Compare Algebraic Proof And Geometric Proof, Which One Is...

Compare Algebraic Proof And Geometric Proof, Which One Is...
Compare algebraic proof and geometric proof, which one is stronger? And why?
Human being is thinking reed, this is famous quote by Pascal, one of the greatest mathematician in our history. Thinking, can simply represented as
imagination or waste of time, but in fact, thinking is root of our knowledge, the reason why we was able to be the most developed animal on earth.
By 'thinking' about a thing, we get 'idea' which then leads to 'curiosity'. From that 'curiosity', we found question simply to answer to the things that
were near, such as comparing which land is bigger, or how to measure the time of the day. And from those 'ideas' and 'curiosity', we started study on
group of similar wonders, which we call 'subject' now a day, and the one of ... Show more content on Helpwriting.net ...
The area of the square in the middle then becomes C^2 and the area of four triangle is therefore, the whole area is which proves that . This shows
that the both proofs are available, but for geometric proof, it is impossible to proof the theorem or the idea in every thesis. For example, for
double–angle formulae of the trigonometric identities, the both proofs can prove the formula, but the Geometric proof is only available in acute angle
(algebraic proof uses sine, cosine, tangent, which geometric proof is unavailable). Therefore, for geometric proof, because it is based on geometry,
which is based on algebra, it is not stronger than the geometric
... Get more on HelpWriting.net ...

Pythogoras of Samos Essay examples
Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we
know little about his achievements. There is nothing that is truly accurate pertaining to Pythagoras's writings. Today Pythagoras is certainly a
mysterious figure.
Little is known of Pythagoras's childhood. Pythagoras's father was Mnesarchus, and his mother was Pythais. Mnesarchus was a merchant who came
from Tyre. Pythais was a native of Samos. As a child Pythagoras spent his early years in Samos, but traveled with his father. There are accounts, that
during their travels, Mnesarchus returned to Tyre with Pythagoras, and had him taught there by the Chaldaeans.
"Certainly growing up he was ... Show more content on Helpwriting.net ...
Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there. After a short stay in Crete, Pythagoras found
himself back in Samos. There he discovered a school called the semicircle. This was the site of his own philosophical teaching, spending most of the
night and daytime there and doing research into the uses of mathematics. He tried to use his unique method of teaching, which was similar to the
lessons he had learned in Egypt, but Samians were not very keen on this. Pythagoras saw that the Samians were not giving him the respect and credit
he deserved, so he moved on
Pythagoras left and founded a philosophical/religious school in Croton on the southern tip of Italy. His school practiced secrecy and communalism
making it hard to tell the difference between the work of Pythagoras and work of his followers. Although it did made outstanding contributions to
mathematics. Pythagoras gained many followers there, and became the head of a society with an inner circle of followers known as mathematikoi. The
mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. "They weren't acting as a mathematics research
group does in a modern university. There were no 'open– problems' for them to solve, and they were not in any sense interested in trying to create or
solve mathematical problems. Rather Pythagoras was interested in teaching the principles of

Ancient Greek Science and Astronomy
The Ancient Greek culture has had such an impact on the world that no matter where you look you 're sure to find something Greek about it. Out of all
the areas that the Greek culture is famous for there are two that tend to exert themselves into our own culture even today. That would be their Science
and
Astronomy fields.
If one were to look up in a library books about ancient Greek science and astronomy they would have a mountain of books to sift through. There seem
to be so many individuals who have contributed towards the great scientific and astronomic revelations that the list of names seems to go on and on.
Many of the theories that were structured in the ancient Greek culture are still put to use today.
The goal of ... Show more content on Helpwriting.net ...
One example of Pythagoras 's feelings of personality towards numbers was the number Ten (10). He insisted it was "the very best" number
because it contained the first four integers – one, two, three, and four [1 + 2 + 3
+ 4 = 10]. When written in dot notation these numbers formed a perfect triangle.
Taken directly from Thomas Heath who was a civil servant and also one of the leading world experts on the history of mathematics is a list of
theorems attributed to Pythagoras and his followers: (i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew
the generalization, which states that a polygon with n sides has sum of interior angles 2n – 4 right angles and sum of exterior angles equal to four right
angles. (ii) The theorem of Pythagoras
– for a right–angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to
Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square
constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to
form a square identical to the third square.
(iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a – x) = x2 by geometrical means. (iv)
The

Comparison Of Soccer Balls
Background
Although a soccer ball may look like a sphere is not a real sphere. When you look closely at the actual ball you will see that it is polyhedron made
of different shapes. A polyhedron is a three dimensional object composed of flat polygon sides typically connected at the edges (Wolfram 1999). The
most common shape of modern soccer balls is called a truncated icosahedron. Icosahedron is a shape made up of 20 triangles. To create a more round
object the icosahedron is truncated. Here, truncated applies to cutting off the edges which makes the icosahedron look more round ("Icosahedron,"
2014). The truncated icosahedron has 32 total faces including 12 regular pentagons and 20 regular hexagons with 90 edges and 60 vertices ("Truncated
... Show more content on Helpwriting.net ...
Check that the lengths of all the lines is 5 cm and that the angles are all 108Вє. Carefully cut out the regular pentagon.
12) Now repeat the previous steps 11 more times to create the 12 regular pentagons. Always check that the compass and lines are 5 cm. B) Create a
regular hexagon
To draw a regular hexagon we followed the instructions provided online (Hartley, 2013).
1) Set the radius of the compass to 5 cm. It is critical that it stays at this length. Throughout the process check that the length stays at 5 cm.
2) Draw a line 5 cm long and label the points A and B (.
3) Draw 2 small arcs centered at point A and point B towards the center of the line AB. Label the point where the two arcs cross as point 1. This is
the center of the regular hexagon and the vertices of the hexagon will be on a circle centered at this point.
4) Initially draw a full circle centered at point 1. In future steps you only need to draw arcs near where the vertices occur.
5) Using point B as the center, draw a short arc that crosses the circle. Label the point of intersection point

Parabolica Vs Hyperbola
Ziv Gabay Mr. Diaz 10 Algebra 2 5 June 2016 Parabola vs. Hyperbola Essay A parabola and a hyperbola are two of the same, but also are very different.
The definition of a conic section is a section (or slice) through a cone. By taking different slices through a cone you can create a parabola or a
hyperbola. Each has its own place independently, but some things function very similar to one another. But the graph you will get whether it be a
parabola or a hyperbola, will always be relatively similar. In a parabola, the curvature of the two lines is dictated by the focus point. The focus point is
a set of coordinates [usually (X,0) or (0,Y)] that, by slope, proportionally separate the two lines in a parabola. Parabolas always have a lowest point

Architecture : Architecture And Architecture
Architecture requires deliberate measurements for practicality and aesthetic appeal. Architecture also requires calculating exact angles and areas of
intersection to ensure the components of the structure are stable and safe (By). I decided to explore architecture because I admire art and architecture
combines the principles of art and mathematics to construct beautiful buildings. Additionally, many pieces of architecture have a story and through
mathematics and art, an architect or the person who commissions the building can tell their stories through a somewhat permanent structure. Architects
have to design spaces in a functional manner that is also aesthetically appealing which requires proportion, balance, and conjunction with its
surroundings. This report looks at architecture and how trigonometry and mathematics have been used in developing St. Peter's Basilica and the
Pantheon through sectors and right triangle trigonometry. St. Peter's Basilica is a late Renaissance church located in Vatican City built at the place of
crucifixion of St. Peter the Emperor Constantine at request of pope St. Slyvester I. It was originally built from 315 A.D. – 349 A.D. but, rebuilt from
1451 A.D. – 1625 A.D. with the dome being designed by Michelangelo. For St. Peter's Basilica's dome to remain structurally sound, tension rings were
added to the structure and there is an inner and outer dome. Utilizing the sector formula, the dome can be analyzed in how it maintains its structure

