Krishna Dt. , Andhra Pradesh

Z.P.High School , VAKKALAGADDA

On the Eve of NATIONAL SCIENCE

DAY.

A new approach in teaching and learning of

Mathematics.

Powerpoint presentationOUR PROJECT...

Present day teaching and learningof mathematics.

Teachers would normally start the class…

Let us take a problem from the text book…

Draw the graph of y=3x+1

This is a linear functionIt is in the form of

y=mx+cthe graph is a straight lineGradient is m=3y-intercept is c=1 it does not go through the origin(0,0)

Teachers would normally state thatTake some values for

x and find y

Now plotting the graph we

get….

X-axis Time(min)

Y-axisWater level(c.m)

0

1

1 4

2 7

3 10

0 2 3 4

4

32

1

56

7

89

10

1

Y=3x+1

This is context free and is the traditional way of teaching in most

of our schools

Let us solve another similar problem…….

Does this make any sense to the students ?

Now a days,pupils are taught mathematics in a traditional way, solely from text books.

As a result they tend to think of mathematics as boring and useless for their life .

They view mathematics as a set of rules and procedures to be memorized.

Teachers

Students are unable to

apply a previously learned skill to a new

type of problem.They fail to retain

mathematical knowledge.

what is the

Solution

The one and only solution is to introduce the process of mathematical modelling in

class room teaching.

Mathematical modelling

is a process of scientific enquiry for mathematics and represents a new way

of doing mathematics.

Mathematical world

Outer world(non mathematical situation)

Modelling Process starts with a real world problem and connects mathematics and reality , the interaction is done by using

known mathematical formulae.

is process oriented and the solutions require continual

revision.

Classical mathematics

Applied mathematics

Real world problem

Mathematical problem Make

assumptions

FormulateEquation

Solve Equation

Interpret solution

Compare with data

Process of Mathematical modelling….

Model interpretation

Model refinement

Real worldSolution

MATHEMATICAL MODELLING

IS NOT making ofMathematicalmodels

are limited to mathematical world only and are expressed in mathematical terms such as fomulae,graphs,equations and so on.

contribute to mere problem solving without context

are mostly product oriented with expected answers.

are used for explaining and definining some mathematical concepts in class rooms.

as they…

The linear function or graph of y=mx+c

It may be more interesting to see how such a graph and function can actually

arise from a real practical situation.

Let us consider the following situation. water flows from a tap into a jar at

a constant rate.

we want to show how the water level changes with time and how long it would

take to fill the whole jar.

Mathematical modelling

Now we try to guess the relationship between the water level ‘ y’ and the

time after the tap is turned on i.e ‘ t.’

At any time` t’ ,the level y should be c + some positive number.

Suppose the water from the tap raises water level

@ 3 c.m/minute in the jar and the initial water level is 1 c.m.

Let the initial water level in the jar = c,Time = t

The water level after t = y

Model

1

13

4

7

10

0c After 1

minute y = 3 x 1 + 1

After 2 minutes

y = 3 x 2 + 1

After 3 minutes

y = 3 x 3 + 1

After 4 minutes

y = 3 x 4 + 1

After t minutes

y = 3 x t + c

This somewhat resembles

y = kt + c

X-axis Time(min)

Y-axisWater level(c.m)

0

1

1 4

2 7

3 10

1 2 3 4

1

2

34

0

kt

c

Y= k

t + c5

6

7

8

9

If the initial water level in the jar c= 0Y = kt + c = kt + 0 = kt.

Then the graph will be in the form of

y=mx

The graph is a straight lineGradient m = 3

y-intercept is c = 1 it does not go through the

origin (0,0)

X-axisTime(min)

Y-axisWater level

(c.m)

0 0

1 3

2 6

3 9

1 2 3 4

4

2

6

1

3

5

78

10

9

y=

kt

DESIGNING CARDBOARD BOXES TO ORDER

Mathematical modelling

Here is the Problem of a customer who wants cardboard boxes as per the following Requirements

The box should hold 0.02 cubic meters(20Litres) of powdered material.

