Pythagoras' Theorem

Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a2 + b2 = c2

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But it only works on right angled triangles.)

How Do I Use it?

Write it down as an equation:

Now you can use algebra to find any missing value.

Discovered by Khorshed

Before you do this, you need to know the following:
(a+b) (a-b) = a²- b<sup>2


Here's why.
(a+b) (a-b) = a+b (a-b)
                  = a (a-b) + b (a-b)
                  =</sup> a^{2 - ab + ba -} b^{2
                
                        =}  a^{2 -} b<sup>2


Therefore:
a2 - b2 = (a+b) (a-b)</sup>

Okay, let's start.
1.   Get two numbers that are the same.   Call them a and b.
 a = b

2.   Multiply by a
a<sup>2 = ba


3.   Subtract</sup> b^{2
a2 -  b2 = ba -} b<sup>2


4.  Swap the left side with (a+b) (a-b) and factorise the right side.
(a+b) (a-b) = b (a-b)


5.   Divide by (a-b)
(a+b) = b


6.    Open brackets.
a+b = b


7.   Subtract b from both sides.
a = 0


NOTE: You cannot divide by (a-b) because a-b = 0.   You cannot divide by 0, which it why it cannot be true.


Discovered by Richard</sup>

Circles

The circumference of a circle is the actual length around the circle which is equal to 360°. Pi (π) is the number needed to compute the circumference of the circle. 
π is equal to 3.14.
Pi is greek and has been around for over 2000 years!
In circles the AREA is equal to 3.14 (π) times the radius (r) to the power of 2. 
The formula looks like:
A= πr2.
In circles the circumference is 3.14 (π) times the Diameter. 
The formula looks like:
2πr or πd.

Example:

discovered by Andrew F

Graphing Lines

Line graphs provide an excellent way to map independent and dependent variables that are both quantitative. When both variables are quantitative, the line segment that connects two points on the graph expresses a slope, which can be interpreted visually relative to the slope of other lines or expressed as a precise mathematical formula. Scatter plots are similar to line graphs in that they start with mapping quantitative data points. The difference is that with a scatter plot, the decision is made that the individual points should not be connected directly together with a line but, instead express a trend. This trend can be seen directly through the distribution of points or with the addition of a regression line. A statistical tool used to mathematically express a trend in the data.



Example:


y=2x+1


discovered by Andrew F

Pascal

Pascal’s triangle is made up by adding the two numbers directly above e.g. the highlighted squares 3+1=4

Fibonacci Sequence
Try this: make a pattern by going up and then along, then add up the squares you will get the Fibonacci Sequence. 

(The Fibonacci Sequence starts "1, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

Symmetrical
And the triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.





discovered by Eddy

Fibonacci Sequences in Nature

Fibonacci numbers are made of the two numbers before that by adding them up, that is : Add the last two numbers to get the next.

The first 10 Fibonacci numbers are 1,2,3,5,8,13,21,34,55.

Fibonacci sequence actually has a very unique characteristic. Dividing a number with previous number allows us to get approximately similar result. However, the result will be the same after the 13th number inside the series: 1,618 that is Golden Ratio.

Interestingly, Fibonacci numbers can be found in nature, like plants!

There are many examples of how Fibonacci numbers are found in nature.

Fibonacci numbers can be found on plants like Romanesco broccoli, sunflowers and daisies.

For example, the distribution of seeds in sunflower is spiral. The seeds of the sunflower spiral outwards in both clockwise and counterclockwise directions from the center of the flower. The numbers of clockwise and counterclockwise spirals are two consecutive numbers in the Fibonacci sequence.

However, you have to be careful that even though they are plants which are spirals, it has to be the numbers adjacent to each other in the Fibonacci sequence.

The shells of the snails also follow the Fibonacci sequence. In the same way, the shells of the nautilus follow the same rule. The only difference between these two is that nautilus’ shells grow in a three-dimensional spiral, whereas snails’ shells grow in a two-dimensional spiral.

Pine cones are one of the well-known examples of Fibonacci sequence. All cones grow in spirals, starting from the base where the stalk was, and going round and round the sides until they reach the top.

