Automata Antics: Ball Throwing

I recently played this game with my students, once again found in Solve This by J.S. Tanton. They had a good time exploring a problem to which the answer was unknown and taking a break from the everyday math to play a game. Play went as follows:

A number of students stand in a circle, each with a card showing either “left” or “right”. John begins a ball game by tossing a ball across the circle. If Beatrice catches the ball she throws it back across the circle one place to the left (in her perspective) of John if she is holding left, or one place to the right if holding right. Then Beatrice flips her card over (“right” becomes “left” and vice versa) and waits for another turn. Each person receiving the ball operates in this way.

There is one complication: If Maxine is holding “right” and receives the ball from Christian directly to her right, the rules do not make sense. We make the convention that in this predicament Maxine holds the ball, turns in place to the right more than 180⁰, and throws the ball to the first person she sees (and still changes the word she holds). This is of course, equivalent to Maxine passing the ball to the person directly on her left.

Similarly, if Keone is holding “left” and receives the ball from Shannon directly to his left, he turns left and throws the ball to the person on his right. This is a little confusing at first, but it doesn’t take long to get the hang of it.

Here is the question: In this game is everyone guaranteed a turn? Will the ball eventually reach everyone, no matter the choice of word initially on everyone’s card?

(A lightweight ball is recommended.)

Note: The book with which this game was found often presents similar problems in the same section. There were two related problems presented before this one which involved a grid-like playing board and proposed the situation of an ant walking along that grid alternating vertical and horizontal steps or a grid with random “L” and “R” which dictate which direction the ant is to turn 90⁰. In both the these the question is similar: will the ant walk through every box in the grid, and will the ant ever return to a box from the vertical direction if it originally left that box from the horizontal? Having done these activities first, we might have had an idea of what would happen in the circle, however I chose the circle because I wanted a whole class activity. Have fun exploring any or all of these and report your findings!

My class has yet to discover a "solution," and I'm not sure what we did find out, but many predictions were made, variables changed, and a modeling tool like Excel has yet to be utilized to try multiple cases quickly and efficiently.

Spring 2010 has been a challenge for our Circle to meet, but we have met a couple of times and a new member from Laramie has joined, Julia. We also decided to continue meeting on the 2nd and 4th Wednesdays, but not as a credit-earning class. We'll offer credit again for our July face-to-face workshop at Ranch A near Devil's Tower. Watch for a directions to sign up and recruit for this SOON. For now, mark your calendars for Thurs July 15 through Saturday July 17, 2010.

I promised (myself) to post some of what I have gathered from attending two national meetings, so let me start with the most recent: Circle on the Road at ASU in Tempe this past week. Pam joined me on Monday morning to see a little, and perhaps she'll add to this note. It was inspiring to meet so many people who have had the courage to start circles, and see how they have grown in time. One of my favorite new friends also runs teacher circles: Ginny Bohme from Arizona. I asked Ginny to join us for an Elluminate session, and she's planning on it.

There were quite a few who wanted others to know how rural and small their communities, but none had us beat, and everyone agreed that the beginning is hard. Now so many are ready to share resources, and this workshop has probably encouraged many more to share! With the new found support of the national mathematics groups like MSRI and AIM, the web resources should get better real fast. Of particular interest is perhaps the elementary grade materials. It seems that many parents with Eastern European roots have translated Russian texts for older students, but lots of creative folks, especially parents running circles that were created for their own kids,

There are a few especially interesting ideas I want to share now:
1. When we post problems for use in classes, include a description of the Standards met so that other teachers more readily see how to justify using the problem in their classrooms.

2. A measure of "success" derived from problem solving work include risk-taking, readiness for tests such as the ACT, and increases in proficiency within high-needs areas. No one seems to have much data (yet) but the ASU group seems intent on gathering this from us and anyone else who might join them. The circles in Charlotte NC offered that they have some evidence of the risk-taking and another promised some data on the second. I got the impression that he was one who told of problem solving work breaking through the boredom that under-achieving students find in too any of our classes.

