PARTE 1 1. Operar con fracciones y números decimales. 2. Conocer las propiedades de las potencias y aplicarlas para simplificar operaciones. Resolver problemas. 3. Transformar números decimales en fracciones y viceversa. 4. Operar con polinomios 5. Conocer las propiedades de las progresiones aritméticas y geométricas
PARTE 2 6. Resolver ecuaciones de primer y segundo grado. Resolver sistemas de ecuaciones lineales. 7. Plantear y resolver problemas con ecuaciones. 8. Conocer las relaciones métricas de las cónicas. 9. Aplicar los teoremas de Tales y Pitágoras para encontrar medidas que se desconocen en cálculos de volúmenes. PARTE 3 10. Conocer las característitcas de una función. 11. Interpretar gráficas de funciones. 12. Calcular la ecuación de la recta y representarla. 13. Elaborar tablas y gráficos estadísticos. 14. Encontrar los parámetros de centralización y dispersión de una distribución estadística. 15. Calcular probabilidades aplicando la regla de Laplace.
Probability does not tell us exactly what will happen, it is just a guide
Example: toss a coin 100 times, how many Heads will come up?
Probability says that heads have a ½ chance, so we can expect 50 Heads.
But when we actually try it we might get 48 heads, or 55
heads ... or anything really, but in most cases it will be a number near
50.
OK, why 108 throws? Well, only 36 throws might not give good
results, 360 throws would be great, but take a long time, so 108
(which is 3 lots of 36) seems just right.
So let's multiply all these numbers by 3 to match our total of 108:
Total
Score
Number of
Ways to Get
Score
2
3
3
6
4
9
5
12
6
15
7
18
8
15
9
12
10
9
11
6
12
3
Total = 108
Those are the theoretical values, as opposed to the experimental ones you got from your experiment.
The theoretical values look like this in a bar graph:
How do these theoretical results compare with your experimental results?
This graph and your graph should be quite similar, but they are not likely to be exactly the same, as your experiment relied on chance, and the number of times you did it was fairly small.
If you did the experiment a very large number of times, you should get results much closer to the theoretical ones.
And, by the way, we've now answered the question from near the beginning of the experiment:
What is the most likely total score?
7 has the highest bar, so 7 is the most likely total score.
Los riesgos de Internet y de las redes sociales para los niños
¿Cuáles son los consejos fundamentales que deben seguir los niños en las redes sociales? Algunas de las pautas a seguir por el código son: -
Cuando me haga un perfil social, configuraré adecuadamemnte la
privacidad para que el contenido que publique sólo pueda ser visto por
mis amigos. - Tendré en cuenta que las personas que conozco por internet son desconocidos en la vida real, NO SON MIS AMIGOS. - No publicaré información personal como mi teléfono, dirección etc, en Internet -
Cuando se publica una foto en Internet, se pierde el control sobre su
difusión y duración. Aunque después se borre, no desaparece de la Red. - En Internet hay que comportarse con respeto y educación. No hagas a otros lo que no quieras que te hagan a tí. - Denuncia a las personas o las acciones que perjudiquen a los demás. Si concoes alguien que esté siento acosado, DENÚNCIALO. -
Si adquieres un teléfono movil con internet, desconecta la opción de
geolocalización, asi cuando publiques en tu muro desde el movil, nadie
sabrá dónde estás. - Guardaré las conversaciones del chat, ya que te puede ser útil en caso que haya algún problema. -
Si me ocurre algo que no me haga sentir bien en Internet, se lo
cumunicaré a mis padres. Tengo que saber decir NO a las cosas que no
quiera hacer y contar con mis padres que son mis mejores aliados. - El mejor filtro para Internet es el conocimiento.
¿Cuáles son los riesgos para los niños en Internet y en las redes sociales? De esas nuevas formas de comunicarse han surgido nuevos riesgos o formas de acoso con un impacto fatal como el grooming, una práctica a través de la cual un adulto se gana la confianza de un menor con un propósito sexual. El sexting
o intercambio de fotografías o vídeos con contenido erótico entre los
propios jóvenes con las que luego llegan a extorsionarse causando daños
psicológicos importantes. Por otra parte, existe el conocido como ciberbullying,
o acoso entre menores que en Internet por su carácter global, supone un
alcance que puede llegar a generar mucho daños no sólo al menos sino a
su familia y amigos.
