A while ago I put out a call for problems and situations in which the initial data suggests some pattern that is not actually what’s going on. The idea was, most students both fail to appreciate the need for proof in math and also are hamstrung in their ability to actually create proofs, by the fact that they’re used to patterns in math class always working. They see the pattern work twice and that’s good enough for them. We exacerbate this situation whenever we treat one or two examples as sufficient grounds for believing something, without a conversation about why they’re working. (Which is an exceedingly common practice that I too have been guilty of; but people, we need to stop, immediately and completely.) This goes on for 10 years and then they get to geometry class, where suddenly we ask them to start “writing proofs,” which they can’t do and don’t see the point of, since up until this point a few examples have been treated as sufficient grounds to believe. (I’ve written in much more detail about these issues here and here.)
So, I wanted to create a repository of problems and mathematical situations that could be used to freak students out and snap them out of the “it works twice it must be true” stupor, by suggesting a pattern that actually fails. I proposed a few; JD2718 suggested a few; and the real jackpot came from James Tanton, who devoted an issue of his newsletter to such situations back in 2006. Here is a pdf of the newsletter. Thank you Sue VanHattum for forwarding me this. Also, although I’d heard about James Tanton’s awesomeness from Bob and Ellen Kaplan, I hadn’t actually seen his website before today and it is clearly fresh, so check it out.
Here are my favorites from the repository as it stands now:
*The points on a circle problem. I described it in the same post as the call for problems. Here is Tanton’s visual:
This is still the best one.
*JD2718’s powers of eleven:
Rows of the Pascal’s Triangle?
I like this one as a counterpoint because the pattern is not completely wrong.
*Tanton’s powers of 3 idea:
|three to the … power||equals||number of digits|
Could you use this pattern to predict the number of digits in a high power of 3?
All three of the above examples are awesome not only because the obvious pattern fails but because the real story is within the scope of many high school classes. Most of what follows does not have this virtue; but it’s still exciting to me because the patterns that fail, and the questions you could ask about them, are still accessible.
Various gems from Tanton:
|6th root of||equals|
Look at the pattern in the first number after the decimal point: 1, 2, 2, 3, 3, 3, … According to Tanton (I haven’t done the calculation myself), the pattern continues for a while: four 4’s, five 5’s, six 6’s.
*Factors of factorials:
1! has 1 factor
2! has 2 factors (1 and 2)
3! has 4 factors (1, 2, 3 and 6)
4! has 8 factors (1, 2, 3, 4, 6, 8, 12, 24)
5! has 16 factors (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120)
*First Fibonacci sneakiness:
How many ways can n be represented as a sum of 1’s, 3’s and 5’s, distinguishing different orders?
1 – only one way: 1=1
2 – only one way: 2=1+1
3 – two ways: 3=1+1=3
4 – three ways: 4=1+1+1+1=1+3=3+1
5 – five ways: 5=1+1+1+1+1=1+1+3=1+3+1=3+1+1=5
6 – eight ways: 6=1+1+1+1+1+1=1+1+1+3=1+1+3+1=1+3+1+1=3+1+1+1=1+5=5+1=3+3
(Challenge question: how many ways can n be represented as a sum of any odd natural numbers, distinguishing different orders? Does this fix the Fibonacci pattern? Can you prove or disprove it?)
*Second Fibonacci sneakiness:
For n≥5, how many ways can you write n as a sum of natural numbers including at least one 5, this time disregarding order?
5 – only one way: 5=5
6 – only one way: 6=1+5
7 – two ways: 7=1+1+5=2+5
8 – three ways: 8=1+1+1+5=1+2+5=3+5
9 – five ways: 9=1+1+1+1+5=1+1+2+5=2+2+5=1+3+5=4+5
(Tanton claims that 10 can be so written 8 ways, but that the pattern breaks after this. I only find that 10 can be written 7 ways. Don’t such representations of n correspond in a bijective way with partitions of n-5? And p(10-5)=p(5)=7, right? In any case, with a class of kids who know the Fibonacci sequence, 1, 1, 2, 3, 5 should be enough to get them stuck on it.)
Funny business with primes:
These ones are easy to come up with because the primes just don’t behave. They are also somewhat harder to make into workable lessons because testing primality can be hard for students, even with numbers in the 2-digit range.
*I made a brainstorm for a lesson based on trying to find formulas for primes here. The punchline is Euler’s beautiful n^2 – n + 41, which Tanton also gives a version of, which produces primes for n=1,…,40 but fails (why?) for n=41.
*In the pdf linked above, Tanton gives several “prime-generating formulas.” None of them fail before you reach numbers too large for students to test primality, so I don’t think they could be used easily in school. However, two of them are just too cute not to mention:
31, 331, 3331, 33331, 333331, and 3333331 are all prime!
3!-2!+1! = 5
4!-3!+2!-1! = 19
5!-4!+3!-2!+1! = 101
6!-5!+4!-3!+2!-1! = 619
7!-6!+5!-4!+3!-2!+1! = 4,421
Lastly, JD2718 gives an idea that is inspiring my creativity in this vein:
*Even numbers like 10 and 32 have an even number of factors, while odd numbers like 49 and 81 have an odd number of factors. Right??
I like this because it actually gives us a template for making up many more such fake patterns: it doesn’t have to start at the beginning and the break on the way up, as all of the above do; maybe you just need to pick the numbers carefully. (E.g., 33, 69, 90… multiples of 3 must have all their digits be multiples of 3, right?)
* * * * *
A while back I mentioned that the awesome Catherine Twomey Fosnot was working on a book about algebra with mathematician Bill Jacob. It’s out. I just ordered it. (I tried to get a friend who went to NCTM to pick me up a copy at the publisher’s booth, but it was sold out before she got there.) It seems to be focused on the early grades, like the rest of the Young Mathematicians at Work series, which is a little disappointing to me since I was hoping for something directly applicable to Algebra I. But I’m sure it’s very thought-provoking and look forward to its arrival.