Cartesian Coordinate Equations
As the name suggests, cylindrical coordinates are 3–dimensional and involve polar coordinates. It combines the 3–dimensional Cartesian coordinate
system (x, y, z) with the polar coordinate system (r, Оё). The "r" stands for radius while the "Оё" stands for theta (the angle usually represented in
radians). The x and y are replaced with the polar coordinate system while z stays the same.The coordinates are shown in this manner: (r, Оё, z).
Graphing cylindrical coordinates is easy as long as you know how to graph in polar coordinates and in the 3D Cartesian system. First you look at only
the xy–planes to locate (r, Оё). Start at the positive x–axis and go in a counterclockwise direction however many radians or degrees are stated. Then
use r to know how far from the pole, center, to go. If r is a negative number, then it is in the opposite side of theta (so Оё=бґЁ/4 becomes Оё=5бґЁ/4)
and r away from the pole. The conversion from (x, y, z) to (r, Оё, z) only requires the polar coordinate to Cartesian coordinate equations.Therefore the
following equations are used: cos Оё=x/r, sin Оё=y/r, tan Оё=y/x, r2=x2+y2, and z=z (Zill,... Show more content on Helpwriting.net ...
Anytime there is a circle on the xy–plane, it is stated with the radius of the circle, for instance rв‰¤4 would be a solid circle with radius 4 since the
radius is anywhere within the distance of 4 from the center. Now to make a circular cylinder function, you would need a solid circle in the xy–plane
and a possible domain for z (either going on for infinity or specify an interval such as 0в‰¤zв‰¤5 where z is only in the interval from 0 to 5). A
cone is not much different from a cylinder, but in a cone the radius and z are either directly or inversely proportional, so as z changes, the radius
changes with it. An equation for a cylinder then could be written as rв‰¤4, z=r. This states that the solid circle on the xy–plane is the biggest when
z=4 and shrinks down to 0 as z goes to

The Impact Of The Great Pyramids Of Egypt And The Pyramids
What makes up a pyramid? 12 lines, 5 faces, 4 triangles, 1 base, and 5 vertexes and in the case of the Great Pyramids of Giza about 2.3 million
stone blocks that weigh an average of 2.5 to 15 tons. That according to the time it took to build them they would have had to place and set a stone
every 2 to 2 and a half minutes. To put that in perspective some weigh as much or more than an armed military cargo truck. So, imagine dragging a
cargo truck with nothing but some strong rope and some other people. Now if you're like most people you can lift maybe 100 pounds. However, the
average powerlifter can lift 350 to 400 pounds and that's just lift not pack around or drag across acres of land. Now how did the Egyptians build the
Great Pyramid of Giza with blocks that weigh up to 15 tons with the technology they had then? It's clearly very obtuse to think the pyramids were
built by the Egyptians and the Egyptians alone. The question is, who helped them? I'll tell you who helped them. It was the beings that have been
erased from human history, the ones hiding amongst the stars at all this time watching us. The ones that stop by to build their structures, while
casually using our resources. The ones that built Stonehenge, the Pyramids, the face on Mars, Easter island, and the ones that make the crop circles,
the Aliens. The aliens have been there all along ever so subtly leaving us messages too advanced for us to understand helping the hard workers build
their so called "sacred structures". Were the aliens just being neighborly trying to help or did they have more incentive? The aliens had more than
enough incentive to help build these structures, especially the pyramids. They took advantage of the Egyptians need for a sacred place to bury their
kings and the obtuse triangle formed by Orion's Belt. They lined up the structures with the stars of Orion's Belt to create the energy column. Then, they
created a complex tunnel system, so no one could ever find their transporter and then they left through the triangular column that forms between the
pyramids and Orion. Only to be able to return or communicate at night when the column reforms. Who do they communicate with, you may ask? The
governments,

Maths Exam
Score: ______ / ______
Name: ______________________________
Student Number: ______________________
| 1. Elsie is making a quilt using quilt blocks like the one in the diagram.
a. How many lines of symmetry are there? Type your answer below.
b. Does the quilt square have rotational symmetry? If so, what is the angle of rotation? Type your answers below.
| | 2. Solve by simulating the problem. You have a 5–question multiple–choice test. Each question has four choices. You don't know any of the answers.
What is the experimental probability that you will guess exactly three out of five questions correctly? Type your answer below using complete
sentences. | | 3. Use the diagram below to answer the following questions. ... Show more content on Helpwriting.net ...
| | 8. Caitlin had $402 in her bank account. She withdrew $15 each week to pay for a swimming lesson. She now has $237.
a. Write an equation that can be used to find the number of swimming lessons that she paid for.
b. How many swimming lessons did she pay for?
c. At the time she had $237, the cost of a lesson rose to $19. How many lessons can she pay for with her remaining $237? | | 9. Is a triangle with sides of
length 6 ft., 21 ft., 23 ft. a right triangle? Explain. | | 10. Identify the number as rational or irrational. Explain. 291.87 | | 11. Is the sequence 5, 9, 15,
... an arithmetic sequence? Explain. Type your answer below. | | 12. Suppose a computer virus begins by infecting 8 computers in the first hour after
it is released. Each hour after that, each newly infected computer causes 8 more computers to become infected. The function y = 8x models this
situation. Make a table with integer values of x from 1 to 4 in the space below.
| | 13. For the function y=1x, make a table with integer values of x from 0 to 4. Then graph the function (you do not need to submit the graph). Does
the graph of show exponential growth, exponential decay, or neither? Explain your thinking. | | _____ 14. Tell whether the sequence 13, 0, 1,–2... is
arithmetic, geometric, or neither.
Find the next three terms of the sequence. Type your answer in the blank to the left. A. neither; 7,–20, 61 | |

Relationship Behind The Pythagorean Theorem
Pythagorean Theorem
Introduction
The Pythagorean Theorem is a relation in Euclidean geometry among the three sides of an right angle. Pythagoras, a greek philosopher is credited for
the discovery, but it is unsure who and therefore theorem is named after him. The formula is a2 + b2 = c2.
History of the Mathematician behind the Pythagoras Theorem and the Pythagorean relationship.
People are unsure whether the relationship was made either by Pythagoras or the Pythagoreans first proof, Pythagoras is Greek from the Samos island
(570–495 B.C.). The Founder of the Brotherhood of Pythagoreans is Pythagoras. Pythagoreans have their own religion where, "Number rules the
universe".They are devoted to geometric proofs and keeping them secret. The ... Show more content on Helpwriting.net ...
Conclusion
The Pythagorean Theorem is a2 + b2 = c2. We learnt the history of the pythagorean relationship, like who made it and how it is made.We also found
proofs of the relationship and learned to understand why this relationship is true. We also learned that there are some rare relationships that exists
between Pythagorean numbers. We are very familiar with the pythagorean theorem and now we can solve complex questions involving the pythagorean
theorem now.
Reflection
Our group has had many fights but in the end we managed to complete our work very well. It was difficult to cope with having a new member join
our group and one of our members was put into another group, but we still managed to do very well in the end and had a lot of fun doing the project.
Bibliography http://www.mathsisfun.com/pythagoras.html http://www.geom.uiuc.edu/~demo5337/Group3/hist.html http://ualr.edu/lasmoller
/pythag.html http://www.ck12.org/book/CK–12–Middle–School–Math–Grade–7/section/9.3/

