The Appel Academy?
For Appelbaum.
Let's explore some fascinating properties of numbers, probability, and geometry. These pages are an attempt to introduce young learners - grades 3 through 8 - to Number Theory, Probability, and Geometry by way of algorithms. A challenge is how to present graduate level mathematics to 8 - 13 year olds and then top it off by adding computer programming to the mix. MIT and Cal Berkeley, through the programming languages Scratch and BYOB, provide the tools to meet the second challenge. The math? As far as I can tell, there is no text targeting this age group, and so I have been developing material as I go. I use a combination of my text books from when I was in Grad School at Arizona and RPI, and papers I used at my earlier jobs and find on the Internet. I also receive links to papers from my "old" colleagues. I take what I can from these sources to share with the students.
The investigations described here and those in development are not presented in any special order. Since they all include a computer programming activity or at least a computer-assisted activity, I try to layer new concepts on material already covered; however, there are no guarantees that I will succeed in this. You can always contact me with comments and questions.
Quick personal point. Everyone thinks that I spend all my time at the computer. This is not true. I believe it is very important to balance one's sitting time with physical activities. In fact, I do some of my best problem solving when walking or working out at the gym.
Just call me full-rack Appelbaum!
Please note: The Scratch and BYOB code is not intended to be the best one can write. Sometimes, I purposely go for clarity over programming elegance. However, I am starting to include some efficiency ideas into some of the activities. I am also beginning to include Python; I see Python as a viable segue into Java. It's also "real world" in that major tech companies use Python for much of the support-level software.
While there is a mapping to the Common Core Math Standards, the primary point of these activities is to provide insights into mathematical theory and to try thinking a little like mathematicians. For the concepts behind operators, sets, numbers, logic, etc., check out these pages. The initial set of 25 activities (some of the activities are broken up into two or three sections, each counting as an activity) have been performed in some manner with select students in grades 3 - 8. Approximately one activity - math investigation, computing challenge, computer programming insights - is added to the collection each week or two.
Personal note - from where (and whence) do I come? Obviously, I am a major believer in teaching real mathematics in our schools, not just recipes. My experience is in modeling information starting in the days before Google, Yahoo!, Bing, and AltaVista (remember them?). I was one of the tech founders of Verity, Inc., which was bought by Autonomy, which was bought by HP (no comment on current legal issues there). I have started two companies, one of which was acquired by Edgar Online, recently acquired by RR Donnelley & Sons. I have a Masters Degree in Math (University of Arizona - Go Wildcats!) and a Masters in Computer Science (Rensselaer Polytechnic Institute).
For the challenges, see if you can do the activity before looking at my solution.
The philosophy behind these pages is excellently laid out in Mathematics by Investigation, Plausible Reasoning in the 21st Century, 2nd Edition, Jonathan Borwein, David Bailey, A K Peters/CRC Press, 2003. The premise is that for many years computers have been used for simulations in the sciences supporting the discovery process, so why not in mathematics?
Here are a couple of the properties (number types) that we will investigate:
Take the integer 623. Reorder the digits in ascending and descending order, then subtract the ascending from the descending:
632 - 236 = 396.
Do the same with 396:
693 - 369 = 594.
Repeat with 594:
954 - 459 = 495.
Repeat with 495:
954 - 459 = 495.
Interesting. Except for triplets (111, 222, 333, ...) that go to 0, all three-digit numbers will lead to sequences that end up at 495. This is called the Kaprekar sequence (series). Two-digit numbers and four-digit numbers also go to respective fixed points. Once you get to five-digit numbers, it gets a bit more complicated in that there are two fixed points. We will also look at how many steps it takes to get to the fixed point.
By the way, the Silicon Valley Math Initiative uses the above property in one of their Problems of the Month. I see many of these activities as complimentary to the SVMI problems; I believe we share common goals but I add a bit of technology to allow for some deeper investigations. Also, the visualization of mathematical principles is pretty much impossible without the computer: check out the activity
Visualizing Kaprekar Constants for an example.
One more. Take the integer 23 and sum the squares of its digits.
4 + 9 = 13.
Do the same for 13:
1 + 9 = 10.
Ditto for 10:
1 + 0 = 1.
Ditto for 1:
1 = 1.
If you can sum the squares of the digits of an integer and it equals 1, the number is called a happy number. Furthermore, if the sum of the digits is not 1 but leads to 1 when you repeat the process, the number and all the intermediate values are happy. In our example, 23, 13, and 10 are all happy. During this process, you may run into a loop; that is, the sum of the squares repeat. Try 89:
89 --> 145 --> 42 --> 20 --> 4 --> 16 --> 37 --> 58 --> 89 --> 145 --> ...
So sad.
We will investigate the following types of numbers:
Harshad Numbers
Kaprekar Numbers
Happy Numbers
Kaprekar Series (same mathematician but not connected to "his" numbers)
Perfect Numbers
Fibonacci Sequence - we will look at the convergence of the ratio of consecutive terms
Prime numbers - patterns
Divisors - patterns
Probability
More ...
The Appel Academy by Lee Appelbaum is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.