Consider the following table displaying the first four positive integer multiples of π rounded to 4 places:
So what's the challenge here? You will be a π-multiple investigator!
(1) Determine the EIGHT positive integer multiples of π, up to 1000π, which, when rounded to THREE places, produce an integer result. For example, the decimal 17.9996 when rounded to 3 places results in the integer 18.000, which we will regard as an 'integer'. Unfortunately, this decimal is not an integer multiple of π so have fun searching! Do you notice any pattern in your results? Describe it! Can you explain this pattern? [This requires more than a Yes/No response!]
(2) In fact, the "pattern" you may have found in (1) continues beyond 1000π. Show that you can go up to SIXTEEN multiples of π before the pattern breaks down. Why do you think the pattern eventually ends?
(3) Which of your results in (1) would produce an integer when rounded to FOUR places?
(4) Can you think of any practical application for finding multiples of π which are very close to an integer? Be creative! Responses may depend on how advanced your math background is.
Comments:
- Do your students know how to program their graphing calculator to do the search? OR in Java or C++ or Python or Perl?? This would certainly facilitate the search! I may display the code for the TI-83 or -84. However, students may also find a creative way to use the TABLE feature on their graphing calculators to save time without programming. Have fun!
- Please post feedback if you use this in your classes. I will not post answers yet in case students find this post in their 'searching'!
- For most students, the full significance of this innocent looking search problem will not be apparent. You might want to give them a hint. Perhaps we should call this post:IS π ALMOST RATIONAL?
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11 comments:
I think irrationality measures are too complicated for middle school and high school students. The proof that a number is irrational if it is can be approximated too well by rationals would just be too difficult.
Eric--
This is a first! You're suggesting that I am setting bar too high!! However, I will need to clarify this activity so that it makes more sense for the younger student.
First of all, if we want a multiple of pi to produce an integer with 2-place accuracy, students should observe that 7pi is approx. 21.9911 which demonstrates that 7pi is approx 22 or pi is approx 22/7, accurate to two places! I should have asked for this first but I jumped over this and went right to 3- and 4-place accuracy.
As it turns out, my goal was to guide them toward the fascinating discovery that 113pi is approx 354.9999699 so that pi is approx 355/113 accurate to more than 4 places -- SIX places in fact!!
My goal was not to prove irrationality, but rather to develop a procedure for approximating irrationals by rationals. I started out planning to develop convergents to pi using continued fractions and ended up in a different place.
An aside here is also the issue of numerical analysis and writing efficient computer searches. I wrote a fairly simple program for the graphing calculator that produces these multiples to any desired accuracy (within limits of course!). If anyone wants me to post the code for this program, let me know...
Ah. You aren't going to develop the best approximations; you're going to going to start with an exhaustive search. I'm not sure that counts as a procedure. It's hardly useful for general use.
Absolutely, Eric! Inefficient, not generalizable, impractical BUT:
(1) Kids from Middle School on (and even younger) love exhaustive searches!
(2) The more advanced student will use her ingenuity to play with the graphing calculator and discover other ways (e.g., use of the TABLE feature).
(3) Some will want to know how I programmed this and will pick up on it easily (or already know how to program in some languages)
(4)Some may ask or can be led to ask about the numerical limitations of this 'method' and the resulting errors
(5) Most importantly, Eric, I already demonstrated a little piece of this activity to a group today and they found the idea of 'almost-integer' multiples of pi leading to a cool fraction like 355/113 (or the more familiar 22/7) to be intriguing.
You never know, Eric, until you try. I'm sure you experimented and took risks when developing lessons. If one of my ideas bomb (they find it too boring or too difficult), I don't abandon the idea necessarily. I think about revising it to work better. To me, that is the 'practice' of teaching.
How about making a sequence of observations like:
pi is close to 3. Well, it's a little more.
So what's pi - 3? Is it a nice fraction?
Maybe if we look at 1/(pi - 3) we'll discover something.
Hey, it's a bit more than 7! So 3 + 1/7.something is pi, so 3 + 1/7 is a bit more than pi. (Useful learning there, too, in the class argument about whether it's more or less.)
OK, so let's look at 1/(pi - 3) - 7. Hm, that's pretty small. Is it a fraction?
Oh, it's about ... so ...
With a very small amount of tweaking, you can produce a continued fraction this way.
Even if kids aren't led all the way to continued fractions, they get a lot of useful ideas, like understanding a small number by looking at its reciprocal, and seeing that 1/7.something is a little less than 1/7 so that 22/7 is a little too big for pi, and so on.
Also, you can find 355/113 by this method, and it seems somehow less "random" that it turns out to approximate pi so closely. You can see how close it is when you take 1/(1/(pi - 3) - 7). Wow, that's REALLY close to 16!
So pi = 3 + 1/(7 + 1/16)), well, almost = anyway.
I think my suggestions here would work in parallel with yours, especially if you also ask about good approximations to some fraction with a big ugly denominator, and maybe the golden ratio too.
