P y t h a g o r a s ´ s s i s t e r

7 Habits of Highly Successful Teens

For teens, life is not a playground, it's a jungle. And, being the parent of a teenager isn't any walk in the park, either. In his book,The 7 Habits of Highly Effective Teens, author Sean Covey attempts to provide "a compass to help teens and their parents navigate the problems they encounter daily."

How will they deal with peer pressure? Motivation? Success or lack thereof? The life of a teenager is full of tough issues and life-changing decisions. As a parent, you are responsible to help them learn the principles and ethics that will help them to reach their goals and live a successful life.

While it's all well and good to tell kids how to live their lives, "teens watch what you do more than they listen to what you say," Covey says. So practice what you preach. Your example can be very influential.

Covey himself has done well by following a parent's example. His dad, Stephen Covey, wrote the book The 7 Habits of Highly Successful People, which sold over 15 million copies. Sean's a chip off the old block, and no slacker. His own book has rung in a more than respectable 2 million copies sold. Here are his seven habits, and some ideas for helping your teen understand and apply them:

Be Proactive

Being proactive is the key to unlocking the other habits. Help your teen take control and responsibility for her life. Proactive people understand that they are responsible for their own happiness or unhappiness. They don't blame others for their own actions or feelings.

Begin With the End in Mind

If teens aren't clear about where they want to end up in life, about their values, goals, and what they stand for, they will wander, waste time, and be tossed to and fro by the opinions of others. Help your teen create a personal mission statement which will act as a road map and direct and guide his decision-making process.

Put First Things First

This habit helps teens prioritize and manage their time so that they focus on and complete the most important things in their lives. Putting first things first also means learning to overcome fears and being strong during difficult times. It's living life according to what matters most.

Think Win-Win

Teens can learn to foster the belief that it is possible to create an atmosphere of win-win in every relationship. This habit encourages the idea that in any given discussion or situation both parties can arrive at a mutually beneficial solution. Your teen will learn to celebrate the accomplishments of others instead of being threatened by them.

Seek First to Understand, Then to be Understood

Because most people don't listen very well, one of the great frustrations in life is that many don't feel understood. This habit will ensure your teen learns the most important communication skill there is: active listening.

Synergize

Synergy is achieved when two or more people work together to create something better than either could alone. Through this habit, teens learn it doesn't have to be "your way" or "my way" but rather a better way, a higher way. Synergy allows teens to value differences and better appreciate others.

Sharpen the Saw

Teens should never get too busy living to take time to renew themselves. When a teen "sharpens the saw" she is keeping her personal self sharp so that she can better deal with life. It means regularly renewing and strengthening the four key dimensions of life – body, brain, heart, and soul.

Taken from: http://www.education.com/magazine/article/Ed_7_Habits_Successful/

TEENAGERS AND DRUGS

by F Ruric Vogel PhD
Counselling Psychologist

Realiza un KWL

View in: http://www.be.up.co.za/UserFiles/Forensic-Division-report_Teens_Eng.pdf

Read the text and write an abstract (between 150-250 words)

THE NEW WORLD OF MATHEMATICS

by George A. W. Boehm

Never before have so many people applied such abstract mathematics to so great a variety of problems. To meet the demands of industry, technology, and other sciences, mathematicians have had to invent new branches of mathematics and expand old ones. They have built a superstructure of fresh ideas that people trained in the classical branches of the subject would hardly recognise as mathematics at all.

Applied mathematicians have been grappling successfully with the world's problems at a time, curiously enough, when pure mathematicians seem almost to have lost touch with the real world. Mathematics has always been abstract, but pure mathematicians are pushing abstraction to new limits. To them mathematics is an art they pursue for art's sake, and they don't much care whether it will ever have any practical use.

Yet the very abstractness of mathematics makes it useful. By applying its concepts to worldly problems the mathematician can often brush away the obscuring details and reveal simple patterns. Celestial mechanics, for example, enables astronomers to calculate the positions of the planets at any time in the past or future and to predict the comings and goings of comets. Now this ancient and abstruse branch of mathematics has suddenly become impressively practical for calculating orbits of earth satellites.

Even mathematical puzzles may have important applications. Mathematicians are still trying to find a general rule for calculating the number of ways a particle can travel from one corner of a rectangular net to another corner without

Now that they have electronic computers, mathematicians are solving problems they would not have dared tackle a few years ago. In a matter of minutes they can get an answer that previously would have required months or even years of calculation. In designing computers and programming them to carry out instructions, furthermore, mathematicians have had to develop new techniques. While computers have as yet contributed little to pure mathematical theory, they have been used to test certain relationships among numbers. It now seems possible that a computer some day will discover and prove a brand-new mathematical theorem.

