Session 04. the Value of Mathematical Discoveries

SESSION 04. THE VALUE OF MATHEMATICAL DISCOVERIES MR. ROLDAN S. CARDONA, PNU-NL

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THE VALUE OF MATHEMATICAL

DISCOVERIESSession 04


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OBJECTIVES

To determine the different mathematical discoveries; and To show appreciation of the importance of these mathematical discoveries.


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ACTIVITY The Pythagorean Theorem Game


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REVIEW ON PYTHAGOREAN THEOREM Pythagorean theorem, also known as Pythagoras's theorem, is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Sources: https://www.mathsisfun.com/pythagoras.html; http://platformninenthreequarters.blogspot.com/2014/05/may-7th-pythagorean-theorem-video-links.html; http://en.wikipedia.org/wiki/Pythagorean_theorem

https://www.mathsisfun.com/pythagoras.html

https://www.mathsisfun.com/pythagoras.html

http://platformninenthreequarters.blogspot.com/2014/05/may-7th-pythagorean-theorem-video-links.html




http://en.wikipedia.org/wiki/Pythagorean_theorem




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REVIEW ON PYTHAGOREAN THEOREM Pythagorean Triples are measures of a right triangle which are integer. Example is the 3-4-5 triangle. That is, a2+b2 = 9+16 = 25 =c2


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GET READY!

Materials

• The Pythagorean Theorem Game master

• The Pythagorean Theorem Game Board master

• The Pythagorean Theorem Game Cards masters

• index cards

• scissors

• tape or glue

• 2 number cubes per group

• 4 different colored counters per group


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GET SET!

Make a copy of The Pythagorean Theorem Game master on for each student in the class. Photocopy The Pythagorean Theorem Game Board master onto card stock for each group. Make a copy of The Pythagorean Theorem Game Cards masters for each group. Have students cut out the game cards, tape or glue them to the index cards, and draw a “?” on the reverse side.


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GO!

A player rolls both number cubes and substitutes the numbers into the Pythagorean Theorem for the lengths of the legs. Then the player moves around the board a distance that is closest to the value of c. For example, if a player rolls a 1 and a 2, he or she would determine how many spaces to move as follows.

222 21 c241 c25 cc5c.2


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GO!

When a player lands on a space with a question mark, a question card is read to the player whose turn it is. If the player answers correctly, he or she can roll one number cube and advance that number of spaces. If the player answers incorrectly, the turn moves to the next player.


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GO!

To finish the game, the players must answer a question card correctly. If answered incorrectly, the player must go back to the space from which he or she started that turn. The first group who goes the farthest wins.


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PLAY PROPER


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ANALYSIS Insights Sharing


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ANSWER THE QUESTIONS

What challenges you encountered in playing the game? How did you address those challenges? How important the theorem is finding the measure of the hypotenuse? To what real-life activities can you apply the theorem?


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VIDEO PRESENTATION

Source: https://www.youtube.com/watch?v=eJ6ky97LaBc

Instruction: Watch and take note of the most important points?

https://www.youtube.com/watch?v=eJ6ky97LaBc




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What insights did you gain from the video presentation?


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ABSTRACTION


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PARAMETER OF THIS SESSION Pi Euler Number or the e Zero Pythagorean Theorem


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PYTHAGOREAN THEOREM, WHERE DID IT COME FROM? Often associated with PythagorasLived 5th Century B.C.Founder of the Pythagorean BrotherhoodGroup for learning and contemplation


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WHERE DID IT COME FROM?

Found in ancient Mesopotamia, Egypt, India, China, and even GreeceKnown in China as “Gougo Theorem”Oldest references are from India, in the Sulbasutras, dating from sometime the first millenium B.C.

The diagonal of a rectangle “produces as much as is produced individually by the two sides.”


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IT WASN’T PYTHAGORAS?

A common discoveryHappened during prehistoric times

Theorem came “naturally”Independently discovered by multiple culturesSupported by Paulus Gerdes, cultural historian of mathematicsCarefully considered patterns and decorations used by African artisans, and found that the theorem can be found in a fairly natural way


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PROOFS OF PYTHAGOREAN THEOREM Whole books devoted to ways of proving the Pythagorean Theorem

Many proofs found by amateur mathematiciansU.S. President James Garfield He once said his mind was “unusually clear and vigorous” when studying mathematics

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EUCLID’S PROOF The idea is to prove that the little square (in blue) has the same area as the little rectangle (also in blue) and etc.

