2ND SPELL - X-CLASS- Maths - Dec,1 to 4 Periods - Computers (2)

7/30/2019 2ND SPELL - X-CLASS- Maths - Dec,1 to 4 Periods - Computers (2)

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APSWREISOCIETY J.TIRUPATI RAOLESSON PLAN FORMAT PGT MATHS

PERIOD: 01 APSWRSCHOOL/JC, PVP

MONTH: NOVEMBER LINEAR PROGRAMMING

Date and Time: class: X Subject: Maths Medium: English


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Contentanalysis/concepts/sub

concepts

Activities indicating behavioral/learning out comesAids/experiments to bedone /demonstrations

etc.

method Evaluation

By teacher By Pupil

What do you call a first degree

equation in x and y?

What is the general form of

linear equation?Give an example for a linear

equation?

How many points do you need

to draw a line?

Can you say any two points on

the line 2x+3y=6?

How many parts do the line

divides a plane?

What are they?

If ax+by+c=0 is a line on the

plane what are the three parts?

What do you call ax+by+c>0

and ax+by+c0 and

ax+by+c++


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Recapitulation: 1) what is half plane divided by the line.2) What represents a linear in equation?

3) What is the convenient point to decide the half plane?

4) What is the solution set of system of linear in equations?

Behavioral changes: 1) pupil recognize the region form by the in equations.

2) Pupil finds the solution set of system of in equations.

Assignment: Exercise -1 problems 4 &5


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Contentanalysis/concepts/sub

concepts

Activities indicating behavioral/learning out comesAids/experiments to be

done /demonstrations etc.

method Evaluation

By teacher By Pupil

Convex set: X is a subset of

a plane. If XQP,

XPQ then X is calledConvex set.

A closed convex polygon in

the set of all points within

and on polygon with a finite

number of vertices.

1. What is a plane?

2. What are the closed figures in the

board?

3. We take a closed figure X in the

plane. How is the set of points of X

to the plane?

In the closed figure X, take any twopoints P, Q. Join the line segment

.PQ is it contains in X?

Is figure (2) convex set?

Why?

What is solution set of systemlinear equations?

What is the solution set of system

of linear in equations?

Draw the system of linear in

equations 5,0,0 + yxyx

What is the solution set?

Is it closed polygon?

Set of infinite points

X is subset of the plane

Yes. XPQ

No

XPQ/

points

OAB

yes

yes

(2)

Synthetic

Method.

What is LPP?


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Recapitulation: What is convex set?What is closed convex polygon?

What is open convex polygon?

What is objective function?

What is LPP?

What is the fundamental theorem on LPP?

Behavioral changes: pupil recognizes the convex set, closed convex set and open convex polygon?

Pupil recalls objective function and fundamental theorem.

Assignment: Find the maximum and minimum values f=4x+y at the vertices O(0,0),A(3,0),B(2,4),C(0,8).


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C l i / / b Aid / i


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Content analysis/concepts/subconcepts Activities indicating behavioral/learning out comes

Aids/experimentsto be done

/demonstrations

etc. method Evaluation

By teacher By Pupil

Maximize f=2x+3y subject to the

constraints0,0;93,5 ++ yxyxyx

What is a closed convex polygon?

What is an objective function?

What is the maximum value f=x+y

at the points

O(0,0),A(2,0),B(3,2),C(0,6)

What is linear programming

problem?

What is the fundamental theorem

on LPP?

Now we learn the graphical

method for solving LPP:

What is given?

What to do?

For maximize objective function

what to do first?

0,0 yx Which region we have

the sol tion?

The set of all points within and on

some polygon with a finite number

of vertices.

A function f=ax+by which is to beminimize or maximize is the

objective function.

6

A linear programming problem

consists of minimize or maximize a

function subject to certain linear in

equations.

The minimum or maximum of

f=ax+by occurs at least one of the

vertices of the polygon.

0,0;93,5 ++ yxyxyx

To maximize f = 2x +3y

We draw the graph of the given

system of in equations

1st quadrant

0,0 yx

So that system

has the solution

in 1st quadrant.

Points table for

x+y=5

5 0

Synthetic

method

What isgeneral form

of objective

function?


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Recapitulation: what are the steps to solve LPP by using graph?

Behavioral changes: pupil solves the linear programming problems

Pupil understands the graphical method to maximize the objective function.

Assignment: Exercise 2. Q.1, 2, 4, 5 and 10.


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C t t l i / t / b Aid / i t t b


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Aids/experiments to bedone /demonstrations etc.

method Evaluation

By teacher By Pupil

Maximize yxf 4+= subject to

the constraints

0,0

1234;4058

++

yx

yxyx

I thi LPP th l ti t i

What is LPP?

What are the steps to find tothe solution of LPP?

Now we learn the solution of

LPP using this graphical

method:

What we do first?

What is the region

represented by 0,0 yx ?

For Shading the half plane4058 + yx

What is the boundary line?

Tell me any two points on the

line?

Draw the line on the graph?

what is the solution set of?4058 + yx

Similarly what is the solutionset of 1234 + yx ?

What is the solution set of

given system?

Since solution set is closed

convex polygon, how do you

maximize?

Wh t th ti f

An LPP consists of

minimize or maximize a

function subject to the

constraints.

Draw the graph and find

vertices and maximize f

We draw the graph of in

equations

1st quadrant

4058 =+ yx

(5,0),(0,8)

Region which contains(0,0)

Region which does not

contains (0,0)

Closed convex polygon

ABCD

0,0 yx

Solution set in 1stquadrant.