Manipulation In The Lycaon Tale
In the second century B.C. A Greek geographer called Pausanias, tells of a man, called Lycaon famous, in so much as, he ritually murders a child,
consequently, after which, he turns into a hybrid Human, Wolf. Hence, since this ancient time, the ruling elite, has used this type of story board,
correspondingly, for Public psychological manipulation.
The philosopher Voltaire stated "Those who can make you believe absurdities, can make you commit atrocities." Consequently, since their conception,
these words have rung true over the centuries in conjunction with titles such as Witches, or Werewolf, and Jew.
Governing powers who lose credibility, to maintain their power, in consequence, with the Lycaon tale, construct, mythologies combined with
allegorical interpretations, of perceived diabolical ... Show more content on Helpwriting.net ...
He noticed that facing him, was a wood panelled, silver door Otis elevator.
On its top panel a reversed isosceles triangle direction indicator, which pointing down glowed with a soft florescent orange.The right, hand side panel.
Sported a circular, black lift call button, backlit displaying a soft green halo.
Before entering the elevator. Sheik Valsaud studied the lift design features. Mentally noting, the thoughtful placement, of the call button. Subsequently
concluding, that in his opinion, it being positioned just, below shoulder height, in clear view, was excellent. His manicured finger pressed lift, call.
An electric hum filled Dawn Oil entrance hall, thus, opening the sliding doors. Consequently, releasing an aroma of new, carpet, into the foyer
environment. Thereupon,Valsaud stepped into a plush,floor and wall,dark charcoal, carpeted elevator, with smoked mirrors,that gave the meter
squared,lift a sense of

Ptolemy's Theorem Essay
Geometry has always appealed to me, perhaps because of the illustrations provided for every question or the way that it's seemingly unsolvable, yet if
you understand and recognise the properties of that figure, you would then be able to solve the question with no difficulties. Therefore, I find it is
important to grasp the concept of every property which is why I choose to delve further into the Ptolemy's Theorem.
The Ptolemy's Theorem provides a relationship between the four side lengths and the two diagonals of a cyclic quadrilateral, an inscribed figure whose
vertices lie on a common circle. The theorem states that when the product of the two pairs of opposites sides are added together, it is equals to that of
the product of the diagonals ... Show more content on Helpwriting.net ...
1
Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. Claudius Ptolemy born around
90 ce in Egypt was a well–known astronomer, mathematician, astrologer and mathematician. Although much was of his life was lost to history,
Ptolemy was famous for his ground breaking work The Almagest. In his lifetime, Ptolemy explored a motley array of disciplines, ranging from
astronomy and geography, to mathematics, philosophy, literature and even poetry. He was also the discoverer of the above mathematical theorem now
named after him, the Ptolemy's Theorem. There is also the Ptolemy's inequality, to non–cyclic quadrilaterals.
There are multiple ways that we can approach the figure to proof that the Ptolemy's Theorem is indeed correct and consistent when the requirements are
fulfilled, and below, the cosine law and the geometric method are just one of the many.
Proof of Ptolemy's Theorem (Cosine Law)
This proof makes use of cosine law to solve. After equating the diagonals to the sides using the cosine law, we can then use substitution to get one
single equation for each diagonal. Obtaining the equations of the diagonals would then allow us to multiply it to conclude and proof the existence of
the Ptolemy Law. This law is long and requires multiple steps to complete which might result in careless errors along the

Questions and Answers Regarding Equilateral Triangles
Since the centroid of a triangle lies along each median, 2/3 of the distance from the vertex to the modpoint of the opposite side, we have:
в– (t=2/3в€™в€
љ3/2в€™c=c/в€
љ3;@u=2/3в€™в€
љ3/2в€™b=b/в€
љ3.)
And in this case (1) becomes (2):
гЂ–3sгЂ—^2=b^2+c^2–2в€™bв€™cв€™cos(A+60В°).
Expanding the cosine of the sum, we have (3): cos(A+60В°)=cos(A)/2–(в€
љ3в€™sin(A))/2. Substituting (3) into (2) yields to (4):
гЂ–3sгЂ—^2=b^2+c^2–bв€™cв€™cos(A)+в€
љ3в€™bв€™cв€™sin(A).
Now we will apply the Law of Cosines to в€†ABC (5): a^2=b^2+c^2
–2в€™bв€™cв€™cos(A). And recall (6):
2в€™S_в€†ABC=bв€™cв€™sin(A).
Substituting (5) and (6) into (4) gives us (7):
3s^2=(a^2+b^2+c^2)/2+2в€™в€
љ3в€™S_в€†ABC
As (7) is symmetrical in a,b and c, it follows that the triangle connecting the three centroids is equilateral:
IH=GH=GI=s=в€
љ((a^2+b^2+c^2)/2+2в€™в€
љ3в€™S_в€†ABC )/в€
љ3=в€
љ((a^2+b^2+c^2)/6+(2в€™S_в€†ABC)/в€
љ3).
The Napoleon's triangle is equilateral. Proved.
Answer: Proved.
Rotating by the point:
Fix the point I as the center of rotation and rotate the entire figure by 120В°, and superimpose the rotated copy on the original figure.
Uder rotation the в€†CAF maps to itself (C maps to A, A maps to F, F maps to C and I maps to itself). Points BB, DD, EE, GG and HH are the images
of the points B, D, E, G and H, respectively. Because of this he triangle в€†A.EE.BB= в€†BCE.
Connect D to EE and G to HH. By the rigidity of the rotation, в€†GHI=в€†GG.HH.I. In particular GH=GG.HH.
Consider the six triangles that coverge on point A. Triangles в€†ABD, в€†ACF and в€†A.EE.BB are equilateral.
The angles of triangles sum to 180В° and

Nt1310 Unit 3 Research Paper
%addcontentsline{toc}{section}{Activity 1.14}
section*{Activity 1.14}
Given ABCD is a square and rhombus EFGH is inscribed in the square. We want to prove that EFGH will be a square.
includegraphics[scale=0.46]{Pictures/actvity1_14_pic.PNG}
noindent
$EH = HG = GF = FE$hfill ––– since $EFGH$ is a rhombus.
$AB = BC = DC = DA$hfill ––– since $ABCD$ is a square.
$BF = DH$hfill since $AD = BG$ and $EF = GH$ and $EH = GF$.
Similarly, $AE = GC$ and $AH = BE = FC = DG$.
The four triangles $(AEH; EBF; FCG; DHG)$ are congruent by $SSS$.
Each angle in the corner of $ABCD$ is $90^{circ}$hfill ––– since $ABCD$ is a square.
$Ahat{E}H + Ehat{H}A = 90^{circ}$hfill ––– since $Ehat{A}H = 90^{circ}$.
Similarly $Bhat{E}F ... Show more content on Helpwriting.net ...
I only prove that $Ehat{H}G = 90^{circ}$, for the reason that the prove of the other angles will be similar. When I was asked to prove if the
inscribed rhombus is a square, I simply say to myself, ``it means I have to prove that the corners of the rhombus are right angled ($90^{circ}$)"
since the properties of a rhombus are such that opposite angles are equal, all sides are equal, and opposite sides are parallel, the diagonals bisect
the angles, and the diagonals are perpendicular bisectors of each other. Therefore, the inscribed rhombus can only be a square if the angles are right
angles. I observe that all four triangles are congruent by SSS, therefore from $Ahat{E}H + Ehat{H}A = 90^{circ}$ and $Ehat{H}G + Ahat{H}E
+ Dhat{H}G = 180^{circ}$ and $Ghat{H}D = Ahat{E}H$, it follows that $Ahat{H}E + Dhat{H}G = 90^{circ}$, hence $Ehat{H}G =
90^{circ}$. There is a need to draw the picture since it gives more logical reasoning and the understanding of how the angles and sides are defined,
without drawing the picture we may not understand what is happening, therefore it is very crucial that as teachers we make use of pictures when we
dealing with this kind of

