Title:Ryan's primes
Personal Author:Juraschek, William A.; Evans, Amy S.
Journal Name:Teaching Children Mathematics
Source:Teaching Children Mathematics v. 3 (May 1997) p. 472-4
AUTHOR: Bill Juraschek and Amy S. Evans
TITLE: Ryan's Primes
SOURCE: Teaching Children Mathematics v3 p472-4 May '97
The magazine publisher is the copyright holder of this article and it is reproduced with
permission. Further reproduction of this article in violation of the copyright is prohibited.
"Why is 1 not a prime number?" This is not a question that is on the minds of most third
graders, but Ryan was not your average third grader. Throughout the school year, he had
impressed his teacher, Mrs. Evans, with his understanding of the mathematics they studied.
The class had been exploring multiplication and factors. After making rectangular arrays of
various numbers of square tiles and seeing the connection between the dimensions of the arrays
and the factors of the numbers, the class had noted those numbers for which only one array is
possible. Evans told the class that these numbers were called prime numbers and invited her
students to describe prime numbers verbally. After some discussion, the students fashioned this
statement: "A prime number has only 1 and itself as factors." Later, when Mrs. Evans informed
the class that 1 was not considered a prime, Ryan was the first to object.
"But 1 is only 1 times 1," contested Ryan.
"What do you mean?" responded Evans.
"Well, 1 is just like 3 and 5," said Ryan. "Three is only 1 times 3, and 5 is only 1 times 5; 1 is
only 1 times 1. So 1 should be a prime. Its only factors are itself and 1."
"You have a good point, Ryan. I am not sure why 1 is not a prime, but I do remember my
math professor saying it isn't," Evans acknowledged.
"Well, your professor is wrong!" insisted Ryan.
At this point Evans suggested that Ryan write his ideas in a letter to Bill Juraschek, and she
would deliver it when she went to class. Ryan wrote the letter shown in figure 1.
After discussing why 1 is excluded from the set of primes, Evans and Juraschek brainstormed
how to convince Ryan. They had been exploring ways to teach mathematics by emphasizing
pattern recognition; this situation would be a good chance to try out a technique using carefully
designed worksheets. To begin, Juraschek would construct a worksheet to guide Ryan to
discover that every composite number can be expressed as the product of primes. To add some
pizzazz and stretch the exploration over a few days, Juraschek would correspond with Ryan by
fax. His first letter to Ryan is shown in figure 2; the worksheet follows in figure 3.
Ryan was excited as he walked to the school library to pick up his fax. After returning to the
classroom, he eagerly began the worksheet. As Ryan worked, he detected patterns and shared his
observations with Evans. "It looks like all numbers can be made with prime numbers," he said.
He also noticed that no matter how you start factoring, you always end with the same prime
factors for a number. "That is really cool," he concluded. "But that doesn't mean 1 isn't a prime."
Evans relayed Ryan's comments to Juraschek; a few days later Ryan received the following fax.
Dear Ryan,
Mrs. Evans told me that when you did the worksheet about factoring composite numbers you
noticed an important pattern. You noticed that every composite number can be written as the
product of prime numbers. And you also noticed that no matter how you factor a number, you
always get the same prime numbers for factors. For example, 20 = 2 × 2 × 5 or 2 × 5 × 2 or 5 × 2
× 2, but we always get two 2s and one 5 as final prime factors.
Just as you did, mathematicians noticed this pattern, but they also saw a possible problem with
1. As you know, 1 behaves a special way when we multiply with it. The product of 1 and any
number is the number. So we can say 20 = 2 × 2 × 5 or 2 × 2 × 5 × 1 or 2 × 2 × 5 × 1 × 1 or 2 × 2
× 5 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1. If we call 1 a prime number, there are many different
ways to write a number as the product of prime numbers. [The number] 20 could be the product
of two 2s, one 5 and one 1; or two 2s, one 5 and two 1s; or two 2s, one 5, and nine 1s.
Now, mathematicians think this situation is messy. One way to clean it up is simply to say that
1 is not a prime number. If 1 is not allowed to be a prime number, we can say "Every composite
number can be expressed as the product of prime numbers in only one way." (For what we are
doing here, we don't consider 2 × 2 × 5 different from 5 × 2 × 2.) This pattern is one of the most
basic patterns in arithmetic.
Suppose a bunch of third graders are playing basketball, and a big tenth grader wants to play
with them. The third graders know that the tall tenth grader will make the game unfair, and it
won't be much fun for them. So, they make a rule that only third graders can play in their game.
This is just like the mathematicians making a rule that 1 is not a prime number. If they let 1 be a
prime number, it messes up the game.
Another nice thing about saying 1 is not a prime number has to do with the way we define
prime numbers. We can define a prime number as a number with exactly two factors. This makes
the definition of a prime number very easy to state. Instead of saying, "A prime number is a
number whose only factors are 1 and itself," we can say more simply, "A prime number is a
number with exactly two factors." Mathematicians like definitions to be as simple as possible.
Well, Ryan, I hope this makes sense to you. Bye for now.
Dr. Juraschek
Ideally, the story would end here, with Ryan knowingly agreeing that it makes sense to
exclude 1 from the set of primes. Not so. Ryan seemed to understand what we were trying to tell
him but did not find it convincing. Evans discussed the situation with Juraschek. It was time to
use their ace in the hole: a magic trick.
The next day, Evans asked Ryan to take a handful of red, yellow, blue, and green colored
cubes from a nearby container. "Let red be worth 2, yellow be worth 3, blue be worth 5, and
green be worth 11," she said. "Now pick any four of the cubes, but don't let me see them. Then
calculate the product of their values and tell me the product. For example, if you had three reds
and a blue, you would multiply 2 times 2 times 2 times 5 to get 40." When Ryan said that his
product was 90, Evans paused and then informed him that he must have used one red, two
yellows, and one blue cube. Ryan was impressed and wanted to try again. He took a new
collection of cubes and soon said that his product was 550. Again Evans could figure out exactly
which cubes he used: one red, two blues, and one green.
