Jeez. When I was in fifth grade we got to take off our shoes to count to 20. Guys could count to 21.
Seriously, I don’t think I encountered Fibonacci until high school (or college?), because the applications of things like this just don’t arise at a lower level. Maybe it is a trend in ElEd to take these quirky and interesting things about numbers and present them earlier to try to spark some interest before the kids get to the point that they actually need to use them.
It looks to me like this statement: “Every 3rd number of the sequence is odd” should be “Every 3rd number of the sequence is even”. Any wiki specialist care to address this?
The Wikipedia article starts the sequence at 0, rather than 1.
Note that the two starting numbers are conventionally 1 and 1 (or 0 and 1, which is equivalent), but one can define the sequence with any two starting numbers. I’ve managed a proof that the square of the nth element minus the product of the (n+1)th and (n-1)th elements is a constant (multiplied alternately by plus or minus 1), which depends on the two initial values, and which is 1 for the “normal” sequence.
Still working on the proof that the difference of squares of the (n+1)th and (n-1)th elements is in the sequence.
Also note that the ratio of consecutive elements approaches the Golden Mean as n goes to infinity, independent of the starting values.
Fun stuff!
-Rick
Looks like the question has been answered, so I just thought I’d ask if anybody else thinks of the parrot from MathNet, the show at the end of Square One, that said, “1, 1, 2, 3”, and they were able to solve some mathematician’s death based on that clue. Every darn time I hear of the Fibonacci sequence, I think of that stupid parrot, and I must have seen that show 20 years ago.
Oh man, I loved that show! 
-FrL-
ETA: Oh dammit, to this day whenever I multiply anything by nine for any reason this song goes through my head. Now that I’ve seen it again (probably for the first time in over 15 years) it’s sure never to leave my head, ever.
Yeah, the Fibonacci sequence is inextricably linked with Mathnet for me as well.
If you mean specifically the case of 5x5 - 2x2, well, that’s clearly 21, which is easily checked to be part of the Fibonacci sequence.
If you mean, more generally, a proof that a difference of squares of Fibonacci numbers two apart is always a Fibonacci number, well, there are of course a million ways to do it, but here’s one boring, fairly mechanical way:
Let F(n) denote the nth Fibonacci number [with the convention that F(0) = 0 and F(1) = 1]. Let G(n) denote the sum F(0)^2 + F(1)^2 + … + F(n)^2.
As a first lemma, we shall prove that F(n) * F(n+1) always equals G(n), by induction on n. The base case (n = 0) is readily manually verified. As for the inductive step, assume this holds for n and then, to show it holds for n+1, note that G(n+1) = G(n) + F(n+1)^2 = [by the inductive hypothesis] F(n)*F(n+1) + F(n+1)^2 = (F(n) + F(n+1))*F(n+1) = F(n+2)*F(n+1).
As a second lemma, we shall demonstrate that F(n+2)^2 - F(n)^2 = G(n+1) + G(n). This is simple; note that F(n+2)^2 - F(n)^2 = (F(n+2) + F(n)) * (F(n+2) - F(n)) = (F(n+2) + F(n)) * F(n+1) = F(n+2) * F(n+1) + F(n) * F(n+1) = [by the first lemma] G(n+1) + G(n).
Finally, we shall prove that both F(n)^2 + F(n+1)^2 = F(2n+1) and that F(n+2)^2 - F(n)^2 = F(2n+2), by simultaneous induction. The base case (n= 0) is readily manually verified. As for the inductive step, assume the two conditions hold for n; we have to know show that both conditions hold for n+1. First, note that F(n+1)^2 + F(n+2)^2 = (F(n+2)^2 - F(n)^2) + (F(n)^2 + F(n+1)^2) = [by both conditions in the inductive hypothesis] F(2n+2) + F(2n+1) = F(2n+3) = F(2(n+1)+1). This establishes the first condition; as for the second, note that F(n+3)^2 - F(n+1)^2
= G(n+2) + G(n+1) [by the second lemma]
= (G(n+2) - G(n)) + (G(n+1) + G(n))
= (G(n+2) - G(n)) + (F(n+2)^2 - F(n)^2) [by the second lemma again]
= F(n+2)^2 + F(n+1)^2 + (F(n+2)^2 - F(n)^2) [by the definition of G]
= F(2n+3) + (F(n+2)^2 - F(n)^2) [by the first condition just previously established]
= F(2n+3) + F(2n+2) [by the second condition of the inductive hypothesis]
= F(2n+4)
= F(2(n+1)+2). This completes the proof; as F(n+2)^2 - F(n)^2 = F(2n+2) for all n, we know that the difference of squares of Fibonacci numbers two apart is always another Fibonacci number. Q.E.D.
(Of course, if they expect the average fifth grader to be able to demonstrate why the result holds true, accurately even if non-rigorously, then there must be some basic insight I’m missing. But perhaps they mean something much less when they ask the students to “Explain.”)
The problem is that fifth-graders don’t know algebra. If you know algebra, it’s easy to prove that the difference between the squares of two Fibonacci numbers that are two apart in the sequence equals another Fibonacci number. If you don’t, the only thing that’s possible on this problem is to do a bunch of examples and guess that this means that it’s generally true. I would explain this in fifth-grade terms as follows:
Here’s the first twelve numbers of the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Now let’s look at the differences between the squares of two Fibonacci numbers that are two apart:
(22) - (11) = 4 - 1 = 3
(33) - (11) = 9 - 1 = 8
(55) - (22) = 25 - 4 = 21
(88) - (33) = 64 - 9 = 55
(1313) - (55) = 169 - 25 = 144
So what do we notice about the differences? Well, they’re all Fibonacci numbers. In fact, if you look at the numbers, they’re every other one in the Fibonacci sequence. That it, they are the numbers I’ve marked with an asterisk in the sequence:
1, 1, 2, 3*, 5, 8*, 13, 21*, 34, 55*, 89, 144*
So it looks like there some sort of a general rule here. In fact, you can prove that there’s a general rule. You have to know algebra to prove it though. Indistinguishable has proved it above using algebra.