Solid Mensuration
CPR
(MATH13– B10)
Members: C06 Wrenbria Ngo C07 Julie – Ann ParaГ±al C08 Dani Patalinghog C09 Marino Penuliar C10 Michael Sadsad
CPR
(MATH13– B10)
Members: C06 Wrenbria Ngo C07 Julie – Ann ParaГ±al C08 Dani Patalinghog C09 Marino Penuliar C10 Michael Sadsad
Prof. Charity Hope Gayatin
Prof. Charity Hope Gayatin
Homework 1.1
#15. Find the sides of each of the two polygons if the total number of sides of the polygons is 13, and the sum of the number of diagonals of the
polygons is 25.
Assume: ... Show more content on Helpwriting.net ...
Find the lengths of the three sides if the area of the triangle is 576cm2 .
Soltion : c/17=9/10=b/9
A= ss–as–b(s–c) s= a+b+c2 s= 10a+9a+17a20b= 9(40)10 s= 95ab=36
576= 18a250; c= 17(40)10
9a2= 14400c= 68 a2= 1600 a=40 Answer: 40cm, 36cm, 68cm
#15. Given triangle ABC whose sides are AB=15in., AC=25 in., and BC= 30in. From a point D on side AB, a line DE is drawn to a point E on side
AC such that angle ADE is equal to angle ABC. If the perimeter of triangle ADE is 28 in., find the lengths of the line segments BD and CE. Given:
?AD 30in 15in

B E C
Required: BD =? ; CE =?
Solution: For BD P ADEP ABC = ADAB P ADE=28in Answer: The length of segments BD and CE is 9in and 10in AD = 15in( 28in)70in P ABC=70in
AD = 6in P ADEP ABC= AEAC BD = 9in AE = 25in (28in)70in AE = 10 in
#17. What is the sum of the areas of the two triangles formed in number 16? Given: 3

Cardinality in Blind Children
Aim, objective and Hypothesis
The current study set out to explore the generalisability of counting behaviour and the understanding of the cardinal principle in blind children. To date,
research in this area has focused mainly on typically developing children. Some researchers have undertaken studies in the atypical population;
however, this is limited to disorders such as Down syndrome (Caycho, Gunn & Siegal, 1991), mental retardation (Baroody, 1986) and severe learning
difficulties (Porter, 1998). Therefore, children with visual impairment have not been given much attention in this field of research.
The aim of the first experiment will be to observe whether children are able to count objects with varying degree of manipulation and mode of input, to
assert the numerosity of the count. These entail counting objects than can and cannot be separated from those that have been counted, as well as the
counting of sounds. The second experiment is designed to confirm the findings of experiment one; hence, children who understood the cardinal
principle will be able to detect error made by others in counting. A further aim of the study will be to investigate whether smallernumber words are
acquired before larger ones, as it is the case in typically developing children. The researcher hypothesised that blind children's development will be
slightly delayed in the younger age group, but will not show significant differences in the older age group when compared to typically

Identifying Human Hair and Animal Fair by Laser Diffraction
1. Introduction:
1.1 Overview
The ability to identify an animal by examining its fur is a useful one for taxonomists and biologists. The hair of prey is often not digested by a
predator, and so can be found in the predators scat and measured to ascertain its diet (Anwar et al., 2011). An electron microscope can be used to
obtain a lot of data from hair, including its width and the shape and texture of its surface, however electron microscopes are expensive and time
consuming to operate (Sessions et al., 2009). Lasers are much cheaper and can be used to easily measure the width of narrow objects by measuring the
interference pattern created when laser light is diffracted around the object. As a result, laser diffraction may be a more preferable method for identifying
hairs based on their width. This experiment hopes to show that laser diffraction is an accurate enough method for obtaining the width of narrow objects,
such as animal and human hairs, to allow comparison between them.
1.2 Wave interactions
1.2.1 Diffraction:
Diffraction is a physical phenomenon that occurs when waves encounter an obstacle or gap between obstacles. Depending on the wavelength of the
wave and the size of the obstacle or gap waves bend and spread out. Waves tend to diffract more around objects or gaps of a similar size to their
wavelength. How a wave will diffract when it meets an object can be modelled using Huygen's construction. Huygen's construction is based on
Huygen's principle of light

Explaining Ap Statistics
Part 3: Writing and Explaining AP Statistics 1.The quadratic formula is to solve for x. The formula is x=–b +/–в€
љbВІ–4ac 2a. It also helps solve the
standard equation AxВІ+bx+c=0. 2. The rule in algebra about adding exponents is to ONLY add the base numbers if they are the same. If the base
numbers aren't the same then you have to make the numbers the same before you continue. 3. To draw a two dimensional representation of a cube,
first you must draw a square. Then draw three diagonal lines going from left to right. These three lines should extend from the top right corner, the top
left corner, and the bottom right corner. Next you have to draw one more diagonal line from the bottom left corner, but this line must be a dotted line.
Then draw

How Did Plato's Beliefs Contribute To The Study Of Philosophy
Philosophy can be described as coming to understand through means of deep thoughts. Two famous examples of practitioners of this field include Plato
and Parmenides. Though these two philosophers taught about many different things, they both taught and agreed on one thing: what it meant to be wise.
Plato and Parmenides both provide an account of what it means to be wise, and the central agreement between them is that at its core, wisdom is
realizing that truth lies beyond the base senses and hubris of the human body.
Plato believed that wisdom consists of gaining knowledge and understanding through the forms, and by reducing dependency on the bodily senses, and
the material realm that they observe. Plato is called a dualist, meaning that he thought there are two separate and ... Show more content on
Helpwriting.net ...
The knowledge that they gained through understanding the eternal and unchanging form is something they could have never been made aware of
if they had tried to gain knowledge through observing the material triangles only with the use of their bodily senses. In his dialogue, Republic,
Plato described the process of using the forms to gain knowledge and wisdom through his famous Allegory of the Cave. In this thought experiment,
Plato described a cave, in which there were multiple prisoners, who for their entire lives have been bound and shackled in such a way that they
were made to look at the wall of the cave. On this wall, several shadows of statues of animals, plants, and animals are cast by a fire behind the
prisoners. These prisoners, if not told otherwise, would live their entire lives believing that the shadows were the only things in existence; not being
aware that they were mere images of something which is more real. However, one of the prisoners were freed, they would be able to turn around and
see that the shadows were less real than the statues,

Exploring The Role Of Hipparchus In Greek Mythology
Hipparchus was born in 190 B.C. in Nicaea, Bithynia, and died in 120 B.C. Due to this , there is not a lot information on his early life. He
approximately started working as an astronomer in his 30's. Hipparchus was best known to be an astronomer, aside from that he was also a Greek
mathematician and geographer. Most of the work that he did is now lost , but "Only one work by Hipparchus has survived, namely Commentary on
Aratus and Eudoxus and this is certainly not one of his major works. It is however important in that it gives us the only source of Hipparchus's own
writings".(1) Nevertheless, most of his research and calculations are found in 'Almagest' by Claudius Ptolemy, which is how mathematicians can see
how he came across the beginnings of trigonometry.
First of all , one of Hipparchus' greatest contributions to mathematics was the creation of the trigonometric table , which allows to tabulate the values
of some or all six of the trigonometric functions for various angles. Additionally, he was also known for discovering the... Show more content on
Helpwriting.net ...
All of the information regarding mathematics that he already knew was based on Babylonian and Egyptians. In order to discover the precession of
equinoxes, Hipparchus had to first find a way to figure out the angles and measurements of the moon and sun. Therefore, he created a table of chords,
which later became a trigonometric table, which could be seen one of the first calculators.When creating the table ,"He considered every triangle as
being inscribed in a circle, so that each side became a chord" , he then originated dividing a circle into 360 degrees, same as Hypsicle's had done with
the ecliptic , and he was the first