Another notable example is human body. In human body, the ratio of the length of forearm to the length of the hand is equal to 1.618, that is, Golden Ratio. Another well-known example on human body is The ratio between the length and width of face

- Ratio of the distance between the lips and where the eyebrows meet to the length of nose

- Ratio of the length of mouth to the width of nose

- Ratio of the distance between the shoulder line and the top of the head to the head length

- Ratio of the distance between the navel and knee to the distance between the knee and the end of the foot

- Ratio of the distance between the finger tip and the elbow to the distance between the wrist and the elbow
The same sequence exists on the leaves of poplar, cherry, apple, plum, oak and linden trees.

discovered by Qi Le

Fibonacci

Fibonacci is a sequence that can be found in nature, and a lot of other things; from buildings, to sea shells.
Fibonacci is a sequence of numbers, which adds up from the previous number.
Example: 0,1,1,2,3,5,8,13,21,34, etc.
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.
Fibonacci is related to the golden spiral, which is a sequence of squares that gets bigger and bigger, in a spiral.

Here is an example of fibonacci found in a building:

This ancient greek temple first precicely inside the golden spiral. It's fibonacci pattern is: 1,1,2,3,5,8,13. You can see from the squares, representing the numbers; the larger the square, the bigger the number.

Plants have no way to know the Fibonacci numbers, but they develop in the most effective way. Thus, many plants have leaves arranged in a Fibonacci sequence layout around the stems. Some pine cones follow a layout on Fibonacci numbers, and also the sunflower.
Rings on the trunks of palm trees meet the Fibonacci numbers. The reason for this is to achieve an optimum, the maximum efficiency. Thus for example, following the Fibonacci sequence, the leaves of plants can be arranged so as to occupy a small space and obtain as much sun.
The idea in this leaves arrangement starts from the consideration of the golden angle of 222,5 degrees, divided by the entire 360 degrees will result in the number 0.61803398 ..., known as the Fibonacci sequence ratio.

discovered by Vincent

The Y Trick

This is a strategy I discovered when I did one of the Otago tests.

What you do is everytime you place 3 consecutive numbers on the ends of the sides, you move in towards the middle by one. You then repeat.

This only works when you use consecutive numbers, and when the number of bubbles/blocks on each side (not including the middle for each side) is a multiple of 3.

First you place the first number on the end of a side. Nest you put number 2 & 3 on the other two sides. If you went clockwise (as I did in the diagram above), you have to continue going in the same direction.

After that, you put Number 4 into the bubble closest to the end on the same line as Number 3. Put Number 5 in the bubble closest to the end on the same line as Number 1. Do the same thing to the Number 6 and place it on the same line as the Number 2. (You will notice you are still going clockwise).

Now repeat this for the Numbers 7, 8, & 9. Once you have put the numbers in, you will be lift with the middle bubble. That is where the biggest number, Number 10, goes.

At the end when you add up the numbers on each side, you should come out with the same total for each side.

This works for any group of consecutive numbers.

discovered by Janice

Mind Reading Trick

1. Think of a numeral between 1-10.
2. Multiply it by 2.
3. Add 10.
4. Divide by 2.
5. Subtract the number you first thought of.
6. You should end up with 1/2 the number you added in step 3.

NOTE: In step 3, you can change the number to any multiple of 2, over 10.

discovered by Cecil

A Quick Way to multiply with 11

Have you ever had trouble multiplying with big numbers like 11? Well now those days are over!

Let's take this equation - 223x11
Look at the 223 part. Working from left to right, take each number and add it to the number on left. The first number is 3 and as there is no number on the left, just write down 3.
= 3
The next number is 2, add it to the number on the left - 3, and you get 5.
= 53
The next number is 2, add it to the number on the left - 2, and you get 4.
= 453
The next number is 0 (there is no number), add it to the number on the left - 2 and you get 2.
= 2453
Therefore 223x11 = 2453

discovered by Sujeetha

Area of a Circle

1) Figure the length of the radius (diameter divided by 2) and then multiply it by itself (r squared).

