Find all combinations of 3 distinct primes whose average is 13

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Just an isolated middle school mini-challenge to get the day started? Perhaps...

Those of you who are familiar with this blog know that MathNotations is dedicated to providing activities/investigations for middle and high school teachers to use or modify (provided proper attribution is given of course). In this post, I will demonstrate how one can build an extended or richer activity from a math contest or standardized test problem.

It is important to remind our readers here that these kinds of activities and problems do not constitute a curriculum. Students need to first develop proficiency with skills and procedures. These explorations are only intended to extend and enrich student learning. They can be used in part or in whole, as a long-term project outside of class, a team activity in the classroom or a myriad of different ways. All of this is at the discretion of the educator.

First of all, the problem in the title, in its present format, would not be an SAT or a standardized test question, unless the standardized test included free-response or open-ended questions.

In SAT format, the question might be changed to:

Which of the following can be expressed as the sum of three distinct primes?
(A) 6 (B) 9 (C) 12 (D) 15 (E) 17

Not a particularly challenging problem, but some students would struggle with comprehending the wording or paying attention to details ('distinct') or because of lack of knowledge about primes. This type of question is fairly common.

Let's return to the original question:

Find all combinations of 3 distinct primes whose average is 13.

I've administered this type of question to students and observed their methods. Sadly, some do not immediately recognize that the problem is equivalent to:

Find all combinations of 3 distinct primes whose sum is 39.

Most students do see this at once, but there are a few in middle and high school who have not developed sufficient conceptual understanding of averages or have simply not been exposed to enough problems.

As far as methods and approaches go, I'm always surprised that many middle and high school students use fairly random listing methods rather than a systematic approach. After all the years now of instruction in problem-solving techniques, one should expect that students would make an organized list as follows:

2,2,35 Discard this for two reasons! Would most students recognize the logic behind concluding that 2 cannot be one of the three primes?

3,5,31
3,7,29
3,11,25 (discard)
3,13,23
(I'll let the reader finish the list!)

If I were to assess the value of this single question, I might give it a 7 on a scale of 10. I'm sure some would rate it as 1 or 2 since some perceive these kinds of questions as useless. However, my feeling is that the question does develop mathematical thinking and there's something to be said for attention to detail and a systematic approach.

But this is not the end. Suppose the educator finds this problem in a book or math contest or online. How can one extend it to a richer experience for all students, not just the accelerated, honors or gifted child? Although it may appear at first that the primary intent of the question is to encourage a systematic approach (making an organized list) or reviewing ideas about averages or primes, the content of the question is essentially about writing a number as a sum of 3 primes, distinct, in fact. Is this an important question that has occupied the minds of our greatest mathematicians for years? Uh, actually, yes! Look here!

Students need to be encouraged to ask more questions after the problem is solved. The instructor guides this exploration by modeling some of the questions students need to ask: Is there anything special about 13? Can every prime be written as a sum of three distinct primes? Every odd? Three primes, not necessarily distinct? Does the original number 13 have to be prime or even odd for that matter? Why are we using three primes in the sum? Why not two? Your turn, boys and girls!

You get the idea. This isolated problem becomes a springboard for deeper mathematical research. Here is one possible assignment:

Write your own challenge problem of this type? Make sure you can solve it and be prepared to present it to the class!

What would you expect your students to come up with? You can't be sure until you try it of course, but can you anticipate some of the responses?

By the way, I have already heard most of the arguments for why this type of research is impractical in a math classroom:

"My students don't even know their basic facts and you want them to become mathematicians!" "This is for the math team geniuses."
"I don't have time for this - I have a real curriculum to cover and if this not going to be tested..."
"Teach children the basics, not this 'fuzzy' math!"

Oh well, enjoy it anyway!