Addmath project for Simultaneous equation and swing in Taman Lestari

1. SMK KUBANG PELIA
2018
Name : NORLIYANA BINTI KAYISHIYAH
Class : 5 ALIBABA
Teacher: Mrs. TEOw NIU SHEN
Icnumber: 011011-01-5781
1 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

2. CONTENT
• Acknowledgement……………..………………………….……………
• Objective……………………………..…………………….……………..
• Introduction…………………………..…………………….……………..
• Part A……………………………………………………….……………..
• Part B…………………………………………………………..…………..
• Part C……………………………………………………….……………..
• Reflection………………………………………………….……………..
• Conclusion……………………………………………………………….
2 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

3. ACKNOWLEDGEMEN
T
First of all, I would like to say Alhamdulillah, thanks to God for giving me
the work. Their advices, which I really needed to motivate myself in
completing this project. They also had been supporting and encouraging
me to complete this task soon as I could so that I would not procrastinate
in doing so.
Then, I would like to thank my Additional Mathematics teacher, Mr Teow
Niue Shen, who had been the one that did nothing to guide me and even
the whole class throughout this project as I have to do it completely by
myself. Even though I had some difficulties in finishing this task, I managed
to finish it well with reference from books as he never taught me a single
pages. He just sit there in the office waiting for my projects done with one
week schedule given. What a shame!
Also, I would like to express my feeling to my tuition center’s teacher that
cannot even give me hints and tips to answer this project. Some of the
teachers in there may not score 10 straight As before in their SPM so they
cannot help me in this simple project. Indeed, it is a waste of money to
put my future for this kind of tuition teachers.
Thanks to my friends who had been always supporting me. Even though
this project had to be done individually, we discussed with each other on
anything that was related to this project via Twitter and text messages. We
shared ideas and methods to answer those asked questions correctly.
Last but not least, thanks to anyone who had been contributed either
directly or indirectly in completing this project work. Without them, I
believed, this project work could not be done in such a good way.
Not forgotten, thanks to my parents for providing everything, such as
laptop and internet connection, which really was a big help to me in
finishing this project up as I could surf the net to find information and
guidance for me to make it
3 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

4. OBJECTIVE
The aims of carrying out this project work are:
• To apply and adapt a variety of problem-solving strategies
to solve problems.
• To improve thinking skills.
• To promote effective mathematical communication.
• To train me to be better person for not trusting any school
teacher or even those profit minded tuition center’s
teachers.
• To develop mathematical knowledge through problem solvi
ng in a way that increases students’ interest and
confidence.
• To use the language of mathematics to express mathematic
al ideas precisely.
• To learn not to trust any school teacher and tuition
teacher as they may not even score straight As in SPM
before.
4 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

5. • To provide learning environment that stimulates and enhan
ces effective learning.
• To develop positive attitude towards mathematics.
5 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

6. INTRODUCTIO
N
History of Equations.
It is often claimed that the Babylonians (about 400 BC) were the first to solve quadratic
equations. This is an over simplification, for the Babylonians had no notion of 'equation'. What
they did develop was an algorithmic approach to solving problems which, in our terminology,
would give rise to a quadratic equation. The method is essentially one of completing the square.
However all Babylonian problems had answers which were positive (more accurately unsigned)
quantities since the usual answer was a length.
The Arabs did not know about the advances of the Hindus so they had neither negative
quantities nor abbreviations for their unknowns. However al-Khwarizmi (c 800) gave a
classification of different types of quadratics (although only numerical examples of each). The
different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each
devoted to a different type of equation, the equations being made up of three types of
quantities namely: roots, squares of roots and numbers i.e. x, x2
and numbers.
1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers, e.g. x2
+ 10x = 39.
5. Squares and numbers equal to roots, e.g. x2
+ 21 = 10x.
6. Roots and numbers equal to squares, e.g. 3x + 4 = x2
.
Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic
formula given for a numerical example in each case, and then a proof for each example which
is a geometricalcompleting the square.
Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his
book Liber embadorum published in 1145 which is the first book published in Europe to give the
complete solution of the quadratic equation.
A new phase of mathematics began in Italy around 1500. In 1494 the first edition of Summa de
arithmetica, geometrica, proportioni et proportionalita, now known as the Suma, appeared. It
was written by Luca Pacioli although it is quite hard to find the author's name on the book, Fra
Luca appearing in small print but not on the title page. In many ways the book is more a
summary of knowledge at the time and makes no major advances.
Pacioli does not discuss cubic equations but does discuss quartics. He says that, in our
notation, x4
= a + bx2
can be solved by quadratic methods but x4
+ ax2
= b and x4
+ a = bx2
are impossible at the present state of science.
6 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

