Direct Proportions - Lesson Plan - Maths (B.Ed. - Bangalore University)

College: Vijaya Teachers College (CTE)

Batch: 2019-21

Student Teacher Name: Nagashwini N

SL.No:1	Subject: Maths	Date:
Name of School:	Unit: Direct Proportions	Time:40 min
Standard:8	Topic: Direct Proportionality

 

 

 

1. Teaching Points

1. Meaning of direct proportionality

2. Explain constant of proportionality

3. Problems on direct proportionality

 

2. Content Analysis

 

1. Two quantities are said to be in direct proportion if increase in one quantity produces similar increase in the other quantity.or Two quantities are said to be in direct proportion in decrease in one quantity produces similar decrease in the other quantity. Two quantities are said to be in direct proportion if the ratio between those quantities remain constant.

 

2. Explain with an example the formula for direct proportionality or Explain constant of proportionality

 

Two quantities have a proportional relationship if they can be expressed in the general form y = kx, where k is the constant of proportionality. In other words, these quantities always maintain the same ratio. That is, when you divide any pair of the two values, you always get the same number k.

The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.

 

3. Problems on direct proportionality

 

Modal Problem:

 

The fuel consumption of a car is 15 liters of diesel per 100 km. What distance can the car cover with 5 liters of diesel?

 

Fuel consumed for every 100 km covered = 15 liters

Therefore, the car will cover (100/15) km using 1 liter of the fuel

If 1 liter => (100/15) km

What about 5 liters of diesel

= {(100/15) × 5} km

= 33.3
Therefore, the car can cover 33.3 km using 5 liters of the fuel.

 

Drill Problem:

 

Train travels 200 km in 5 hr, how much it will take to cover 600 km?

 

ICA Problem:

 

How much it will take to cover 800 km, If train travels 200 km in 5 hr?

 

3.

 

	Instructional objective	Task Analysis
1	Explain when 2 quantities are in direct proportion	Identify in a family there are 4 members then consumption of milk is 1 packet
		Identify in a family there are 6 members then consumption of milk is 2 packets
		Recognize that when family members are increased the consumption of milk packets are also increased. Thus these 2 quantities are in direct proportion.
		Explain that if books in the bag are removed the weight of bag decreases and when books are added to bag the weight of bag increases so here 2 quantities are bag and books and they are in directly proportion.
		Identify the 2 quantities in this example which are in direct proportion
2	Explain the term direct variance	Identify that the cost of one book is given as 10 Rs
		Calculate the cost of 2 books
		Calculate the cost of 2 books just by multiplying 2 by 10 and get the answer as 20
		Write the ratios for number of books by total cost in this case is 2/20
		Identify number of books given are three
		Calculate the cost of 3 books
		Calculate the cost of 3 books just by multiplying 3 by 10 and get the answer as 30
		Write the ratios for number of books by total cost in this case is 3/30
		Identify number of books given are four
		Calculate the cost of 4 books
		Calculate the cost of 4 books just by multiplying 4 by 10 and get the answer as 40
		Write the ratios for number of books by total cost in this case is 4/40
		Recoginze the situation where an increase in one quantity causes a corresponding increase in the other quantity, or a decrease in one quantity results in a decrease in the other quantity.
		Recognize that there is relationship between 2 quantities that is number of books and cost of books.
		Recognize that in all the above cases or examples the book by cost ratios are same, that is 1/10 = 0.1
		Recognize that all the 3 examples the values of the last column are constant or same.
		Identify that this is called direct variation
		Define the term Direct proportion/variance.
3	Explain with an example the formula for direct proportionality
		Identify the 2 quantities number of books and cost of books in terms of variables
		Recognize that number of books be y
		Recognize that cost of books be x
		Identify y/x= 1/ 10 which is a constant
		Write the constant as k, the equation becomes y/x = k
		Recognize y/x = k can also be written as y = k x, where k is constant
		Identify that y is directly proprotional to x, and is written as y ∝ X, when the equal sign (=) is replaceed with the proportionality sign (∝)
		Identify if x1 = 1, y1 = 10 and x2 = 2, y2 = 20 then this can be written as x1/y1 = x2/y2
		Define the term Direct proportion/variance as the relationship between two variables whose ratio is equal to a constant value.
		Recognize the condition 1: Two quantities are said to be in direct proportion in increase in one quantity produces similar increase in the other quantity. OR Two quantities are said to be in direct proportion in decrease in one quantity produces similar decrease in the other quantity.
		Recognize the condition 2: The proportional relationship between any two quantities can also be defined as the quantities whose ratio is constant.
4	Solve the problem to find the missing quantity, when 2 quantities are in direct proportion	Read the problem statement of modal problem:   The fuel consumption of a car is 15 liters of diesel per 100 km. What distance can the car cover with 5 liters of diesel? (teacher solves on board)
		List the given values
		Identify unknown values to be found out
		Substitute the values in the formula
		Solve the problem
		Read the problem statement of drill problem:   Train travels 200 km in 5 hr, how much it will take to cover 600 km?(student solves on board)
		List the given values
		Identify unknown values to be found out
		Substitute the values in the formula
		Solve the problem
		Read the problem statement of drill problem:   Train travels 200 km in 5 hr, how much it will take to cover 800 km?(student solves on board)
		List the given values
		Identify unknown values to be found out
		Substitute the values in the formula
		Solve the problem

