Pythagoras' Mystery Tablet
Posted June 4, 2001 as a Math Forum EPoW

Introduction: In this problem, students used an applet to compute the area of a square based on its side length. They then used this information to determine whether it was possible to get exact side lengths for certain areas.

Where’s the Math: The goal of this problem was to get students to investigate the concept of irrational numbers through a familiar concept like area. Students realized that certain areas, like 4, had an exact side length, while others, like 2, did not. This led to questions about the categorization of such numbers, based on how "easy" it was to get an exact area. Without actually introducing the mathematical nomenclature, students discovered the existence of certain irrational numbers. Standards: Number & operations Role of Components: Standard Java text field components display both the fractional and decimal representation of the side length. The growable area box is just a Java text area component with a fancy border. The area display at the bottom uses more text field components, but with editing disallowed. Finally, the Tablet display on the side is just a ".gif" image on the applet's HTML page. An invisible "model" component handles the arbitrary precision arithmetic.

Try the applet!

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Sample submitted solutions:

From:  Sharon, age 13
School:  Issaquah Middle School, Issaquah, Washington

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 length 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.

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