f[sub]n+1[/sub][sup]2[/sup] - f[sub]n-1[/sub][sup]2[/sup] = f[sub]n[/sub][sup]2[/sup] + 2f[sub]n[/sub]f[sub]n-1[/sub]
For example, for the subsequence 2, 3, 5 we have
Etcetera, etcetera… (to paraphrase Yul Brenner as the King of Siam)
(ETA) This is not what I would call an immediately obvious result, but it is true for any such subsequence in the series.
Well Wendell Wagner, i’m not sure why i brought it up either. maybe i’m too conservative but i guess i just feel that a 10-year-old should not be subjected to posts like #26.
i may be underestimating the ability of the average 5th grader but imho even your example in post #37 might be a tad confusing (especially for pupils not familiar with English yet) , whereas i would be comfortable with showing my examples (as questions with suitable numbers blanked out) in post #36 to any pupil from P3 onwards who has grasped the basic four arithmetic operations. but that’s just off topic, so i’ll stop the hijack now.
This is an excellent post, and should be kept in mind when wondering if kids are doing the math involved in the posts shijinn listed. I don’t think they are held to such a high level of ability, but a basic introduction and being asked for any insights or understandings they can glean from it is not out of the ordinary.
Post #26 wasn’t for the fifth grader, of course. It was for the various other people in this thread who might be interested in seeing a rigorous proof of what the fifth grader was probably expected to simply assume to hold true through generalization from sufficient observation.
I don’t have anything to contribute, really. I just wanted to say that, although I’ve always considered myself pretty good at math, this is damn confusing at first! I took Algebra 1 and 2, and Geometry/Pre-Trig in high school, and we never learned this at all. (Stupid hick school.)
Ignorance fought, though. I learned something today! does happy nerd dance
Not to hijack this after all these posts, but **freekalette’s ** note reminds me of a famous (in the edubiz) video of an interview of Harvard grads, in which 18/23 of those interviewed - including a know-it-all professor - either could not explain the reason for the seasons, or the phases of the moon. One of them reports that he had taken Planetary Motion, Astrophysics, etc. “Covering” a topic has very little to do with teaching it, and many disciplines are made of courses that make no room for wonder, exploration, questions, inquiry, or departure from the canon. Consequently, most of us report that we took such and such a course, and can’t figure out why we didn’t learn about xyz. There’s work to be done, folks, and it ain’t via NCLB. xo, C.
I don’t think the teacher is trying to point out any special properties of 2, 5, and 21 or the fibonacci series; I think they are just trying to teach the kids about the distributive property and foil… i.e. shijinn has it right.
Actually, I’m terrified. I have no idea why I was interested in this kind of stuff as a kid. I have no recollection of any teacher or parent pointing me to this stuff. I have no recollection of learning it in school. I don’t know if I just checked out the right library book on accident one day or what.
And so the reason I’m terrified is, I have no idea how to help make sure my kids are into this (or at least, in the position to understand what there is to be into should one’s inclinations incline one that way). And it looks like I can’t trust the schools to do it so, I’m scared.
I have a feeling that sitting them down and showing it to them myself isn’t exactly the right approach. That would be forcing it.
Having the right books laying around could be a good idea, though, I suppose.
At the top of the article, there will be a tab called “History”. Click it, and you will be able to see all the old versions of the article, see which edits were made when, etc.
Ouch. Wow. I don’t think I started looking at series until high school, and I don’t think I really understood the material until I got my masters degree…
Regarding my homework: I am pressed during the day and really don’t have any business posting at all. But I really would like to participate.
I don’t think it’d be quite right to say the kids are learning about series, as a general topic. Rather, their being shown how some sequences of numbers you get by simple algorithms can have some interesting “number-magic” type properties. Nothing high level there, AFAICT.
If they were having the kids construct inductive proofs or something, that’d be cause for amazement.
I hope you didn’t take my post too seriously. I was just kidding with you. I jump in on threads I haven’t read carefully practically as a matter of policy. Most people seem to, in fact.