Beacon Lesson Plan Library

Numbers Beyond Reason!

Cylle Rowell


Students use the story of the “discovery” of irrational numbers to learn about the different classes of numbers, the different ways in which numbers may be represented, and how to classify different numbers into their particular class.


Associates verbal names, written word names, and standard numerals with integers, rational numbers, irrational numbers, real numbers and complex numbers.

Understands that numbers can be represented in a variety of equivalent forms using integers, fractions, decimals, and percents, scientific notation, exponents, radicals, absolute value, or logarithms.


-Pencil and paper for each student
-Basic calculator for each student
-“Numbers in the Sand” from [Marvels of Math], by Kendall Haven, Teachers Idea Press, 1998.
-Overhead of questions from reading (or copies could be made for each student) See Associated File
-Large display calendar


1. Become familiar with story “Numbers in the Sand” prior to lesson.
2. Run copies of story “Numbers in the Sand” (if students will be reading themselves).
3. Gather materials for lesson.


1. Lesson Introduction: Hold up a large calendar of current month. Ask the students to write down three things they see on the calendar (possible answers will be numbers, days of the week, a grid, lines, columns, rows, vertical lines, horizontal lines.) Most of the students will say “numbers”. Explain to the students that what they see on the calendar are not actually numbers, but numerals. Explain that a number is just an idea that can be represented by a simple word or a numeral, but can also be represented in other ways. If today’s date was September 12th, ask the students if they can think of another way to represent “twelve”. For example, 11 + 1, 15 – 3, 4X3, 12 /1, 24/2, 12.0, 1200%, and the square root of 144, are just a few other ways to represent the numeral 12. Have the students brainstorm with a partner to come up with 5 more ways to represent 12 (or whatever the current date is). Some other possible answers are using tally marks, Roman numerals, exponents, fractions. Students should discover that there are an infinite (endless) number of ways to represent the idea of the number twelve. After students have had a few minutes to brainstorm with one another, have the students share some of their answers with the class. Display as many answers as feasible on the board or overhead. This may be a good time to briefly review the following terms: fraction, decimal, percent, exponents, and square roots.

2. Ask the students if they know what we would call a number such as twelve. (A whole number or a counting number) Why do we call numbers such as 1,2,3,etc.., whole numbers? (Those types of numbers always represent so many “wholes” of something; no parts) Then, ask the students if we would consider the number twelve as a fraction? The students will probably think it is not a fraction, but point out to the students where we represented twelve in fraction form such as 12/1 and 24/2. Therefore, twelve can be represented as both a whole number and a fraction! Ask the students if there are any numbers that can be represented as a fraction but NOT a whole number? (any proper fraction or improper fraction not equal to 1). Get some examples from the class and write them on the board.

3. Have the students work in their groups and convert the fractions on the board to their decimal representations. For example, one-half can also be represented as .5. Two-thirds may be represented as .6666….. Two-ninths is .181818….. Review with the students how to change a fraction to a decimal number using the division button on the calculator. For example, to convert one-half, the student will punch in 1, then division key, then 2 and enter to get a result of .5. Ask the students about the difference in the decimal representation of one-half and the decimal representation of two-thirds and two-ninths. Review the terms of terminating and repeating decimal forms.

4. Then, ask the students if they know of any numbers that CANNOT be written as a fraction. Let the students brainstorm again with a partner or group to see if they can find a number that cannot be represented as a fraction. This is a good time visit each group of students to informally assess the students’ previous knowledge of irrational numbers. Most students will NOT know that an irrational number is the only kind of real number which cannot be represented in fraction form (other than approximations). After a few minutes, see if anyone came up with a correct answer. This activity will lead into the story of the discovery of irrational numbers.

5. Tell the students that for many years the top mathematicians only knew about numbers that could be represented as a fraction. In fact, one of the most famous of all mathematicians, Pythagoras, only knew about positive whole numbers and fractions. He called these types of numbers “rational numbers.” In fact, Pythagoras believed that these numbers held all the secrets of the then universe. Point out that rational means reasonable and also contains the word “ratio” (another way of representing a fraction). But Pythagoras’ world of rational numbers was severely upset when his students discovered numbers that did not fit into the neat, rational world of Pythagoras. Either read or have the have the students read the story “Numbers in the Sand.” Individual copies may be copied for use in the classroom. Have the students answer on paper the following questions after they complete the reading. Students may work in groups to answer questions on the reading. All questions are to be answered using complete sentences and correct grammar!

1. When and where was Pythagoras born? (On the Greek Island of Samos around 572 BC)
2. What did Pythagoras believe could describe everything in the world? (whole numbers or fractions)
3. What are whole numbers? (The counting numbers we all first learn as small children 1,2,3,4,ect.) What are fractions? (A ratio of two whole numbers such as ½ or ¾) Give three examples of each.
4. What is the Pythagoras’ great theorem on right triangles? (The Pythagorean Theorem. Students may need to use a reference book)
5. What “number’ did Philoclease and Dionesa discover using Pythagoras’ theorem? How did they discover it? (The square root of 2; They explored a right triangle with the lengths of the two sides both equal to one unit in length.)
6. Find the square root of 2 on a calculator and convert to a decimal number. What is the result? Does the result “appear” to terminate, repeat, or neither? (1.414213…; it neither terminates or repeats.)
7. According to the story, what is “special” about the square root of 2? (It cannot be represented as a fraction or ratio of whole numbers) What do we call these types of numbers today? (Irrational) What are some examples of other numbers like square root of 2? (square root of 3,5,6, etc., , or any non-terminating or non-repeating decimal number such as 0.1010010001 or 112123123412345…)
8. Why do you think that numbers that cannot be represented as a fraction are called irrational numbers? (They are beyond REASON!)

5. After students have had enough time to complete the reading and to answer questions, discuss the questions as a whole group. Bring the lesson to a close by summarizing with the class the different kinds of numbers that exist in our real number system. A Venn diagram could be used to illustrate how the set of whole and counting numbers are also part of the set of rational numbers, but not all rational numbers are whole numbers. The set of irrational numbers would not be included in either set. Integers have not really been discussed in this lesson (Pythagoras also did not “believe” in the existence of negative numbers), so you might ask the students in which class of numbers (rational or irrational) would they place negative whole numbers or fractions.

6. Hand each student eight index cards. On the cards, each student is to write two whole numbers, two numbers that are only rational, two numbers that may be whole OR rational, and two irrational numbers. Students are to write the number representation on one side of the index card and the number class/es on the other side. Challenge the students to include terminating and repeating decimal forms that might “stump” their classmates. Put the students in groups of four. Have the groups combine their flashcards into one pile and shuffle. Have the students take turns picking a card and then classifying the number representation. If the student correctly classifies the number, they keep the card. If they classify the card incorrectly, the card goes back into the pile. Students continue until all cards have been drawn and correctly classified. Winner is student with most correct cards.


Assess students formatively at the end of the lesson through the written assignment. Provide additional practice on classifying and different number representations with teacher-made worksheets if needed. Then, assess students summatively at the end of a unit on number sense and concepts by asking the same types of questions on a paper-and-pencil test.

1. Name and describe (in words) the three classes of numbers about which we read today.
2. What famous mathematician believed that numbers were the key to understanding the universe? State three facts about this mathematician.
3. Classify the following numbers into their correct class. If the number can be placed into more than one class, put into both classes.

5, square root of 5, 15/3, 5.2, -2/3, .2727.., 0.232332333...., square root of 4, pi

Attached Files

A copy of the questions that can be reproduced for classroom use.     File Extension: pdf

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