TC Class -- Day 1 -- Sept 14, 2009

What a great day. After housekeeping was handled, and everyone knew how to communicate within our Elluminate class, we started sharing. Kirk shared a GREAT source of inexpensive equipment: Radio Shack. Gigaware: 2 headsets and 2 web cams for $14.95?!

1. Bertine is organizing math events to mesh with the Science Olympics meetings that already happen in Gillette.

2. Cindy has started a program she's calling "Math in Action" and she told of her students roaming the Sheridan area, capturing pictures of things related to Cartesian coordinates.

3. Dave, Kendall, and Ariane told of problems they've started posing their students, and they also shared a question about "what makes a good question". In part their interest is in guiding their own search for effective problems. Part too is the need to appeal to colleagues and have a good cogent statement about these problems we've come to appreciate. Note that on the blog mentioned later, a man named Larry Zimmerman proposes that these (good ) problems have the following elements:
1. a goal (construct, prove, maximize, minimize, classify, compare, compute)
2. given information
3. special rules (sometimes)

Dave told us about his upcoming geometry meetings using Zome tools that he's purchased.

4. We shared which of the problems from our Jackson days have served us since. These are ones we recommend as starters: the human tangle, Conway's tangle of ropes, Mark's numerical problems, the pennies problem, the pills. When Ariane & Kirk discussed have a simply stated purpose with problems like the ropes, Kirk volunteered that for him it's the inverse aspect. Lynne said it was about feeling that a solution is possible without knowing what it will be. Kendall suggested persistence with math challenges. Kirk suggested problems using chess or Go moves. Steve shared a rubric for ranking problems. The 8 categories were whether the task:

1. is relevant to experiences of target population;
2. can unfold over time and is not sequential and closed;
3. is cooperative, drawing on current knowledge;
4. has outcome accessible to self-verification;
5. has accessible extensions;
   
6. has a problem solving trajectory that can be shared;
7. has a solution involving fundamental math;
8. involves learning that's detectable on standard ed measures.

5. We looked at a problem based on one from Marilyn Burns: Number Bracelets. There is a great discussion at http://www.geom.uiuc.edu/~addingto/number_bracelets/number_bracelets.html. Note that under the link What's going on here? you'll find a lot of the math extensions!!! There is also a great blog called "Math Be Brave" that mentions this game: http://mathbebrave.blogspot.com/2009/07/larry-zimmerman-plus.html. Any one can join their blog, and most interesting!!!! They discuss the nature of and how to establish good problems: check out the old post on Larry Zimmerman II. It's a very nice discussion of how to invent good problems.

6. Kendall shared his first attempts using Excel to search for number bracelets. The command that he used reveals that this is a "mod problem". Kendall placed the first 2 numbers in A1 and A2, then A3=mod( sum(A1,A2),10) was the formula he put in the column of derived values. Mod(???,10) gives the last digit (base 10) of the value ???.


By the end of class we had resolutions and ideas of where to go. Most importantly, coming together seems a positive and helpful effort. We want to continue helping ourselves and some of the ways will be to

• try some problems with kids and share -- for now we all try the bracelets problem
  
• work some problems in class
• post problems on blog --- rate them with Steve's rubric or one that we evolve
• find a couple of new problems on our own to share & invite others to share before next mtg
• look for problems with a "theme" like Kendall & Lynne looked for mod problems
• get feedback on how problems "roll"
• bring problems we find and get suggestions for implementing & extending

Lynne & Kendall will be in contact with Josh about Zome Tools, the Math Forum, and extensions to the bracelet problem.

Math Teachers' Circle Network

Hello, everyone! I'm Brianna Donaldson, and I'm the Director of Special Projects at the American Institute of Mathematics and the program manager for the national Math Teachers' Circle Network. Lynne, Mark, and Kendall brought a team from Wyoming to one of our "How to Run a Math Teachers' Circle" workshops last summer. Lynne shared your group's blog with me, and it's exciting to read about your plans for the coming year!

As your Math Teachers' Circle gets started, I wanted to make sure that everyone is aware of the community of Math Teachers' Circles around the country that is currently developing. You can read more at the Math Teachers' Circle Network website, www.mathteacherscircle.org. We currently have some resources for Math Teachers' Circle session leaders posted (we always welcome contributions!), and we are in the planning stages of a teachers' area of our site for sharing lesson plans and ideas. I noticed that some teachers mentioned being interested in adapting and developing Math Circle-type problems for classroom use, which is wonderful. If you come up with any that you'd eventually like to share with the Math Teachers' Circle community, or if you have ideas or suggestions for me about our website (always a work in progress), please feel free to contact me at brianna(at)aimath.org.

