"Pythagoras' Mystery Tablet" Teacher Support

Pythagoras' Mystery Tablet Archived PoW || Student Version

Pythagoras' Mystery Tablet is no longer the current ESCOT Problem of the Week. The student version allows teachers to use the problem with their students without giving the students access to the archived answers. Teachers can use the link to the archived problem to get ideas about student thinking.

[Standards]  [Activities]  [Lessons]  [Student Resources]  [Other Resources]

This problem is meant to motivate the need for a different type of number other than rational numbers; thus work on this problem should be a conceptual introduction to irrational numbers.

The problem is designed to have students explore the relationship between the area of a square and the length of a side as a means for discovering that it is difficult to find a number expressed as a fraction that is a square root of a number (e.g., square root of 2, square root of 5/36). The students are asked to place the side lengths of the squares into two categories and to describe the characteristics of those numbers. The side lengths that were difficult to find will constitute the category of irrational numbers since they cannot be expressed as a ratio of p/q with p and q both whole numbers.

The pre-activity is designed to have students develop the idea of square numbers and review the relationship between the side of a square and its area.

There are also two suggested possible post-activities. In the first activity, students can further develop their understanding of irrational numbers by using the pythagorean theorem to find the length of an hypotenuse on an isosceles right triangle with legs of length 1 inch. In the second activity, students explore the relationship between circumference of a circle and its diameter to discover the famous irrational number pi.

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Alignment to the NCTM Standards

Number & Operations
- understand numbers, ways of representing numbers, relationships among numbers, and number systems
Whereas middle-grades students should have been introduced to irrational numbers, high school students should develop an understanding of the system of real numbers. They should understand that given an origin and a unit of measure, every point on a line corresponds to a real number and vice versa. They should understand that irrational numbers can only be approximated by fractions or by terminating or repeating decimals. They should understand the difference between rational and irrational numbers. Their understanding of irrational numbers needs to extend beyond Pi and the square root of 2
Geometry
- use visualization, spatial reasoning, and geometric modeling to solve problems

Measurement
- select and apply techniques and tools to accurately find area to appropriate levels of precision

Representation
- create and use representations to organize, record, and communicate mathematical ideas
- select, apply, and translate among mathematical representations to solve problems

Problem Solving
- solve problems that arise in mathematics and in other contexts
- monitor and reflect on the process of mathematical problem solving

Communication
- communicate their mathematical thinking coherently and clearly to peers, teachers, and others
- use the language of mathematics to express mathematical ideas precisely.
Possible Activities:
Pre-Activity
Have students investigate several squares with different side lengths and review the relationship between the side length and the area of the square.

Give students the following square pattern and ask them to study the pattern.

If this pattern were to continue, for any number of tiles on the side of a square, we should know what number of tiles is needed to build the entire area of the square.  Fill in the table below.

 Number of tiles on side Number of tiles in area 1 1 2 4 3 9 4 5 6 7 64 81 100

How many tiles would be on the side if there are 324 tiles in the area?
How many tiles would be on the side if there are 841 tiles in the area?

Based on what you learned, what is the relationship between the side length of a square and its area? If you know the area, how do you find the side length?

Post-Activity:
Explicitly discuss the definition of an irrational number why several of the side lengths from the Pythagoras problem are irrational numbers.

Activity 1: Use the pythagorean theorem to find the hypotenuse of a right triangle with each leg equal to 1 foot. Explore this problem both algebraically using the formula as well geometrically by building squares on each side of the triangle. Several Web sites below can be used for students to further explore the irrationality of the square root of 2.

Activity 2: Explore the relationship between circumference and diameter of many circles. Have students use string and a ruler to approximately measure the circumference and diameter of several circles and investigate any relationship between them. The students should eventually suggest that the circumference looks about a little more than 3 times the diameter. By collecting all the class data and comparing, they can discover that C/d is about equal to 3.14. Several Web pages suggested below can be used to have students further explore the irrationality of Pi.

If you have access to dynamic geometry software (such as The Geometer's Sketchpad), the students can explore both these activities using the construction and measurement tools in the software.

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Related Lessons Online:

- Difference between rational and irrational numbers - lesson plan from Province of British Columbia, Ministry of Education Web site

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Resources to help students:

- Integers, Rational Numbers, Irrational Numbers
- Irrational Numbers by Jim Loy
- Millions of Digits of Common Irrational Numbers
- Square root of 2, including a calculator to compute the square root of any number
- Lots of links to resources about Pi

Ask Dr. Math Archives
- Golden Ratio and Golden Rectangle (Elementary/Golden Rectangle)
- Irrational Pi (High School/Transcendental Numbers)
- The Number e (High School/Transcendental Numbers)
- Meaning of Irrational Exponents (High School/Algebra)
- Irrational Powers (College/Modern Algebra)
- Proof that Sqrt(2) is Irrational (High School/Square Roots)
- Proving the Square Root of 3 Irrational (High School/Square Roots)
- Are Transcendentals Irrational? (High School/Transcendental Numbers)

Previous POWs related to irrational numbers
- Accuracy Please - posted February 7, 2000
Considering precision, accuracy, and appropriateness in finding the volume of a cone.
- Partial Area - posted February 5, 2001
Find the area of the portion of the circle that is not included inside the inscribed square.
- The Length of Larry's Rectangle - posted January 3, 2000
Larry wants to know the length of his rectangle. Can you help him?
- Stained Glass Window - posted May 10, 1999
Find the size of a stained glass window for my front door.

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Historical Information about Irrational Numbers

- What is "How Many?"
- Pi Mathematics, History and Lesson Plans
- Finding the value of Pi
- A history of Pi

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