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I tend to teach 5th graders math ever so often just so they can be "friendly" with math in a playful manner, instead of being afraid.

However, one question that I constantly struggle with is this: Why should we care if a number is prime or not?

Coming from a computing background I try my best to explain to them the use of prime numbers in cryptography and how primes are related to factorization (as kid friendly an explanation as possible). However, they "sigh" and move on with the belief that I'm telling the truth. But, they still don't seem to get excited about it.

Answers for questions like these: Real-world applications of prime numbers? don't seem to be well suited for 5th graders.

What are some interesting ways/examples that one can use to help 5th graders understand why the study/knowledge of primes is useful? Bonus if they can "see the use" sooner in their 10 year old lifespan instead of waiting till college.

I'm okay even conjuring "games" to help them learn/understand. For example, currently I'm trying to use something like Diffie Hellman Key Exchange to make a game for them to encode messages and see if "eavesdroppers" (i.e., other students) can guess the message. Something on the lines of Alice wants to send Bob a number that she's thinking about. Other students have to guess what that number is. The number can be 'encoded' (loosely speaking) as a manipulation of numbers similar to the key exchange protocol, but that 5th graders can play around with. Hopefully the 'decoding' process shows them why it's better to choose primes. However, this could be rather abstract. That's the best I've got for now.

Any other ideas?

Community
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PhD
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    I believe this question is better suited for Matheducators.SE – MathematicsStudent1122 Dec 29 '15 at 19:48
  • @MathematicsStudent1122 - I didn't even know we had such a site! I'm okay if it's moved there – PhD Dec 29 '15 at 20:01
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    I'd just say, nowadays, "computers make our data safe using prime numbers." – Thomas Andrews Dec 29 '15 at 20:01
  • @ThomasAndrews - that's the direction I tend to take quite often – PhD Dec 29 '15 at 20:05
  • There are many interesting theorems regarding the primes that your students can understand. For instance, the Green-Tao theorem states that the primes contain arbitrarily long arithmetic progressions. Moreover, the fundamental theorem of arithmetic as mentioned below!

    I recall something called the Ulam spiral, and it might be a useful visual aid for your students. Sort of `mysterious'.

    But definitely post in the math educators stack exchange. (Is it still in it's beta?)

     – Nobody Dec 29 '15 at 20:07
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    I would have thought that computations with fractions (least common denominator, etc.) would be plenty enough for them. Maybe not for why people spend their lives researching prime numbers, but certainly for why primes are important to them. – Dave L. Renfro Dec 29 '15 at 20:27
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    Here's a nice game called "Prime Climb", which my grandson plays with his dad: http://mathforlove.com/games/ – Lee Mosher Dec 29 '15 at 21:27
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    I have a feeling you might also have to explain this to bureaucrats, what with No Child Left Behind and Common Core. – Bob Happ Dec 29 '15 at 21:37
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    Can you give an example of a math subject on which the 5th Graders students get excited? Perhaps the "problem" are the 5th Graders students rather than the prime numbers (in the sense that it is very hard to teach mathematics at this level - at least in my country.) – Pedro Dec 30 '15 at 10:35
  • What's really happening when you're trying to get your students to study prime numbers? Are they questioning you from the very start saying "Why do we need to study prime numbers?" Are they struggling to figure out how you want them to study prime numbers? If they're struggling, what is it that you want them to learn how to do that they're struggling to figure out how to do? When I was in elementry school, I was introduced to the concept of prime number but wasn't made to study hard stuff. I believe I was just taught the definition of a prime number and that was it. I believe they did not – Timothy Feb 25 '20 at 21:38
  • introduce the fundamental theorem of arithmetic which is good. I believe they did not make me do something I would have found too hard at that time, of studying prime numbers and coming up with theorems about them all on my own. I probably would have been so confused what they are trying to get me to do at that time. – Timothy Feb 25 '20 at 21:41
  • My personal opinion is I actually agree with them "Why study prime numbers?" I'm now interested in more general number theory unlike before. Prime numbers are a narrowly focused topic. It now would come to me naturally to see and notice the answer to the question "Did you get the result by breaking everything down to the basics?" for stuff like the fundamental theorem of arithmetic. Sometimes something more entertaining for me to think about is the associative law of natural number addition. The associative law of natural number addition is simpler and less focused so I like it better. – Timothy Apr 28 '20 at 00:43
  • However, since the world works differently, we might as well plug along with it. – Timothy Apr 28 '20 at 00:43
  • I think school is teaching too much. They aren't learning to give up on getting the students to explore everything they're teaching them in great depth which they can't do. The students could be pondering more of what they were already taught in greater depth because they may be good at doing things their own way but they're taking away attention from that by instead making them struggle to learn how to show their work. They could do their own thinking and see how if you think of things a certain way, then what the teacher is teaching means something but instead they're being tested on their – Timothy May 12 '20 at 03:09
  • ability to do things the teacher's way and only their way which is a struggle. Finland doesn't give standardized tests. They use a student centered approach. If it weren't for being given so much material, maybe they would devote a lot of attention to this and explore it in more depth in their head. You could say what a prime number is and then state that for any odd prime p, 2 is a square modulo p if and only if p is congruent to 1 or 7 modulo 8. That might really fascinate them. Then you could state starting from the integers, crossing off the multiples of 2 gets you the odd numbers, – Timothy May 12 '20 at 03:15
  • crossing off the numbers that can be expressed as 3 times and odd number gets you those that are not a multiple of 2 or 3, crossing off those that can be expressed as 5 times a number that is not a multiple of 2 or 3 gets you those that are not a multiple of 2, 3, or 5, and crossing of those that can be expressed as 7 times a numbers that cannot be expressed as a multiple of 2, 3, or 5 gets you those that are not a multiple of 2, 3, 5, or 7. This also shows that once you cross off multiples of a prime number, the second multiple you cross of is its square. The numbers that be expressed as 5 – Timothy May 12 '20 at 03:21
  • times a number congruent to 1 or 5 mod 6 are 30 apart and those that can be expressed as 7 times a number congruent to 1 or 5 mod 6 are 42 apart. Now it's not that complicated to get those that are not a multiple of 2, 3, 5, or 7. It's simple to describe modulo 30. Now of them, we know that the ones under 11^2 are all prime. Now it's straight forward to show that 11^2 = 5^2 + 60 + 6^2 = 6 \times 4 + 60 + 6^2 + 1 = 6 \times 5 \times 4 + 1 = (30 \times 4) + 1. They might be fascinated by the method of showing what all the prime numbers under 11^2 are. – Timothy May 12 '20 at 03:29

