Saturday, February 9, 2008

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?

3,11,25 (discard)
(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!


jd2718 said...

Fun game to play. I have a feeling my classes would work it sort of quickly.

But then the follow up questions... I could warm them up with your questions and then I could see challenging kids to rewrite it. And I could also see specifying a target - The answer is 5, write the question.

If I do, I'll let you know.

But you know what I am really after here. How do I make the strongest kids scowl without making the weakest kids quit. These sorts of investigations are great for that.


mathmom said...

To be honest, this isn't one of my favorites of your middle-school-accessible investigations. As you know I love to do this kind of research in class, and your follow-up questions are interesting, but I think the original problem is fairly tedious.

One of my favorite things to teach about averages is how to deal with them in terms of "offsets" from the mean, without having to add everything up. When creating the sets of triples for this activity, you could think about it like this:

Instead of saying: I have 5 and 11 and I need (13x3) - (5+11) = 39-16 = 23

I like to teach students to say: I have 5 and 11, which are 8 below and 2 below the required mean. That means I need a number that is (8+2)=10 above the mean. 10+13=23

(Which I realized looks more complicated when I write it out like that, but I do believe it is quicker in many cases, though perhaps not this one per se.)

I was at MATHCOUNTS yesterday with my 11yo son and 3 of his classmates. During the "Countdown" round, where time is of the essence, there were several mean-related questions that I felt were more quickly solvable using that approach. By which I mean to say that I beat the 11-13yos to the answer. Which, sadly enough (for me, but happily for the future of our nation), was not always the case, but I'm pretty sure I did beat them on all the "mean" problems. (I can't give any of the actual problems, as they are still being used across the country throughout the rest of the month.)

Dave Marain said...

I agree. That's why I only rated it a 7!

However, the problem itself wasn't my main focus. I was trying to demonstrate how an educator can develop a more detailed investigation from a single math problem. This far transcends the immediate question. However, I gave this question to a strong group of students yesterday and they struggled with it. They didn't see it as particularly tedious as there are only 6 sets of solutions. But many did not get it.

What's far more interesting to me is how this question is related to Goldbach's conjecture. Take a look at the link I gave. Your students may get turned on by all those unsolved problems! If you do try it with your students, let me know...

Thanks for the support. The questions I suggested were just models. I would hope that students would ask their own after we suggest just one or two possibilities.

jd2718 said...

And my quickie solution?

37, too big
31, need 8, so 3 and 5...

Because there is less freedom at the beginning, it felt a bit more directed than moving up from small numbers.


Joshua Zucker said...

Hi Dave,
I think you contradict yourself a bit in your discussion here.

First you say "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."

I strongly disagree with this. Problems like these can motivate kids to want to learn and practice those skills (though I agree with mathmom that this is not one of your most engaging). For instance, suppose I think that my class needs some practice with adding one and two digit numbers. Well, this problem might be a fine way to give them a bunch of practice with that skill. If I said "first, let's do a worksheet with twenty one- and two-digit addition problems" they would be bored or frustrated or both. With this problem, they'll practice that skill automatically. Of course, you have to choose your problems carefully to ensure that the appropriate skills get practiced, and you also have to be careful not to put too much problem-solving up front before they can start experimenting. One of the big plusses of this problem of yours is that it's pretty easy for students to jump in and start experimenting, adding some small numbers, right off the bat.

At the end of your post, you write '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!"'

which makes it seem more like you agree with my comment above, and at least to me sounds like you're contradicting your original disclaimer of "only do this kind of stuff if your students are already masters of their basic skills".

jd2718 said...

I would say that these activities do not substitute for learning the algorithms, facts, etc.

But, frankly, I would rather other activities to help build addition strength... here incomplete knowledge would serve to frustrate.

Dave Marain said...

I've learned not to respond to thoughtful comments such as yours after 4 PM. My brain shuts down after that! In the clear light of day, freezing tho' it may be this morning, I will try to respond as thoughtfully.

First, you and I have known each other, virtually, for over a dozen years on various message boards and forums. I know that you are an exceptionally dedicated and talented teacher with a broad and deep understanding of the subject matter. I also know that you have developed many activities for your students of high quality. You also know me. You know that I have not wavered or waffled from a BALANCED perspective from Day One.

In looking back over your comments, I agree that I contradicted myself by implying that there is some sequence to learning in which skills must be mastered BEFORE children should engage in problem-solving or other activities. That was unintended and I appreciate your pointing that out.

In all my years in the classroom, I varied my teaching techniques on a regular basis. I absolutely used these kinds of questions to motivate and inspire students of all ages and abilities. I sometimes provoked students with questions prior to their having proficiency with all of the necessary skills, frustrating though that might have been. I sometimes used these problems, as you do, to reinforce and develop arithmetic or algebra skills.

BUT, I've learned a simple truth. The best manipulatives, visualizations and problem-solving activities DO NOT REPLACE the need for students to practice in traditional ways.

I'm afraid we may simply have reached the point where we will agree to disagree: IMO, students need both engaging activities that develop understanding AND the 'five-finger exercise' of repetition and drill. Now there will always be some children who can develop both proficiency and understanding with minimal practice, but, in my experience, the majority of learners do not easily TRANSFER their learning from problem-solving, activities and games to accurate and efficient computational performance on a piece of paper when they need to do so. I have also seen some of the brightest students at the high school level who have the number sense and algebra sense to see through a sophisticated calculus problem, fail to complete the problem accurately because of some basic skill weakness. We can make traditional learning more fun, we can motivate children in so many wonderful ways, but, in the end, there are no short-cuts.

I felt the need for that paragraph near the top of my post because many districts have adopted problem-based curricula that neglect the other side of learning. This is not BALANCED.

Similarly, I added the paragraph at the end for the staunch traditionalists who do not see the need for rich engaging activities.

So, while it may appear that I'm contradicting myself, I do not believe I am. Maybe, a balanced perspective is an extremely difficult one to defend these days!

Sorry to pontificate like this, but every few months or so, I have to do so, since this is the central theme of the math wars!

mathmom said...

Joshua, I think this would be an odd activity for the practice of single/double digit addition. It would seem to me that most students who are at a point of understanding primes and who are ready to attack a problem via a "make an organized list" method, do not really need much practice with single/double digit addition (and conversely, those who do need basic addition practice would be totally lost with a problem like this.)