Table for

8x+5y=40

x 5 0

y 0 8

?4058 + yx Represents half plane

which contains (0,0)

Boundary line of1234 +

?

Sysnthetic

method

What isobjective

function?


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Recapitulation: what are iso profit lines?What is the general form of iso profit line determined by objective function f=ax+by

What are the steps to solve LPP by general method?

Behavioral changes: Students understanding the difference of solving LPP methods.

Students draw the iso profit lines.

Students maximize or minimize LPP by general method if the solution set is not closed convex polygon.

Assignment: solve exercise 2 in the text book.


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Content analysis/concepts/sub Aids/experiments to be


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Aids/experiments to bedone /demonstrations etc.

method Evaluation

By teacher By Pupil

Minimize f=4x+y subject to

the constraints

0,0

,82,6

++

yx

yxyx

What is the solution set of

system of linear in equations?

Are sure, it must be closed?

What is the feasible region?

What are iso profit lines?

If we draw iso profit lines

moving away from the origin

then how is value of f?

For finding minimum value of

f, how do you draw iso profit

lines?

What is the general method to

maximize f= ax+by subjet to

constraints.

Now we minimize f subject

to the constraints

What is the objective function?

What are the constraints?

What is the region denoted by0,0 yx ?

What are the boundary lines of

th h lf l

Convex set

No, either closed or open

The solution set of LPP is a

convex set called feasible

region.

Parallel lines determined by

objective function are callediso profit lines.

Value of f increases.

Moves closed to the origin

Graph the system and draw

iso profit line and move

towardsOrigin.

f=4x+y

0,0

,82,6

++

yx

yxyx

1st quadrant

82;6 =+=+ yxyx

0,0 yx Represents in

1st quadrant.

x+y=6

x 6 0

y 0 6

Synthetic

method

How iso profit

is lines eachother?

What is profit

line?


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Recapitulation: Describing the general graphical method to maximize f=ax+by?How do you draw iso profit lines?

Behavioral changes: Students understand the general method to minimize LPP.

Students find the method to various LPP problems.

Assignment: exercise 2 questions 6 ,7 &9.


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Content analysis/concepts/sub Aids/experiments to


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Aids/experiments tobe done

/demonstrations etc.

method Evaluation

By teacher By Pupil

A manufacturer makes two

models A and B of a product.

Each model should be processedby two machines. To complete

one unit of model A, machines I,

II must work one hour and 2

hours respectively. To complete

1 unit of model B machines I, IImust work 4 hrs and 2 hrs

respectively. Machine I may not

operate more than 8 hours per

day. And machine II not more

than 10 hours per day. If the

profits on models A and B per

unit are Rs.200 and Rs.280respectively. How many units of

each model should be

manufacturer produce per day to

maximize his profit?

What is an objective function?

What is the LPP?

What are the steps to maximize f?

LPP problems arise in Business,

industry and transports. Now we

learn some application of LPP.

How many models are produced?

What are they?

To complete each model what

machines is used?

What is required in the problem?

Now we convert the given

conditions into in equations

Let the number of units A is x,

and units of B is y

Can number of models x and y be

negative?

H d d if

f=ax+by which to be

minimized or maximized.

Problem consists of

minimizing an objectivefunction f=ax+by subject to

constraints

Draw the graph. Find the

vertices of polygon; determine

the value of f of each vertex.

2

A & B

I & II

To maximize the profit of howmany units of each model

should be produced.

No

0,0 yx

Number of units of

A=x

No. units of B=y

X and y are nonnegative

0,0 yx

-------(1)

To make x , y units,

the machine I should

be work 1,4

hrs.respectively and

not more than 8 hours.

84

8)4()1(

+

+

yx

yx

--------(2)

A

N


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Recapitulation: What is given? What is required?What are the conditions to maximize the profit?

Maximize P=200x+280y subject to above constraints?

Behavioral changes: Students understand that LPP problems are used in industry.

Students apply their knowledge into realistic problems.

Assignment: exercise 3. Q1


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Content analysis/concepts/sub Aids/experiments to be


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y pconcepts Activities indicating behavioral/learning out comes

pdone /demonstrations

etc.

method Evaluation

By teacher By Pupil

A shop keeper sells not morethan 30 shirts o each colour. At

least twice as many white ones

are sold as green ones. If the

profit on each of the white be

Rs.20, and that of green beRs.25, how many of each kind

be sold to give him a

maximum profit?

What is the LPP?

What we do in LPP?

Where do LPP problems arise?

Now we learn an LPP

What are given conditions in

this LPP?

What is required?

What kind shirts are sold?

Let No. white shirts are x No.

green shirts are y

Can x, y be negative?

Represent symbolically?

What are the conditions be

given?

He sells x, y and not more than

thirty. Can you express this in

in equation?

What is next condition?

Problem consists of

minimizing an objective

function f=ax+by subject to

constraints

Maximize are minimize an

objective function

In business, industry

How many kind of eachcolored shirts be sold to give

maximum profit

White and green

No

0,0 yx

He sells not more than 30

totally.

30+ yx

At least twice as many white

ones are sold as green.

Let No. white shirts arex No. green shirts are y

x, y are non negative0,0 yx

---------(1)

Sum of x an y not

more than 30

therefore

30+ yx

--------(2)At least twice as many

synthetic

method

A

N

If x is non

negativerepresent

symbolically?


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Recapitulation: What is the maximizing function in this problem?What are the conditions to maximize the profit?

What are the exact linear in equation for given data?

Behavioral changes: Students apply the concept of LPP into the real life.

Assignment: exercise 3 question NO.3

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2ND SPELL - X-CLASS- Maths - Dec,1 to 4 Periods - Computers (2)

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