Different Types of Variations in Mathematics

Different Types of Variations

1. Variation Grade 9- Mathematics

2. 1. Illustrate situations that involve the following variations: (a) direct; (b) inverse 1 2. Reflect on how can the idea of variation be relative to real-life. 2 Objectives 3 3. Solve problems involving variation

3. The Emoji Challenge Instructions: Guess the following names of places found in the Philippines using the emoji presented on the screen.

4. Variation A variation is a relationship between two or more variables in such a way that as one quantity changes, the other quantity also changes in a definite way.

5. Variable A variable in Mathematics is defined as the alphabetic character that expresses a numerical value or a number. In algebraic equations, a variable is used to represent an unknown quantity. These variables can be any alphabets from a to z. Most commonly, ‘a’,’b’,’c’, ‘x’,’y’ and ‘z’ are used as variables in equations.

6. Two Types of Variable

7. Independent It is a variable that stands alone and is not changed by the other variables you are trying to measure . It is rather considered the cause or reason of an observed effect. It typically acts as the “x” of the variation. (Note: you can also use other letters aside from x )

8. Dependent It is a variable that depends on another variable you are trying to measure. It is considered as the effect or outcome in a situation. Dependent variable acts as the “y” of the variation. (Note: you can also use other letters aside from y )

9. Relationship between independent and dependent variable

10. Relationship between independent and dependent variable

11. Relationship between independent and dependent variable

12. Identify the dependent and independent variables. 1. Jonas can’t decide if he should go to school because the LGUs don’t yet announce if the class is suspended due to the alleged Typhoon. 2. Sir Brenzy wants to give good grades, but the students do not comply with the subject requirements. 3. If Gray will not attend the class, then so does Yvonne.

13. Identify the dependent and independent variables. 4. God will provide, let us don’t lose hope. 5. I forgot to review, that’s why I got low scores with my quizzes and post rants about my teacher on social media.

14. Is there a point of your life where you felt dependent/independent (except from your family)? Elaborate. Answer the following question above on a 1/2 sheet of paper

15. Types of Variation

16. 1. Direct Variation

17. Direct Variation A direct variation is a variation in which the quotient of two variables is constant. “In direct variation, as one variable increases/decreases, the other variable also increases/decreases respectively.”

18. Example of Direct Variation Time 2 5 10 20 Distance 100 ? ? ? Miwah and Penny walk to go to school. At a constant rate, it takes them 20 minutes to reach the school in time for their first subject. How long is the distance they have to travel in 5, 10, and 20 mins?

19. Direct Variation Step 1: Distinguish which is the independent variable (x) and dependent variable (y).

20. Direct Variation Time 2 5 10 20 Distance 100 ? ? ? Miwah and Penny walk to go to school. At a constant rate, it takes them 20 minutes to reach the school in time for their first subject. How long is the distance they have to travel in 5, 10, and 20 mins? x y

21. Direct Variation Step 1: Distinguish which is the independent variable (x) and dependent variable (y). Step 2: Use the formula to find the constant(k). k = y/x k = 100/2 k = 50 Step 3: Use the formula to find the value of y. y = k(x)

22. Direct Variation Time 2 5 10 20 Distance 100 ? ? ? Miwah and Penny walk to go to school. At a constant rate, it takes them 20 minutes to reach the school in time for their first subject. Given the table below, how long is the distance they have to travel in 5, 10, and 20 mins? x y y = k(x) y = 50(5) y = 250

23. Miwah and Penny walk to go to school. At a constant rate, it takes them 20 minutes to reach the school in time for their first subject. Given the table below, how long is the distance they have to travel in 5, 10, and 20 mins? Direct Variation Time 2 5 10 20 Distance 100 250 ? ? x y y = k(x) y = 50(10) y = 500

24. Miwah and Penny walk to go to school. At a constant rate, it takes them 20 minutes to reach the school in time for their first subject. Given the table below, how long is the distance they have to travel in 5, 10, and 20 mins? Direct Variation Time 2 5 10 20 Distance 100 250 500 ? x y y = k(x) y = 50(20) y = 1000

25. Miwah and Penny walk to go to school. At a constant rate, it takes them 20 minutes to reach the school in time for their first subject. Given the table below, how long is the distance they have to travel in 5, 10, and 20 mins? Direct Variation Time 2 5 10 20 Distance 100 250 500 1000 x y

26. Graphing Direct Variation Time 2 5 10 20 Distance 100 250 500 1000

27. Summary: Direct Variation As one variable (x) increases, the other variable (y) also increases; or As one variable (x) decreases, the other variable (y) also decreases. Formula for constant (k): Formula for dependent variable (y) or the equation of the variation : Formula for independent variable (x):  

28. 1.5: Direct Power Variation

29. Direct Power Variation A direct power variation is a variation where one quantity y varies directly as the power n of the other quantity x wherein the formula for the constant (k) is k = y/x n where k ≠ 0 . “In direct power variation, as one variable increases/decreases, the other quantity increases/decreases in power respectively.”

