Incorporating History of Mathematics In The Classroom

History in Mathematics is a difficult topic to incorporate into the curriculum. When people think of this, the almost immediate reaction is to try to come up with a project for students to research an historic mathematician and present it to the class. This is difficult to incorporate into the curriculum considering the amount of content that is currently being taught. The easy way out is always in the "Independent Study" area of the course. However, it may not be the most appropriate way.

One way to overcome the obstacle of trying to make it an entire unit, is to constantly include it in the class and curriculum. A simple way is to use problems historic in nature during a regular curriculum topic. Below you’ll find fantastic problems for use in many grades for topics as simple as Area, Perimeter, Systems of Equations, and so on.

Students identify with the past to make connections to the future. Using these in the classroom as examples or on tests is an excellent way for students to come to the realization that mathematics has been in existence for an eternity, and has been used in many forms depending on date and location.

Problems In Curriculum

The following problems were taken from the book of essays: Learning From The Masters, edited by Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson and Victor Katz.

  1. The partial sums of the series 1 + 2 + 3 + … are usually referred to as triangular numbers. In the seventeenth century, the mathematician Gottfried Wilhelm Leibniz (1646-1716) was challenged to develop a rule for the sums of the reciprocals of these partial sums. Develop a rule you think Leibniz might have concluded.

  2. In his Ars Magna of 1545, Gerolamo Cardano proposed the following problem: "Find two numbers whose sum is ten and whose product is 40." He then remarked, "Obviously this is impossible, but nevertheless let’s operate." Try to "operate" on this question to develop a conclusion. What do you find?

  3. The following problem was first posed over 2000 years ago in China. It is the fifteenth problem in the ninth chapter of Juizhang Susanshu (Nine Chapters on Mathematical Art): Given a right triangle with legs of length a and b and hypotenuse of length c, what is the length x of the side of the largest inscribed square utilizing the right angle as one of its vertices?

    4.

  4. A beam of length 30 feet stands against a wall. The upper end has slipped down a distance of 6 feet. How far did the lower end move? (Babylonia, c. 1800 B.C.) (answer: 18 ft.)

  5. The height of a wall is 10 feet. A pole of unknown length leans against the wall so that its top is even with the top of the wall. If the bottom of the pole is moved 1 foot farther from the wall, the pole will fall to the ground. What is the length of the pole? (China, 300 B.C.) (answer: 50.5 ft.)

  6. There is a 30 foot ladder leaning against a 30 foot tower. If the foot of the ladder is 18 feet from the foot of the tower, how far from the top of the tower does it reach? (Italy, A.D. 1300) (answer: 6 ft.)

  7. I have added the area and 2/3 the side of my square and it is 35/60. What is the side of my square? (Babylonia, 2000 B.C.) (answer: x=0.5 units)

  8. A gentleman received $4 a day for his labor, and pays $8 a week for his board; at the expiration of 10 weeks he has saved $144; required the number of idle and working days. (post-Civil War America) (answer: 14 idle days and 56 working days)

  9. Three hundred pigs are to be killed for a feast. They are to be killed in three batches on three successive days with an odd number of pigs in each batch. How can this be accomplished? (England, A.D. 775)

    10. The following problems were taken from the book: A History of Mathematics: For Secondary Schools, by H.A. Freebury.

  10. An area A, consisting of the sum of 2 squares, is 1000. The side of one square is 2/3 of the side of the other square, diminished by 10. What are the sides of the square? [copied from a Babylonian tablet, 1950 B.C.][answer: 10 units]
  11. Ahmes, an Egyptian scribe, wrote on the Rhind papyrus,"Heap, its seventh, its whole, it makes nineteen." "Heap" in Egyptian is a large number (we can call this number x). The object of this problem is to find a number such that the sum of it and 1/7 of it shall equal 19. [1650 B.C.]
  12. "Heap, together with its 1/5 makes 21. What is it?" [Ahmes/Rhind papyrus, 1650 B.C.]
  13. Show how Thales [Greek mathematician, 640-550 B.C.] measured the height of the pyramid, using the diagram below and the following proposition:

    14. The sides of similar triangles are proportional.

     

     

    The following problems were taken from the book: History of Mathematics, by Arthur Gittleman.

  14. A coin is tossed three times. A and B each bet $1, and the first to win two of the three tosses wins the bet. Suppose A chooses heads (for all three tosses) and wins the first toss, but then has to leave. How should the bet be divided? [modernized version of type of problem solved by Pascal and Fermat][answer: A should get $1.50 and B should get $0.50]

  15. Problem 62 of the Ahmes papyrus asks for the amount of each precious metal in a sack which contains equal weights of gold, silver, and lead. The sack is bought for 84 sha’ty. A deben of gold is worth 12 sha’ty, a deben of silver worth six sha’ty, and a deben of lead worth three sha’ty. [answer: 4]

  16. In Problem III.6 of his Arithmetic, Diophantus finds three numbers such that their sum is a square and the sum of any pair is a square. In effect, he finds numbers w, y, z, a, b, c, and d such that

w + y + z = a2

w + y = b2

y + z = c2

w + z = d2

  1. Following Diophantus, let a = x+1, b=x, c=x-1. Then show that w+z=6x+1.

  2. Choose d to be any number, say 11. Then find x, and from it find w, y, and z.

  3. Find a solution different from that found in part b.

  1. Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 39 dou. Two sheafs of good, three mediocre, and one bad are sold for 34 dou; and one good, two mediocre, and three bad are sold for 26 dou. What is the price received for a sheaf of each of good crop, mediocre, and bad crop? [first problem of chapter VIII in the Chinese Nine Chapters on the Mathematical Art][answer: 9 and ¼ dou for good, 4 and ¼ dou for mediocre and 2 and ¾ for bad]

    2. The following problems are Indian mathematics problems from the Vija-Ganita of Bhaskara (1150 A.D.):

  2. "[Arjuna], the son of Pritha, exasperated in combat, shot a quiver of arrows to slay Carna. With half his arrows he parried those of his antagonist; with four times the square root of the quiver-full, he killed his horse; with six arrows he slew Salya; with three he demolished the umbrella, standard and bow; and with one he cut off the head of the foe. How many were the arrows, which Arjuna let fly?" [Hint: Let x2 represent the total number of arrows.]

  3. "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 they in all?"

  4. "The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninths of the whole swarm; a female is buzzing to one remaining male that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees." [Hint: Let 2x2 be the number of bees.] {Indian math problem from the Vija-Ganita of Bhaskara (1150 A.D.)}

 

 

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Last updated January 31, 2000 by Annamae Lang and Nancy Yan