He does so using basic facts about triangles, parallelograms, and angles.


http://en.wikipedia.org/wiki/Image:Illustration_to_Euclid's_proof_of_the_Pythagorean_theorem.svg


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APPLICATIONS OF PYTHAGOREAN THEOREM

Computing distances between two points, such as in

navigation and land surveying.

Another important application is in the design of ramps.

Ramp designs for handicap-accessible sites and for

skateboard parks are very much in demand.


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PI

http://holon.newgrounds.com/news/post/918235

The mathematical constant Π is an irrational real number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter in Euclidean geometry.Π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209...


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First Introduced by William Jones in “Synopsis Palmariorium Mathesios” on 1706. Made Standard by Leonard Euler Greeks, Babylonians, Egyptians and Indian: slightly more than 3 Indian and Greek: Madhava of Sangamagrama: Ahmes: Babylonians:

1 1 12

1

2

114

k k kk

81256

825

HISTORY OF PI

2rA

http://upload.wikimedia.org/wikipedia/commons/3/36/Pi_eq_C_over_d.svg


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PI AND THE BIBLE

A little known verse of the Bible reads:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about (I Kings 7, 23).


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BITS OF INFORMATION ABOUT PI Archimedes c. 250 BC proved

A = π r2.

John Lambert 1761 proved

π is irrational.

Ferdinand von Lindemann 1882 proved

π is transcendental.


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PI JOKES

Q: What do you get when you take the sun and divide its circumference by its diameter?

A: Pi in the sky.

Said the Mathematician, "Pi r squared.“

Said the Baker, "No! Pie are round, cakes are square!"


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PI MNEMONICS (WORD LENGTH) How I wish I could calculate pi.

How I wish I could enumerate Pi easily, since all these horrible mnemonics prevent recalling any of pi's sequence more simply.

How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics. One is, yes, adequate even enough to induce some fun and pleasure for an instant, miserably brief.

Do you see a problem with mnemonics?


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ARCHIMEDES CALCULATION

Perimeters of 96-sided polygons inscribing a circle and inscribed by it

π is between 223⁄71 and 22⁄7. The average is 3.1419.


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ISAAC NEWTON

Isaac Newton was able to calculate 15 decimals in the pi series. It is said that Newton spoke about his work with pi saying he was ashamed how long it took him just to calculate the 15 digits without working on anything else.


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DEVOTED HIS LIFE TO 35 DIGITS OF PI By 1610, the German mathematician Ludolph van Ceulen computed the first 35 decimal places of π.

He was so proud of this accomplishment that he had them inscribed on his tombstone.


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LOVE FOR PI

In 1706 John Machin developed a converging series for pi that calculated up to 100 decimal places.

By the year 1949 digital computers were invented and could calculate 100’s of decimal places of pi in hours. In 1949 John von Nuemann calculated 2037 digits of pi in 70 hours.


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CURRENT RECORD

December 2002, Yasumasa Kanada of Tokyo University correctly computed π to 1.24 trillion digits


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DO WE NEED THAT MANY DIGITS? While the value of pi has been computed to billions of digits, practical science and engineering will rarely require more than 100 digits. As an example, computing the circumference of a circle the size of the Milky Way with a value of pi truncated at 40 digits would produce an error margin of less than the diameter of a proton. But we can use them for testing computers


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CELEBRATING PI DAY

Pi Minute – 3/14 1:59

Pi Moment – 3/14 1:59:27

In Europe Pi Approximation Day – 22 July.

Albert Einstein’s birthday

Happy Pi Day!

PI IN THE PROFESSIONS.

Agricultural professionals may use pi to determine the area covered by a pivot irrigation system or storage facility. The would use the formula

Architects and construction works would both use the formula for area extensively. hey also use the formula for volume extensively to fill columns of concrete and to know the space a building would take up


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PI IN THE PROFESSIONS Engineers use advance formulas that include pi.

These are just some of the formulas an engineer would use.

Moving around structures such as landmasses and buildings would require the use of some of these formulas.