3. Models for student math circles in more urban and established places include summer "camps" and math Saturdays. Many of the latter stay in contact with the students directly via email. Elementary age circles are shorter, and often held in a person's home near the schools. Those running these young-age circles believe in having a few visitors, but mostly agreed that having one main person is important for the younger children. The Kaplans run an amazing number of circles in one week, and they describe the importance of a non-competitive "congenial setting". One of the Bay Area parents arranges with teachers from neighboring schools to transport students to her home.

4. Some practical advice is available for parents who might be anxious during the initial weeks, and slides from various parent presenters should become available on the MSRI or TC sites soon.

5. Others urge "recreational math" as the content so that every problem is new and no one needs to catch up. Others have quarterly or yearly themes, but agree that adjacent sessions are best when they stand alone. Those running circles for older children break many of these rules.

6. Lots of people are creative in a humorous way with the problems they pose to children. A favorite was from a Mom who asked "If 1 kid can annoy 1 Mom in 1 hour, how many hours can 10 kids annoy 10 Moms?" Another person introduced us all to the resources from NACLO, the North American Computational Linguistics Olympiad. If you haven't seem their sort of challenging, go look soon! Many agreed that these challenges met everyone's criteria for engaging and entry-level.

7. In some places, Circles partner with other groups such as the Boys & Girls Club and government agencies that have an outreach mission.

Guess what! Another problem solving group -- http://www.mathlesstraveled.com/ -- also showcased the Number Bracelet problem, and they have added a cool visual showing all possible bracelets in all mods through 12: http://www.mathlesstraveled.com/?p=520. This also leads to a couple of suggestions for presentation software.

mastermind

My game suggestion from last class was Mastermind, and the interactive version available from NLVM: http://nlvm.usu.edu/en/nav/frames_asid_179_g_2_t_1.html?open=instructions. We didn't have much time to consider it, but perhaps a next step is to consider a book, MASTERMIND® Mathematics Logic, Strategies, and Proofs that teases out the higher level thinking skills.

Tootsie Roll Game (“Iterated Sharing” from Solve This)

A group of student sits in a circle, each with a pile of wrapped candies. (Wrapped candy is used because each piece will be handled by many people before being eaten.) Some people have 20 or more, others none, and the rest some number in between. The distribution is arbitrary except for the fact that everyone has been given an even number of pieces. A reserve supply is set aside.
The group now follows these instructions: Give half of your candy to the person on your left and half to the person on your right (and hence receive a supply of candy from the people on your left and right). Do this simultaneously. Now recount your candy supply. If you now have an odd number of pieces, take an extra piece of candy from the reserve supply. This boosts your pile back up to an even number of pieces and everyone is ready to perform the maneuver again.
What happens to the distribution of candy among the group if this maneuver is performed over and over again? Will people be forever taking extra pieces from the center, so everyone’s amount of candy will grow without bound? Or will the distribution stabilize or equalize in some sense? Will one person end up with all the candy? Might “clumps” of candy move around the circle with each iteration or some strange oscillatory pattern emerge? Is it possible to predict what the result will be?
Taking it further: What happens if instead of adding pieces, you eat any odd piece of candy to bring your pile back down to an even number?
Hint: Based on the investigations my class performed, and the patterns they found, I encourage you to examine what the range of distributions is. For example, if you started with 0 – 18 pieces (a range of 18), what happens? And is it different if you started with 0-20 pieces (a range of 20)?
My classes enjoyed this activity, though my advanced class looked into in much further than my lower level class. They tried several rounds to test and prove conjectures. It was a difficult game to play with a class larger than 12 students. I tried putting them into two groups, but this was difficult since I was facilitating both groups. It was also difficult to play with desks in a circle. Playing on the floor or at a large table would have made things easier.

Math Circle Documents

I added a Math Circle Documents Link ....if you have documents that you would like to share - send them to me and I'll post them.

