"Wasan" geometry
or traditional Japanese mathematics
flourished under the Tokugawa
shogunate around the Edo period
and expressed itself in a unique
way through mathematical votive
pictures called San Gaku, written
in formal Sino-Japanese language
and displayed in temples and
public places.
Beautiful wooden tablets of many
sizes and shapes would outline
in pleasant and colorful way mathematical
problems mostly left for the viewer
to solve. This tradition slowly
disappeared and today less than
a thousand Sangaku have survived
abandon or destruction.
If people do not believe that maths is simple, it is only because they do not realize how complicated life is.
Tuesday 20 March 2012
Sunday 18 March 2012
How to Study Math: Learn From Your Errors
This is probably one of the more important sections here and also one of the most over looked. Learning from your mistakes can only help you.
Review Homework. When you get your homework back review it looking for errors that you made.Review Exams. Do the same thing with exams.
Understand the Error. When you find an error in your homework or exams try to understand what the error is and just what you did wrong. Look for something about the error that you can remember to help you to avoid making it again. Get Help. If you can find the error and/or don’t understand why it was an error then get help. Ask the instructor, your tutor, or a classmate who got the problem correct.
Rushed Errors. If you find yourself continually making silly arithmetic or notational errors then slow down when you are working the problems. Most of these types of errors happen because students get in a hurry and don’t pay attention to what they are doing.
Repeated Errors. If you find yourself continually making errors on one particular type of problem then you probably don’t have a really good grasp of the concept behind that type of problem. Go back and find more examples and really try to understand just what you are doing wrong or don’t understand.
Keep a List of Errors. Put errors that you keep making in a “list of errors”. With each error write down the correct method/solution. Review the list after you complete a problem and see if you’ve made any of your “common” error
Introduction to Algebra
x | + | 5 | = | 12 |
Start with: | x + 5 = 12 |
What you are aiming for is an answer like "x = ...", and the plus 5 is in the way of that! If you subtract 5 you can cancel out the plus 5 (because 5-5=0) |
|
So, let us have a go at subtracting 5 from both sides: | x+5 -5 = 12 -5 |
A little arithmetic (5-5 = 0 and 12-5 = 7) becomes: | x+0 = 7 |
Which is just: | x = 7 |
Solved! | |
(Quick Check: 7+5=12) |
Ships
The Puzzle: There are 5 ships in a port:
1. The Greek ship leaves at six and carries coffee.
2. The Ship in the middle has a black chimney.
3. The English ship leaves at nine.
4. The French ship with blue chimney is to the left of a ship that carries coffee.
5. To the right of the ship carrying cocoa is a ship going to Marseille.
6. The Brazilian ship is heading for Manila.
7. Next to the ship carrying rice is a ship with a green chimney.
8. A ship going to Genoa leaves at five.
9. The Spanish ship leaves at seven and is to the right of the ship going to Marseille.
10. The ship with a red chimney goes to Hamburg.
11. Next to the ship leaving at seven is a ship with a white chimney.
12. The ship on the border carries corn.
13. The ship with a black chimney leaves at eight.
14. The ship carrying corn is anchored next to the ship carrying rice.
15. The ship to Hamburg leaves at six.
Which ship goes to Port Said? Which ship carries tea?
The Solution . . .
The Spanish ship goes to Port Said and the French ship carries tea. However, tea can be carried by the Brazilian ship, too. If you understood position 'to the right' to mean anywhere on the right side from the given point (not only right next to).
|
Wasan geometry
During
the greater part of the Edo
period (1603-1867) Japan was
almost completely cut off from
the western world. Books on
mathematics, if they entered
Japan at all, must have been
scarce, and yet, during this
long period of isolation people
of all social classes, from
farmers to samurai, produced
theorems in Euclidean geometry
which are remarkably different
from those produced in the
west during the centuries of
schism, and sometimes anticipated
these theorems by many years..
These theorems were not published in books, but appeared as beautifully coloured drawings on wooden tablets which were hung under the roof in the precincts of a shrine or temple.
These theorems were not published in books, but appeared as beautifully coloured drawings on wooden tablets which were hung under the roof in the precincts of a shrine or temple.
The
following is an extended series
of variations on Sangaku problems
- a tribute to mathematicians and
artists alike who graced the walls
of many temples in eighteenth century
Japan. This work would not have
been made possible without the interest
and dedication of mathematicians Dr. Fukagawa,
Pedoe, Rothman, Kotera and
the many other who provided inspiration,
guidance and support throughout
this adventure.
The artworks featured
in the galleries are printed 20'x20',
220g., archival fine art paper.
and available for purchase, individually
or in series. Contact the artist
for more information.