Essay about Acoustic Theory and Synthesis
Acoustic Theory and Synthesis
Frequency:
Frequency means the number of cycles per second and depending on the amount of cycles per second determines how high or low pitched the sound
is and the time that it takes to complete one cycle is called the period. Frequency is measured in Hertz (Hz). And An average human is able to hear
sounds between 20Hz and 20,000Hz.
As the cycles per second increases, the smaller the wavelengths become, therefore there is a higher frequency which will cause the pitch of the sound to
increase or get higher.
If there is frequency of say 20KHz then it is going to be a much higher pitch than a 20Hz because there is a lot more cycles per second.
Fundamentals:
A fundamental is the lowest frequency ... Show more content on Helpwriting.net ...
Sustain: This is how long the volume is sustained for during the main event of the sound. Release: The is how long it take for the sound to fade
out. An Example of a slow and fast release can be found on track 7 and 8 of my logic session which is with my assessment. Filters: Filters can either
emphasize or reduce some frequencies from a signal. There are different types of filters that can be used. There are different types of filters that can
be used. There are also filters that can reject frequencies within a certain frequency band e.g. band bass or band reject.
High Pass Filter: A High Pass filter lets through all the frequencies above the cutoff and eliminates the ones below the cutoff frequency. An example
of a HPF can be found on track 10 of my logic session. !
Low Pass Filter: A Low Pass filter lets through the frequencies below the cutoff frequency and eliminates the ones above it. An Example of a low
pass filter can be found on track 9 of my logic session which is with my assessment.
!

ESM
The ESM synthesizer is a very basic synth to create simple bass and lead sounds that can emulate the Roland – TB 303 synth because it is a
monophonic synth and you cant play chords . With the ESM you can only choose between a saw tooth and square wave and it only

How Did Egypt Use Triangles
The Greeks and the Egyptians used triangles as early as 3500 BCE. They used these triangles as rules of thumb. They could apply these rules to
specific applications. For example, the Egyptians knew that the 3:4:5 ratio was a right triangle. They could derive this because for them to create a
right triangle the Egyptian land surveyors used a rope divided into twelve equal parts, creating a triangle with three pieces on one side, four pieces on
the second side, and five pieces on the last side. The right angle was found where the three–unit side came together with the four–unit side. This was a
very efficient way to create right triangles. It's a mystery as to how the Egyptians came up with this, but this was later used by Pythagoras (c.571 –

Koch Snowflake Investigation Angus Dally
Koch Snowflake Investigation Angus Dally
Background:
In 1904, Helge von Koch identified a fractal that appeared to model the snowflake. The fractal was built by starting with an equilateral triangle and
removing the inner third of each side, building another equilateral triangle where the side was removed, and then repeating the process indefinitely. The
process is pictured below, showing the original triangle at stage zero, and the resulting figures after one, two and three iterations.
Method:
Let Nn=the number of sides, Ln=the length of a single side, Pn= the length of the perimeter and An= the area of the snowflake, all for the nth stage.
Using an initial side length of 1, create a table that shows the exact values of N_n,гЂ– ... Show more content on Helpwriting.net ...
Consequently, the area of each of these triangle is 1/9 of the original triangles.
Therefore the total area added is:
Area added=3Г—1/9
=1/3
Therefore, to find the area of snowflake n=1, these two area ratios must be multiplied by the area of the original triangle at n=0.
A_1=в€
љ3/4 (1+1/3)
A_1=в€
љ3/3
For snowflake n=2, twelve additional triangles are added onto the initial triangle, with side lengths of 1/9. Consequently, the area of each of these
triangle is 1/81 of the original triangles.
Therefore the total area added is:
=12/81
=4/27
Therefore, to find the area of snowflake n=2, this area ratio, as well as the previous two, must be multiplied by the area of the original triangle at n=0.
A_2=в€
љ3/4 (1+1/3+4/27)
A_2=(10в€
љ3)/27
At n=3, following the previous pattern:

48 triangles with side length 1/81, and area 1/729 are added.
=16/243
Therefore, to find the area of snowflake n=3, this area ratio, as well as the previous three, must be multiplied by the area of the original triangle at n=0.
A_3=в€
љ3/4 (1+1/3+4/27+16/243)
A_3=(94в€
љ3)/243
A_n Data Table
IterationAn n=0 в€
љ3/4 n=1 в€
љ3/3 n=2 (10в€
љ3)/27 n=3 (94в€
љ3)/243
The area ratios being added in each iteration yields a pattern:
1/3,4/27,16/243...= 4^((1)–1)/3^(2(1)–1)

How Did Diophantus Contribute To Algebra
Diophantus was exceptional mathematician born between the the years of 200 and 214 BC.the area in which he spent most of his life was
Alexandria .at the time Alexandria was going through the silber age in which it was the center of much greek culture and knowledge.As great of a
mathematicain in which he was ,not much information is regarded towards his own personal life only thing known about him was interpreted in the
form of mathematical puzzle which describes his life in his years throughout his life. The contributions in which Diophantus made to mathematics was
exceptional in his time but one of his major works is 'Arithmetika',which is said to have the most influence to algebra in the history of greek.earning
him the title as the Father of Algebra.These series of books greatly influenced the development of the number theory.Arithmetica contained a total of
13 books which in total had at least an average of 130 problems which gave solutions to determinate equations or in his era would be called
diophantine equations.this of course was only a small stepping stone to a person of his grandeur.He had made exceptional advanced in the... Show more
content on Helpwriting.net ...
This equation is then used by his mathemal puzzle which described his life through the years and could be solved through this manner with the
integers in which he was very skilled in dealing with.many of these problems would then be seen in his books not being relevent in the first book of
three mainly including simple word problems, but in the later books where the introduction of problems involving indeterminate.Thourhgout his books
he had used the word number to represent positive or even rational numbers.IN books 5 to 7 he then starts to complicate the basic methods by
introducing problems of higher numbers which can then be made smaller to a binomial equation.The purpose of his books were created in order for
the reader to learn and experience the mathematical skills and techniques that were being expressed in the

Eight Interlocking Theoretical Concepts On Personality And...
Eight Interlocking Theoretical Concepts
Bowen's theory is influenced by eight interlocking theoretical concepts to explain the family unit's emotional functioning (H. Goldenberg & L.
Goldenberg, 2013). The interlocking concepts explain the process of emotions within the family system as they are developed throughout the
generations (Wineck, 2010). One must fully understand each concept individually in order to fully grasp the impact within the family unit (H.
Goldenberg & L. Goldenberg, 2013). Concepts one through five were Bowen's original theoretical viewpoints (Baege, 2006). According to Bowen
(1976), his sixth concept was rooted in Walter Toman's Family Constellation: It 's Effect on Personality and Social Behavior publication from 1961. In
1975, the final two concepts were developed and added to complete the entire eight interlocking theoretical concepts (Bowen, 1976). Bowen never
published or created any work on a ninth concept; however, spirituality was mentioned as a possibility for a ninth concept. Spirituality concept is an
undeveloped concept with promise, particularly within the field of addiction studies. Underneath every concept is the basic assumption that chronic
anxiety prompts the development of each concept premise (H. Goldenberg & L. Goldenberg, 2013). These interlocking constructs are known as
differentiation of self, triangles, nuclear family emotional system, family projection, process, emotional cutoff, multigenerational transmission process,
sibling