Of course they do. Second graders know algebra, it’s just that instead of using x’s and y’s, they use blanks and duckies. 
(But yes, of course, they don’t know proofs.)
I don’t know what math program the OP’s school uses, but there is a strong math program now out of the University of Chicago which, if used from day one and continued through graduation, is a really fantastic program. Unfortunately, all its strengths turn into weaknesses when you switch into or out of the program midstream. The general idea is that you don’t divide maths into “mathematics, algebra, geometry, trigonometry, calculus” and teach them sequentially, but you dabble here and there, showing them some basic fundamentals and then some interesting stuff. It’s all put together so that one concept feeds another, and almost everything is done by Real World Application (I don’t think they call them Word Problems anymore, because Word Problems are boring and hard.) They also teach different algorithms than the classic style, which is great for the kids who grok them, but really hard for the parent to help with unless the school gives out the Parent Letters (which, in my child’s school, they didn’t, because of cost issues.)
This isn’t the only program which would include Fibonacci numbers by 5th grade, but it’s one of them. Everyday Mathematics
ok, grade 5 is like primary 5 right? we don’t teach Fibonacci numbers in primary schools but the ability to see number patterns are taught in general, so this is an alternative answer if the topic is not specifically about Fibonacci numbers.
for 3a) the key numbers here as hinted in the question are 2,5 and 21.
2+5=7
5-2=3
7x3=21
it works with any number you put in.
WhyNot writes:
> Of course they do. Second graders know algebra, it’s just that instead of using
> x’s and y’s, they use blanks and duckies. (But yes, of course, they don’t know
> proofs.)
Show me any sense in which fifth graders know algebra that would help with this problem. I thought I was helping by showing Leaffan’s daughter the pattern which she might be able to see herself. If you have something to show her that might help her, do so.
I don’t, for instance, see how this by shijinn helps:
> for 3a) the key numbers here as hinted in the question are 2,5 and 21.
>
> 2+5=7
> 5-2=3
> 7x3=21
>
> it works with any number you put in.
What’s the pattern here? I don’t know what you’re trying to show, and I’m a mathematician. Give us several examples so that it’s clear what pattern you’re trying to show us.
I am not smarter than a 5th grader. 
But, at least I know where to send her if my future 5th grader ever has a math problem I can’t help her solve. Great job guys!
Since this thread is about the Fibonacci numbers, perhaps a link to the Staff Report What’s up with Fibonacci numbers? would be appropriate.
I think he’s trying to show that x^2 - y^2 = (x+y) * (x-y). I don’t really think it’s what Leaffan’s daughter’s assignment is looking for, though; they definitely want her to point out that 21 is in the Fibonacci sequence, and everything else everyone else has said. I think you were right when you said they probably just want the students to notice the pattern that doing this difference of squares of Fibonacci numbers two apart thing in order gives every other term in the Fibonacci sequence, with the students expected to basically assume it works out all the time once they see it work out for enough examples with no counterexamples.
You are correct, sir. Back in the day, kids learned arithmetic. Today, while it’s usually called mathematics, it’s still arithmetic - basic arithmetic. Except where a teacher is intelligent enough to realize that numbers are really cool and kids deserve to learn this, so they truly broaden the scope of math instruction to include lots of things that are not simple computations. Now, the edubiz has been advocating this approach for a very long time, but teachers are legendary for swearing fidelity to a reasonable and relatively progressive methodology while they’re in college, but reverting to the conventional approaches when they get into the classroom. So this kid is experiencing a still relatively rare approach - her teacher is trying to engage the kids in learning. The question could use some tweaking IMO, but the point you make is valid. Has been for a long time, explaining why Mark Twain said, “I never let my schooling interfere with my education.”
what Indistinguishable said . just offering a different point of view.
5 – 5 – 25
2 – 2 – 4
7 – 3 – 21
8 – 8 – 64
4 – 4 – 16
12 – 4 – 48
12 – 12 – 144
7 – 7 – 49
19 – 5 – 95
23 – 23 – 529
16 – 16 – 256
39 – 7 – 273
The problem is, shijinn, that what you’re trying to show isn’t about the Fibonacci numbers. What you’re trying to show is just a fact about numbers (in fact, about any real numbers, not even about just integers). It’s an interesting fact about numbers, but I’m not sure why you brought it up. The way that you’ve written it isn’t very clear either. This is a better way of showing it:
Take any two numbers. Look at the difference of the squares of those two numbers. Then look at the product of the sum of the two numbers and the difference of the two numbers:
For instance, let the two numbers be 5 and 2:
(55) - (22) = 25 - 4 = 21
(5 + 2) * (5 - 2) = 7 * 3 = 21
So the two results are the same.
And so on for all your other examples.
On re-reading your post, I see that you knew that it wasn’t a fact about Fibonacci numbers. I assumed that you were talking about Fibonacci numbers in this thread.
Immaterial. Look at the F #s listed: 0 1 1 2 3 5 8 13 21 34 55 79 144 233 377 610 987 1597 2584 4181 6765. Even(?), odd, odd, even, odd, odd, even, odd, odd, even, … Every 3rd number of the sequence is even.
The Wikipedia article history reveals that it had long stated “Every 3rd number of the sequence is even”, up until about a week ago, and, in fact, it says that again. Just a transient blip of vandalism you happened to spot, apparently.
A discussion of Fibonacci numbers is often a case of splitting hares.