Mathematical patterns
June 20, 2016 6:35 PM   Subscribe

Does this math pattern have a name? How can I find more like it?

My son, 9 years old, loves math. Last year, he told me that he was thinking about math while lying in bed one night before sleep – you know, as you do!-- and this pattern had occurred to him:

n^2 + (n+n-1) = (n+1)^2

(He described it to me in 9-year-old words, not an equation, for what it’s worth.) And then, the other day, he told me that he’d figured this out:

n^2-1 = (n-1)*(n+1)

This kind of blows me away, because my brain just doesn’t work that way and I have to sit down and really think this stuff through—but it comes effortlessly to him.

So, two questions:

1. Do either of those patterns have a name? I know about, like, the Fibonacci Sequence and I think it would tickle him to know that he’s stumbled onto Someone’s Theorem or whatever.

2. What are some resources that can encourage / expand this kind of thinking? Obviously this is not my thing so I’m having trouble even finding the words to describe what I’m after. Remember that he’s just a little kid, so he’d probably lose interest if it’s dry and complicated. He just really digs the way that numbers are so orderly.
posted by oceanmorning to Science & Nature (17 answers total) 6 users marked this as a favorite
That first equation isn't valid...

These are quadratic equations.
posted by mr_roboto at 6:39 PM on June 20, 2016

The first one should just be + 1 on the left instead of - 1. That is n2 + n + n + 1 = (n + 1) 2

I don't know of any term for those kinds of things. They're provably equivalent algebraically, but I don't think there's nothing mathematically special about them, other than one side being satisfyingly simple.
posted by aubilenon at 6:49 PM on June 20, 2016 [1 favorite]

The -1 should +1 in the first equation but otherwise it's correct. Your son is learning algebra.
posted by rdr at 6:49 PM on June 20, 2016 [2 favorites]

I'm not sure what you're seeing as a pattern here. The first is technically called a binomial expansion (with a typo as stated). The second is the answer to a polynomial multiplication problem, albeit a nice tidy answer instead of the all-terms salad that usually comes out.

I'm guessing he might be interested in Pascal's triangle, which is a super-cool pattern of numbers that will also produce the coefficients for any-power binomial expansion. He might see the pattern in how that comes about (that the simple pattern in how the triangle is constructed gets replicated when you multiple two binomials together). I've linked just to the GIF; in the tradition of Wikipedia math articles the main article presupposes the reader comes equipped with a PhD in set theory and a week of free time.
posted by range at 6:50 PM on June 20, 2016 [1 favorite]

Sounds like he understands the geometry behind multiplying numbers (you can think of the first one like adding a new row and a new column to a square to get a larger square, and the second like making a row into a column to form a rectangle from a square).

Math is all about generalizing and finding patterns. What is (x+1)3 equal to? Maybe you can use the geometry of a cube to figure it out! What about (x+1)4? Is there a pattern as the numbers go up?
posted by panic at 6:55 PM on June 20, 2016 [4 favorites]

He might enjoy the fun short math videos of Vi Hart. (Some of these miiight have some bad language in them? I can't remember, would be worth previewing maybe.)
posted by LobsterMitten at 6:57 PM on June 20, 2016 [3 favorites]

As far as finding more stuff like this, he might enjoy Numberphile videos.
posted by brentajones at 6:58 PM on June 20, 2016 [1 favorite]

The second one is an application of difference of squares. More generally (A^2 - B^2) = (A+B)(A-B). Even more generally, x^n - 1 = (x-1)(x^(n-1) + x^(n-2) + ... + x^2 + x + 1), where the larger polynomial on the right is called a cyclotomic polynomial.

The first one has a nice visual illustration, showing that the square numbers are sums of consecutive odd integers. Basically you've got that the recursive sequence a_n = 1, a_(n+1) = a_n + 2n+1 is just the perfect squares.
posted by Elementary Penguin at 7:15 PM on June 20, 2016 [1 favorite]

The first one would be true if it were n^2 + n + n +1 = (n+1)^2. This is actually a special case of a more general rule (as mentioned above, called a "binomial expansion") (n + m)^2 = n^2 + 2nm + m^2, which might be fun for him to figure out for himself - what happens if you take (n + 2)^2...? (n + 3)^2...? (n + m)^2?

A cool way to discover the pattern (and what Elementary Penguin said about squares being sums of odd numbers) in an approachable way that I got to do as a kid is to use graph paper to first shade in one square, then (in a different color) shade in enough more squares to make a 2 by 2 square, then (in a different color than the previous one) enough more squares to make a 3 x 3 square - and have him count up how many squares he's shading each time. (So how many more little squares does it take to make a square one row bigger on each side? A square two rows bigger on each side? How does this change as the squares get bigger?)

