Evelyn B. Christensen - Author & Puzzle Creator

Pythagorean Triples Served for Supper

After an enjoyable, but exhausting, week of teaching, my husband Ralph and I sit down with our children to Friday supper with that "let go" sense of relief that the weekend has finally arrived. As the meal begins, our eighth grader casually, with almost studied indifference, remarks, "I figured out a new math idea at school today." Martin is an inquisitive, motivated student, but he is definitely very reserved.

We express interest. "Do you want to tell us about it?"

Giving us more credit for being on his wavelength than is our due, Martin plunges in. "For Pythagorean triples, if a is odd, then a² = b + c." [Pythagorean triples are three positive integers a, b, and c such that a<b<c and a² + b² = c².]

Does this make sense? Clearly, our minds are still too focused on spaghetti and buying new shoes tomorrow. We have no idea what he is talking about.

He tries again. "You know the Pythagorean theorem." We nod agreement. "If you take any odd number a, you can get a Pythagorean triple by squaring a, dividing that answer by 2, and taking the whole numbers on either side as b and c."

I happen to be a visual, paper-and-pencil kind of person and am still lost. Ralph and our daughter Susan do not seem to be much more enlightened. Martin exasperatedly says, "I'll show you," and leaves the table. He returns a minute later with a spiral notebook. He turns to a page covered with mathematical notations and points to a list of numbers grouped in triples.

"My geometry teacher told us that on our next test she might ask us to list all the Pythagorean triples we know. Since I was interested in these anyway, I started making a list of them. Then I noticed this pattern." He points to (5, 12, 13). "See, if you square 5, it equals the sum of 12 and 13. The thing works for (3, 4, 5): 3² = 4 + 5."

I am beginning to understand. "But what were you saying about dividing by 2?"

"Well, I figured out that if I have any odd number, for example, 7, and I want to generate a triple with that number being a, then first I square it; 7² = 49. Then I divide by 2; 49/2 = 24.5. Then I take the whole numbers on either side of it. In this case, that gives me 24 and 25. Then (7, 24, 25) is a Pythagorean triple."

Ahh! This explanation does make sense. My husband, Ralph, is already wondering why this method works algebraically. He and Martin start jotting down the Pythagorean equation on a piece of scrap paper.

a² + b² = c²

a² = c² - b²

a² = (c + b)(c - b)

Since Martin's method of generating c and b requires them to have a difference of 1, the second factor c -b will always be 1. This result leaves us with

a² = c + b,

which was Martin's original observation.

Ralph is delighted to have the mystery solved. Strangely, Martin is disappointed. "Oh, the idea I had is nothing special. It's just part of the original equation."

We try to explain that just because the results can be verified mathematically with the equation does not mean that his idea is any less interesting or creative. In the meantime, I am wondering, "What happens if you start with an even number instead of an odd number?"

Martin immediately says, "It won't work. The square of an even number is even, and you can't have two consecutive whole numbers that will sum to an even number."

"Yes, but couldn't we still try taking the numbers that are 0.5 more and 0.5 less than the square divided by 2?" persists twelve-year-old Susan, who has just finished making sense of all this discussion. "They won't be whole numbers, so it won't be a Pythagorean triple, but will they work?"

She and Martin try an example: 6² = 36. 36/2 = 18. The question is: Does (6, 17.5, 18.5) fit the Pythagorean theorem? They get a calculator and check it out, coming up with an affirmative answer.

By this time supper is long over and dishes need to be cleared. Later I suggest to Martin that he might like to write up his idea to share with other people in a publication. He is negative about the suggestion, saying, "It wouldn't be worth sharing because it's not useful for anything."

Ralph tries to explain. "An idea doesn't have to be useful in the ordinary sense to be worth sharing if it's a new and creative way of looking at something."

"Yes," I agree, "but besides that, your idea can be useful. When a teacher is first introducing students to the Pythagorean theorem, she usually likes to give examples that have easy numbers, ones that don't get too 'messy' in their calculations. Your idea is a great way to generate those. Even the ones like (6, 17.5, 18.5) aren't as messy as most sets of numbers picked randomly."

Unfortunately, as the reader might suspect, we were not convincing enough, which is why my name is on this article. In concluding, I would like to make four observations about this experience:

Many middle school students are at an age when egos are very fragile. Sharing a creative idea or insight beyond the regular assignments takes courage for them and may not happen unless an atmosphere of acceptance and support is generated in the classroom and in the home.
Martin has a family that could understand and share in his discovery. Many middle school student are not that fortunate. Listening to and exploring students' ideas take time. Often, as teachers, we may feel so pushed for time that students' comments that seem a bit "off track" may not be given the attention they deserve.
In our efforts to make mathematics highly practical and relevant to real life, we may be leaving students with the false impression that unless an idea has immediate usefulness, it does not have much value.
I suspect that most middle school students believe that within their capabilities, very few new ideas in mathematics are left to be discovered. Giving them a sense that their discoveries, although not new to others, are worthwhile and exciting is important and may influence their future motivation.

Postscript

After I had written this article, our family continued to explore questions about the Pythagorean triples. Perhaps the reader or the students he or she knows would also like to explore some of them.

What happens if the difference between b and c is 2 instead of 1? Is it still possible to generate Pythagorean triples?
If it is possible to generate Pythagorean triples with c - b = 2

Is it possible to generate Pythagorean triples with c - b = 3, with c - b = 4, . . .? If it is, will these sets contain "new" triples?
Choose a Pythagorean triple (a, b, c) with a an even number. Do not choose a triple that is a multiple of another (a, b, c) with a even. Does c - a have any special characteristic? Is this result always true? Can you find a counterexample?

Mathematics Teaching in the Middle School, (September, 1997): 60-62. Copyright: National Council of Teachers of Mathematics. (703) 620-9840/ www.nctm.org. Reprinted with Permission.