All the best,

Brianna

Casting Out Nines

Paul described the idea of casting out nines. He reminded us of the rule that the sum of a number’s digits equals the remainder when that number is divided by nine. It was here that he introduced to us the concept of modular arithmetic. He then used this to explain why the sums work, and followed this by showing us a trick. That is, if you write a number (the more random and complicated, the better), then reverse its digits and subtract the two numbers, you can figure out any missing digit in the answer by omitting zeros, thinking in terms of mod 9, and hearing the rest of the digits. He modeled this trick by standing with his back to the board and showing us how he could figure out the number.

This activity not only showed us the fun of clever math tricks to play on our students, but also gave a brief but thorough explanation of the relationship between any number and nine. By knowing this trick we can “Make it easier” to solve various problems or tricks involving these relationships.

The downfall of this activity was that it happened very quickly. There was a lot of explanation to happen in a very short time. I think many of us who had never seen modular arithmetic still find the details quite vague and even those of us who are familiar with modular arithmetic find much of the casting out nines concept to be fuzzy. Also, although we had several opportunities to experience modular arithmetic in later problems, we never again encountered a problem that would use casting out nines, so I think it would be helpful for us to know what other problems this strategy would be used for. I think it would also be good for us to discuss or at least reflect on our own how we could explain this to our students without using ‘mod’ language. More specifically, I think we could talk in terms of remainders to use language our students are familiar with to encourage their understanding of nines. It would also be helpful for us to practice these tricks with each other so that we feel comfortable and can be more fluid with them in front of our students.

Bay Area Circle for Teachers resources

Hi Everyone,

I had a great time with you all in Jackson. Now I am sitting in the UW library, amazed that there is free wireless, an electrical plug, coffee, and even sushi!

I want to share with you the URL for the Bay Area Circle that I attended just a few days before the WY circle: http://bact.mathcircles.org/summer09. On this page there are links to all the activities, a book list, and also on this page are some tips that I wrote for getting a zome collection started.

Have a great rest of the summer!

-Paul

How to Post Documents on the Blog

Good evening all,

Turns out that we CAN post a document using Google Documents as the storage site. We can thank Bryan Shader for the following steps:

1. Go to Google Documents -- you could log into your Google account and then click on Docs
2. Upload a file -- select the Upload tab at the upper left jut under the words Google Docs
3. Share the file with yourself -- you'll have to list yourself with a non-Google email address in the text box that appears, and then click on Invite Viewers
4. An email will be sent to you with a link to the PDF file
5. Place this link in the blog.

--Bryan

Update on reimbursement and enrollments

Good afternoon again,

I've submitted your reimbursement paperwork this afternoon, and I am about to walk to Outreach to submit both your enrollment cards and the evals. We'll let you know how to sign up for the Fall 2009 course as soon as it's up. I've also just updated the contact list that Mark started to include us all, and emailed that. Please check it over and if you wish to change something about your contact info let me know.

Lynne

The Game of Take Away

Directions
To play, begin with a set of 16 pennies or other small objects. Two players take turns removing pennies. At each turn a player must remove anywhere from 1 to 4 pennies (inclusive). The winner will be the last player who makes a legal move.

In the classroom, allow students to play for awhile. Ask them to look for patterns and come up with strategies without telling other classmates. When you feel that some students have ideas, allow them to discuss what they have noticed with the class. See where this takes you!! Have fun!

Explanation of the Game and How It Works
Of course the winner takes the last 4-1 pieces and wins the game by leaving zero pieces.
In order to do this the winner must present their opponent with exactly five pieces. The opponent can take anywhere from one to four pieces leaving you with a win.
So are there more options than leaving five pieces? Yes! All multiples of five could be presented.
This idea can be shown as a list of oasis and deserts. Because the winner must leave zero pieces, an oasis ≡ 0 (mod 5). Remember that a player can move from desert to oasis (winning spot), but never from oasis to oasis. In other words, the winner always wants to present their opponent with an oasis amount of pieces, leaving themselves in a desert.

0 1 2 3 4 5 6 7 8 9 10 11 12 . . .

Now that students understand the idea of oasis and deserts make the game more difficult by changing the rules. What if a player can only remove 2-4 pieces or 3-4 pieces. What will the oasis and deserts look like?

Hint: Removing 2-4 pieces OASIS ≡ 0, 1 (mod 6)
Removing 3-4 pieces OASIS ≡ 0, 1, 2 (mod 7)

Change the rules around and encourage your students to develop new strategies. Take Away Poison requires the winner to leave only one piece. You can also change the number of pieces at the beginning of the game as well as the number of pieces that can be removed on each turn to change the game.