Some kids would show the calculator skill, and other kids would be very quick at seeing some continued fraction pattern and getting some great fractions really quickly!
Very nice, Joshua! That's the original direction my post was taking but for some reason I became intrigued with 'almost integral' multiples of pi, partly because it was somewhat different from other approaches I've seen.
I do think it is reasonable for a student to wonder: "What if pi is a rational, say, M/N?" Then N(pi) = M, so let's search for multiples of pi. Well, look at how close 7pi is to an integer, hmmm? The goal here is not to prove the irrationality of pi, but rather to consider a different view of approximations.
Yes, you definitely took the 'randomness' out of 355/113! Your series of questions would be challenging but accessible to strong middle schoolers and would lead to a more profound understanding of rational numbers. Now I feel motivated again to develop the ideas of continued fractions.
It would be interesting to break up the students into 2 groups and have them work with one of these approaches, then the other. But of course who has the time for that when we have to cover more 'important' content!
There's lots of opportunities to introduce continued fractions.
Say, solving x^2 + 4x = 3 for x. This is what I do on the first day I teach quadratics.
What's the "wrong" way an algebra student would approach this? If she's good at algebra, but has never solved a quadratic before?
She would probably factor, x(x+4) = 3, and then "solve" for x,
x = 3 / (4+x).
Then I point out, we can't evaluate the left side unless we know what x is. But wait! We do know what x is! It's 3 / (4+x)!
So, x = 3 / (4 + 3 / (4+x)).
And so on! So you get an infinite continued fraction, x = 3 / (4 + 3 / (4 + ...))
And you can approximate it on the calculator and see that it works pretty well.
The fun question, to do later: where does the second root go when you use this process?
Also, what if you want the continued fraction for sqrt(2)?
I was taught basically the same method that I showed for pi. Subtract 1, reciprocal, rationalize denominator, etc.
But:
x^2 = 2
x = 2/x ... doesn't get you anywhere useful when you substitute for x.
Try again:
x^2 = 2
maybe difference of squares?
x^2 - 1 = 1
(x+1)(x-1) = 1
x-1 = 1/(1+x)
x = 1 + 1/(1+x)
and now we're cooking!
A wonderful development, Joshua, which could be an excursion for a group of motivated students - how fortunate your students are!
I am very comfortable with the number theory here and we should collaborate on something like this for an article for a Middle/HS Math Journal.
Believe it or not, in the wanderings of my mind, I've been considering a "NetZine" of classroom ideas for math teachers with precisely the kind of development you're describing for continued fractions. I know there must be others out there who would like to share their innovative approaches in the math classroom. Currently, one would submit an article to the Mathematics Teacher or College Math Journal and wait for approval or request to rewrite from the referees. These days, educators may want an easier way to submit their ideas.
Unless it's already out there and I missed it, I think the time has come for an online publication in a different format. The Carnivals and what Denise at Let's Play Math is doing are wonderful, but these do not constitute a K-12 or 7-14 online math magazine consisting of more extensive articles.
It would be intense getting this off the ground, but the editors themselves would not have to create the content (although, in the beginning, they might), rather, they would compile. For example, a blogger (or math teacher) wanting more exposure might be interested in submitting such an article or having his readers redirected to this 'magazine.'
Interested?
Joshua,
I should add that each publication of the magazine would have a theme such as Number Theory Investigations for Grades 7-12. I would imagine that a quarterly publication would be feasible as it would take time to reach out to readers for submission and to compile and edit.
I'm beginning to think that such a publication (perhaps an online version of a print publication) is already in existence. I do know there is Connect Magazine but its focus is more on the K-8 end and addresses more than just math.
Writing is a lot of work! And writing in an online journal sounds like a lot of work for no pay. But I suppose blogging or blog commenting, or, for that matter, submitting to a print journal, is also a lot of work for no pay.
So, then, I guess the question is whether it will be enough fun to be worth doing!
I'm sure I can find something to submit if you are doing all the work of starting the journal.
But I still wonder if it wouldn't be better to submit to the usual print journals, which are mostly full of junk (I won't name names for risk of offending someone). I think we could do a lot to improve those journals by submitting, rather than starting our own journal.
Joshua,
Yes it would be a lot of work for no compensation at first. If it catches on, ads, sponsorships and/or subscriptions would defray expenses and provide some limited revenue. I have submitted to print journals in the past and I have mixed feelings about those publications (for some of the same reasons you may have).
I think we can do something different from existing journals. Certainly, your continued fractions idea could be a powerful contribution to this kind of effort. I would need to establish guidelines for submission of course. For example, some of my blog posts could be revised to conform. There would need to be a complete solutions page to any problems posed. What I like about an online journal is that links can be provided as resources for background or further extensions of the problem or activity.
I would need a real tech person to help me with technical details but the biggest obstacle may be to reach enough people to let them know about this effort and to invite submissions. I'm not sure that NCTM would promote a publication that might be seen as competing in some way with the Arithmetic Teacher or the Mathematics Teacher publications!
I envision 7-8 articles on a particular theme with additional resources such as a problems page, helpful links.
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