Math in Smartphone
Are you bored of doing the same math question with the same method of doing it.?Well lets join us and play Math Maniac.Math Maniac is a very fun and addictive game.You can find this app on your smartphone.
The goal of Math Maniac is simple.In 10 seconds,player have to combine numbers to equal the number at in the left bottom corner.
Hope this app will help you think that math is not just in your test paper but also in your communication device.
Saturday 17 March 2012
Kirigami and Mathematics
Kirigami is an art which is a combination of origami i.e. paper folding and paper cutting.In Japan the word Kirigami has been in use for a long time because "kiru" means to cut and "gami" means paper.Making paper snowflakes is an example of Kirigami.
You can explore many hidden symmetry patterns and write on mathematical interpretations.This way we would not only start loving mathematics but surely ask for more of it.
Fibonacci Sequence
It was first observed by the Italian mathematician Leonardo Fibonacci in 1202
He was investigating how fast rabbits could breed under ideal circumstances.He made the following assumptions:
Begin with one male and one female rabbits.Rabbits can mate at the age of one month,so by the end of the second month,each female can produce another pair of rabbits.The rabbits never die.The female produce one male and one female every month.He calculate how many pair of rabbits would be produce in one year.
This flower is one of the art of Fibonacci...
Quadratic Equations
3 Basic Techniques in Solving Quadratic Equation Questions
In this chapter we will learn 3 most basic techniques on how to:- Solve the quadratic equations
- Form a quadratic equation
- Determine the conditions for the type of roots.
Generally, is the quadratic equation, expressed in the general form of , where a=1, b=- 6 and c=5. The root is the value of x that can solve the equations.
quadratic equation only has two roots.
Example1: What are the roots of ?
Answer: The value of 1 and 5 are the roots of the quadratic equation, because you will get zero when substitute 1 or 5 in the equation. We will further discuss on how to solve the quadratic equation and find out the roots later.
1) Solve the quadratic equations
There are many ways we can use to solve quadratic equations such as using:1) substitution,
2) inspection,
3) trial and improvement method,
4) factorization,
5) completing the square and
6) Quadratic formula.
However, we will only focus on the last three methods as there are the most commonly use methods to solve a quadratic equation in the SPM questions. Let’s move on!
Factorization
Factorization is the decomposition of a number into the product of the other numbers, example, 12 could be factored into 3 x 4, 2 x 6, and 1 x 12.Example 2: Solve using factorization.
Answer: We can factor the number 12 into 4 x 3. Remember, always think of the factors which can be added up to the get the middle value (3+4 = 7), refer factorization table below,
So we will get ( x + 3 )( x + 4 ) = 0,
x + 3 = 0 or x + 4 = 0
x = – 3 or x = – 4
Example 3: Solve using factorization.
Answer: Rearrange the equation in the form of
So we will get (4x – 3)(2x – 1)=0,
4x – 3 = 0 or 2x – 1 = 0
x = or x =
Completing the square
Example 4: Solve the following equation by using completing the square method.Quadratic formula
Normally when do you need to use this formula?
1) The exam question requested to do so!
2) The quadratic equation cannot be factorized.
3) The figure of a, b, and c of the equation are too large and hard to factorized.
Example 5: Solve using quadratic formula.
2) Form a quadratic equation
How do you form a quadratic equation if the roots of the equation are 1 and 2? Well, we can do the work out like this using the reverse method:We can assume:
x = 1 or x = 2
x – 1 = 0 or x – 2 = 0
(x-1)(x-2)=0
x2-2x-x+2=0
x2-3x+2=0
So the quadratic equation is x2 – 3x + 2=0. This is the most basic technique to form up a quadratic equation.
Let’s assume we have the roots of and :
In other words, we can form up the equation using the sum of roots (SOR) and product of roots (POR). If the roots are 1 and 2,
SOR = 1+2
= 3
POR = 1 x 2
=2
Sometime we need to determine the SOR and POR from a given quadratic equation in order to find a new equation from a given new roots. In general form,
Let’s look at the example below on how the concept above can help us solve the question.
Example 6: Given that and are the roots of , form a quadratic equation with the roots of ( - 5 ) and ( - 5 ).
3) Determine the conditions for the type of roots
Refer back to example 2, we know that has two different roots (-3 and -4) by solving using factorization method. However, how are we going to determine the types of roots of without solving the equation? The trick is we can use .is called a discriminant. Remember, when the value is greater than 0, we have 2 different roots, when it is 0, we have 2 equal roots, and when it is less than 0, we have no roots.
From the quadratic equation, , , we have 2 different roots since the discriminant is greater than zero. Refer table below.
Example 7: A quadratic equation has two equal roots. Find the possible values of h.
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