ADDITIONAL MATHEMATICS PROJECT WORK

2015

NAME : Previndrarasan S/O Krishnamutty

CLASS : 5 Gamma

I/C : 980313-06-5865

TEACHER’S NAME : Puan Fatimah Binti Uyub

OBJECTIVES

We students taking Additional Mathematics are required to carry out a project work

while we are in Form 5.This year the Curriculum Development Division, Ministry of

Education has prepared two tasks for us. We are to choose and complete only one task

based on our area of interest. This project can be done in groups or individually, and I

gladly choose to do this individually. Upon completion of the Additional Mathematics

Project Work, we are to gain valuable experiences and able to:

To apply and adapt a variety of problem-solving strategies to solve problems.

To improve thinking skills.

To promote effective mathematical communication.

To develop mathematical knowledge through problem solving in a way that

increases student interest and confident.

To use the language of mathematics to express mathematical ideas precisely.

To provide learning environment that stimulates and enhance effective

learning.

To develop positive attitude towards mathematics.

We are expected to submit the project work within three weeks from the first day the

task is being administered to us. Failure to submit the written report will result in us not

receiving certificate.

ACKNOWLEDGEMENT

First of all, I would like to say thank you to my friends, teachers and parents for giving

me their full support in making this project successful.

Not forgotten to my family for providing everything, such as money, to buy anything

that are related to this project work and their advise, which is the most needed for this

project. Internet, books, computers and all that act as my source to complete this project.

They also supported me and encouraged me to complete this task so that I will not

procrastinate during doing this project work.

Next, I would like to thank my teacher, Puan Fatimah Binti Uyub for guiding me and

my friends throughout this project. We had some difficulties in doing this task, but she

taught us patiently and gave me guidance throughout the journey until we knew what to

do. She tried her best to help us until we understand what we supposed to do with the

project work.

Besides that, my friends who were doing the same project as mine and shared our

ideas. They were helpful that when we combined and discussed together, but however, this

task was done individually.

Last but not least, any party which involved either directly or indirect in completing

this project work. Thank you everyone.

INTRODUCTIONTHE HISTORY OF BRICK PAVING

There are reports of brick pavements having been used for at least 5000 years. The

earliest evidence of such pavements comes from Mesopotamia. Here, because of lack of

local stone, pavements were surfaced with bricks.

In the USA, as in Europe, the selection of brick thickness was largely empirical. In

the American mid-west experience showed that brick thicknesses as little as 60 mm could

successfully withstand motor vehicles. To verify this, accelerated trafficking tests were

initiated in 1926 by the Bureau of Public Roads. Here the objective was toe examine the

feasibility of using brick thicknesses of 75 mm or less. These tests appear to be the first

test-track evaluations of a segmental pavement and, indeed, must represent one of the first

evaluations of any pavement made using scientific methods.

As in the case of stone setts, bricks were generally installed on a sand bed placed

either directly on the sub grade or on a broken stone base. The joints were filled with sand

or, more commonly, with a bituminous material applied hot. As an alternative to this, trials

using premoulded expansion strips to seal the joints were made in the USA but do not

appear to have been successful.

The principal problem associated with brick pavements was their propensity to

surface damage. This was manifest as cracking and cobbling pavers. In an attempt to

strengthen brick pavements experiments were conducted in Holland and America whereby

both transverse and longitudinal steel reinforcement was laid in joints which were then

mortared. Tests conducted in Illinois in the 1930s showed such pavements could carry

heavy traffic with little or no maintenance. A more conventional approach was developed

in Hungary. Here very high-quality paving bricks were made from clay with a high lime

content which was molded in steel forms under high pressure and fired at high

temperatures. This process yielded bricks with strengths similar to basalt setts and enabled

pavement lives of about 30 years under traffic to be achieved. However, in general, brick

pavements normally had an effective life of less than 20 years.

PART A

(a)(i)

Estimation Method 1: (Area)

Area of pavement =

Area of a tile = 25 x 10

Number of tiles =

2 m1 m

Circular pavement

Plants

Estimation method 2: ( Using concept of Arithmetic Progression)

Starting from the interior, the number of tiles for the first layer

The number of tiles for the second layer

The number of tiles for the third layer

This is an arithmetic progression with

Number of tiles

= 980

*Alternative Method 3:

From the diagram on the right,

and

Hence, for the innermost layer, the number

of tiles

For the second layer,

and

Number of tiles

For the third layer,

and

Number of tiles

For the fourth layer,

and

Number of tiles

This is an arithmetic progression: 25.3, 27.8, 30.3, 32.8, …

a = 25.3, d = 2.5,

hence, the total number of tiles

= 981

(a)(ii)

Method 2 is a more accurate estimation compare with method 1. Method 2 takes

into consideration the spaces between tiles whereas Method 1 does not.