(from The New World of Mathematics, Faber and Faber, London, 1959)

Taken from: http://www.uefap.com/reading/exercise/tewufs/tewufs15.htm

Phi, Pi and the Great Pyramid

Phi, Pi and the
Great Pyramid of Egypt at Giza

The Great Pyramid of Egypt is based on Golden Ratio proportions

There is still some debate as to whether the Great Pyramid of Giza in Egypt, built around 2560 BC, was constructed with dimensions based on phi, the golden ratio. Its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with certainty.

There is compelling evidence, however, that the design of the pyramid embodied these foundations of mathematics and geometry:

§ Phi, the Golden Ratio that appears throughout nature.

§ Pi, the circumference of a circle in relation to its diameter.

§ The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c².

First, phi is the only number which has the mathematical property of its square being one more than itself:

Φ + 1 = Φ²

1.618… + 1 = 2.618…

By applying the above Pythagorean equation to this, we can construct a right triangle, of sides a, b and c, or in this case a Golden Triangle of sides √Φ, 1 and Φ, which looks like this:

This creates a pyramid with a base width of 2 (i.e., two triangles above placed back-to-back) and a height of the square root of Phi, 1.272. The ratio of the height to the base is 0.636.

The Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet). This also creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy. If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.53567, so the difference of only 0.3567 meters appears to be just a measurement or rounding difference.

The Great Pyramid has a surface ratio to base ratio of Phi, the Golden Ratio

A pyramid based on a golden triangle would have other interesting properties. The surface area of the four sides would be a golden ratio of the surface area of the base. The area of each trianglular side is the base x height / 2, or 2 x Φ/2 or Φ. The surface area of the base is 2 x 2, or 4. So four sides is 4 x Φ / 4, or Φ for the ratio of sides to base.

The Great Pyramid also has a relationship to Pi

There is another interesting aspect of this pyramid. Construct a circle with a circumference is 8, the same as the perimeter of this pyramid with its base width of 2. Then fold the arc of the semi-circle at a right angle, as illustrated below in “Revelation of the Pyramids”. The height of the semi-circle will be the radius of the circle, which is 8/pi/2 or 1.273.

This is less than 1/10th of a percent different than the height of 1.272 computed above using the Golden Triangle. Applying this to the 146.5 meter height of the pyramid would result in a difference in height between the two methods of only 0.14 meters (5.5 inches).

Its near perfect alignment to due north shows that little was left to chance

Some say that the relationships of the Great Pyramid’s dimensions to phi and pi either do not exist or happened by chance. Would a civilization with the technological skill and knowledge to align the pyramid to within 1/15th of a degree to true north leave the dimensions of the pyramid to chance? If they didn’t intend the precise 51.83 degree angle of a golden triangle, why would they have not used another simpler angle found in divisions of a circle such as 30, 45, 54 or 60 degrees? If the dimensions of the pyramid were not based on both phi and pi, would it not be most reasonable to assume that phi was used since it is based on the visible base of the pyramid and not an invisible circle with the same circumference as that base?

Other possibilities for Phi and Pi relationships

Even if the Egyptians were using numbers that they understood to be the circumference of the circle to its diameter and the golden ratio that appeared in nature, it’s difficult to know if they truly understood the actual decimal representations of pi and phi as we understand them now. Since references to phi don’t appear in the historical record until the time of the Greeks hundreds of years later, some contend that the Egyptians did not have this knowledge and instead used integer approximations that achieved the same relationships and results in the design.
A rather amazing mathematical fact is that pi and the square root of phi can be approximated with a high degree of accuracy using simple integers. Pi can be approximated as 22/7, resulting in a repeating decimal number 3.142857142857… which is different from Pi by only 4/100′s of a percent. The square root of Phi can be approximatey by 14/11, resulting in a repeating decimal number 1.2727…, which is different from Phi by less than 6/100′s of a percent. That means that Phi can be approximated as 256/121.

The Great Pyramid could thus have been based on 22/7 or 14/11 in the geometry shown about. Even if the Egyptians only understood pi and/or phi through their integer approximations, the fact that the pyramid uses them shows that there was likely some understanding and intent of their mathematical importance in their application. It’s possible though that the pyramid dimensions could have been intended to represent only one of these numbers, either pi or phi, and the mathematics would have included the other automatically. We really don’t know with certainty how the pyramid was designed as this knowledge could have existed and then been lost. The builders of such incredible architecture may have had far greater knowledge and sophistication than we may know, and it’s possible that both pi and phi as we understand them today could have been the driving factors in the design of the pyramid.

