
ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary 
Please keep in mind that this is a research project, and there may sometimes be glitches with the interactive software. Please let us know of any problems you encounter, and include the computer operating system, the browser and version you're using, and what kind of connection you have (dialup modem, T1, cable).Pythagoras was an Ancient Greek mathematician who loved to put all the numbers he encountered into different categories. Archeologists recently uncovered Pythagoras' island of Samos and found several puzzling artifacts that indicated he was working on a major new category of numbers.
Your job is to understand what kind of numbers Pythagoras was trying to categorize. One artifact found on the island was the following tablet with three columns. The second and third columns were badly damaged. The first column shows the areas of different sized squares and the second shows the lengths of the sides of the squares. The third column has some strange symbols in it that may point to the new categories of numbers Pythagoras was making.
Pythagoras' Mystery Tablet
The square areas on the tablet are:
4 5/36 3 1/9 2.25 9 2 25/16The tablet shows that Pythagoras was looking at some relationships in a square to discover a category of new numbers. Use the following applet to continue his work.
Questions:
 Use the applet to answer these questions:
 If side length = 4, area = ?
 If side length = 3/4, area = ?
 If side length = 2.5, area = ?
 What is the relationship between the side length and area of a square?
 Use the applet to find the corresponding side lengths for each area on the mysterious tablet. Describe your strategy for finding each side length (or explain why you weren’t able to find one).
 If area = 9, side length = ?
 If area = 3, side length = ?
 If area = 1/9 (0.111111...), side length = ?
 If area = 5/36 (0.138888...), side length = ?
 Think about the difficulty of computing some of the side lengths versus others. Explain what Pythagoras was doing in the third column when he put the side lengths into two categories of numbers.
Bonus Question:
How would you translate the two category symbols (~ and *) in the third column into English?
Most had no trouble with questions 1 and 2, although I have a feeling that many simply used a calculator and did not attempt to use the applet. We wanted you to play with the applet so you could see what you'd have to do to compute a square root without the use of a calculator. How do you think Pythagoras computed them?The theme of rational and irrational numbers existed throughout the answers for both questions 1 and 2, but the terms were often used incorrectly. This, combined with not using the applet made question 3 and the bonus difficult for students to solve correctly. The applet should have shown how difficult it was to find some square roots, but the use of a calculator allowed some kids to never try and to simply write, "no square root, it is irrational."
For problem 3 and the bonus, almost all of the submitters wrote that Pythagoras divided the numbers using rationality. This appeared correct for most of the solutions, but not all. I think most thought that irrational numbers included repeating decimals, and thus some fractions. It was difficult to get the kids to see the problem from an angle that didn't include irrational numbers without explicitly telling them what to focus on or that any number that can be written as a fraction must be rational. The idea of an irrational number was a huge problem for a lot of people.
It's not clear that the applet helped much for those who didn't already understand the concept. For example, I found that some kids stopped searching for the square root after one decimal place  e.g., 0.4 was close enough for an area of 2.

1. Use the applet to answer these questions: a. If side length = 4, area = 16 b. If side length = 3/4, area = 9/16 c. If side length = 2.5, area = 6.25 d. What is the relationship between the side length and area of a square? The square of the side lengh equals the area of the square 2. Use the applet to find the corresponding side lengths for each area on the mysterious tablet. Describe your strategy for finding each side length (or explain why you weren’t able to find one). a. If area = 9, side length = 3 The square route of 9 is 3. b. If area = 3, side length = Could not find one because the side length was not rational. A approximate answer would be 1.73. Since 1^2 is 1 and 2^2 is 4, The square route of 3 is between 1 and 2. It would be closer to 2 because 3 is closer to 4, so I plugged in the number 1.7. Then seeing that the square of 1.7 was close to three I added another digit to the right until the square of that number was pretty close to 3 but not over. c. If area = 1/9 (0.111111...), side length = 1/3 since the square route of 9 is three, the square route of 1/9 is 1/3 d. If area = 5/36 (0.138888...), side length = Couldn't find one because the side length was not rational. Though 36's square route is 6, 5's square route does not repeat and cannot be converted to fraction form. An approximate answer would be 1/3, since 36's square route is 6 and 5's square route is around 2. 3. Think about the difficulty of computing some of the side lengths versus others. Explain what Pythagoras was doing in the third column when he put the side lengths into two categories of numbers. At that time he maybe just thought of which solutions on the tablet he could solve and which he could not solve and then wrote down symbols representing that category of numbers. These categories would also be numbers that terminated or can be displayed by fractions and which numbers did not terminate or cannot displayed by fractions. Bonus Question: How would you translate the two category symbols (~ and *) in the third column into English? ~ means numbers that have rational square routes (areas that have rational side lengths)numbers that terminate or repeat. * means numbers that have irrational square routes (areas that have irrational side lengths)numbers that do not terminate or repeat Perhaps ~ also meant that the solutions were easy to solve (or solvable) and * meant that the solutions were very difficult to solve (or not being able to solve at all)

1. Use the applet to answer these questions: a. If side length = 4, area = 16 b. If side length = 3/4, area = 9/16 c. If side length = 2.5, area = 6.25 d. What is the relationship between the side length and area of a square? The area of a square is equal to the side length squared. A=x^2 2. Use the applet to find the corresponding side lengths for each area on the mysterious tablet. Describe your strategy for finding each side length (or explain why you weren’t able to find one). a. If area = 9, side length = 3 b. If area = 3, side length = 1.732... (square root of 3) c. If area = 1/9 (0.111111...), side length = 1/3 d. If area = 5/36 (0.138888...), side length = .372... (square root of 5, divided by 6) To find the side lengths, I found the square roots of the areas. When I calculated this, I used neither the applet or a calculator, using instead previously memorized values of sqrt(3) and sqrt(5). Another method of finding the value of x, using the applet, would be to enter more and more accurate values for the side length and recording the area for each side length, and to continue as long as one wishes. 3. Think about the difficulty of computing some of the side lengths versus others. Explain what Pythagoras was doing in the third column when he put the side lengths into two categories of numbers. It is easier to obtain a value for side lengths when working with rational numbers. Pythagoras was grouping numbers into rational numbers, which can be expressed as a/b, with a and b as integers and with b not equal to 0, and irrational numbers, which cannot be expressed in that way. It is far more difficult to calculate irrational numbers without a calculator. Bonus Question: How would you translate the two category symbols (~ and *) in the third column into English? I would translate ~ as rational, and * as irrational. These symbols refer to the side lengths. sqrt(3) is an irrational number, while the lengths 2 and 3/2 are rational.
Alyssa D., age 13  Caroline Davis Intermediate School, San Jose, CA
Andy H., age 14  Issaquah Middle School, Issaquah, WA
Sharon L., age 13  Issaquah Middle School, Issaquah, WA
Jacob M., age 15  Pembroke Hill School, Kansas City, MO
Shannon M., age 13  Caroline Davis Intermediate School, San Jose, CA
Andrew W., age 14  Issaquah Middle School, Issaquah, WA
Shelby Y., age 14  Issaquah Middle School, Issaquah, WA