(a)(iii)

Method 1 is quick and simple and therefore easy to understand. This is a major

reason why it is being practiced by most masons with some modification. Due to their

experience, they deduct a certain number of tiles from the total number required by the

area calculation to compensate for the spaces between tiles, and they are quite accurate.

PART B(b)(i) Octagonal pavement

Method 1 : (Area)

Area of one trapezium =

Area of octogonal pavement =

Number of tiles =

= 905

Method 2 : (Arithmetic Progression)

Referring to the diagram

on the right,

and this is an arithmetic

progression.

How many layers are required?

MM1 = M1M2 = 10 cm

Number of layers

and

Number of tiles required

Total number of tiles for the octagon pavement = 8 x 106

= 848

*Alternative Method 3: (Scale Drawing)

Using a scale of 1 : 20 and draw one eighth of the octagon.

3 tiles

Estimation from the scale drawing:

There are 18 complete layers.

Number of complete whole tiles needed = 18 x 3 + 15 + 12 + 9 + 6 + 3

= 99

To fill up the empty spaces, tiles will be cut into smaller parts and the estimated

number of tiles = 3 x 3

= 9

Total number of tiles = 8 x (99 + 9)

= 864

There is quite a difference between Method 1 and Method 2. This is probably due

to the number of layers of tiles. Method 2 only considers 18 complete layers which does

not cover the whole area of the octagon pavement, whereas Method 1 considers the

whole area.

(b)(ii)

Comparing the circular design and the octagonal design, the octagonal design will be easier

to construct because of its straight line layers and management of the few empty spaces

between the tiles. To lay the tiles in circles will be quite a task and there empty spaces to fill

in between every tile.

FURTHER EXPLORATION

(a) Two circular plots pavement.

Refer to the diagram beside.

Method 1: (Using Area)

=1.172 radian

Area of segment KLMN

cm2

Number of tiles required

Number of tiles required for the pavement

2.5 m

3 m

= 1920

Method 2:

To calculate the number of tiles required for the arcs passing through points K1, K2, K3

and K4.

(i) Arc passing through point K1

Arc length

Number of tiles required

(ii) Arc passing through point K2

Arc length

Number of tiles required

(iii) Arc passing through point K3

Arc length

Number of tiles required

(iv) Arc passing through point K4

Arc length

Number of tiles required

(v) Total number of tiles required for the pavement

(b) Refer to the diagram beside.

Area of triangle EFG

Total area of pavement

cm2

Number of tiles required

(c)

I will choose the overlapping double octagonal design as shown in Diagram 3. The

alternative design that I will suggest is an overlapping double rombus design. This is

because the number of tile used is lesser and it makes better landscape.

(d)

It is not practical for using aluminium tins and sand to lay the pavement. Although brick

is very heavy, but it costs cheaper than other materials and gives good cold and heat

insulation, it is also waterproof and fireproof. However, aluminum tin shouldn’t use at

pavement which can even cause injuries to people who step on it when there is a sharp edge

on the aluminium tin.

CONCLUSION

After I accomplished this project, I have found that the additional mathematics is fun

and very useful in daily life. I have learnt the important of perseverance as time will be

inverted to ensure the completion and excellence of this project.

On the other hands, I have learnt the virtue to making together as I have helped and

received help from my fellow peers in the production of this project. I realised the important

to be thankful and appreciative during completing this task. This is because I able to apply

my mathematical knowledge in daily life and appreciate the beauty of additional mathematic.

This project is a several training stage for me to prepare myself for the demands of my

future undertaking in the university and work life.

REFLECTION

While I conducting this project, there is a lot of information that I have found. I have

learnt how the area of the pavement is measured. Apart from that, this project encourages the

student to work together and share their knowledge. It is also encourages student to gather

information from the internet, improve thinking skills and promote effective mathematical

communication. Not only that, I had learned some moral values that I can practice. This

project had taught me to responsible on the works that are given to me to be completed. This

project also had made me felt more confidence to do works and not to give easily when we

could not find the solution for the question. I also learned to be more discipline on time,

which I was given only 3 weeks to complete this project and hand in to my teacher just in

time. I also enjoy doing this project as I need to spend my time with friends to complete this

project and it had tightens our friendship.

REFERENCES

http://www.bosspaving.co.za/history-of-paving.html

http://www.cut-the-knot.org/arithmetic/rapid/rapid.shtml

http://www.pavingexpert.com/geometry_01.htm

http://www.mathsteacher.com.au/year8/ch06_ratios/06_scale/draw.htm