Note: There has been a revision in one of the properties of 17 below - I'm sure you already caught the error! Joshua caught another error involving the 4th powers - both have now been corrected.
For middle or high school students:
Number sleuths -- get into your detective groups. No calculators for the first 5 minutes.
You will have a total of 15 minutes to uncover as many interesting or fascinating facts as you can about the number 97.
One member of the group must record these and report back. Make two columns.
The first labeled: Discoveries found without the calculator.
The other column: Discoveries found with the calculator.
Ah but you're wondering what makes some fact interesting or fascinating. That's pretty subjective, right? Let's model one:
17
- Prime number
- Both of its digits are odd
- The product of its digits is prime
- Can be written as a sum of 2 consecutive integers: 17 = 9+8
- If its digits are reversed, the resulting number, 71, is also prime and the difference of 71 and 17 is 64 or 82; oh, and 8 just happens to be 1+7!!
Note: This is clearly incorrect - 3-point penalty!! - Can be written as a sum of squares: 17 = 42 + 12
- Can be written as a difference of squares: 17 = 92 - 82
- Can be written as a sum of a perfect square and a perfect cube: 17 = 32 + 23
- Can be written as a sum of 2 consecutive fourth powers: 17 = 14 + 24; in fact, 17 is the least integer which can be written as the sum of two distinct nonzero 4th powers
- You can get your driver's license in some states on your 17th birthday (this is clearly the only fact that's interesting to an adolescent - I had asked one of my students what's special about March 14 and he replied, "It's my birthday and it's also the Junior Prom.")
- 17 is the hypotenuse of a Pythagorean triple; 8-15-17 [high school level?]
- 17 is part of another Pythagorean triple: 17-144-145 [high school level?]
- Is there any end to this list?
You will have 15 minutes to complete your investigation of the number 97.
To rate each of your lists, we can assign a point value to each fact. For example, the first four facts listed above for the number 17 could each receive one point. Some of the other facts could receive 2, 3, 4 or even 5 points (max) for being more difficult to find or just more amazing. If any fact is incorrect, 3 points are deducted, so you'd better do independent fact-checking on your team! We will then determine the top 3 lists from this rating system and those teams will receive worldwide recognition by having their results appear on You Tube (just their lists, not names or faces for reasons of confidentiality of course). This will surely go viral in 17 seconds or less. Oh, so you'd rather receive 97 bonus points on your next test?
Oh, I forgot to mention. As a super extra credit bonus project/assignment, do the same this evening for the number 153. You will need to email me with your individual lists by 10 PM (yes, yes, we all know this is unfair to students who do not have access to email or whose parents may not give permission for this or for some other reason, so this is just an option!).
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17 comments:
Okay 15 minutes of fun:
Is a Prime number
Is the 25th prime number
Is a happy prime number
Is the largest two digit prime number
Its digits are two consecutive odds
The reverse 79 is also a prime number
Sum of the digits of the product of the digits: 9*7 = 63, 6+3 = 9
Sum of the digits of the sum of the digits: 9+7 = 16, 1+6 = 7
Sum of the product and the sum of the digits: 63+16 = 79
Can be weitten as the sum of consecutive primes: 97 = 29+31+37
Can be written as a sum of two squares: 97 = 9^2+4^2 = 9^(9-7) + 4^(9-7)
Can be written as a sum of two consecutive quadrics: 97 = 3^4+2^4
The 2nd digit of 97: 97 mod 9 = 7
The 2nd digit of 97: 79 mod 9 = 7
The first digit of 97: 97 mod 9 + 79 mod 7 = 7 + 2 = 9
The first digit of 97: 79 mod 9 + 79 mod 7 = 7 + 2 = 9
The mean of the digits: 97 mod 7 + 79 mid 7 = 6 + 2 = 8
The following sequence finds the first digit of 97:
97-79 = 18
81-18 = 62
62-26 = 36
63-36 = 27
72-27 = 45
54-45 = 9
The following sequence finds the 1st digit of 97:
97 mod 79 = 18
81 mod 18 = 9
Nice, Florian!
I suspect there will be a few more discoveries...
PLS NOTE THE CORRECTION TO ONE OF THE PROPERTIES OF 17 - I PENALIZED MYSELF THREE PTS!
Dave dont be so hard on yourself math is supposed to be fun ;)
17 is not the smallest sum of two nonzero fourth powers: 2 is.
Ok, another 3 pt deduction for me, Joshua! That's why I would pick you to be on my team! I'll correct that one too.
Guess I meant, distinct or unequal 4th powers...
Hopefully your students will be more careful than me, but then validation and fact-checking may be an important benefit of this activity!
So, has anyone 'uncovered' a few more fascinating facts about 97 to add to Florian's excellent list?
By the way, 17 is the canonical random number. Any time you ask someone to choose a number at random, he'll choose 17. Look it up on Wikipedia or Google. Or look at the work of Michael Spivak.
On the "randomness" of 17:
http://cosmicvariance.com/2007/02/09/the-power-of7/
My own research (taken on the first day of school in the stats classes I teach over the last few years) gives me 18 seventeens out of 103 students.
The rationalizations are the fun part "lots of people choose 17 because they are 17 years old" -- then I show them that this same phenomenon occcurs in college students, not choosing 18 or 19 there. On the other hand, it seems that globally 7 is the second most popular, while in my class 18 did surprisingly well ...
I also heard that when asked to pick a random prime number, a significant fraction (maybe a simple majority) chose 91 !!!
TC
As is well known, 91 is the smallest Zucker pseudoprime (defined as a number that Zucker has been known to think is prime, but which is actually composite -- kinda like Carmichael numbers). So it's not surprising that a lot of people would choose it when asked to name a prime; it just shows that their primality testing software uses the Zucker algorithm rather than a more sophisticated pseudoprimality tester (Carmichael pseudoprimes start at 561, I guess, which gets you at least a little farther than Zucker).
