by three. Once we know the weight of one packet we can work out the weight of any number of packets.

84. 3 packets of crisps weigh 84 g. How much do 7 packets weigh? 3 packets weigh 84 g. ÷ 3 ÷ 3 1 packet weighs 28 g. × 7 × 7 7 packets weigh 196 g. This is called using a unitary method . Direct proportion problems

85. A A A A 3.4 Inverse proportion Contents 3.3 Direct proportion 3.1 Ratio 3 Ratio and proportion 3.2 Dividing in a given ratio

86. Inverse proportion It takes one person 1 hour to put 150 letters into envelopes. The more people there are, the less time it will take. 5 people will take a fifth of the time to put the same number of letters in the envelopes. One person takes 1 hour so 5 people take of an hour. 1 5 of 60 minutes = 1 5 12 minutes The number of people and the time they take are said to be inversely proportional . How long would it take 5 people, working at the same rate, to put 150 letters into envelopes?

87. Inverse proportion Two quantities are said to be inversely proportional if, as one quantity increases, the other quantity decreases at the same rate. For example, the speed that a car travels is inversely proportional to the time it takes to cover the same distance. If the car doubles its speed it will take half the time to cover the same distance. If the car trebles its speed it will take a third of the time to cover the same distance. If the car halves its speed it will take double the time to cover the same distance.

88. It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate?

89. As the number of people goes up, the painting time goes down. As the number of people goes down, the painting time goes up. We can use: t = k/n Where: t = number of hours k = constant of proportionality n = number of people

90. It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate? The number of men is inversely proportional to the time taken to do the job. Let t be the time taken for the 8 men to finish the job. 4 × 6 = 8 × t 24 = 8t t = 3 hours

91. 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it?

92. 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it? "4 people can paint a fence in 3 hours" means that t = 3 when n = 4 3 = k/4 3 × 4 = k × 4 / 4 12 = k k = 12 So now we know: t = 12/n And when n = 6: t = 12/6 = 2 hours

93. A school wants to spend ₱ 10 000 on mathematics textbooks. How many books could be bought at ₱ 50 each? Clearly 200 books can be bought. If the price of a textbook is more than ₱ 50, then the number of books which could be purchased with the same amount of money would be less than 200.

94. A school wants to spend ₱ 10 000 on mathematics textbooks. How many books could be bought at ₱ 50 each? Clearly 200 books can be bought. If the price of a textbook is more than ₱ 50, then the number of books which could be purchased with the same amount of money would be less than 200. Price of each book (₱) ₱ 50.00 ₱ 100.00 ₱ 200.00 ₱ 500.00 ₱ 1 000.00 Number of books that can be bought

95. PARTITIVE PROPORTION Is a proportion that describes the total amount being distributed into two or more unequal parts The sum of two numbers is 27. Their ratio is 1:2. What is the larger number?

96. Mario and Norie shared an amount in the ratio of 3:5 to buy a gift for their Mother. If the gift was worth Php 320, how much did each child share?