Construct your own pyramid to the same proportions as the Great Pyramid

Use the template below in gif or pdf format:

Taken from: http://www.goldennumber.net/phi-pi-great-pyramid-egypt/

ABSTRACT

La más antigua de las 7 maravillas del mundo:

En el maravilloso mundo del antiguo Egipto encontramos la gran pirámide de Giza (Gizeh) construida alrededor del año 2560 a.c con 2.300.000 bloques de piedra, en la cual ¿TODO ES ORO?

En base a las investigaciones de arqueólogos de todos los tiempos, se puede conjeturar que la gran pirámide fue construida con un diseño que encarna la proporción áurea, que se rige por el número phi=

, llamado habitualmente número de oro, ya que está presente en toda la naturaleza, el cuerpo humano, la música y otras artes.
También se dice que en la pirámide está presente el número PI y el Teorema de Pitágoras.

Muchos dicen que las dimensiones proporcionales a PHI o PI, o no existen, o que la construcción fue por casualidad, pero esta última idea se refuta dando infinidad de pruebas donde se ejemplifica la presencia del número de oro en la pirámide.

Otros afirman que los egipcios al no tener instrumentos que permitieran construir basándose en el número PI y el número de oro, construyeron valiéndose de aproximaciones racionales:

Los constructores de esta increíble obra de arte pueden haber tenido mucho conocimiento y tecnología, pero nos preguntamos: ¿cómo adquirieron ese conocimiento y qué pasó con él? ¿Se perdió?

También te invitamos a construir tu propia pirámide de Giza a escala, que conserva las proporciones originales.

DONALD IN MATHMAGIC LAND

Taken from: https://www.youtube.com/watch?feature=player_embedded&v=cvstoNWxJKU

Here the subtitles:

0:59 - 1:01

Very strange.

¶
1:13 - 1:17

Huh, that's an odd-looking creature
1:28 - 1:32

What kind of a crazy place is this?
1:46 - 1:49

Well, what do you know? Square roots!
1:51 - 2:00

Pi is equal to 3.141592653589747 etc. etc. etc.
2:02 - 2:05

Hello?
2:06 - 2:08

Hello, Donald.
2:08 - 2:11

That's me! Where am I?!
2:11 - 2:14

Mathmagic land.
2:14 - 2:17

Mathmagic land? Never heard of it.
2:18 - 2:21

It's the land of great adventure.
2:21 - 2:23

Well, who are you?
2:23 - 2:27

I'm a spirit, the true spirit of adventure.
2:27 - 2:30

That's for me! What's next?
2:30 - 2:33

A journey through the wonderland of mathematics.
2:33 - 2:37

Mathematics? That's for eggheads!
2:37 - 2:39

Eggheads? Now hold on, Donald.
2:39 - 2:42

You like music don't you?
2:42 - 2:43

Yeah.
2:43 - 2:46

Well, without eggheads, there would be no music.
2:46 - 2:47

Bah.
2:48 - 2:55

Come on, let's go to ancient Greece, to the time of Pythagoras, the master egghead of them all.
2:55 - 2:56

Pythagoras?
2:56 - 2:59

The father of mathematics and music.
2:59 - 3:00

Mathematics and music?
3:00 - 3:05

Ahh, you'll find mathematics in the darndest places.
3:05 - 3:06

Watch
3:07 - 3:09

First we'll need a string
3:09 - 3:10

Hey!
3:10 - 3:13

Stretch it good and tight; pluck it!
3:14 - 3:17

Now divide in half. Pluck again.
3:18 - 3:22

You see? It's the same tone, one octave higher.
3:22 - 3:25

Now divide the next section.
3:25 - 3:27

And the next.
3:27 - 3:32

Pythagoras discovered the octave had a ratio of two to one.
3:32 - 3:39

With simple fractions, he got this [major triad]
3:39 - 3:48

And from this harmony in numbers, developed the musical scale of today. [major scale]

Taken from: http://www.amara.org/es/videos/MPL5usrEqq0a/en/18983/

ANSWER MULTIPLE CHOICE QUESTIONS:

1) ¿Dónde está Donald?
- En en planeta de las calculadoras
- En la tierra de las matemáticas
- A 3,14159265 Km de la Tierra

2) Según Donald, la matemática es para...
- Los intelectuales
- Los huevos
- Los locos

3) Sin matemática no habría ....
- Leyes
- Música
- Ropa

4) ¿Quién es el padre de la matemática y la música?
- Thales de Mileto
- Eratóstenes de Cirene
- Pitágoras de Samos

5) ¿Qué proporción hay en una octava musical?
- 2 a 1
- 8 a 1
- 4/3

Second Degree Equation

Solving Quadratic (2nd degree) Equations

There are 3 widely used methods for solving quadratic equations. A quadratic (or second-degree) equation is an equation in which the variable has an exponent of 2.