Best Practices Of Mathematics Instructions
Best Practices in Mathematics Instructions
RATIONALE
As a future educator, it is important to introduce the subject matter and set objectives that will grasp the learners' interest and help them to connect with
the problem solving and questioning techniques. A unit plan should contain objectives, state standards, a summary of duties, and goals. Moreover, it
should contain the types of material needed for students to accomplish the task. There should be a breakdown of the unit by day or week. Teacher
should include the lecture and any quizzes, tests, or other assignments that will occur. Mathematics units are designed to help learners know what to
expect and become familiar with the state standards that are set before them. It is the ... Show more content on Helpwriting.net ...
To add, after learners have gained a deeper understanding of the unit, the teacher assigns matrices handouts for homework.
Lesson two introduces pupils to Geometry. In this unit learners will become familiar with circles and volumes. They will select the appropriate
theorem or formula to find the solution to the problem. In addition, learners will prove that all circles are similar and manipulate and use the volume
formula to find the volume of a cone. Activities may be differentiated to address the needs of learners who may have difficult in breaking down the
steps in the problems. Teacher will group learners according to their abilities cooperatively and allow learners to work independently on circles and
volumes. Step by step breakdown of the problem will be modeled by the teacher. Computerized interactive software and graphing tools will be used to
enhance learners' comprehension of the technique.
Lesson three introduces learners to Algebra. In this unit, learners will interact with linear and exponential functions. Pupils will understand how to
graph equations in two variables and plot their answers on a coordinate plan that often forms a curve or line. In addition, pre–assessments will be given
to determine learners' knowledge of linear equations and exponential functions. Teacher will model and guide learners into the steps in learning the
concepts. Then, students

What Students Should Know?
What Students Should Know According to the learning progressions report, coming into third grade, students know how to analyze, compare, and
classify two–dimensional shapes by their properties. When students do this, they relate and combine these classifications that they have made (The
Common Core Standards Writing Team, p. 13). Because the students have built a firm foundation of several shape categories, these categories can be
the "raw material" for thinking about the relationships between classes. Students have learned that they can form larger, superordinate, categories, such
as the class of all shapes with four sides, or quadrilaterals, and recognize that it includes other categories, such as squares, rectangles, rhombuses,...
Show more content on Helpwriting.net ...
15). In fourth grade, students learn to represent angles that occur in various contexts as two rays, explicitly including the reference line, which is a
horizontal or vertical line when considering slope or a "line of sight" in turn contexts. They understand the size of the angle as a rotation of a ray on the
reference line to a line depicting slope or as the "line of sight" in computer environments (The Common Core Standards Writing Team, p. 15). In fourth
grade students also might explore line segments, lengths, perpendicularity, and parallelism on different types of grids, such as rectangular and
triangular (isometric) grids. Finally, students also learn how to reason about these above mentioned concepts (The Common Core Standards Writing
Team, p. 16). According to the learning progressions report, coming into fifth grade, students should know how to represent angles that occur in
various contexts. They should also already understand the size of the angle as a rotation of a ray. Finally, they should have developed explicit
awareness of and vocabulary for many concepts they have been developing, including points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines (The Common Core Standards Writing Team, p. 17). In fifth grade, students develop competencies in shape
composition and decomposition, and especially the special case of spatial structuring of rectangular

C. E. Willis: A Short Story
It started on a nightly walk back home from work. Brogan E. Willis, worked at the small town hall, just outside of Elbridge, where she lived. Willis
always started her night with a small stroll back into town, where she was permitted to park her vehicle. The trail however, was a very vigorous path,
especially in her small black leather pumps. It was a muggy night, the dirt on the trail, damp and loose, her clothes slightly wet from the fog. The smell
of rain radiating around the dark woods slightly soothed the sharp feeling of someone watching her ahead in the undergrowth.
She focused on the slight movement ahead in the bush. She had barely noticed the slight drizzle of rain around her, as she was tranquilized by the
figure. It was small, ... Show more content on Helpwriting.net ...
Take this...it's ruined my life man!" The man threw down what looked to be a talisman.
Brogan quickly picked up the object. It was a brown stone circle, with carvings of "3w" on the front. It had three gems encrusted in a triangle shape,
inside that triangle was a Sun–Moon symbol. Willis quickly shoved the object in her pocket.
"You must destroy it...It's dangerous" The man backed away from the women, before sprinting back into the woods.
Brogan quickly stood up, running to her car this time. She slammed the door after sitting in the front seat. She then threw her purse, and the strange
object in the back seat, before speeding off onto the highway.
Once reaching her driveway, she couldn't wait to be in bed. Brogan had a long, hard, and very strange day. Her arm had started hurting from her
encounter about an hour ago. She stepped out of her car, and ran barefoot inside the house where she knew she was safe. Slamming the door once
inside, she heard the muffled sounds of her family throughout the house. She tore off her white trench coat, and threw it in front of the wash. She
hoped that she would eventually be able to get that mud off the sleeve from the man touching her. It was quiet for a bit, until her family finally realized
mom was home. That gave her a few minutes to scrutinize the object. She took it into the kitchen, where the light was

Designing A Tower Under The Clock
For this project our group was tasked with designing a tower under the parameters of a 40 cm height minimum and 10 x 10 cm tower base minimum
using bass wood and wood glue. The overall aim was ultimately to design a tower with the greatest mass–to–weight ratio. The following consists of the
procedures, results, and conclusions of our designing, constructing, and testing stages.
After the tower was completed the mass of the structure was calculated to be at around 29.2 g (0.06437498 lbs). Through the use of sand and metal
weights (i.e. 8 lbs, 15 lbs, and/or 25 lbs) the structure withheld a force of around 102 lbs until the upper half of the tower snapped. The weight held by
the structure was divided into 62 lbs of metal weights and 40 lbs ... Show more content on Helpwriting.net ...
The building was created as a triangle since it would require less material and would be quicker to create than building a four sided building.
The building stayed the same height through the sections and we felt that that would be the most stable since the building would be able to compress
upon itself instead of having to spend more time to try and create a stable design that could withstand the forces. Even though we may be able
distribute weight with a larger base it would weigh more with a ratio that's not as good as what we have or a ratio that has increased but compared
with the effort and materials that we would need it wouldn't be worth it.
The most difficult part of the construction was trying to connect the wall at a 60В° angle without letting it fall on itself and keeping an even space
through the tower. On the first attempt we unfortunately broke the first section of the building and noticed the glue didn't glue at connection where the
wood would meet but around the wood. After connecting the two pieces together it was simple to finish the tower but it involved a lot of waiting time
since we weren't able to move on when we connected the wall since it was unstable since the glue wasn't dry.
Another major difficulty we had was the large amounts of time spent waiting for the glue to dry. The tedium of having to deal with not being able to
work on anything else until the structure had finished

Freen Friend Number Two Circle
In my opinion friend number one is correct. Friend one is saying the circles are concentric, meaning every ray you draw from the center of the
circles will cut the big circle once and the little circle once. This makes a correspondence between the points of the one circle and the points of the
other circle. So they have the same amount of points. If you look below this, I have taken the circle and drew a few lines on it. Showing you how
each ray, or line, cuts the big circle one and the little circle once. Looking at the picture you see that each ray cuts each, big and little, circle once.
Making friend number one's statement correct, because they do have the same amount of point on the grid. Friend number two's statement was
correct to a certain point. They had said, "If you draw a horizontal line through the center. At first, the line cuts both circles in two places, but as
move the line up or down you will see line will cut the big circle twice, but only just touch the smaller circle at one point. And if you get farther away
than that, the line will touch the bigger circle (still twice!), but not touch the little circle at all. All points on the smaller circle are related to some
point on the bigger one, but some points on the bigger one are not related to point on the smaller one. So the larger circle has more points."... Show
more content on Helpwriting.net ...
No matter how far up or down they move (still touching the small circle) the line it is going to cut the small circle two times. If they would have
said "At first the line cuts both circles in two places, but as move the line up or down and the farther away you get from the small circle, the line will
touch the bigger circle (still twice!), but not touch the little circle at all." Then they would have also been correct. Looking at this circle below I have
shown you how it makes two cuts on each circle, expect when the line is not even touching the small

