[Today’s class: Maya arithmetic, especially subtraction (use toothpicks and small candies---or pencil and paper),
review of Pythagorean Theorem and its converse,
Puzzle Proofs of Pythagorean Theorem activity,
Proofs of Pythagorean Theorem via area and algebra (see Pythagorean Theorem activity),
Historical applications of Pythagorean Theorem]
Mathematics 115
Homework Assignment #4
Due Monday, January 14, 2002
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Prof. Beery's office hours this week: Thursday 1/10 10:30 a.m.-12:30 p.m., 4-5 p.m.
Friday 1/11 1:30 - 3:30 p.m.
Monday 1/14 10:30 a.m.-12:30 p.m., 4-5 p.m.
and by appointment, Hentschke 203D, x3118
Tutorial session: Sunday, Jan. 13, 4 - 5 p.m., Hentschke 204 (Jody Cochrane)
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Read: "No Stone Unturned (Early Southern California math artifacts?)"
"Kernel revealing history of humans in the New World"
"Mayan Arithmetic" (you may skip Section 4, Division)
"Mayan Head Variant Numerals"
"Ancient Mayan Symbols Yield New Meaning"
All of these selections are in the Mathematics of the Americas section.
The following articles are in the Mathematics of Ancient Greece section.
“Pythagoras of Samos” (pages 399-401 in “The Greeks”)
“Pythagoras’ Trousers”
“The Pythagorean Problem” (pages 103 and 104 and top of page 105)
Do: Exercises 1 - 8 from the "Pythagorean Theorem Problems" handout
Answer the following questions about the "Mayan Arithmetic" essay.
1. What's the difference between the two Mayan number systems? List at least
four (4) differences.
2. Write the numbers 12 bak 16 kal 9 hun and 8 bak 6 kal 16 hun using Mayan
symbols, then add them together. Show all steps.
3. Write 12 kal, 6 hun and 5 kal, 18 hun using Mayan symbols, then
subtract the second number from the first number. Show all steps.
4. Write 12 bak 16 kal 9 hun and 8 bak 6 kal 16 hun using Mayan symbols,
then subtract the second number from the first number. Show all steps.
Monday's class will begin at 1:15 p.m. and will end at the usual time.
Monday's quiz (50 points) will cover all class work, reading, and homework since
the start of Interim!
Read ahead: "Hydraulic Societies" and "Prime Numbers" (way up front)
Pythagorean Theorem Problems
1. Use the Pythagorean Theorem to find the length of the third side of the right triangle.
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1a.
1b.
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1c.
2. The converse of the Pythagorean Theorem says: If a triangle has sides of lengths a, b, and c and a2 + b2 = c2, then the triangle is a right triangle with legs of lengths a and b and hypotenuse of length c. Use the Pythagorean Theorem and its converse to determine whether or not each triangle is a right triangle.
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2a. 2b.
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2c.
Exercise 3 is from a medieval European military handbook.
3. (Johnson and Mowry, 364) An army wishes to scale a 24-foot high castle turret. If the turret lies across an 18-foot wide moat, how long must the army’s ladder be?
Exercise 4 is from the Cairo Papyrus, an Egyptian mathematical papyrus dating from about 300 BCE. Of its 40 mathematical problems, nine require use of the Pythagorean Theorem.
4. (Burton, 76) A ladder of length 10 cubits leans against a wall. If the foot of the ladder stands 6 cubits from the wall, how high up the wall does the ladder reach?
Exercise 5 is from an ancient Mesopotamian clay tablet from the Old Babylonian period (1900-1600 BCE).
5. (Bunt, et al, 59) A beam of length 30 units stands vertically against a wall. As the top of the beam slides down the wall, the bottom of the beam slides away from the wall. If the top of the beam slides 6 units down the wall, how far does the bottom of the beam slide away from the wall? (Hint: The beam becomes the hypotenuse of a right triangle.)
Exercise 6 is from the Lilavati, written by the Indian mathematician and astronomer, Bhaskara II, in about 1150 CE.
6. (Plofker, 23) In a
lake with flocks of geese and cranes, the tip of a lotus bud stands 1/2 hasta
above the surface. The wind slowly pushes the bud until it is submerged at a
distance of exactly 2 hastas from its original position. What is the
depth of the water? What is the height of the lotus plant?
Exercises 7 and 8 are from the Chinese text, Nine Chapters on the Mathematical Art, which originally was composed in about 200 BCE, and rewritten and expanded in about 250 CE by Liu Hui. Like Euclid's Elements, the Nine Chapters was a compilation of the mathematics known to that day (Swetz and Kao, 17, 26-27). The subject of Chapter 9 of the Nine Chapters is the Pythagorean Theorem.
7. (Katz, 42) A pole leans against a 10 chi high wall so that the top of the pole is even with the top of the wall. If the bottom of the pole is moved just 1 chi further from the wall, the pole will fall flat on the ground without sliding any further from the wall. What is the length of the pole? (Hint: Begin by writing the length of the hypotenuse in terms of x.)
8. (Swetz and Kao,
42-44) A measuring rod has the same length as the diagonal of a rectangular
door. If the measuring rod is 4 chi longer than the width of the door
and 2 chi longer than the height of the door, find the dimensions of the
door. (Hint: What are the lengths of the legs of the triangle in terms of x?)
References:
Bunt, Lucas N.H., Phillip S. Jones, and Jack D. Bedient, The Historical Roots of Elementary Mathematics,
Dover Publications, New York, 1988.
Burton, David, The History of Mathematics: An Introduction (3rd edition), McGraw-Hill, New York, 1997.
Johnson, David, and Thomas Mowry, Mathematics: A Practical Odyssey, PWS Publishing, Boston, 1995.
Katz, Victor J., A History of Mathematics: An Introduction, Addison-Wesley, Reading, Massachusetts, 1998.
Plofker, Kim (translator), The Lilavati of Bhaskara II (unpublished manuscript), Brown University,
Providence, R.I., 2000.
Swetz, Frank J., and T.I. Kao, Was Pythagoras Chinese?: An Examination of Right Triangle Theory in
Ancient China, National Council of Teachers of Mathematics, Reston, Virginia, 1977.