(This is also fun to do with Legos or Duplos or any other toys in different colors you might have around. This blog post has a picture of the pattern!)

The second one is also a special case, this time of a "difference of two squares": n^2 - m^2 = (n+m)(n-m), which might also be fun for him to figure out for himself - what's (n+2)(n-2)? (n+3)(n-3)? How does the pattern generalize?

A way to explore this one I also learned from my family with Legos/Duplos/etc.: start by making a perfect square, for example, a 5 by 5 square, with the blocks. Then make one side one row longer and one side one row shorter - so now you have a 4 x 6 rectangle. How many fewer blocks did you need to make this? Now make one side one row longer and one side one row shorter - now you have a 3 x 7 rectangle; how many fewer blocks did that take? And so on.

Another thing he could think about after he does this is that each of the rectangles have the same perimeter: that is to say, if we imagine that each rectangle is a field, you'd need the same amount of fence to go around the edge of the 5 x 5 field as the 4 x 6 field, but the 4 x 6 field has less grass on the inside! So if you have a certain amount of fence, what's the best rectangle to fence your field to have as much grass inside as you can? (And then, what's the best fence if it doesn't have to be a rectangle and can be any shape you want?)

Also, when I was about his age, some of the books I loved that explore concepts like these in a fun, playful way were The Number Devil and The Adventures of Penrose, the Mathematical Cat - lots of cool things for exploration there. So cool to hear he's excited about this! And of course whatever direction and pace he wants to take with all this is awesome - I would definitely follow his lead about what he's excited about or not.
posted by beryllium at 8:39 PM on June 20, 2016 [8 favorites]

I would call these algebraic identities, and, interestingly, they were studied a lot by early algebraists, who themselves were just figuring this kind of pattern out! For example, early Indian mathematicians developed certain algebraic rules, and early Islamic mathematicians developed really nice proofs of the sort of algebraic identities that your son developed: for example, to show that (a+b)^2 = a^2 + ab + ab + b^2, you construct a big square, which is length a+b on each side, and divide the square into pieces between the length a and the length b on both sides, to make 4 pieces, two squares and two rectangles:

| ****|***********|
(bad ascii, sorry)---so then, the whole thing has area (a+b)^2, but it can be subdivided into smaller pieces. The upper left square is a x a = a^2, then there are two axb = ab rectangles, and finally the lower right square is b x b = b^2, so the total area must also be equal to a^2 + 2ab + b^2. If you click through the link to the islamic algebraists, you will see this exact picture, only better drawn!

Some other traditional high-school algebraic identities:
(a-b) = a^2 - 2ab + b^2
a^3 + b^3 = (a+b)(a^2- ab + b^2)
(a+b)^3 = a^3 + 3a^2b+3ab^2+b^3
---this one is cool because it generalizes to a formula for expanding (a+b)^n, which uses combinatorial counting formulas (how many ways are there to choose k things from a set of r things) and the coefficients of (a+b)^n are what's listed as the nth row of Pascal's Triangle, referenced by several commenters above. (This is why I have memorized how to expand out binomials up to the 5th power, because that's how many rows of the triangle I've memorized. Mind you, I do this professionally.)

A couple of my favorite identities/patterns:
(1) What can you say about the sum of the first n odd numbers? (that is, if you compute
1+3 = ?
1+3+5 = ?
1+3+5 +7 = ?
1+3+5+...+(2n-1) = ?

Can you/he figure out an expression for the right-hand side, in terms of n? What's the pattern?

(2) A classic: What can you say about twice the sum of the first n integers? What is 2(1+2+...+n) = ?

These are standard problems to give when learning a kind of proof called mathematical induction, for what it's worth.

It's possible that some of the activities on this page might interest your son. Math enrichment might be a good search term. MeMail me if you want to chat some more---I'm not a elementary math teacher (I'm a college professor who teaches mathematics) but I do have two elementary-aged kids.

(Also, my Barnes & Noble has a bunch of summer workbooks for kids, including a series on "Math for the gifted student". I've got the grade 3 version in front of me, which is way too easy for your son, I'm sure, but it's got a good assortment of interesting problems. So that might be worth checking out. I'm guessing he's going into grade 5?)

(There's whole branches of mathematics based on people "dig[ging] that numbers are so orderly", such as number theory, algebra, and combinatorics! And some lucky people---such as myself---get to have as part of their work showing new ways that numbers, and other mathematical objects, work! It's cool!)
posted by leahwrenn at 10:16 PM on June 20, 2016 [12 favorites]

I think your son might enjoy some of the things you find when you google "proofs without words".