Simple Math Working Models
You Can prove that radius to the point of contact of a tangent is perpendicular by
Take two Iron Rings with a radius of a pin that you have like a stitching needle.. The iron rings should be at a thickness of 1 cm... Join the two rings by
superimposing but not exactly, without gap. just leave a gap between the two circumference of the circle such that there is parallel gap throughout the
two rings then join them at two points{one at any where and another at straight opp to the other} using m~seal. no find the center by drawing a circle
with same radius with compass.(no need to measure the radius just estimate the radius by having the ring)then keep the rings on it and then take a wire
make it as the diameter with the contact points as ... Show more content on Helpwriting.net ...
To place a number of marks or stations in any given direction the horizontal plates are clamped together, the telescope pointed in the desired direction
and then moved vertically and focused on each mark as required.
Although the theodolite looks like a difficult instrument to use, its basic concept is very simple. The surveyor would begin by picking a distinct point
in the distance. After centering the theodolite over the primary point of interest, the surveyor would use the eye piece to align the sight axis with the
point of interest in the distance. Next the surveyor would zero the horizontal and vertical axes graduated circles. Finally the surveyor would move the
sight axis using the eye piece to the final point of interest and determine the horizontal and vertical angles between the points. With this known angle
and the triangulation principle, the distance between each of the three points may be determined.
See, told you that if you understand it is quite easy to use. One major difference between the clinometers and theodolite is that theodolite is used in
professional using ( NOT THIS ONE ::: THIS ONE IS HOMEMADE). THEY USE THE DIGITAL ONES|
U can go for anything.
1) Make use of thin sticks or Straws. Have papers cutting fix between two sticks to show angle. Decorate it according to yourself.
2) since you are in 10(C.B.S.E) your study of Trigonometry is concerned to 2–D only . even then you can try 3–d. Following is a description:

The Pythagorean Theorem Was Discovered And First Proven By...
Background:
The Pythagorean Theorem was discovered and first proven by the Greek mathematician, Pythagoras. The Pythagorean Theorem states that the sum of
the squares of the two legs of a right triangle equals the square of the hypotenuse of the triangle. In simpler words, when looking at the right triangle
below, aВІ+bВІ=cВІ. This major discovery in the history of mathematics lead to the accomplishments of many other basic things we do in life. The
Pythagorean Theorem does not just stop at the famous equation of aВІ+bВІ=cВІ, but it has many other aspects. Whole numbers that can fit into this
theorem are commonly referred to as Pythagorean Triples. The Pythagorean Theorem includes numerous amounts of poofs, from the basic proof to the
Once the concept of the Pythagorean Triples was discovered, mathematicians were eager to validate the concept. Euclid created a widely accepted
formula in deriving a Pythagorean Triple, which can be proved with the help of the unit circle and algebra.
Euclid's Formula and its Proof
Euclid's formula for a Pythagorean triple is: a = 2mn, b = m^2 – n^2, c = m^2 + n^2
The variables: m and n: stand for positive, rational integers. It is important to remember that m has to be greater than n, otherwise the statement valuing
variable b on a Pythagorean triangle would be false because b cannot
This can be understood in terms of the rational points on the unit circle. A unit circle is a circle with the radius of 1.
We can draw a right triangle: a and b are the legs c is the hypotenuse
In the next part, we must use the trigonometric identity of sin2+cos2=1. For representational purposes, we must look towards the following diagram for
how the triangle would be represented in this proof.
The next step in formulating this next equation would require us to find the sine and cosine of triangleABC. The sine formula is opposite/hypotenuse,
which would equate to a/c. Additionally, the formula for cosine is adjacent/hypotenuse, equating to b/c. Knowing the trigonometric identity, we can
derive the new eqution:
For the next step, we will need to know the equation of a [unit] circle: x2 + y2 = 1
We know that

Evaluation Of The Elementary School
This lesson was taught November 12, 2015 at Rosa Taylor Elementary School in Miss. Baggott's fifth grade class. That week the students went to
see Miller Middle School November 10th and were out of school for Veteran's Day on November 11th. This week in math the students were learning
about polygons and their characteristics. Triangles have a first and last name describing the sides and types of angles. There are different types of
quadrilaterals and other polygons with five for more signs. For the summative assessment given to the students by the cooperative teacher, the students
had to identify the shapes by their name, number of sides, number of angles, and (for triangles) their "first and last names." My lesson was titled
"review stations" where the class was broken up into groups of three or four. There were six stations that included: polygon collage, the shape game,
foldable focus group, triangle board game, tangrams, and computer games–provided by the teacher. Each station had a red folder that consisted of
everything the student needed for that particular station. The folder had written directions either on the sheet of paper or on the folder itself. For the
mini–lesson, we collectively wrote a chart in their interactive journals for notes. The chart consisted of five columns, including: prefix, name, number
of sides, number of angles, and shape. Their assessment for Friday asked questions about number of sides and number of angles a particular polygon
had. Each

Wave Patterns
The Effect Mixing Has on the Wave Patterns of a Song
Anna Rezhko
Grade 10
Woodbridge High School
January 16, 2014
Miss Cooper
Abstract The purpose of the experiment was to find the objective difference between an original song and an acoustic or a remix version of the same
song. The expectation was that the difference lies in the amount of contrast between the highest and lowest amplitude. Sound waves are created when an
object vibrates back and forth and they have amplitudes, wavelengths, and frequencies. Sixty songs were analyzed in Audacity, amusic program. Using
the output, their amplitudes were measured and compared with other versions having clean audio and the same artist. Most of the data supported the
hypothesis by stating ... Show more content on Helpwriting.net ...
The outliers to this was Lips Are Moving, Viva la Vida, Take me to Church, and Ice Ice Baby. A possible reason for this is differences between
recording circumstances. All artists have different equipment and that impacts the sound quality. Unfortunately, this is a variable that couldn't have
been manipulated due to lack of resources and remains an issue in the experimental design. The recording of Lips Are Moving especially had
unusably low Greatest Amplitude readings: 0.20, 0.17, and 0.18 cm. This is probably a flawed recording, not a program error, because all three trials
were outliers and the rest of the songs had different results. Ideally, in future experiments, all the artists should be brought in to one studio and had them
record in the exact same circumstances with the same

The Evolution Of The Topic
SubjectTopicCreate a written narrative of the evolution of the topic. Include significant contributions from cultures and individuals. Describe important
current applications of the topic that would be of particular interest for students.
Number Systems Complex NumbersThe earliest reference to complex numbers is from Hero of Alexandria's work Stereometrica in the 1st century AD,
where he contemplates the volume of a frustum of a pyramid.
The proper study first came about in the 16th century when algebraic answers for roots of cubics and quartics were revealed by Italian mathematicians
Tartaglia and Cardano. For example, Tartaglia's formula for a cubic equation x^3=x gives the solution as 1/в€
љ3 ((в€
љ(–1))^(1/3)+1/(в€
љ(–1))^(1/3) ).
For example, the treatment of resistors, capacitors, and inductors are unified by combining them in a single complex number called the impedance,
which is the measure of the opposition that a circuit presents to a current when a certain voltage is applied.
AlgebraThe Quadratic FormulaEarly methods for solving quadratic equations were purely geometric.
Babylonian tablets contained problems which could be reduced to solving quadratic equations. The Egyptian Berlin Papyrus (2050–1650 BC) contains
the solution to a two–term quadratic equation.
Euclid (300 BC) used geometric methods to solve quadratic equations in his book Elements.
In Arithmetica, Diophantus (250 BC) solved quadratic equations with methods which more closely resembled algebra. However, his solution only gave
one root, even when both roots are positive.
Brahmagupta (597–668 AD) explicitly described the quadratic formula in words instead of symbols in Brahmasphutasiddhanta in 628 AD. His solution
of ax^2+bx=c equated to the formula: x=(в€
љ(4ac+b^2 )–b)/2a
In the 9th century, Persian mathematician al–Khwarizmi solved quadratic equations algebraically.
The quadratic formula which covered all cases was first described by Simon Stevin in 1594.
The quadratic formula that we know today was published by Rene Descartes in La Geometrie in 1637.