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IN LITERATURE

Math Adventure Books by Cindy Neuschwander

Sir Cumference and the …

… Dragon of Pi

… First Round Table

… Sword in the Cone

… Great Knight of Angleland


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CHARACTERS

Sir Cumference

Wife: Lady Di of Ameter

Son: Radius

Friend: Vertex

Sword: Edgecalibur

Carpenters: Geo and Sym of Metry


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When Sir Cumference drinks a potion which turns him into a dragon, his son Radius searches for the magic number known as pi which will restore him to his former shape


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ZERO

http://sarathc.com/zero-emissions-zero-accidents-zero-ownership.html

0: zero, null, nill. Invented about 825 by Al-Kwarizmi


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VIDEO PRESENTATION

Watch the video presentation entitled Discovery of Zero – BBC India.


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VIDEO LEARNING

What were the important points emphasized in the video presentation?


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E OR THE EULER NUMBER

http://www.gogeometry.com/software/word_cloud_sw_e_euler_number.html

An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. It models a variety of situations in which a quantity grows or decays continuously: money, drugs in the body, probabilities, population studies, atmospheric pressure, optics, and even spreading rumors!


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THE NUMBER E

The number e is known as Euler’s number. Leonard Euler (1700’s) discovered its importance. The number e has physical meaning. It occurs naturally in any situation where a quantity increases at a rate proportional to its value, such as a bank account producing interest, or a population increasing as its members reproduce.

y = e

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n

yn


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Euler eventually related all five of math’s most important numbers in his famous “Euler’s Identity”:

E = 2.71828 18284 59045 23536 …

e can be expressed as The constant was first discovered by Jacob Bernoulli when attempting a continuous interest problem

Was originally written as “b”

Euler called it “e” in his book Mechanic

Is also called Euler’s number

One of the five most important numbers in mathematics along with 0, 1, i, and pi.

THE NUMBER E - DEFINITIONn

1 2

2 2.25

5 2.48832

10 2.59374246

100 2.704813829

1000 2.716923932

10,000 2.718145927

100,000 2.718268237

1,000,000 2.718280469

1,000,000,000 2.718281827

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n

n

0

11

n

An

1, 1

n

As n en

The table shows the values of as n gets increasingly large.

n As , the approximate value of e (to 9 decimal places) is ≈ 2.718281827


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WHY DOES EULER’S NUMBER MATTER

xe

ietie

)( ctxie

)(xe ti

Describes things that grow/decay

Describes things that oscillate

Alternating current

Radio/sound wave

Quantum mechanical wave packet

kteAA 0kteQQ 0ktPeA


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COMBINING THEM…

n

ntinectu 2)(

deFtu ti

)(2

1)(

Fourier series:

sound synthesisers, electronicsFourier transform:

Image processing, crystallography, optics, signal analysis

2// 222 dxdgexdheef xixiyi


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COMPOUND INTEREST

The formula for compound interest:

( ) 1

ntr

A t Pn

Where n is the number of times per year interest is being compounded and r is the annual rate.


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COMPOUND INTEREST - EXAMPLE Which plan yields the most interest? Plan A: A $1 investment with a 7.5% annual rate compounded monthly for 4 years

Plan B: A $1 investment with a 7.2% annual rate compounded daily for 4 years

A:

B:

12(4)0.075

1 1 1.348612

365(4)0.072

1 1 1.3337365

$1.35

$1.34


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INTEREST COMPOUNDED CONTINUOUSLY

If interest is compounded “all the time” (MUST use the word continuously), we use the formula

where P is the initial principle (initial amount)

( ) rtA t Pe


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( ) rtA t Pe

If you invest $1 at a 7% annual rate that is compounded continuously, how much will you have in 4 years?

You will have a whopping $1.32 in 4 years!

(.07)(4)1* 1.3231e


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YOU DO!

You decide to invest $8000 for 6 years and have a choice between 2 accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?

YOU DO ANSWER

1st Plan:

2nd Plan:

0.0685(6)(6) 8000 $12,066.60P e

12(6)0.07

(6) 8000 1 $12,160.8412

A


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APPLICATION


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TASK 1

Provide a reflection on the value of these mathematical discoveries in your life as a student, a future teacher and as a citizen of the country. Include in the reflection more applications and uses of these concepts in everyday living. Indicate the reflection in your mathematics journal.


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TASK 2

Compile five application problems for each of the mathematical concepts: pi, e, zero, and Pythagorean theorem. Indicate it in your math journal. Cite sources.

Session 04. the Value of Mathematical Discoveries

Documents

Transcript of Session 04. the Value of Mathematical Discoveries