4 Answers4

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I may be a bit jaded, but I don't think there are a whole lot of applications that will impress someone 10 years old. I tell college students (in a class for future educators, no less!) that their ability to shop safely online depends on prime numbers, and hardly get a reaction.

So, I'd take a different approach: Mystery and intrigue.

It's hard for anyone who hasn't studied math seriously to understand what mathematics is about, but most people believe we have it pretty well figured out (and if not, the Bigger and Better Computer of Tomorrow (TM) will surely have it all straightened out in a few years, right?). Which is why it should be surprising that we really don't understand prime numbers very well!

That's a bit of a stretch, of course. We know a lot about prime numbers, but the biggest (or at least most famous) open question in all of mathematics, the Riemann Hypothesis (I wouldn't even mention the name, let alone give any details!) is a belief about prime numbers. Another big-hitter (again, fame-wise; I can't fathom why it would be important to anyone) is the Goldbach Conjecture, another belief concerning prime numbers. This one could easily be stated to 5th graders, and they could verify that it's true for any numbers they pick out.

If the million dollar bounty for the Riemann Hypothesis is still in effect and you knew everything there was to know about prime numbers, you'd walk away with a cool million dollars! That's how much more we want to know about prime numbers, because we just don't know certain things!

The point is this. It's easy to define a prime number, and easy to work with prime numbers. But when we start asking certain questions, we just don't know. Nobody does. A handful of incredible mathematicians know a bit more than most, but even the most well-informed people on earth only know incrementally more than your students, when it comes to prime numbers. (Again: obviously a stretch. This really applies to isolated statements about prime numbers, but we're trying to sell here, not be pedantic).

I'll also mention that when we talked about the Sieve of Eratosthenes for finding prime numbers (again in the class for future educators), I remarked that this method is over two-thousand years old (older than many popular Western religions). Fast forward to now, and our best methods for listing all prime numbers in a certain range are only incrementally better. Cooler still, these better methods all use this basic sieving technique at their core! So we're better at listing primes because we're better at sieving, but not that much better -- in two thousand years!

Your 5th graders could easily sieve, and use the primes they find to verify Goldbach's Conjecture for tons of numbers. They'd be playing the game of mathematics then, getting their hands dirty in a completely self-sufficient way. And it can be phrased as a challenge: "I bet you can't write 138 as a sum of two primes!"

So the big moral of the story is that, to mathematicians, primes are mysterious, shiny objects. I wouldn't focus on their shininess, only the mystery. They have such mysterious facets that, in some ways, we're not much better at understanding them than we were thousands of years ago.

pjs36
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    I'm not sure it's even necessary for school to make them study prime numbers at all. Sometimes they learn the things they're interested in learning better. Study of prime numbers can be moved to job specific training. – Timothy Apr 28 '20 at 00:55
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I can tell, without doing any calculations, that

$$31\cdot 23\neq 37\cdot 19.$$

It follows from the unique factorization of integers into prime numbers, and I know that all four numbers are primes.

Mankind
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    Let's step into the shoes of a 5th grader for a bit. The possible reaction I see is this: "So what?" Perhaps a counter example would be helpful to add to the answer :) – PhD Dec 29 '15 at 20:03
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    @PhD Make a $31\times 23$ grid and a $37\times 19$ grid of Pokemons, and demand that they tell you, if the two grids contain the same number of Pokemons or not. Seriously though, I would've thought that if your $5$th grader was somewhat interested in mathematics, and I claimed that I could tell that $31\cdot 23\neq 37\cdot 19$ with only a glance at the numbers, then they would demand to know how. To be honest though, I don't know what makes a $5$th grader tick nowadays. – Mankind Dec 29 '15 at 22:47
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Donald Knuth said, "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations"

When I try to explain the so what in what I do, I often start there. The reason I study algorithms and mathematics is because I find it beautiful and interesting but it has a practical effect of making my computer do something a lot faster. Everyone likes faster computers.

To be honest, sometimes the beauty is found in the speed alone.

amcalde
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Here's another suggestion. Have some of the students (maybe $5$ or so) play a again. The purpose of the game is to end up with the lowest possible score.

Each student pick a number between $30$ and $40$. Maybe they pick $31, 32, 35, 37, 39$.

The game is to go through all the integers from $1$ to $40$, and each time an integer divides a student's number, that student gets a point. Of course the students with prime numbers will end up with only two points and will win the game.

This does not answer the question of why we should care about prime numbers (oh, to win the game of course!), but might lead to a better understanding or a bigger interest in primes.

Mankind
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