 

 

 

4. Instructional Strategy

 

Inducto deductive method

Problem solving approach

 

5. Instructional Material

 

Chart having picture of train and car problem

Chart having picture of family and milk packets problem

Chart having formula of direct proportionality

Problem statement cards

 

6.

 

Content Analysis		Learning Experience
		Teachers Activity	Pupils Activity
Engaging		Teacher greet students
		Teacher give instructions to students
		How many members are there in your house?	4
		Do you drink milk?	yes
		So, if a family consists of 4 members, they require 2 packets of milk. Now your uncle will come to your house, at that time do you need extra packets of milk or not?	Need extra packets of milk.
		Now tell me if number of persons increase the milk packets required increase or decrease?	More milk packets are required
		Here milk packets and number of persons are 2 quantities and we are comparing these 2 quantities.
		Now suppose I give you 2 books to keep it inside the bag, the weight of increases or not?	Weight of bag increases
		If you remove 3 books from your bag, the weight of the bag decreases or increases?	Weight of bag decreases
		What are the 2 quantities involved in this example	Bag and book
		Which are the 2 quantities we are comparing here	Book and Bag
		How many of you travelled in train?	Students will raise their hands
		Look at the picture of train and tell me students do you know to calculate the time taken by train that travelled certain distance?	Looks at the picture
ICA		How much it will take to cover 800 km, If train travels 200 km in 5 hr?	No response
Topic Declaration		In today’s class we will find the answer to this question
Development		Identify in a family there are 4 members then consumption of milk is 1 packet	Looks at the picture
1. Two quantities are said to be in direct proportion if increase in one quantity produces similar increase in the other quantity.or Two quantities are said to be in direct proportion in decrease in one quantity produces similar decrease in the other quantity. Two quantities are said to be in direct proportion if the ratio between those quantities remain constant.		Identify in a family there are 6 members then consumption of milk is 2 packets
Taking an example of family members and milk packets scenario		Recognize that when family members are increased the consumption of milk packets are also increased. Thus these 2 quantities are in direct proportion.
		when family members are increased the consumption of milk packets are also increased
		The values of 2 quantities are related in such a way that changes in one results in a corresponding change in other. Such 2 quantities are said to be directly related. That is directly proprotional or in direct variance.
Taking an example of books and bag member scenario		If books in the bag are removed the weight of bag decreases there number of books and weight of bag are said to be in direct or indirect proportion	Direct proportion
		Direct proportion is also called direct variance
		Explain when 2 quantities are in direct proportion	Two quantities are said to be in direct proportion in decrease in one quantity produces similar decrease in the other quantity.
2. Explain with an example the formula for direct proportionality or Explain constant of proportionality   Two quantities have a proportional relationship if they can be expressed in the general form y = kx, where k is the constant of proportionality. In other words, these quantities always maintain the same ratio. That is, when you divide any pair of the two values, you always get the same number k. The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
		Identify that the cost of one book is given as 10 Rs
		Calculate the cost of 2 books	20
		So you Calculated the cost of 2 books just by multiplying 2 by 10 and get the answer as 20
		What is the ratios for number of books by total cost in this case ?	2/20
		How many number of books given here showing the picture	three
		Calculate the cost of 3 books	30
		So you Calculated the cost of 3 books just by multiplying 3 by 10 and get the answer as 30
		What is the ratios for number of books by total cost in this case?	3/30
		How many number of books given here showing the picture	four
		Calculate the cost of 4 books	40
		So you Calculated the cost of 4 books just by multiplying 4 by 10 and get the answer as 40
		What is the ratios for number of books by total cost in this case	4/40