¿A partir de qué edad entran los niños en las redes sociales? En España, la edad mínima para acceder a una red social, excepto a las específicas para menores, es de 14 años.
Actualmente, Tuenti está trabajando con los menores 14 que quieren
acceder a la red social a través del consentimiento paterno. Por ello,
la Red Social solicita el permiso paterno antes de permitir que se lleve
a cabo el registro del perfil.
¿A qué edad los niños disponen ya de su propio teléfono móvil? El desarrollo de la telefonía móvil y de los conocidos como Smartphones permite que los jóvenes puedan tener acceso a Internet en su bolsillo.
Según el Estudio sobre seguridad y privacidad en el uso de los
servicios móviles por los menores españoles, elaborado en 2010 por
INTECO y Orange, la edad media de inicio en la telefonía móvil por parte de los menores españoles se sitúa entre los 10 y los 12 años. Además, la mayoría de los menores accede a internet en su casa o en la calle.
¿Cómo advertirles de los peligros de esta nueva forma de comunicación?Los
pequeños tienen que valorar la comunicación física por encima de la que
pueden entablar a través de las nuevas tecnologías. Las ventajas de
conocer al interlocutor frente a los riesgos de no hacerlo. Su educación es la base de un futuro Internet más seguro. Así, es importante decirles que tienen que tener en cuenta que hablar habitualmente con un desconocido en internet, no le convierte en conocido. La
adicción a las nuevas tecnologías es un problema en alza que los padres
no deben descuidar y darle la importancia que tiene. El tiempo que
pasen sus hijos en Internet tiene que estar limitado, según sean las
motivaciones de uso de la red y la edad.
When
working with equations that have two (1)__________, the coordinate plane is an
important tool. It's a way to draw pictures of equations that makes them easier
to understand.
To
create a coordinate plane, start with a sheet of graph paper. Next, draw a
(2)__________ line. This line is called the x-axis and is used to locate values
of x. To show that the axis actually goes on forever in both directions, use
small arrowheads at each end of the line. Mark off a number line with
(3)__________ in the center, positive numbers to the (4)__________, and
negative numbers to the (5)__________.
Next
draw a (6)__________ line that intersects the x axis at zero. This line is
called the y-axis and is used to locate the values of y. Mark off a number line
with zero in the center, (7)__________ numbers going upwards, and (8)__________
numbers going downwards. The point where the x and y axes intersect is called
the origin. The origin is located at (9)__________ on the x axis and zero on
the y axis.
Locating
Points Using Ordered Pairs
We
can locate any point on the coordinate plane using an ordered pair of numbers.
We call the ordered pair the coordinates of the point. The coordinates of a
point are called an ordered pair because the order of the two (10)__________ is
important.
The
first number in the ordered pair is the (11)__________ coordinate. It describes
the number of units to the left or right of the origin. The second number in
the ordered pair is the (12)__________ coordinate. It describes the number of
units above or below the origin. To plot a point, start at the (13)__________
and count along the x axis until you reach the x coordinate, count right for
positive numbers, left for negative. Then count up or down the number of the y
coordinate (up for (14)__________, down for (15)__________.)
Quadrants
To
make it easy to talk about where on the coordinate plane a point is, we divide
the coordinate plane into four sections called quadrants.
Points
in Quadrant 1 have positive x and positive y coordinates.
Points
in Quadrant 2 have (16)__________ x but (17)__________ y coordinates.
Points
in Quadrant 3 have (18)__________ x and (19)__________ y coordinates.
Points
in Quadrant 4 have (20)__________ x but (21)__________ y coordinates.
2.Match each word with its translation in
Spanish.
COORDINATE PLANEEJES
CARTESIANOS
DEPENDENT VARIABLEVARIABLE
DEPENDIENTE
INDEPENDENT VARIABLEFUNCIÓN
FUNCTIONPAR
ORDENADO
SET
OF POINTSVARIABLE
INDEPENDIENTE
3.Complete the Crossword.
Across:
1
The steepness or slant of a line
3
The horizontal number line on a coordinate plane
4
The vertical number line on a coordinate plane
5
The value of y at the point where the line
crosses the y axis
1.Read
the following text about the Origins of Mathematics.