The first appearance of the general solution in modern mathematical literature was in an 1896 paper by Henry

Mathematics Of Deciphering A Crime Scene
Amanda Stevenson
IB Mathematics SL
Block 1 – December 2014
Internal Assessment
The Mathematics Involved in Deciphering A Crime Scene
Introduction
Though one may assume that the logistics involved with deducing the events that occurred at a crime scene merely involve the reasonable deduction
skills of an individual, in reality adequate results can only be achieved through the use of certain mathematical methods. For this particular
investigation, the examples deal with a homicide case where it is known that there was only one shooter and one victim. In order to analyze blood
patterns and determine a time of death, the Pythagorean Theorem and Newton's Law of Cooling must be utilized. This exploration is important to my
future as an attorney with a focus on homicide cases. It will be important to discover the uncertainties associated with each of these methods, as they
appear quite simple to begin with.
Proving the Pythagorean Theorem
In an effort to fully understand the Pythagorean Theorem, I must first examine a method used to derive the formula. Whilst I examined several
different methods, I was confused as to how a person could think of such strategies. This method was first derived by Bhaskara. I used this proof,
because it best connects with the crime scene situation at hand. The use of multiple triangles and squares mimic the makeup of a room.
Bhaskara's Proof: I decided to manipulate Bhaskara's method and make it more understandable for my usage.
My interpretation

Friedrich Wilhelm 's Impact On Education
Friedrich Wilhelm August Froebel lived from April 21, 1782 to June 21, 1852. He was a German teacher, and he laid out the foundation for modern
education. His observations and actions were based on the recognition that all children have different needs and capabilities, which at the time was a
milestone in education. Furthermore, he invented the concept of "kindergarten," and he also created the educational toys that are known as Froebel
Gifts. Froebel is from Oberweissbach, Schwarzburg–Rudolstadt, Germany. Shortly after his birth, Froebel's mother 's health began to fail. When he
was nine months old, she died, which greatly impacted his life. In 1792, Froebel moved to the small town of Stadt–Ilm with his uncle. Then, at the age
of... Show more content on Helpwriting.net ...
Later, in 1840, he invented the word for the institute, which is where the term "kindergarten" comes from. Friedrich Froebel continued on to
design his Froebel Gifts, which are educational toys that include geometric building blocks and pattern activity blocks. They were called "gifts"
because they were given to the the children as gifts, and they also act as tools for adults to observe the human "gifts" each individual child
possesses. According to Froebel 's method, one can observe the qualities and ideas that make each child unique when they are able to explore and
create. Froebel 's main goal was to recognise how important it is for children to learn, and show how his Gifts could help. Additionally, he intended to
introduce young children to the adult world to help their minds grow and develop thoughtfully. He introduced the concept of "free work" into teaching
and class work. Some activities that took place in the first kindergarten involved singing, dancing, gardening and playing with the Froebel Gifts. It is
assumed by modern historians that Froebel spent a lot of time observing children and creating the designs for the Gifts. He simply named them as
Gifts One through Ten. These Gifts are different from other materials used in kindergartens, because they have the capability to return to their original
form when they are done being used. An important part of playing with the Gifts is that the Gifts are always presented as a whole form, and when the
time to

Compass And Straightedge
Compass and Straightedge: Basic Constructions and Limitations In Euclid's Elements, Book 1, the very first proposition states, "To construct an
equilateral triangle on a given finite straight–line." (Heiberg, Fitzpatrick, Euclid, pg 8) This proposition is saying that it is possible to construct an
equilateral triangle from a given segment. Euclid was able to perform this construction with just a straightedge and compass. As The Elements was
published in 300 BC (Heiberg, Fitzpatrick, Euclid, pg 4) only the most basic tools had been invented. The straightedge Euclid used the Euclidean ruler,
used only the draw the straight line through any two points (Martin, pg 6). There were no markings on the straightedge, unlike modern rulers. The
compass ... Show more content on Helpwriting.net ...
With modern tools such as a protractor and marked ruler, this question is easy to do. However, it is more difficult with only a straightedge and
compass. For thousands of years people tried to solve this problem and either ended up with an inexact construction or they had to use other tools than
the Euclidean tools. Finally in 1837 Pierre Laurent Wantzel, a French mathematicians, solved the problem (Major, Jost, pg 81). It turns out that in
general it is impossible to trisect any angle, but there are cases, for example 90 degrees and 180 degrees, that can be trisected with a straightedge and
compass (Courant, Robbins, pg 147). The degrees that can be trisected come from the n–gons that are constructible from a compass and straightedge.
An example of this is an equilateral triangle. The angles are 60 degrees, which is a third of 180 degrees. But in general, most angles cannot be
trisected. To prove that in general it is impossible to trisect an angle, it suffices to show that one angle cannot be trisected. For example take an angle
of 60 degrees. Let the angle be denoted by Пґ and cosПґ = y. Need to find x = cos(Пґ/3). Cosine of Пґ/3 is connected to cosПґ=y=4гЂ–cosгЂ—^3(Пґ
/3) – 3cos(Пґ/3) Thus trisecting an angle with cosПґ = y comes to гЂ–4zгЂ—^3–3z–y=0. Then take Пґ =60 degrees, and y = cos(60) = 1/2. Thus
гЂ–8zгЂ—^3–6z= 0. There is no rational number that satisfies z, thus the general holds, and it is impossible to trisect an angle (Courant, Robbins, pg

Rudy's Last Rudy: A Short Story
ripped out of its cuticle were minor concerns she reminded herself to block out the distraction of pain in what she knew to be the battle for her soul
she already felt she was losing. with equal force
They did try to help her, but every time they almost had her, she slipped away from them a tug of war between those two young men and the ocean.
Guess who won. Giselle psyched herself up over the next three rounds, then commanding every atom in her body to thrust herself onto the rocks.
Rudy could not manage to get far enough on the rocks and water to help he tried, but fell on his stomach cut up from chest to knees, he tried to grab
her hand, just as he almost had hold of her, the ocean spews her out only to swallow her back to its depths. ... Show more content on Helpwriting.net ...
(BEGINNING WILL HAVE DARPA AND THEGIANT SQUID, INVISIBILITY, AND MAGNETISM, AND WHAT THESQUID AND ITS
OFFSPRING HAD TO EAT WAS MAGNETIC, ALUMINUM, TIN, METAL, OF THAT SORT.)
Giselle could not recall ever seeing anything change so fast, not even in a film as the land and pilings zoomed into a tiny circle as she shot out to sea,
all she could think of was how hideous he looked with that tank, and BC twisted around his arms The high current pulled her out fast.
She gets herself psyched up forcing her body parallel to the shore her gauges are all wrong, indeed they were, one spinning around like the twilight
zone or just flew through the Bermuda triangle.
Holding the tank still, she won't let go, but at this point, it's because it has some air left in it
She didn't follow the first rule of diving don't panic, and though she technically didn't panic she didn't trust the sea any more, unaware she angered one
of its tenants. And that's how you show respect for the sea you respect it by believing it by knowing your limits knowing that if you don't she knew
that regard, real and 100% percent respect meant she knew the sea would win
Comply with the sea, or it will kick your ass every

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