2) The formula is:
Area equals pi multiplied by (r squared)

3) Example: Calculate the area of a circle with a diameter of 12cm. Divide the diameter by two, so 12 divided by 2 =6. Radius times radius, so 6 times 6, or 6 squared =36. The area equals pi times r squared, which in this case is 36, so 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609... times 36, or, to make it simpler, 3.14 times 36 =113.04 (approximately). So, the answer to this question is... 113.04cm squared (approximately)!!

discovered by Kyle

Multiply Up to 20X20 In Your Head

This is an strategy to multiply from 11 to 20 in your head easily.

Try this:

1. Take 15 x 13 for an example.

2. Always place the larger number of the two on top in your mind.

3. Then add the bigger number with the last digit of the smaller number. So add 15 + 3 = 18

4. Then multiply it by 10 to get 180.

5. Multiply the covered lower 3 from the 13 by the single digit above it the "5" (3x5= 15)

6. Add 180 + 15 = 195.

This works only with numbers from 11-20.

discovered by Helen

Congruent Triangles

Introducing the FANTASTIC:

Congruent Triangles

Congruent Triangles are just a fancy mathematical term for two completely equal triangles this need to be proven for it to work, there are formulas to prove congruent triangles here are them: SSS SAS ASA HL.

Example:

If AB=A'B' and ∠BAC=∠B'A'C' (∠stands for angle)and AC=A'C'

Are triangle A and B congruent Triangles? 

Answer:

Because of the information given above the SAS works for this triangle because S stands for side and A stands for angle. Side Angle Side look at the triangles above AB and A'B' are a side and so are AC and A'C', with the angles BAC and B'A'C' tucked in the middle (that is important not to get any other angles)

HL is a little bit more complicated here is an example:

CB=C'B' AB=A'B'

Answers: The answer to this we can use the formula HL the H is hypotenuse

which is an angle that doesn't touch the right angle and the L stands for leg which is a side that touches the right angle, if a hypotenuse and a leg is equal in right angle triangles then they are congruent triangles.

I have just covered a tiny bit to solving congruent triangle problems, there are more complicated problems that involves other strategies and formulae. Hope you get the basic understanding of what congruent triangles are!

discovered by Ben Z

Division Rules

1: Any number is divisible by one.

2: Numbers that end with 0, 2, 4, 6, or 8 is divisible by two.

3: Add up the sum of the numbers digits, and if that is divisible by 3 then the number is divisible by 3.

4: If the last two digits in the number is divisible by four then the number is divisible by four.

5: Numbers that end with 5 and 0 can be divided by 5.

6: If the number is divisible by 2 and three, the number is divisible by 6.

7: Remove the last digit, double the last digit, then subtract the doubled number from the digits left. If the number is still too big, repeat the process.
eg. If you want to find out if 348 is divisible by 7 take out the last digit which is 8 then double it which become 16. Subtract 16 by 34 (the remaining digits) which gives you 18. Which means 348 is not divisible by 7.

8: If the last 3 digits of the number is divisible by 8 then the number is divisible by 8.

9: If the sum of the number’s digit is divisible by 9 then the number is divisible by nine.

10: Any number that ends with 0 is divisible by 10.

Discovered by Yiwen

A Little Mind Reading Trick

1.   Get a friend to write down any three digit number, but all the numbers must be different.  Tell them not to let you see what they write.
2.   Write the number again backwards.
3.   Minus the small number from the big number.
4.   Ask him/her what the first digit of the final answer is.
5.   If the first digit is a ... 1 - then the final answer they have is 198, 2 - then their final answers is 297, 3 - 396, 4 - 495, 5 - 594, 6 - 693, 7 - 792, 8 - 891, 9 - 99. e.g. 123 > 321, 321-123=198.   198 begins with a ` so their answer must be 198!

TIP: If the first number is 1-8, the second number is always 9.  The first and last number's sum is always 9.
e.g. 198 1+8=9

Amaze your friends!

discovered by Lucia

Welcome to the Mathex class of 2011

We are a new mathex group this year and we are eagerly making our new discoveries.

Follow our maths discoveries here – we hope you are as excited as we are.