Here's a way you might think about those results graphically:

(n + 1)^2 = n^2 + 2n + 1:
Imagine you have a n x n square of dots. How do we expand it to an (n + 1) x (n + 1) square? Well, we start off by adding a column of n vertical dots on the right hand side; now we have n rows and n + 1 columns. Now we fill in the bottom row. We already expanded the number of columns from n to n + 1, so we now have to add n + 1 dots to fill out the square. So, to go from an n x n square to an (n + 1) x (n + 1) square, we started with n^2 dots, then added n, then added n + 1; (n + 1)^2 = n^2 + n + (n + 1) = n^2 + 2n + 1.

(n + 1) * (n - 1) = n ^ 2 - 1:
Imagine again that we have a n x n square of dots (so we have n^2 dots). Take away the one in the bottom right corner, so now we have n^2 - 1. There's a "almost row" of n-1 dots on the bottom. Take it away, flip it 90 degrees to make it a column, and add it to the right-hand side of the rectangle that's left (it matches up exactly; the square started off with n rows, and we've completely removed the bottom one, so it now has n - 1 rows). So now we have a rectangle with n - 1 rows (because we removed the bottom row), and n + 1 columns (because we moved all but one of its dots to be a new column on the right). Summarizing all of this, if you take an n x n square of dots, and take away one of the dots (leaving you n^2 - 1), you can rearrange what's left into a rectangle with n - 1 rows and n + 1 columns. In other words, n^2 - 1 = (n - 1) * (n + 1).

If you / your son like that kind of way of looking at it, I think you'll like the proofs without words websites.

The best "fun but challenging" math books at the middle / high-school level are the "art of problem solving" series. They have a books aimed at younger audiences, but I haven't actually looked at them in as much depth and can't vouch for them as strongly.
posted by The Notorious B.F.G. at 10:20 PM on June 20, 2016 [2 favorites]

for example, to show that (a+b)^2 = a^2 + ab + ab + b^2, you construct a big square, which is length a+b on each side, and divide the square into pieces between the length a and the length b on both sides, to make 4 pieces, two squares and two rectangles:

Here's a pretty version that shows this. You can do it with (a+b)^3 and you get a "binomial cube" which decomposes to similar pieces.
posted by BungaDunga at 11:18 PM on June 20, 2016

panic said it clearly above: these are geometrically-based proofs of algebraic identities. they were used by penrose (a very famous mathematician) in a book (called the emperor's new mind) to argue that artificial intelligence is impossible. i can't find the example from his book, but there's a paper with similar arguments (but different conclusions) here.

sorry, none of that is suitable for an eight year old, but the fact that he already thinks like this is very cool (but may also be related to the way that children are taught maths these days, which seems to play more emphasis on this kind of thing?)
posted by andrewcooke at 6:29 AM on June 21, 2016

Since he has, I surmise, visualized laying out an n x n square, and adding blocks to make it n+1 x n+1, you could challenge him to start with an n x n x n cube. Or to start with an n x m rectangle.
posted by SemiSalt at 6:48 AM on June 21, 2016

This might seem like a weird idea, but: Get him a couple of high school math textbooks. You can get used copies of slightly older editions of the Holt textbooks (e.g. 2007, 2008) for $5-$10 each now that newer editions are out. Algebra 1 and Algebra 2 would be good starting points, given the equations he's figured out. Might as well get Geometry and Pre-Calculus, too, given that they're so cheap and the ideas in them are interesting.

You might think, ugh, textbooks, but many of them are surprisingly well-written, well-illustrated, and engaging. And for a total investment of under $40, there's no harm in having them lying about the house.

Don't be worried whether or not he's able to handle them; if he figured out one of the more interesting quadratic identities on his own, he's already ahead of most of the high school students for whom these textbooks are written.

One of the neatest things my Dad did for me was randomly buy a bunch of old engineering textbooks at an auction. He had no clue about math, but he recognized that I liked math and they contained math, so he bought the boxful and brought 'em home. I consulted one of them, a calculus quick reference book from the 1950s, just the other day.
posted by clawsoon at 11:19 AM on June 21, 2016

Thanks everyone for these thoughtful replies!

My kid was delighted to no end yesterday by the terms 'quadratic' and 'polynomial'. I'm going to look into all of these options for further learning.

(And you're right, I did make a typo-- oops! My mistake, not my kid's.)
posted by oceanmorning at 2:22 PM on June 21, 2016

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