Joshua, 91 is one that I have to think twice about too. And I also have to think twice about 97, because I think may it's really the one that's not prime. ;-)
I posted some of the fascinating facts my students found for 97 and 153 on my blog. One referenced unsummables!
Dave, did you choose those numbers at random, or were there particular fascinating facts you were hoping the students would find.
Mathmom--
Thank you as always for giving these to your middle schoolers, something I am not able to do. I was particularly impressed by someone making the link to unsummables. Congratulate your students for me -- they did an 'awe-sum' job! My experience with older students is the same, BTW. They do not ordinarily consider pairwise differences, comparing the 2 series. This is why youngsters may in fact need a teacher. Some things you need to show them! BTW, I believe students should be exposed to the comparison method BEFORE they get to infinite series techniques in Calc.
As far as my reasons for 97 and 153, I have always been fascinated by these numbers, 153 in particular. That is, ever since, I read that 153 is the smallest positive integer, greater than 1, which equals the sum of the cubes of its digits! This is definitely not a property that middle or secondary students would be likely to consider unless a model were provided. One could write a program in Java or any other language to determine the next such integer. Have fun trying to find it!
I am also a number theory enthusiast. The unsolved problems regarding primes was the principal reason I became a math major and why Algebraic Number Theory was my area of research in grad school. You may recall I published a post on Goldbach's conjecture some time ago (search my blog or the labels for this).
A weak form of Goldbach's conjecture states that every odd number greater than or equal to 9 can be written as the sum of 3 prime, not necessarily distinct. Florian found such a representation for 97, in fact, he found 3 consecutive primes! Cool! There can also be more than one such representation possible.
I suspect if you tried this with another group next year, there would be some amazing new discoveries!
Of course, the traditionalists out there would be quick to criticize this activity as irrelevant to hte curriculum: "Who has time to waste with such silliness!" Hopefully my readers know that, whenen I was in the classroom, I always found time for skills, concepts and higher-order thinking and problem-solving. Just learning how to formulate one's discoveries using correct math terminology makes this worthwhile in my opinion. We know that most students struggle to express their ideas or discoveries properly. Communication is so critical here, not to mention creating an environment of mathematical research. You also know I am a firm believer that these activities are not intended only for the gifted or honors kids. I often tried them out with my 'skills' classes, with surprising results.
The typical reaction I have received from all of this is:
"Dave, if I were to do this once or twice a month, I would not be able to finish the curriculum!" Yup, we're back to Prof. Schmidt's ideas and the report of the National Math Panel. Perhaps 'less is more'...
These investigations help me find the places in the "basic" curriculum that we need to go back to. Of course, teachers on a tight time schedule might be horrified at that thought, but just because they all "got" the distributive law when they were having a lesson on the distributive law does not mean that they (a) retained it, and (b) can use it in appropriate situations when it comes up in the context of some other problem. If they can't do (b), why teach it at all? When teachers are teaching just because the curriculum says so, they are missing the point. It may not be their fault -- I know there is a lot of pressure to "cover" the curriculum, but "coverage" does not equate to useful "teaching" IMO.
Mathmom,
One of my favorites is "rather than cover many topics, I prefer to uncover a few" -- anyone know the source of that one?
I'm going to start a blog with precisely the goals you talk about here, that Dave illustrates so well and mathmom reinforces. I want to find interesting (and fun!) problems at all levels that require some higher-order thinking and problem solving skills to do, but that also provide practice with basic skills.
One thing I hadn't thought of enough is mathmom's point that kids can learn a skill and practice it when they know it's time to use it (in the chapter on whatever) but then when it comes up in some other context, they need to both recognize its appropriateness and recall how to use it. I think most curricula give some training in the recall but hardly any in the recognition part. Thanks mathmom for pointing out to me how important that is!
The curriculum I teach from is a big problem book -- threaded by topic to some extent, but still one that invites students to figure out which tools are appropriate rather than knowing that it's the section on one particular tool. So this does a lot for enhancing student ability to recognize contexts where their favorite old tools will be useful. (In this particular case, it tends to lead to the idea that the Pythagorean theorem solves everything, which is not far from true ...)
Anyway, I've got to organize some thoughts on this, choose a few good problems, and post them on my new blog. http://uncoverafew.blogspot.com/ will be the site ... coming soon!
Good luck with this, Joshua. Considering your knowledge, insights, experience in the classroom and your profound thinking, we are looking forward to this -- what took you so long! I will also promote it in one of my 'Odds and Evens' updates. The investigations I am writing always imply an underlying base of skill, without which the student would have little chance of success. I'm curious about that problem book. Sounds fascinating. Is this book supplemented by more traditional materials? Are other teachers using similar materials? Are your students middle of the road or high achievers?
I love the 'uncover' phrase. I' ve used it myself because it implies to me that in many instances children already have the knowledge, but need to recognize how to apply it. Our role is to lead them toward such awareness. I've always thought that one interpretation of the word 'educate' is helping to lead one out of the darkness into the light.
Mathmom, thank you as always for your insights into the benefits of these investigations. You know I completely agree with your comments.
http://math.exeter.edu/dept/materials/ is the curriculum we use. We have pretty extraordinary students and even in our population, roughly half our students need some pretty heavy supplementation of this with a more traditional textbook-y curriculum.
Thanks for the kind words, Dave -- now I just have to get unlazy enough to start writing things up!
Joshua -- I'm greatly looking forward to your new blog! Your teaching materials sound great. I can't access them from the link you sent, though.
Dave, I do know that you and I agree on this. It's nice to have a little preaching-to-the-choir lovefest around here every so often. ;-)
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