In order to get full credit in the open ended questions, you need to spend some time reflecting upon and writing your answers. In most of these problems, the probability that a one-sentence answer will get full credit is close to 0 (and so will be what you learn from the exercise)
Make sure you show all your work on the problems that require it so. Otherwise, even if you give a correct answer, if you do not explain how you obtained it, you'll get very little or no credit.
It would be great if you discussed ideas with your classmates. The write-up, however, must be done individually.
Recall that the slides of the lectures can be found here.
Egyptian multiplication/division a) Multiply 37 by 40 using the Ancient Egyptian method. b) Divide 1935 by 43 using Ancient Egyptian techniques.
Why base 2 for the algorithm? Try to imitate the Egyptian doubling method but using tripling instead: use powers of 3 to compute 6 × 15. Explain why this approach breaks down. State clearly the advantage of 2 over 3 (or any integer >2) in this algorithm.
Alternative to Problem 2 (if you prefer a proof‑style task): Explain why the Egyptian multiplication algorithm works in terms of powers of 2. Could you design a similar algorithm using powers of 3? Why is base 2 more appropriate? (Hint: investigate whether every positive integer can be written as a sum of distinct powers of 3.)
Rhind Papyrus circle rule A circular field has diameter 9 khet. Compute its area using the Rhind rule. Then generalize: starting from the diameter‑9 example, derive a formula for a circle of diameter d. Using your result, determine the constant the Egyptians used to approximate π and compare it to the modern value.
Short writing (choose one prompt) ~100–200 words Choose one and write two short paragraphs in your own voice: a) What was the purpose of the Rhind Papyrus? b) Your view on Problem 79 of the Rhind Papyrus—practical or recreational? c) Compare Egyptian mathematics as practical vs. pure science.
Chace's transations of the Rhind Papyrus (optional for quick lookup):
here,
here and a third one
here.
Extra Credit. Moscow Papyrus—volume of a pyramid Imagine you are an Egyptian scribe. Write clear step‑by‑step instructions (in English) to compute the volume of a square pyramid with side 9 and height 9. (Hint: follow the ideas of Problem 14 of the Moscow Papyrus.) Deduce algebraically the volume formula of a truncated square pyramid. Why couldn't this derivation have been done in ancient Egypt?
B. Ancient Mesopotamian Mathematics
Sexagesimal conversions Convert 3/10 to sexagesimal (base‑60), showing steps, and then convert to cuneiform.
Square‑root algorithm (Babylonian method) Starting from the guess a = 3, compute one iteration to approximate √10. Explain the steps of the algorithm and include the appropriate pictures illustrating your argument. Make sure to label the length that is your answer.
Coefficient sense‑making In Mesopotamian geometry, a "coefficient" encodes a formula. Give a modern interpretation of 7/8 as the coefficient for the equilateral triangle. Provide a quick numerical example (in Hindu Arabic numerals) of how this coefficient could be used.
Area by Babylonian rule Using the Old Babylonian rule for a trapezoid ("average of sides × average of bases"), compute the area for the figure below. Give your result in Hindu‑Arabic numerals. Then comment briefly: why is this rule not generally correct? Provide a concrete counterexample sketch.
Short interpretive: Plimpton 322 In a few sentences (100-200 words), explain what Plimpton 322 is without relying on modern labels like "trigonometry table." Summarize one scholarly interpretation (e.g., Robson discussed in class) and state why the tablet still demonstrates mathematical sophistication regardless of the debate. Aim for clarity, not length.
Sample Quiz 2: Egypt
Make sure you show all your work. Even if the answer is correct, if you do not explain how you got it, you'll have very little or no credit.
Three of the following four problems about Ancient Egyptian Mathematics:
Egyptian multiplication: Multiply 23 by 35 using the Ancient Egyptian method.
Egyptian division: Divide 1680 by 28 using Ancient Egyptian techniques.
Circle area: A circular field has diameter 12 khet. What is its area according to the Rhind Papyrus method? Show your steps.
Powers of 2: Explain in 2-3 sentences why the Egyptian multiplication algorithm works. Why can every positive integer be written as a sum of distinct powers of 2?
Sample Quiz 3: Mesopotamia
Make sure you show all your work. Even if the answer is correct, if you do not explain how you got it, you'll have very little or no credit.