The standard form of a quadratic equation is
.

The three methods used to solve quadratic equations are: 1) factoring, 2) the square root property, and 3) the quadratic formula. Quadratic equations generally have 2 solutions.

1) Factoring is one method used to solve second-degree and larger equations. First the equation must be written in standard form. This means that the polynomial must be in descending form and set equal to zero. Next, you must factor the polynomial. (You may want to review factoring.) Once the polynomial is factored, set any factor which contains a variable equal to zero and solve (using isolation) for the variable. Check your answer in the original equation. Every quadratic equation has two solutions, although in the case of a perfect square trinomial both of the solutions are the same.

If the equation is factored and set equal to zero as in the example,

, then

and

.

Solve the equation,

. First, you must get the equation in standard form.
This means that you must add 16 to both sides so that the –16 is removed from the left side of the equation and the equation will then be equal to zero …

…

. Now you are ready to factor,

. Set the factors equal to zero. Since

is repeated twice as a factor, there are two solutions, but they are both the same. Thus,

is the only “unique” solution to this problem. This is a perfect square trinomial, which factored into the square of a binomial.

If the problem has a degree of three (in other words the variable in the equation is cubed) then you will find three solutions. Example:

.
First, factor the GCF of +2x from each of the terms of the polynomial …

.
Next factor,

. We now have three factors which contain variables, therefore,

and

. This example of a third degree equation has three solutions

2) The square root property involves taking the square roots of both sides of an equation. Before taking the square root of each side, you must isolate the term that contains the squared variable. Once this squared-variable term is fully isolated, you will take the square root of both sides and solve for the variable. We now introduce the possibility of two roots for every square root, one positive and one negative. Place a sign in front of the side containing the constant before you take the square root of that side.

Example 1:

… the squared-variable term is isolated, so we will take the square root of
each side

                    … notice the use of the  sign, this will give us both a positive and a
                                               negative root
         … simplify both sides of the equation, here x is isolated so we have
                                               solved this equation

Example 2:

                  … again the squared-variable term is isolated, so we will take the
                                             square root of each side
         … again don’t forget the  sign, now simplify the radicals

… this time p is not fully isolated, also notice that 4 are
rational numbers, which means …

and

Example 3:

… squared term is not isolated, add 1 to each side before
beginning

… now take the square root of both sides

… simplify radicals

… radical containing the constant cannot be simplified, solve for the
variable

           … notice the placement of the –1 before the radical on the
                                                            right-hand side, these numbers may not be combined since
                                                           –1 is a rational number and  are irrational numbers

In each of the first 3 examples involving the square root property, notice that there were no first-degree terms. These equations although they are quadratic in nature, have the form . To solve a quadratic equation that contains a first-degree term using the square root property would involve completing the square which is another "trick" that will be explained in another lesson.

3) The third method for solving quadratic equations described uses the quadratic formula ó Bhaskara's Formula
This formula is . If you notice, the right hand side has variables a, b, and c.
These variables are the coefficients of the terms of the quadratic equation. (Remember the standard form is .)

Example 4:

… first, the equation must be in standard form
…move terms to one side

… identify a = 1, b = –8, c = –9

… use the quadratic formula, substitute values

… simplify radical

… solve for r

and … quadratic equations have 2 solutions

Adapted from: http://mbaker.columbiastate.edu/AlgebraTips_pages/Solving%20Quadratic%20Equations.htm

RESPONDE:

Skimming:
1) ¿Cuál es el tema principal del texto?
2) ¿Cuál es la forma estándar o general de una ecuación de segundo grado?

Scanning:
3) Según el texto ¿Cuántos métodos hay para resolver ecuaciones de segundo grado?
4) Generalmente, ¿cuántas soluciones tiene una ecuación de segundo grado?

Reading in detail:
5) Utilizando la fórmula de Bhaskara resuelve: x²-x-2=0

6) ¿Qué método es más conveniente (más rápido) para resolver la ecuación x²+2x=0?