		Thus here in this situation an increase in one quantity causes a corresponding increase in the other quantity, or a decrease in one quantity results in a decrease in the other quantity.
		Explain the term direct variance	Two quantities are said to be in direct proportion if increase in one quantity produces similar increase in the other quantity.or Two quantities are said to be in direct proportion in decrease in one quantity produces similar decrease in the other quantity. Two quantities are said to be in direct proportion if the ratio between those quantities remain constant.
		What are the common characteristic you see in the above table?	x/y = 0.1 in all 3 examples
		When one quantity increased other quantity increases or decreases?	increases
		Tell me is there relationship between 2 quantities that is number of books and cost of books.	yes
		Two quantities are said to be in direct proportion if increase in one quantity produces similar increase in the other quantity. This is condition 1
		What do you infer from this table	Book by cost ratios are same, that is 1/10 = 0.1
		Recognize that in all the above cases or examples the book by cost ratios are same, that is 1/10 = 0.1
		In this 3 examples the values of the last column are constant or same?	Constant
		Thus Relationship between these quantities is in direct variation
		Suppose 2 quantities are number of books and cost of books are represented in terms of variables
		So number of books be y
		So that cost of books be x
		Identify y/x=	1/ 10
		which is a constant
		Write the constant as k, the equation becomes y/x = k
		So y/x = k can also be written as y = k x, where k is constant
		So that y is directly proprotional to x, and is written as y ∝ x, when the equal sign (=) is replaced with the proportionality sign (∝)
		And if x1 = 1, y1 = 10 and x2 = 2, y2 = 20 then this can be written as x1/y1 = x2/y2
		Thus If, x/y = k, where k is a constant, then x and y are said to vary directly.
		Another condition for 2 quantity to be in Direct proportion/variance is
		It is the relationship between two variables whose ratio is equal to a constant value. This is condition 2.
		Let’s say if y1 and y2 are the values of y corresponding to the values x1 and x2 of x, respectively, then: x1/y1 = x2/y2 This is the formula for direct variance.
		Explain the condition for 2 quantities to be in direct proportion	Condition 1: Two quantities are said to be in direct proportion if increase in one quantity produces similar increase in the other quantity. OR Two quantities are said to be in direct proportion in decrease in one quantity produces similar decrease in the other quantity.
			Condition 2: The proportional relationship between any two quantities can also be defined as the quantities whose ratio is constant.
		Explain the term direct variance with formula	If, x/y = k, where k is a constant, then x and y are said to vary directly. Let’s say if y1 and y2 are the values of y corresponding to the values x1 and x2 of x, respectively, then: x1/y1 = x2/y2 This is the formula for direct variance.
modal problem:		The fuel consumption of a car is 15 liters of diesel per 100 km. What distance can the car cover with 5 liters of diesel?
		What is given?	Fuel consumed by car for every 100 km covered is given
		How much Fuel consumed for every 100 km covered?	15 liters
		How much the car will cover using 1 liter of the fuel	(100/15) km
		What about 5 liters of diesel =?	{(100/15) × 5} km
		Therefore, how much distance the car can cover using 5 liters of the fuel.	33.3 km
Drill Problem:		Train travels 200 km in 5 hr, how much it will take to cover 600 km?	Student solves it on the board
			200/5 = 600/y2 y2 = 5 x 600 / 200 = 15 hr
Revisit ICA		How much it will take to cover 800 km, If train travels 200 km in 5 hr?	Student solves it on the board
			200/5 = 800/y2 y2 = 5 x 800 / 200 = 20 hr
Application		Give 2 examples of direct proportionality	The cost of the food items is directly proportional to the weight.   The fuel consumption of a car is proportional to the distance covered.
Closure		In this class we have learnt definition of direct proportionality solved problems on direct proportionality. Now I will end the class. Before that take the home work.
Home Assignment		Two ratios are said to be in proportion when the two ratios are equal.   Proportion means a mathematical comparison between two numbers.   Define the term direct proportionality   Give 2 examples of direct proportionality