Should have a square base & top

Double thickness top and bottom.

Cardboard cost could be Rs.15/- per square meter.

The most economical size is to

be decided.

The box should be designed like this

W and h are to be calculated

w

h

2 Step Two

Making Formulae

Ignoring the thickness of the cardboard

Volume W W h

W2h

As per the requirement. the volume should be 0.02 cubic metres

W2h 0.02

m

3

Areas:

Area of the 4 sides

W

h=

4 X w X h

=4 w h

Area of Double Tops and Bases

W

W

= 4 X w X w

= 4 w 2

Total cardboard needed:

= Total area of Cardboard

= 4 w h + 4 w 2

= 4 w ( h + w )

Step Three: 3 single formula for cost

Cost = 15.00 × Area of Cardboard

= 15.00 x 4 w ( h + w )

= 60 w ( h + w )And that is the cost when we know width and

height.This is a function with two variables

But we can make it simple

Volume = w2 h = 0.02 Cubic metres

h=0.02 / w2

…that can be put into cost formula…..

Cost= 60 w0.02

w+ w

With a l i t t le s impl ifi cat ion we get :

Cost = 600.02

+ w

w

And now the cost is related directly to width only.

2

3

Step 4

Plotting and fi nding minimum cost

Cost = 60 (0.02 + w ) / w 3 The formula only makes sense

for widths greater than zero .

For widths above 0.5m. the cost just gets bigger and bigger.

So here is a plot of cost formula for widths between 0.1 m and 0.5 m:

WIDTH (m)X-axis

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

COST(Rs.) Y-axis

12.60 9.35 8.40 8.55 9.40 10.78 12.60 14.82 17.40

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

2

4

6

8

10

12

14

16

18

20

12.6

9.35000000000001

8.4 8.55

9.4

10.78

12.6

14.82

17.4

X

x

Y

Width

Cost y = 60 (0.02 + w ) / w 3

Just by eye, I see the cost reaches a minimum at about (0.225,8.3).

Recommendations Any width between 0.2 m and 0.225m would be fine.

In other words:When the width is about 0.225m(x-

value), The minimum cost is aboutRs.8.30

per box(y-value).

suggested improvements to this model:

Include cost of glue/staples and assembly

Include wastage when cutting box shape from cardboard.

The thickness of card board is also to be taken into account

Anthropology² Modeling, classifying and reconstructing skullsArcheology² Reconstruction of objects from preserved fragments² Classifying ancient arti¯cesArchitecture² Virtual realityArti¯cial intelligence² Computer vision² Image interpretation² Robotics² Speech recognition² Optical character recognition² Reasoning under uncertaintyArts² Computer animation (Jurassic Park)Astronomy² Detection of planetary systems² Correcting the Hubble telescope² Origin of the universe² Evolution of stars

Some of the areas where M.M is widely used

² Population dynamics² Morphogenesis² Evolutionary pedigrees² Spreading of infectuous diseases (AIDS)² Animal and plant breeding (genetic variability)Chemical engineering² Chemical equilibrium² Planning of production unitsChemistry² Chemical reaction dynamics² Molecular modeling² Electronic structure calculations

Meteorology² Weather prediction² Climate prediction (global warming, what caused the ozone hole?)Music² Analysis and synthesis of soundsNeurosciencee² Neural networks² Signal transmission in nervesPharmacology² Docking of molecules to proteins² Screening of new compoundsPhysics² Elementary particle tracking² Quantum ¯eld theory predictions (baryon spectrum)² Laser dynamicsPolitical Sciences² Analysis of electionsPsychology² Formalizing diaries of therapy sessionsSpace Sciences² Trajectory planning² Flight simulation² Shuttle reentry

Materials Science² Microchip production² Microstructures² Semiconductor modelingMechanical engineering² Stability of structures (high rise buildings, bridges, air planes)² Structural optimization² Crash simulationMedicine² Radiation therapy planning² Computer-aided tomography² Blood circulation models

Sandh

ya.