30. Example of Direct Power Variation Length 1 2 3 4 Area 4 ? ? ? Kharlos created a lot to plant his crops. If the length is 1 meter, which makes its area into 4 square meters, what would be the area of the lot if the length will be increased into 2, 3, 4 meters?

31. Direct Power Variation Step 1: Distinguish which is the independent variable (x) and dependent variable (y).

32. Direct Power Variation Length (m) 1 2 3 4 Area (m 2 ) 4 ? ? ? Kharlo created a lot to plant his crops. If the length is 1 meter, which makes it area into 4 square meters, what would be the area of the lot if it will be increased into 2, 3, 4 meters? x y

33. Direct Power Variation Step 1: Distinguish which is the independent variable (x) and dependent variable (y). Step 2: Use the formula to find the constant(k). k = y/(x) n k = 4/1 2 k = 4 Since were looking for the values area, therefore we will use n=2 Step 3: Use the formula to find the value of y = k(x) n

34. Direct Power Variation Length (m) 1 2 3 4 Area (m 2 ) 4 ? ? ? Kharlo created a lot to plant his crops. If the length is 1 meter, which makes it area into 4 square meters, what would be the area of the lot if it will be increased into 2, 3, 4 meters? x y y = k(x) n y = 4(2) 2 y = 16

35. Kharlo created a lot to plant his crops. If the length is 1 meter, which makes it area into 4 square meters, what would be the area of the lot if it will be increased into 2, 3, 4 meters? Direct Power Variation Length (m) 1 2 3 4 Area (m 2 ) 4 16 ? ? x y y = k(x) n y = 4(3) 2 y = 36

36. Kharlo created a lot to plant his crops. If the length is 1 meter, which makes it area into 4 square meters, what would be the area of the lot if it will be increased into 2, 3, 4 meters? Direct Power Variation Length (m) 1 2 3 4 Area (m 2 ) 4 16 36 ? x y y = k(x) n y = 4(4) 2 y = 64

37. Kharlo created a lot to plant his crops. If the length is 1 meter, which makes it area into 4 square meters, what would be the area of the lot if it will be increased into 2, 3, 4 meters? Direct Power Variation Length (m) 1 2 3 4 Area (m 2 ) 4 16 36 64 x y

38. Graphing Direct Power Variation Length (m) 1 2 3 4 Area (m 2 ) 4 16 36 64

39. Summary: Direct Power Variation As one variable (x) increases, the other variable (y) also increases in power; or As one variable (x) decreases, the other variable (y) also decreases in power. Formula for constant (k): Formula for dependent variable (y) or the equation of the variation : Formula for independent variable (x):  

40. 2. Inverse Variation

41. Inverse Variation Also called indirect variation, is a variation in which the product of two variables is constant. “In inverse variation, as one variable increases/decreases, the other variable do the opposite.”

42. Workers 5 10 20 25 Days 20 ? ? ? Inverse Variation If 5 workers can complete a job in 20 days. How many days can 10 workers/20 workers/ 25 workers finish the same job?

43. Inverse Variation Step 1: Distinguish which is the independent variable (x) and dependent variable (y).

44. Workers 5 10 20 25 Days 20 ? ? ? Inverse Variation If 5 workers can complete a job in 20 days. How many days can 10 workers/20 workers/ 25 workers finish the same job? x y

45. Inverse Variation Step 1: Distinguish which is the independent variable (x) and dependent variable (y). Step 2: Use the formula to find the constant(k). k = y(x) k = 5(20) k = 100 Step 3: Use the formula to find the value of y. y = k/x

46. Inverse Variation Workers 5 10 20 25 Days 20 ? ? ? If 5 workers can complete a job in 20 days. How many days can 10 workers/20 workers/ 25 workers finish the same job? x y y = k/x y = 100/10 y = 10

47. If 5 workers can complete a job in 20 days. How many days can 10 workers/20 workers/ 25 workers finish the same job? Inverse Variation Workers 5 10 20 25 Days 20 10 ? ? x y y = k/x y = 100/20 y = 5

