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These are a few of my favorite things..... Using History to “Spice” up a Math Class Shawnee McMurran mcmurran@math.csusb.edu

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Algebra: Completing the Square Abu Ja'far Muhammad ibn Musa Al-Khwarizmi Born: about 780 in Baghdad (now in Iraq) Died: about 850

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Al-jabr The Arabic word al-jabr is the source of our word algebra. In Arabic al-jabr means “restoring”.

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x x10

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x x

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x x5 5

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x x5 5

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Writing across the curriculum

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From Mahāvīra (9 th century India) One night, in a month of the spring season, a certain young lady was lovingly happy along with her husband on the floor of a big mansion, white like the moon, and situated in a pleasure garden with trees bent down with the load of the bunches of flowers and fruits, and resonant with the sweet sound of parrots, cuckoos, and bees which were all intoxicated with the honey obtained from the flowers therein.

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Then on a love quarrel arising between the husband and the wife, that lady’s necklace made up of pearls became sundered and fell on the floor. One-third of that necklace of pearls reached the maid-servant there; one-sixth fell on the bed; then one-half of what remained (and one-half of what remained thereafter and again one-half of what remained thereafter and so on, counting six times in all) fell all of them everywhere; and there were found to remain (unscattered) 1161 pearls. Give out the measure of pearls in that necklace.

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Word Problems: It’s a tradition Bhāskara (7 th century) The eighth part of a troop of monkeys, squared, was skipping in a grove and delighted with their sport. Twelve remaining monkeys were seen on the hill, amused with chattering to each other. How many were there in all?

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Calculate like an Egyptian

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Circles on the Nile Cut a piece of string to be the diameter of your circle. Mark a starting point on your circle. Carefully lay your diameter string in the valley of your circle. Mark your new start at the end. Repeat until you reach the original “start”. How many diameters fit in your circle? Answer: 3 and a little more (i.e. “ ”) Using your string as a radius, draw your circle in the sands of the Nile.

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A Basket for Pi Calculating Pi via the surface area of a “basket” (hemisphere) that is 4½ units in diameter Answer: 32 Method: d(8/9)(8/9)d

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Babylonian Problem Sets Formula for the area of a circle on a Babylonian Tablet (c.1700 BCE) A=C 2 /12 or A=5C 2 since 5 is the reciprocal of 12.

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The People behind the Math

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Leonhard Euler (1707-1783) 886 papers and books! (fills about 90 volumes) and 13 children!

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Guillaume De l'Hôpital 1661-1704 Analyse des infiniment petits pour l'intelligence des lignes courbes (1692) was the first text-book to be written on the differential calculus

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Guillaume De l'Hôpital 1661-1704 Analyse des infiniment petits pour l'intelligence des lignes courbes (1692) was the first text-book to be written on the differential calculus L'Hôpital’s Example for his rule: L'Hôpital’s Rule is really Johann Bernoulli’s rule. (Unfortunately Bernoulli failed to read the fine print.)

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William Sealey Gosset 1876-1937 The “Student” of the t distribution Gosset was a Guinness Brewery employee who needed a distribution that could be used with small samples. The brewery did not allow the publication of research results, hence the pseudonym Student.

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Derivatives, Second Derivatives and Recent History

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Look at the CPI for the last 50 years – some sample questions Does the data supply a possible reason for Jimmy Carter’s failure to win a second U.S. presidential term? Does the data supply a possible reason for the success of George Bush, Sr. in winning the U.S. presidency? Does the data supply a possible reason for the results of the latest presidential election? Explain

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Max or Min? How should the aircraft be proportioned to maximize range? The derivative of range with respect to volume is zero when the volume is almost all in the wing. Oops – the second derivative is positive under these circumstances!

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Math Victorian Style Down to our own time many persons have believed that an educated mother would be “in danger of deserting her infant for a quadratic equation.” (Arthur Gilman quoting Sydney Smith – 1888)

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Philippa Fawcett Took the Cambridge Tripos exam in 1890 The exam was 36 hours long spread over 6 days (140 problems) She beat the senior wrangler by 400 pts, (13 percentage points)

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Some of those exam questions Show that the sum of the products three together of the first n odd numbers is: Show that the equation has at most three real roots. Find the chance of throwing 8 at least once in n throws of a dice.

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Hail the triumph of the corset Hail the fair Philippa Fawcett Victress in the fray Crown her queen of hydrostatics And the other mathematics Wreathe her brow in bay. If you entertain objection To such things as conic sections Put them ought of sight Rather sing of the essential Beauty of The differential Calculus tonight. Worthy of our approbation She who works out an equation By whatever ruse Brighter than the rose of Sharon Are the beauties of the square on The hypotenuse. Curve and angle let her con and Parallelopipedon and Parallelogram Few can equal, none can beat her At eliminating theta By the river Cam. May she increase in knowledge daily Till the great professor Cayley Owns himself surpassed Till the great professor Salmon Votes his own achievements gammon And admires aghast.

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BSc/HND IETM Week 9/10 - Some Probability Distributions.

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