Ryan was totally hooked. He had to know how the trick worked. Evans told him to think about
it a while. He soon noticed that the values of the cubes were prime numbers. Evans simply
expressed his product as the product of primes and that told her which cubes he used: 90 = 2 × 3
× 3 × 5, so one red, two yellows, and one blue were used. He wanted to try it himself and soon
mastered the procedure.
Now for the clincher. Evans informed Ryan that she was adding black cubes, which were
worth 1. After choosing her cubes, Evans announced that their value was 88. Ryan thought for a
moment and then said, "You have three reds and one green." Evans slowly opened her hand.
Ryan was stunned to see that he was wrong. He pondered for a while, and suddenly the light
dawned. With a grin, he said, "I want to choose some cubes this time."
Ryan chose some cubes and announced that the product was 3. Evans told him that she knew
he had one yellow cube but that she could not tell him anything else. Ryan opened his hand to
reveal one yellow and several black cubes. Filled with excitement, he said, "Aha, it doesn't work
anymore. I could have a million black cubes, and you would never know!" Ryan was thoroughly
convinced that 1 should not be called a prime. He finally observed, "[The number] 1 does meet
our definition of prime numbers, but it doesn't fit the mathematicians' other definition."
Ryan eagerly asked if he could present the game to his classmates. The other students were
impressed and amazed, but because they had not been involved in the discovery, it was merely a
"trick" to them. It held little mathematical importance. For Ryan, however, the trick had
significant meaning. He understood why it worked, and, in learning the underlying mathematics,
he had communicated ideas, constructed knowledge, and satisfied his mathematical curiosity. In
short, Ryan experienced the pleasure of mathematical power. Why is 1 not called a prime
number? Ask Ryan; he knows.
Added material
Bill Juraschek, wjurasch@carbon.cudenver.edu, teaches at the University of, Colorado,
Denver, CO 80217. He also spends one day each week working with teachers. When this article
was written, Amy Evans was living in Germany at Zimmerstrasse 46,06667 Weisenfels,
Germany. She was teaching at the Colorado Academy in Denver when this experience occurred.
She is interested in developing students' understanding through communication and cooperative
problem solving.
FIGURE 1 Ryan's letter
Ryan and his classmates revisit his third-grade investigation of primes.
Photograph by Rene Galindo; all rights reserved.
FIGURE 2 JURASCHEK RESPONDS TO RYAN
Dear Ryan,
You wonder why 1 is not considered a prime number? It does seem strange. I will send you
several faxes in the next few days to try to explain.
What you have discovered so far about prime numbers is true. A prime number has only 1 and
itself as factors. The other whole numbers are called composite numbers. The word "composite"
comes from "compose." Composite numbers are composed of prime numbers, like a song is
composed of notes. The problems on the next page should show you what it means to be
composed of prime numbers.
I will send another fax Friday. Bye for now.
FIGURE 3 EXERCISE TO SUGGEST A PATTERN WITH COMPOSITE NUMBERS
COMPOSITE NUMBERS
Let's explore the whole numbers that are not prime numbers. Continue the pattern and fill in
the blank squares.
4 = 2 X 2
6 = 2 X 3
8 = 2 X 4 = 2 X 2 X 2
9 = 3 X 3
10 = 2 X 5
12 = 2 X 6 = 2 X 2 X 3
12 = 3 X 4 = 3 X [Graphic Character Omitted] X [Graphic Character Omitted]
14 = [Graphic Character Omitted] X [Graphic Character Omitted]
15 = 3 X 5
16 = 2 X 8 = 2 X 2 X 4 = [Graphic Character Omitted] X [Graphic Character
Omitted] X 2 X 2
16 = 4 X [Graphic Character Omitted] = 4 X [Graphic Character Omitted] X
[Graphic Character Omitted] = [Graphic Character Omitted] X [Graphic
Character Omitted] X 2 X 2
18 = 3 X [Graphic Character Omitted] = 3 X [Graphic Character Omitted] X
[Graphic Character Omitted]
20 = 2 X 10 = 2 X [Graphic Character Omitted] X [Graphic Character Omitted]
20 = 4 X 5 = [Graphic Character Omitted] X [Graphic Character Omitted] X 5
21 = [Graphic Character Omitted] X [Graphic Character Omitted]
22 = 2 X 11
24 = 2 X [Graphic Character Omitted] = 2 X [Graphic Character Omitted] X
[Graphic Character Omitted] = [Graphic Character Omitted] X [Graphic
Character Omitted] X [Graphic Character Omitted] X [Graphic Character
Omitted]
24 = 3 X [Graphic Character Omitted] = 3 X [Graphic Character Omitted] X
[Graphic Character Omitted] = [Graphic Character Omitted] X [Graphic
Character Omitted] X [Graphic Character Omitted] X [Graphic Character
Omitted]
24 = 6 X [Graphic Character Omitted] = 6 X [Graphic Character Omitted] X
[Graphic Character Omitted] = [Graphic Character Omitted] X [Graphic
Character Omitted] X [Graphic Character Omitted] X [Graphic Character
Omitted]
24 = 12 X [Graphic Character Omitted] = [Graphic Character Omitted] X
[Graphic Character Omitted] X 2 = [Graphic Character Omitted] X [Graphic
Character Omitted] X [Graphic Character Omitted] X [Graphic Character
Omitted]
Now, you complete the rest up to 36, and then discuss the patterns you notice.
WBN: 9712100445001