48. If 5 workers can complete a job in 20 days. How many days can 10 workers/20 workers/ 25 workers finish the same job? Inverse Variation Workers 5 10 20 25 Days 20 10 5 ? x y y = k/x y = 100/25 y = 4

49. If 5 workers can complete a job in 20 days. How many days can 10 workers/20 workers/ 25 workers finish the same job? Inverse Variation Workers 5 10 20 25 Days 20 10 5 4 x y

50. Inverse Variation Workers 5 10 20 25 Days 20 10 5 4

51. As one variable (x) increases, the other variable (y) also decreases; or As one variable (x) decreases, the other variable (y) also increases. Formula for constant (k): Formula for dependent variable (y) o r the equation of the variation : Formula for independent variable (x):   Summary: Inverse Variation

52. Activity no. 1: Direct Variation If y varies directly as x Find the constant (k) Find the equation of variation Find the value of x when y = 12 y = 32, when x = 16 y = 8, when x = 1/2 or 0.5 y = 3/4 or 0.75, when x = 6

53. Activity no. 2: Direct Power Variation If y varies directly as square of x Find the constant (k) Find the equation of variation Find the value of y when x = 4 y = 36, when x = 3 y = 8, when x = 2 y = 100, when x = 5

54. Activity no. 3: Inverse Variation If y varies inversely as x Find the constant (k) Find the equation of variation Find the value of y when x = 4 y = 6, when x = 3 y = 8, when x = 2 y = 100, when x = 1/5

55. 3. Joint Variation

56. Joint variation is a variation where a quantity varies as the product of two or more other quantities. “In joint variation, as either or both two variable increases/decreases, the other variable also increases/decreases.” Joint Variation

57. In joint variation, there will be two (2) independent variables , the x and the y , and one (1) dependent variable , which is the z . Joint Variation

58. Sab and Gab have a collection of rectangular frames, one measuring 3m in length and 2m in width, with an area of 6m 2 . If we increase the length by 3 more units, what happens to the width if the area will be 12m 2 ? If we increase the width by 2, what happens to the length if the area will be 12m 2 ? If we increase both the length and width by 5, what happens to the area? Joint Variation

59. Joint Variation Length (m) 3 Width (m) 2 Area (m 2 ) 6 Situation No. 1: If we increase the length by 3 more units, what happens to the width if the area is 12m 2 ?

60. Joint Variation Length (m) 3 3 + 3 = 6 Width (m) 2 Area (m 2 ) 6 Situation No. 1: If we increase the length by 3 more units, what happens to the width if the area is 12m 2 ?

61. Step 1: Distinguish which are the independent variable (x) and (y), and the dependent variable (z). Joint Variation

62. Joint Variation Length (m) 3 6 Width (m) 2 Area (m 2 ) 6 12 Situation No. 1: If we increase the length by 3 more units, what happens to the width if the area is 12m 2 ? x y z

63. Step 1: Distinguish which are the independent variable (x) and (y), and the dependent variable (z). Step 2: Use the formula to find the constant(k). Step 3: Use the formula to find the value of y:   Joint Variation

64. Joint Variation Length (m) 3 6 Width (m) 2 Area (m 2 ) 6 12 Situation No. 1: If we increase the length by 3 more units, what happens to the width if the area is 12m 2 ? x y z  

65. Joint Variation Length (m) 3 6 Width (m) 2 Area (m 2 ) 6 12 Situation No. 1: If we increase the length by 3 more units, what happens to the width if the area is 12m 2 ? x y z

66. Joint Variation Length (m) 3 6 Width (m) 2 2 4 Area (m 2 ) 6 12 12 Situation No. 2: If we increase the width by 2, what happens to the length if the area is 12m 2 ? x y z

67. Step 1: Distinguish which are the independent variable (x) and (y), and the dependent variable (z). Step 2: Use the formula to find the constant(k). Step 3: Use the formula to find the value of x:   Joint Variation

68. Joint Variation Length (m) 3 6 Width (m) 2 2 4 Area (m 2 ) 6 12 12 Situation No. 2: If we increase the width by 2, what happens to the length if the area is 12m 2 ? x y z  

69. Joint Variation Length (m) 3 6 3 Width (m) 2 2 4 Area (m 2 ) 6 12 12 Situation No. 2: If we increase the width by 2, what happens to the length if the area is 12m 2 ? x y z