Tallies and Tablets - The Origins of Mathematics By Colleen
Messina
1 Have you
ever wondered what a teraflop is? No, it is not a clumsy, prehistoric fish. A
teraflop is a unit that measures the speed of computer calculations. One
teraflop is 1 trillion calculations per second, and today there are computers
that can sustain speeds of 35.86 teraflops! These incredible electronic
calculations originated with the idea of numbers and counting. Most of us take
math for granted, but numbers and counting have taken thousands of years to
develop. So how did it all begin? 2 Tens
of thousands of years ago, our ancestors found their food by hunting for meat
and gathering wild plants. Survival was a constant struggle. Little did they
realize that some mathematics could vastly improve their lives. For example, if
they knew when certain berries were ripe, they could save themselves a lot of
wandering time by only going to the berry thickets at precisely the right
moment. The hunters and gatherers of ancient times needed something constant in
their environment to help them track time. 3 The
most constant thing in their world was the sky since the landscape changed
through the seasons of the year. Early peoples observed the geometry in nature,
the cycles of the seasons, and the splendor of the Milky Way. Our ancestors
noticed the moon's pattern of becoming full, then slender, then full again in a
recurring thirty-day cycle. This cycle gave them a key to solving the dilemma
of tracking time. With this knowledge, they observed that sour, green berries
took approximately a full cycle of the moon to ripen, so they began to cut
notches in a tree or a stick to keep track of the days of the lunar cycle.
Harvesting the berries and other food became much more efficient with this new
system. 4 The
idea of keeping track of the lunar cycle sounds simple, but it was a momentous
event in the evolution of mathematics. Our ancestors were keeping a tally,
for the first time, and they probably began to use this form of counting in
other areas of their lives. The earliest known tallies were carvings in bones
dated approximately 15,000 years ago, which were discovered in the area now
known as the Middle East. Putting pebbles or
shells in a pile was another way of keeping a tally. Keeping track of items by
using simple marks or objects was still a long way off from the invention of
numerals, but it was a big step forward. 5 Another
way that early people kept track of things was by using "body
counting." Different parts of the body represented different amounts of
things. For many thousands of years people counted using their ten fingers, and
some tribes took this idea even further. The Paiela tribe, who lived in the
highlands of Papua New Guinea, counted by pointing to different parts of their bodies
to represent different numbers. For example, their little fingers represented
the number "one." Other fingers, wrists, elbows, shoulders, ears and
eyes all represented different numbers up to twenty. Body counting worked fine
as long as there was no need for large numbers. 6 When
our ancestors became farmers, they needed to keep track of larger amounts of
things. Farming probably started when the hunters and gatherers visited a
campsite where they had lived during the previous season and noticed grain
growing from seeds they had accidentally dropped on the ground. They learned to
save seeds and sow crops rather than gather wild plants. They also learned to
keep sheep, goats, and cows in pens and slaughter them rather than hunt for
wild animals. Life became easier, and villages formed since no one had to
wander around to survive. A better system of counting evolved because the men
who became shepherds had to count their animals, and the men who became farmers
had to keep track of their harvest.
2. Answer the
following questions.
1.
What is
a teraflop?
A unit of measurement for computer calculations
The missing link in man's evolution
A clumsy, prehistoric fish
An archaeological artifact
2.
What
does the word, "lunar," mean in paragraph 3?
Moon
Crazy
Musical
Sun
3.
Which of
the following is not a fact from the article?
Tallies began about 15,000 years ago
A lunar cycle is 6 months long
A written system of numbers was developed about 5000 years ago
(3000 B.C.)
Sometimes pebbles were used to keep a tally
4.
How long
is a lunar cycle?
30 days
3 days
Thousands of years
The article does not say
5.
Where
were the earliest known devices for keeping tallies found?
Africa
The Middle East
North America
South America
6.
Where
did the Sumerians live? South America
Africa
Iraq
Iran
7.
What is
missing in the Sumerian system of numbers?
1
0
60
10
8.
What is
a quipas?
An ancient joke
An ancient numeral
A knot in a cord used for counting
An ancient game
3.
Read the following text about the Age of Discovery.