-Kendall

Bertine started us with teaching strategies for half-life. Steve & Kendall suggest a penny shake: shake, remove heads, shake again, remove heads, etc.

Lynne gave a presentation about the Teacher Circles to the UW Math Club on Friday, Sept 25. The following day at an event for families of prospective students, the Math Club used the ropes and Conway’s Tangles. Kids said, “cool”.

Many had tried the human knot. Bertine found that his mixed gender group was reluctant, so he went to single-gender tangles. Lynne tells of using this same Tangle for UW freshman during Summit 2009, having talked about touching BEFORE doing the tangle. Dave's 5th graders held onto two opposite ends of a shoestring, not hands.

Lynne tells a little of a talk at UW’s Center for Learning, e-volution with Gardner Campbell, "Narrate, Curate, Share: Integrative Learning and Web 2.0".

Steve is shared his “lock boxes” teaching strategy. (Steve has 56 physical lock boxes, and about 130 questions organized by the answers.) He tells his 7th graders that in a problem solving situation, rather than looking at looking for a rule, that they should look all around to figure things out. Steve is working with kids who have difficulty with even basic concepts and words can go out, discover relationships, etc. His example is to find a number that is 6 less than the product of the 2nd and 6th primes. Kids don’t immediately start asking questions --- learned helplessness of 'I don’t get it' --- now they read, investigate, etc. Mary adds that last year with her SPED kids would ask to do a lock box, and helped each other.

Ariane did number bracelets with all of her HS classes. Her point was that the problem of number bracelets engaged high performing and low performing students. One class was having a test under a sub, and the class wanted permission to make number bracelets after their test. A young man said that there were only 100 combos, so he would go home over the weekend, work out all 100 and let the class know what happens.

Lynne posted an NCTM site where they claim they “combine understanding with skill”. We had a small attendance (6), but the conversations were animated as we all face the challenge of implementing richer problems.

Kendall posed "Math Magic": pick 2 positive integers in a vertical fashion.

456

987

1443

2430

3874

Then add the sum to the 2nd addend to get a 4th number. Now add the 3rd and 4th to get a 5th, keep going until you have 10 numbers. How can you quickly get the sum of all 10? Trick: 11 times 7th number = sum of all 10. Kendall’s problem is a challenge for his 13-year-old son Benjamin. Kendall also asks for help to decide favorites from Jackson to take to ND where he’ll present about TCs, and for suggests about other patterns that are like this, or ways to extend this. Maybe there is a similar pattern with 3 numbers to start.

Lynne & Kendall have a brief exchange about how people, even people who don’t think of selves as good in math, are drawn to this sort of problem.

Steve has us view a YouTube clip of Dr Who where we need to enter a Happy Prime:. Had his students exploring digit squares and looking for happy primes and happy numbers. Square the digits of a number, sum and then repeat. If this process reduces to 1, the original number is a happy number. Steve likes this for the training it gives for the 2-step process of squaring and adding. Examples of happy numbers are 13 and 19. Note that permutations of the digits will give another happy number, e.g. 91 or 31.

13 -> 1^2+3^2=1+9=10 -> 1^2+0^2=1

19-> 1^2+9^2=1+81=82 ->8^2+2^2=68 -> 6^2+8^2=36+64=100 ->1

Ariane asks for more information about Kendall’s pattern, and Lynne suggests understanding the pattern using letters instead of specific numbers. Combine with multiplication, and it all works equally well.

Steve relates that there are things we know about finding happy numbers. For example, every number along the path of converging to a happy number is in turn a happy number. For example, in expanding 19 we generated 82, 68 and 100.

Lynne suggests we return to our rubric for judging problems. Steve suggests that last two problems are not relevant to general population, but that it would be good for grades 4-5. Seem to rank well on all other points. Steve suggests that perhaps #8 is not perfect, and maybe we need a point about engagement.

Ariane suggests that there is a challenge in defending problems that don’t directly relate to her standards. Struggle will be to align and then get back to the book = boring things saying that the other part can be fun too. Steve agrees about the importance of some routines.