70. Joint Variation Length (m) 3 6 3 8 Width (m) 2 2 4 7 Area (m 2 ) 6 12 12 Situation No. 3: If we increase both the length and the width by 5, what will be the area? x y z

71. Joint Variation Length (m) 3 6 3 8 Width (m) 2 2 4 7 Area (m 2 ) 6 12 12 Situation No. 3: If we increase both the length and the width by 5, what will be the area? x y z  

72. Situation No. 3: If we increase both the length and the width by 5, what will be the area? Joint Variation Length (m) 3 6 3 8 Width (m) 2 2 4 7 Area (m 2 ) 6 12 12 56 x y z

73. Joint Variation Length (m) 3 6 3 8 Width (m) 2 2 4 7 Area (m 2 ) 6 12 12 56

74. As either or both variables (x & y) increase, the other variable (z) also decreases; or Formula for constant (k): Formula for the dependent variable (z) o r the equation of the variation : Formula for independent variable (x): Formula for independent variable (y):   Summary: Joint Variation

75. Activity no. 4: Joint Variation If z varies jointly as x and y Find the constant (k) Find z if you add one (1) to both x and y Find the value of y when z = 64, x = 8 z = 48, when x = 3, y = 4 z = 56, when x = , y = 14 z = , when x = 6, y = 16 If z varies jointly as x and y Find the constant (k) Find z if you add one (1) to both x and y Find the value of y when z = 64, x = 8 z = 48, when x = 3, y = 4

76. 4. Combined Variation

77. A combined variation is a kind of variation that involves the combination of either the “direct and inverse variation” or “joint and inverse variation.” “In combined variation, as one variable increases, the other two variables also increase and do the opposite, respectively.” Combined Variation

78. Workers 5 10 20 25 Days 20 10 ? 4 Labor 5,000 ? 80,000 ? Combined Variation If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job?

79. In joint variation, there will be one (1) independent variables , the x , and two (2) dependent variables , which is the y (inverse) and z (direct) . Technically, combined variation are two variations, combined. Combined Variation

80. Step 1: Distinguish which are the independent variable (x), the inversely dependent variable (y), and the direct dependent variable (z). Combined Variation

81. Workers 5 10 20 25 Days 20 10 ? 4 Labor 5,000 ? 80,000 ? Combined Variation If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job? x y z

82. Step 1: Distinguish which are the independent variable (x), the inversely dependent variable (y), and the direct dependent variable (z). Step 2: Use the formula to find the constant (k). Step 3: Use the formula to find the value of z:   Combined Variation

83. Workers 5 10 20 25 Days 20 10 ? 4 Labor 5,000 ? 80,000 ? Combined Variation If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job?   x y z

84. If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job? Workers 5 10 20 25 Days 20 10 ? 4 Labor 5,000 20,000 80,000 ? Combined Variation x y z

85. Step 1: Distinguish which are the independent variable (x), the inversely dependent variable (y), and the direct dependent variable (z). Step 2: Use the formula to find the constant (k). Step 3: Use the formula to find the value of y:   Combined Variation

86. If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job?   Workers 5 10 20 25 Days 20 10 ? 4 Labor 5,000 20,000 80,000 ? Combined Variation x y z

87. If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job? Workers 5 10 20 25 Days 20 10 5 4 Labor 5,000 20,000 80,000 ? Combined Variation x y z

88. Step 1: Distinguish which are the independent variable (x), the inversely dependent variable (y), and the direct dependent variable (z). Step 2: Use the formula to find the constant (k). Step 3: Use the formula to find the value of z:   Joint Variation

89. If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job?   Workers 5 10 20 25 Days 20 10 5 4 Labor 5,000 20,000 80,000 ? Combined Variation x y z

90. If 5 workers can complete a job in 20 days, and the expenses for the labor would be Php5,000.00. How many days can the new set of workers finish the job? How much would be the cost of labor if you add more workers for the same job? Workers 5 10 20 25 Days 20 10 5 4 Labor 5,000 20,000 80,000 125,000 Combined Variation x y z

91. Combined Variation Workers 5 10 20 25 Days 20 10 5 4 Labor 5,000 20,000 80,000 125,000

92. As the x increases, the y decreases, and the z also increases. Formula for constant (k): Formula for the independent variable (x): Formula for dependent inverse variable (y): Formula for independent variable (z) o r the equation of the variation :   Summary: Combined Variation

93. Activity no. 5: Combined Variation Rest Well and have a great day ahead

94. THANKS Grade 9 Independent Variables!