7. Scipione dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the University of
Bologna and certainly must have met Pacioli who lectured at Bologna in 1501-2. Dal Ferro is
credited with solving cubic equations algebraically but the picture is somewhat more
complicated. The problem was to find the roots by adding, subtracting, multiplying, dividing and
taking roots of expressions in the coefficients. We believe that dal Ferro could only solve cubic
equation of the form x3
+ mx = n. In fact this is all that is required.
For, given the general cubic y3
- by2
+ cy - d = 0, put y = x + b/3 to get x3
+ mx = n where
m = c - b2
/3, n = d - bc/3 + 2b3
/27.
However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been
able to use his solution of the one case to solve all cubic equations. Remarkably, dal
Ferro solved this cubic equation around 1515 but kept his work a complete secret until just
before his death, in 1526, when he revealed his method to his student Antonio Fior.
Application of Equations.
We now fastforward 1000 years to the Ancient Greeks and see what they made of quadratic
equations. The Greeks were superb mathematicians and discovered much of the mathematics
we still use today. One of the equations they were interested in solving was the (simple)
quadratic equation:
They knew that this equation had a solution. In fact it is the length of the hypotenuse of a right
angled triangle which had sides of length one.
It follows from Pythagoras’ theorem that if a right-angled triangle has shorter sides and and
hypotenuse then
Putting and then . Thus
7 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

8. So, what is in this case? Or, to ask the question that the Greeks asked, what sort of number is it?
The reason that this mattered lay in the Greek’s sense of proportion. They believed that all
numbers were in proportion with each other. To be precise, this meant that all numbers
were fractions of the form where and are whole numbers. Numbers like 1/2, 3/4 and
355/113 are all examples of fractions. It was natural to expect that was also a fraction. The
huge surprise was that it isn’t. In fact
where the dots mean that the decimal expansion of continues to infinity without any
discernible pattern. (We will meet this situation again later when we learn about chaos.)
was the first irrational number (that is, a number which is not a fraction, or rational), to be
recognised as such. Other examples include , , and in fact "most" numbers. It took until the
19th century before we had a good way of thinking about these numbers. The discovery
that was not a rational number caused both great excitement (100 oxen were sacrificed as
a result) and great shock, with the discoverer having to commit suicide. (Let this be an awful
warning to the mathematically keen!) At this point the Greeks gave up algebra and turned to
geometry.
Far from being an obscure number, we meet regularly: whenever we use a piece of A4
paper. In Europe, paper sizes are measured in A sizes, with A0 being the largest with an area
of . The A sizes have a special relationship between them. If we now do a bit of origami,
taking a sheet of A1 paper and then folding it in half (along its longest side), we get A2 paper.
Folding it in half again gives A3, and again gives A4 etc. However, the paper is designed so that
the proportions of each of the A sizes is the same - that is, each piece of paper has the same
shape.
We can pose the question of what proportion this is. Start with a piece of paper with
sides xand y with x the longest side. Now divide this in two to give another piece of paper with
sidesy and x/2 with now y being the longest side. This is illustrated to the right.
8 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

9. The proportions of the first piece of paper are and those of the second are or
. We want these two proportions to be equal. This means that
or
Another quadratic equation! Fortunately it's one we have already met. Solving it we find that
This result is easy for you to check. Just take a sheet of A4 (or A3 or A5) paper and measure the
sides. We can also work out the size of each sheet. The area of a piece of A0 paper is given
by
But we know that so we have another quadratic equation for the longest side of A0,
given by
This means that the longest side of A is given by (why?) and that of A
by . Check these on your own sheets of paper.
Paper used in the United States, called foolscap, has a different proportion. To see why, we
return to the Greeks and another quadratic equation. Having caused such grief, the quadratic
equation redeems itself in the search for the perfect proportions: a search that continues today
in the design of film sets, and can be seen in many aspects of nature.
9 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

10. 10 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

11. 11 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

12. After spending countless hours, days and night to finish this project without helps
from those lazy people surrounding me and also sacrificing my time for video
games and stuffs during thismid-yearschool break, there are several things that Ican
say. I’m going toexpress it through words anyways.
12 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R

13. After doing some researches, answering the given questions, drawing the
graphs and some problem solving, I saw that the usage of quadratic equation is
important in daily life. It is not just widely used in architecture such as
determining the area of a sculpture with curve(s) but we use quadratic
equation in our daily life as well. To be related, determining the area is important
as it can give the exact amount of the needed cost.
But, what is the use of quadratic equation in daily life of normal people like us?
In reality most people are not going to use the quadratic equation in daily life.
Having a firm understanding of the quadratic equation as with most maths helps
increasing logical thinking, critical thinking, and number sense.
We use quadratic equations to determine how to shape the mirror of, say a car
headlight, that is familiar, and where to put the light. If the light is at the focus, as
it should be, all light from the bulb will be reflected straight out.
As a conclusion, quadratic equation is a daily life essentiality. If there is no
quadratic equation, architect won’t be able to create such perfect buildings,
and light from bulbs in front of a car cannot shine brilliantly. Please trust no one
other than yourself and always have the faith that you are better than your any
school teacher (they did nothing to you and expect so much from you) or your
tuition teacher (what they want from you is just your money, even they cannot
help me to finish this project but for sure they will ask for the tuition fees monthly)
Thank you!
13 | A d d i t i o n a l M a t h e m a t i c s P r o j e c t W o r k 1 / 2 0 1 8 K U A L A L U M P U R