The Four types of Variations in Real - life setting
- 1. VARIATIONS Prepared by: SASMER H. ASEJO
- 2. LEARNING COMPETENCY M9AL-IIa-1 The learner illustrates situations that involve the following variations: (a) direct; (b) inverse; (c) joint; (d) combined.
- 3. REVIEW Ratio is a comparison of two numbers with the same units, or different units of the same kind. It is obtained by dividing the first number by the second number. Proportion is a statement indicating the quality of two ratios. If a: b and c: d are two equivalent ratios, then a: b = c: d is a proportion.
- 4. TOPIC & OBJECTIVES Our lesson today is about Variations and its different types and our objectives are: a. Define direct, inverse, joint and combined variation. b. Identify direct, inverse, joint and combined variation. c. Illustrates situations that involves direct, inverse, joint, and combined variations.
- 5. EXAMPLE/ACTIVITY 1 • A local government organization launches a recycling campaign of waste materials to schools in order to raise students’ awareness of environmental protection and the effects of climate change. Every kilogram of waste material earns points that can be exchanged for school supplies and grocery items. Paper, which is the number one waste collected, earns 5 points for every kilo.
- 6. Questions: 1. What happens to the number of points when the number of kilograms of paper is doubled? Tripled? 2. How many kilograms of paper will the Grade 9 class have to gather in order to raise 500 points? Write a mathematical statement that will relate the two quantities involved.
- 7. EXAMPLE/ACTIVITY 2 • You are living 40 km away from your school. Driving a tricycle, the time it takes you to reach school depends on your average speed. Some possible speeds and the length of time it takes you are as follows: • To see clearly the relation of the two quantities, the graph of the relation is shown below. • Base on the graph, how do the speed and time of travel affect each other?
- 8. EXAMPLE/ACTIVITY 3 • The volume of a pyramid varies jointly as the area of the base and the altitude. Questions: • What are the variables in the example? • How will you relate these variables?
- 9. EXAMPLE/ACTIVITY 4 • The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. Questions: • What are the variables in the example? • How will you relate these variables?
- 10. LET’S ANALYZE! Answer the following questions. 1. What have you observed from the four activities/examples? 2. What are the key words that you can create from the different activities? 3. Which of the four activities is an example of direct variation? Inverse variation? Joint variation? Combined variation? 4. What is direct variation? Inverse variation? Joint variation? Combined variation? 5. How can you illustrate a situation that involves direct variation? Inverse variation? Joint variation? Combined variation?
- 11. THE DIFFERENT TYPES OF VARIATION There is Direct Variation whenever a situation produces pairs of numbers in which their ratio is constant. The statements: “y varies directly as x” “y is directly proportional to x” and “y is proportional to x” can be translated mathematically as y = kx, where k is the constant of variation. For two quantities, x and y, an increase in x causes an increase in y as well. Similarly, a decrease in x causes a decrease in y.
- 12. Example: Helen and Joana walk a distance of one kilometer in going to the school where they teach. At a constant rate, it takes them 20 minutes to reach the school in time for their first class. One morning, the two became so engrossed to discuss an incident that happened inside the school that they did not notice that their pace of walking slowed down. This situation illustrates a direct variation with two pairs of variables: time and distance, and distance and rate. As time increases, the distance also increases; and as rate decreases, distance also decreases.
- 13. Inverse Variation occurs whenever a situation produces pairs of number whose product is constant. For two quantities x and y, an increase in x causes a decrease in y or vice versa. We can say that “y varies inversely as x” or y = / . 𝑘 𝑥 The statement, “y varies inversely to x”, translates to y = / 𝑘 𝑥 , where k is the constant of the variation.
- 14. Example: Anna lives 40 km away from the office of ABC Corporation where she works. Driving a car, the time it takes her to reach her workplace depends on the car’s average speed. Some possible speeds and the lengths it takes her are as follows: The situation illustrates an inverse variation. The speed varies inversely as its time. As time decrease, the speed increases. Time in hours 1 4/5 2/3 4/7 ½ Speed in kph 40 50 60 70 80
- 15. Joint Variation is closely the same with direct variation, but joint variation involves 3 or more variables. Statements a varies jointly as b and c means a – kbc, or k = abc, where k is the constant of variation. Example: The area of a triangle varies jointly as its base and its altitude. The statement illustrates joint variation. It involves three variables which are area base, and altitude.
- 16. Combined variation is another physical relationship among variables. This is the kind of variation that involves both the direct and inverse variation. Example: The maximum load of a beam varies directly as the breadth and the square of the depth and inversely as the length. The situation illustrates a combined variation. It is composed of direct and inverse variation.
- 17. LET US ANSWER! Identify the following. 1. What are the 4 types of variation? 2. It is a kind of variation, wherein for two quantities, x and y, an increase in x causes an increase in y as well. 3. It is a variation that occurs whenever a situation produces pairs of numbers whose product is constant. An increase in x causes a decrease in y or vice versa. 4. In mathematical equation y = kx, what variable is constant? 5. What is the formula for inverse variation?
- 18. 7. Which of the following table describes inverse variation? 8. Formulate a real – life situation that illustrates the 4 different variation. One situation for each variation. 6. Which of the following table describes a direct variation?
- 19. WRITTEN TEST State the variation being illustrated in the given situation. 1. The current varies directly as the electromotive force and inversely as the resistance. 2. The amount of gasoline used by a car varies jointly as the distance travelled and the square root of the speed. 3. The number of persons sharing a pie to the number of slices of the pie. 4. The volume of a right circular cylinder varies jointly as the height and the square of the radius. 5. The number of hours to finish a job to the number of men working.
- 20. TRANSFER OF LEARNING! Create a journal writing and portfolio of real – life situations/pictures where concepts of the 4 types of variation are applied.
- 21. THANK YOU!