Dear Distinguished Mathematicans:For your penultimate Learning Adventure, I've pulled a classic out of the archives. It uses MicroWorlds EX, but only as a laboratory for running an experiment using tools I have created for you. You don't have to write a line of code, unless of course you wish to customize the environment.Al of the instructions are in the attached PDF file.Happy Problem Solving!
So with this LA, I have spent a lot of time thinking about what others have done, as well as talking with a few math people. First of all, no one I asked has ever heard of this problem before. It was funny because I guess I thought this was a big deal in math, but apparently no one I spoke with thought so. Now, jsut for clarification, I have not taken a math class since I was in high school- that means 1992 was my last math class which was Prob/Stat and trig second semester. So, umm, yea- it has been a LONG time since I have done any math thinking. According to a friend of mine, we think about math everyday, we just don't know it. I must really not know it.
So, I began my thinking contemplating why I need to know this? Why does this problem in math even matter? Why is this important? This is where I sought out some math experts talking to them about this problem. No one could give me a practical application and this I think is why I struggle sometimes in math. I want to know how and why I have to know something to use it useful to learn. This I am learning about myself.
Next, I just began by working with numbers I liked. I know this is corny, but I wanted to see what happened with my favorite number. Here is the chart I found with using 7
7 : 14 generations (G)
21: 5 G
35: 11 G
49: 22 G
63: 105 G- holy crap my biggest generation wow- why did this happen?
77: 20 G
91: 90 G
I tried to stay with odd numbers hoping to get the biggest generations realizing that by using whole numbers, the generatons would automatically be reduced much faster. Whereas with odd numbers, there aren't pairs and so by multiplying by 3 and then adding one, the system seems riged because eventually we will always create 4, 2, 1.
For my next try at this LA, I decided to try something one of my students had suggested. I took a number and did 2 to the power of that number and then -1. The results were interesting generations. Here is what I found:
Number/ 2n /2n-1 /Generations
3 8 7 14
4 16 15 15
5 32 31 104
6 64 63 105
7 128 127 44
8 256 255 45
9 512 511 59
10 1024 1023 60
11 2048 2047 154
12 4096 4095 155
13 8192 8191 156
14 16384 16383 157
15 32768 32767 127
16 65536 65535 128
17 131072 131071 222
18 262144 262143 223
19 524288 524287 175
20 1048576 1048575 176
Here is what I noticed: looking at the first two, they are separated by one. I am going to call these pairs from now one. Then there is a 90 point jump between the first pair and the next pairing. So, then I decided to look at pairings and see if they were all separated by 90 points. No, that was not the case. The next pair is separated by 60 points and then next was 15 points, 94 points, the finally we get to a large section for numbers 11-14 that are continuous. Why does that happen? Then the pairs start again with being separated by 30 points, then 95 points and down to 47 points. I didn't see any connection or patterning other than the pairs and the one quartet. I am unsure of why this only happens within sets of two or four but it is interesting .
What I learned from this learning adventure is that it is helpful to seek out experts to get some clarification. I had no idea about this LA and it really helped to have people who could get me thinking. Also, I can't find the graphing button. I have no idea how to graph this project and am wondering if it is totally in front of me, but somehow I am just not seeing it.
What did you observe about the learning style(s) of your collaborators?
I think this is one of the portions of our LA’s I enjoy more than anything. It is always really interesting to see how each person approaches the journey. For example, it usually seems that the same people complete the LAs right away but this time we had a new leader. There is something interesting to me about who completes each task first and then watching others come back over and over again to add their two cents and additional interpretations. Tanner jumped all over this assignment and I was truly amazed by his understanding as well as continual support of others to complete this task. It seems that he was quite the leader here helping others figure things out. Additionally, another observation of learning styles is how some of us need direction from others to get us started while others need to complete parts on their own then come back to the fold to gain additional understanding, and finally, there are those that totally operate solo.
Which subject(s) does this project address?
Off the bat I thought about the applications to math and science, but as Gary pushed my thinking I could see the point he was making about a connection into descriptive writing thus a Language Arts class. I also wonder if you could make some connections into art classes with the graphing.
What might a student learn from this project?
A student would learn various things such as problem solving, collaboration, hypothesis, critical thiking, data interpretation, to name a few.