Monday, July 20, 2009
Monday, July 13, 2009
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.
Friday, July 10, 2009
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!
Thursday, July 9, 2009
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.
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.
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.
The Puppies and Kittens Game
For this game, two players alternate turns. We start with a pile of 7 kittens and 10 puppies. A legal move is one of the following three possibilities:
removing any positive number of puppies (but no kittens), or
any positive number of kittens (but no puppies), or
an equal number of both puppies and kittens.
The winner is the last player who makes a legal move. Which player has a winning strategy? For what starting values of k kittens and p puppies will player 1 win? For what values of k kittens and p puppies will player 2 win?
Exploring this game required us to build on to our previous experience of the concept of “oasis” and “desert”. An oasis is a place in the game that we can't lose. For example, if we have just had our turn and we have taken all of the puppies and kittens leaving 0 of both, then we win. So, 0 puppies and 0 kittens is a great spot to leave your opponent.
We then explored different combinations to determine if the situation was an oasis for us thereby leaving our opponent in the desert. What if, there were 2 of one and 1 of the other? The first player could not take all of the animals therefore leaving a final move. So 2 of one and 1 of the other is an oasis point for us to leave. Deciding to organize the data in a graph to look for potential relationships helped us to determine the next couple of oasis points. So in order to win, we need to get to one of the oasis points and keep the game at these optimal points.
But why? Once the values are found and put in a table, a pattern can be discerned. The pattern can be generalized and proved using higher level math involving the Fibonacci Sequence. So if you have kids that can take it that far, excellent. Otherwise, simply find a strategy and play to win. Intermediate and Middle School students can still enjoy finding a strategy to win by simply collecting data and finding a way to win every time.
I tried not to give away any secrets. Grab a colleague, a child, or student and discover the math this game has to offer.
Wednesday, July 8, 2009
1. Host another Math Circle Conference – Week after 4th of July 2010 - Tuesday to Thursday or Wed to Fri
We would like to add a 4th day so we could do a hike or canoe thing …recreational …team building ….measurement activities etc… do it on 2nd or 3rd day
Who gets preference?
1. Past participants that are actively doing math circle like things
2. New participants that come with past participants
3. New participants
2. Fall Class
We meet 8 times for 2 hours virtually – 1 credit classParticipants each take part of the responsibility for organizing and developing the course.
3. Math Science Conference
Credit – Problem solving strand ….at Math Science Conference
Kira – Taking the problems and turning them into larger type projects
Mary- Math Circle problems in classroom
Kirk – A lot will be implemented in the classroom and in remediation ..being willing to go beyond the knowledge of the teacher.
Cindy – Incorporate these problems in a Monday Friday Cycle – introduce problem on Mondays ….debrief on Fridays
Vern – Create a math club centered around the math circle ideas – AMC test
Steve – Look at activities – in particular Zome tools – discover which activities are of most interest
Dave – Problem Solving Center - these problems will enrich the center – open ended explorations…Math academy for elementary – try to incorporate teacher circles in the academy
Ariane – Will be the representative for teacher circles in Laramie – Have a mini math circle at lunch with other teachers…Make problem solving that is part of the class – commit part of your time to a “problem of the week” – maybe try to fit it into the unit.
Mark and Kendall – Incorporate in classes – problem solving….Get more teachers involved. Start mini-math circle in Casper…help organize conference for next summer….
Nancy – Use in online problem solving and in intermediate algebra… 6th grade teacher in Cheyenne that might be interested in collaborating.
Bertine - Would really like to create a math club…not sure how to structure it…Problem of the day in the classroom
Deb – Designing a new class with Vern – Algebra I support class – Algebra I is an indicator for graduation rates….the course is wide open – Kids that are smart …just lost – if we can help them to feel successful – it will be powerful…even just the little games…if they can be successful with these math circle problems …hopefully they’ll be able to had success in Algebra I
Lynn – Do a better job with the middle school course – We could do more with the math club…Promoting the AMC contest and the Circle ….staying in touch with the national circles ….Taking on this Math circle fall course – figuring out how to do it well. Will work with some kids that want to do well in the contest….Writing up a “dialog” for the tangles problem – at a level that is accessible.
Greg – Encourage other UW mathematics faculty members to get involved in these kinds of things.
Put Down is a game where two players alternate taking turns. The last player to make a legal move is the winner. The goal is to place pennies on a bare rectangular surface until no more pennies can be placed. They cannot overlap, but can almost touch each other. The winner is the person who places the last penny.
Using notions of symmetry, player A (the first player) should place a penny in the center of the rectangle. Then after player B (the second player) places a penny anywhere in the rectangle, player A should place the third penny using 180 degree rotational symmetry from the second penny. This guarantees that player A will win. As long as there is room for player B to place a penny, player A will also have a spot for the rotated penny. When player B runs out of spots for pennies, player A will win because he placed the last penny.
Things are not so assured if you are not the first player. (Always good to be the first player!) If the other player doesn’t pick the center, just rotate what he does and you will win. If he picks the center, all you can hope for is that he doesn’t rotate your next move. And even then, you may lose.
Knowing that symmetry is good, it makes sense to pick the center first. The rectangle has 180 degree rotational symmetry as does the penny. Picking the center keeps the rotational symmetry. Then as you mirror each move the other person makes you also maintain rotational symmetry.