Sunday 18 March 2012

How to Study Math: Learn From Your Errors


finger-in-socket-1 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

The Calculator


Maths Pic


Math(S)


Maths Song


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


ShipsThe 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).

French5.00teablueGenoa
Greek6.00coffeeredHamburg
Brazilian8.00cocoablackManila
English9.00ricewhiteMarseille
Spanish7.00corngreenPort Said

A Level


Wasan geometry

pic
     
    "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.
   In Dr. Fukagawa and D. Pedoe own words:
   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.

pic
     
    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.


Just having fun


Creative Thinking


Hahahaha


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:
  1. Solve the quadratic equations
  2. Form a quadratic equation
  3. Determine the conditions for the type of roots.

Generally, {{x}^{2}}-6x+5=0 is the quadratic equation, expressed in the general form of a{{x}^{2}}+bx+c=0, 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 {{x}^{2}}-6x+5=0?
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  {{x}^{2}}+7x+12=0 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,

factorization table
So we will get ( x + 3 )( x + 4 ) = 0,
x + 3 = 0    or   x + 4 = 0
x = – 3  or         x  = – 4
Example 3: Solve 10x-3=8x^2 using factorization.
Answer: Rearrange the equation in the form of 
a{{x}^{2}}+bx+c=0

So we will get (4x – 3)(2x – 1)=0,
4x – 3 = 0    or   2x – 1 = 0
x = \frac{3}{4}   or         x  = \frac{1}{2}

Completing the square

Example 4: Solve the following equation by using completing the square method. 
example 4
example 4b

Quadratic formula

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 a{{x}^{2}}+bx+c=0 are too large and hard to factorized.
Example 5: Solve (x-2)=6x(x+3)  using quadratic formula.
example 5 answer

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 \alpha and \beta:
forming equation explanation
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
{{x}^{2}}-(\text{sum of roots)}x+(\text{product of roots)}=0
{{x}^{2}}-3x+2=0
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,
forming equation explanation2
Let’s look at the example below on how the concept above can help us solve the question.
Example 6: Given that \alpha and \beta are the roots of 5{{x}^{2}}-2x-2=0 , form a quadratic equation a{{x}^{2}}+bx+c=0 with the roots of (\alpha - 5 ) and ( \beta - 5 ). 
example 6 answer part1
example 6 ans part2

3)     Determine the conditions for the type of roots

Refer back to example 2, we know that {{x}^{2}}+7x+12=0 has two different roots (-3 and -4) by solving using factorization method. However, how are we going to determine the types of roots of {{x}^{2}}+7x+12=0 without solving the equation? The trick is we can use {{b}^{2}}-4ac.
{{b}^{2}}-4ac 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, {{x}^{2}}+7x+12=0 , a=1,b=7,c=12,\text{ so }{{7}^{2}}-4(1)(12)=1 , we have 2 different roots since the discriminant is greater than zero. Refer table below. 

discriminant table
Example 7: A quadratic equation {{x}^{2}}+2hx+4=x has two equal roots. Find the possible values of h.
example 7 answer