Three of the following four problems about Ancient Babylonian Mathematics:
Sexagesimal conversion: Convert 7/12 to sexagesimal (base-60) notation. Use this to conver the fraction to cuneiform. Show your steps.
Square root algorithm: Using the Babylonian method, compute one iteration to approximate √10 starting from the guess a = 3. Show your calculation.
Coefficient interpretation: In Mesopotamian geometry, what does the coefficient 1/2 for a square's side mean? Give a numerical example using a square with side 6.
Area calculation: Using the Babylonian rule for a trapezoid (average of sides × average of bases), find the area of a trapezoid with parallel sides 8 and 12, and non-parallel sides both equal to 5. Then explain why this rule is not generally correct.
Grading Rubric
Problem A1: Egyptian Multiplication/Division (10 points)
Component
Points
Criteria
Part (a): Multiplication 37 × 40
4
Correct use of Egyptian doubling method with accurate result
Part (b): Division 1935 ÷ 43
4
Correct use of Egyptian division technique with accurate result
Work shown
2
Clear presentation of steps and calculations
Total
10
Problem A2: Why Base 2 Algorithm (10 points)
Component
Points
Criteria
Attempt at tripling method
3
Shows attempt to use powers of 3 for 6 × 15
Explanation of breakdown
4
Clearly explains why the tripling approach fails
Advantage of base 2
3
States clear advantage of 2 over other integers
Total
10
Problem A3: Rhind Papyrus Circle Rule (10 points)
Component
Points
Criteria
Area calculation (d=9)
3
Correct application of Rhind rule for diameter 9 khet
General formula derivation
4
Derives correct formula for arbitrary diameter d
Egyptian π constant
2
Determines Egyptian approximation of π
Comparison to modern π
1
Compares Egyptian value to modern π
Total
10
Problem A4: Short Writing Essay (10 points)
Component
Points
Criteria
Prompt selection
1
Clearly chooses one of the three prompts
Content knowledge
4
Demonstrates understanding of Rhind Papyrus and Egyptian mathematics
Personal voice
3
Written in student's own voice with clear opinions/analysis
Length and structure
2
Meets 100-200 word requirement, organized in two paragraphs
Total
10
Problem B1: Sexagesimal Conversions (10 points)
Component
Points
Criteria
Sexagesimal conversion
5
Correctly converts 3/10 to base-60 with three places
Cuneiform notation
3
Accurately represents result in cuneiform symbols
Steps shown
2
Clear presentation of conversion process
Total
10
Problem B2: Square-Root Algorithm (10 points)
Component
Points
Criteria
√2 iteration (a=3/2)
4
Correct application of Babylonian method for √2
√5 iteration (a=2)
4
Correct application of Babylonian method for √5
Algorithm explanation
2
Explains the update step in words
Total
10
Problem B3: Coefficient Sense-Making (10 points)
Component
Points
Criteria
Modern interpretation
6
Correctly interprets 1/3 as coefficient for circle's diameter
Numerical example
4
Provides clear numerical example demonstrating the concept
Total
10
Problem B4: Area by Babylonian Rule (10 points)
Component
Points
Criteria
Area calculation
4
Correctly applies Babylonian trapezoid rule to given figure
Hindu-Arabic result
2
Provides final answer in Hindu-Arabic numerals
Rule incorrectness
2
Explains why the Babylonian rule is not generally correct
Counterexample
2
Provides concrete counterexample with sketch
Total
10
Problem B5: Plimpton 322 Interpretation (10 points)
Component
Points
Criteria
Tablet description
3
Explains what Plimpton 322 is without modern labels
Scholarly interpretation
4
Summarizes one interpretation (e.g., Robson) accurately
Mathematical sophistication
2
Explains why tablet demonstrates sophistication regardless of debate
Clarity and length
1
Clear writing within 100-200 word range
Total
10
Extra Credit: Moscow Papyrus Pyramid Volume (5 points)
Component
Points
Criteria
Egyptian scribe instructions
2
Clear step-by-step English instructions for pyramid volume
Truncated pyramid formula
2
Algebraic derivation of truncated pyramid volume
Historical limitation
1
Explains why derivation couldn't be done in ancient Egypt