The Age of Discovery - Gravity and Gauss By Colleen
Messina
1 By the
seventeenth century, mathematics had come a long way from the tallies and abacuses
of the ancient world. Mathematicians had finally adopted the new Arabic
numbers, as well as the symbols for addition, subtraction, multiplication, and
division. Logarithms made difficult problems much easier, and calculus opened
up new possibilities in science. Mathematicians applied these new tools in
exciting ways ranging from world exploration to astronomy. Ships crisscrossed
the oceans to new places, and telescopes scanned the skies and discovered the
elliptical orbits of planets. The understanding of gravity revolutionized
military science. It was truly an age of discovery.
2 The
discovery of gravity especially changed how people viewed the world. Up until
the 16th century, people thought that heavy objects fell faster. A feisty
Italian named Galileo Galilei had another idea. In 1585, he climbed to the top
of the leaning Tower
of Pisa, made sure no one
was down below, and dropped two objects. One object was heavy and the other was
light, but both reached the ground at the same time. Galileo proved that
objects fall at the same rate and accelerate as they fall. Eventually, military
engineers understood that a cannonball shoots out in a straight path, but the
force of gravity makes the cannonball fall downward in a curve called a
parabola. The engineers could then fortify their strongholds in the right
places, and artillerymen could shoot their cannons more accurately. Galileo's
experiment revolutionized military science.
3 Galileo
also did experiments with pendulums that helped clockmakers design accurate
clocks. Seamen needed accurate time-keeping devices to navigate during long
journeys. The weight-driven clocks of the previous centuries were not accurate
enough; now seamen needed to measure minutes and seconds, so the new clocks
were invaluable. Navigators then accurately plotted the daily positions of
their ships on maps that had vertical and horizontal lines of latitude and
longitude. When they connected the dots on these grids, they saw an accurate
record of the ship's journey.
4 Rene
Descartes, a great French mathematician and philosopher, also liked grids. He
had a big nose and a sheath of black hair that came down to his eyebrows. He
always stayed in bed until late in the morning and said that that was the only
way to get ready to do mathematics! Descartes tied geometry and algebra
together by writing equations for a geometric shape, like a parabola, on a
graph. His analytic geometry became the foundation of the higher mathematics of
today, and some people call him the first modern mathematician. The Cartesian
coordinate system is named after Descartes.
5 Another
mathematician who laid the foundation for higher math was a number-crunching
prodigy. In 1779, three-year-old Carl Friedrich Gauss watched his father add up
the payroll for a crew of bricklayers and pointed out a mistake his father made
in the calculations! When Gauss was 14, a wealthy Duke noticed his incredible
abilities and was so impressed that he sponsored Gauss's entire education. This
patronage was well deserved, as Gauss dominated mathematics of the nineteenth
century.
6 Gauss
first became famous when an Italian astronomer discovered an asteroid in 1801.
Joseph Piazzi accidentally found a minor planet and then lost sight of it in
the bright sky near the sun. This new planet, called Ceres, caused a great rush
of excitement all over Europe. When it
disappeared, astronomers were upset because they didn't know how to find the
new planet again. Gauss used the tables of logarithms he had memorized to
predict where Ceres would reappear. The tiny planet showed up on the other side
of the sun just where Gauss said it would! Gauss received many honors from
scientific societies because of this triumph.
4. Answer the following questions.
1.
Which
scientist performed an experiment from the top of the Tower of Pisa?
Galileo
Newton
Descartes
Gauss
2.
What
field was affected by Galileo's experiments with gravity?
Magnetism
Counting machines Electricity
Military science
3.
Check
which discoveries Galileo made from the Tower of Pisa
experiment.
Objects accelerate as they fall.
Objects fall at the same rate regardless of weight.
A scientist should warn people before dropping objects from
great heights.
The Tower leaned too much to make the experiment useful.
4.
What
kind of geometry did Descartes develop?
Lateral
Analytic
Longitudinal
Topographic
5.
Why did
Carl Gauss first become famous?
He located a lost asteroid.
A wealthy duke financed his education.
He loved complicated equations.
He was brilliant at an early age.
6.
What is
unique about a complex number?
It involves negative numbers.
It involves the square root of minus one.
It is very large.
It is used in complicated equations.
7.
What was
John Napier's counting machine called?
Napier's bones
Napier's multiplier
Napier's calculator
Napier's abacus
8.
Who was
the first computer programmer? Albert Einstein
John Napier
Charles Babbage
Ada Lovelace
