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I'm teaching an after school workshop for a few 7th graders. I was having them try to predict the next item in a complicated sequence. After some failed attempts, one of the kids started analyzing the differences between the numbers, then the differences between those differences, and used that to predict the next few items on the list. (This worked because the list was generated by a quadratic function but they don't know this part; they can predict the next list items but cannot write a formula yet).

I'm trying to assess the students and am not sure whether they just discovered that technique or had learned it in school already. In what grade are sequences and common differences usually covered? I'm talking about in the US, specifically NYC.

I'm also interested in pushing the idea further with them, but ideally in a different direction form what will be done in school. How often does the following concept get covered in MS-HS, and when?

Given a sequence $S=(s_1,s_2,s_3,\dots)$, its sequence of common differences is $D(S)=(s_2-s_1,s_3-s_2,s_4-s_3,\dots).$ Now take a polynomial $$f(n)=c_kn^k+c_{k-1}n^{k-1}+\dots+c_0,$$ and form the sequence $F=\big(f(0), f(1), f(2), \dots\big)$. Then $D^k(F)$ is a constant string of the number $c_kk!$, and using this one can determine the polynomial completely.

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  • $\begingroup$ This is typically an application in a first linear algebra course (2nd year after high school), and I suspect you'd only see it in high school as an optional topic or as an "extra for experts" type of exercise, and then probably only for quadratic and cubic polynomials in special cases as an application of simultaneous linear equations. If the students are quite strong (future strong participants in pre-Olympiad math contests, likely math majors in college, etc.), then I'd say sure, have at it. But for the vast majority of students I would avoid doing very much with it. $\endgroup$ – Dave L Renfro Nov 10 '18 at 20:40
  • $\begingroup$ In contrast to @DaveLRenfro 's experience, this topic was covered in the remedial college algebra course I TAed in grad school. (not so very long ago, students mostly from Michigan) Not proved or in full generality, of course, and restricted to at most 2nd order. I'm not sure what previous exposure was, but it didn't seem totally new to them. $\endgroup$ – Adam Nov 10 '18 at 22:21
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    $\begingroup$ This is something that one can figure by themselves: Checking the differences identifies arithmetic series, and trying that but noticing the differences have a pattern is within bounds of a mathematically oriented child. $\endgroup$ – Tommi Brander Nov 11 '18 at 10:12
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    $\begingroup$ A cool approach to this goes by the name "discrete calculus". $\endgroup$ – Steven Gubkin Nov 13 '18 at 13:38
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    $\begingroup$ To contrast Adam's and Dave L Renfro's comment, I learned this in High School Algebra II (10th grade, although 'standard level' students would take it in 11th grade). I found it very interesting to learn this and thought it was an amazing tool. The number of common differences you have to look at tells you what kind of polynomial it is?!?!? Crazy! When I tried showing this to some 12th grade students or even first year college students, it blew their mind as they have never seen it before. $\endgroup$ – ruferd Nov 14 '18 at 14:03
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The linear function component is covered early(ish) in Algebra 1, and quadratic functions are covered towards the end of Algebra 1; so, the former by 7th/8th grade and the latter - if at all - by 8th grade. But, even for students in New York (where I teach) taking Geometry in 9th grade, more general polynomials may not be covered until Algebra 2 in 10th grade. Even then, using (nth order) constant differences to solve for the corresponding (unique, degree n) polynomial is unlikely to be covered: It requires solving n equations in n variables, which is one reason that DL Renfro remarks that it may be saved for a first course in (post-secondary) Linear Algebra.

Personally, I teach it in a Problem Solving & Problem Posing course to 12th grade students who have already taken, or are currently taking, differential Calculus. This is not the norm in the state/city, and/but I do not think that the students would otherwise see such an idea during (middle school or) high school beyond the linear and quadratic cases.

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    $\begingroup$ One aspect of the question that I didn't bring up in my first comment (and which I wasn't particularly interested in doing so, to be honest) is that the question ranges from 7th grade through high school (that's like basic calculus through measure theory), nothing was said about the level of students involved (many students won't see any algebra until high school, and only then very elementary algebra and no simultaneous equations), and the description of the problem suggests a too high mathematical approach (but perhaps that description was only designed for people here). $\endgroup$ – Dave L Renfro Nov 11 '18 at 19:03
  • $\begingroup$ @DaveLRenfro I don't disagree! $\endgroup$ – Benjamin Dickman Nov 11 '18 at 19:23
  • $\begingroup$ @DaveLRenfro Yes, the description of the general idea was for reader on this site, and would not look like that at all when done with students. $\endgroup$ – j0equ1nn Nov 12 '18 at 15:56
  • $\begingroup$ @j0equ1nn: This is an old post, but I think I should point out that in fact no more than basic arithmetic is actually required to understand how to use the iterated difference sequences to easily compute the original polynomial. It is explained in this post. It does not require any linear algebra. From a high-level mathematical view, it is simply a matter of choosing the right basis (binomial coefficients) so that the difference operator becomes a shift-operator on the sequence after transforming to the new basis. $\endgroup$ – user21820 Jul 9 at 13:39
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This is a topic I could imagine not being adequately covered in all U.S. schools, although (as Dave L. Renfro pointed out in a comment), it is listed in the Common Core Mathematics standards under high school functions (HSF.BF.A.2).

I could imagine it being presented successfully anywhere from the 3rd grade up to an advanced college class for mathematics educators. As one example, the NYC community college where I teach probably has at least 5 different math courses where this gets treated, including our lowest-level liberal-arts quantitative-reasoning course; as well as (briefly) in our highest-level terminal course for mathematics and computing majors.

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    $\begingroup$ I can't find the phrase "common difference" or "arithmetic sequence" anywhere in the Common Core Mathematics standards --- Actually, F-BF.A.2 states: "Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms." For what it's worth, I'm somewhat familiar with this as I've written quite a few standardized test items targeting various common core standards. $\endgroup$ – Dave L Renfro Nov 11 '18 at 21:44
  • $\begingroup$ @DaveLRenfro: Thanks for that. I guess having the words "arithmetic" and "sequence" separated like that foiled my search attempt. $\endgroup$ – Daniel R. Collins Nov 11 '18 at 23:27

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