Wyoming Math Teacher Circle

Automata Antics: Ball Throwing

2010-03-25T08:57:00.000-07:00

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.

2010-03-19T10:52:00.000-07:00

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.

2009-12-27T10:21:00.001-08:00

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

2009-11-09T15:19:00.001-08:00

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)

2009-11-03T17:52:00.000-08:00

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

2009-10-26T13:02:00.001-07:00

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

2009-10-12T18:23:00.000-07:00

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.

Bead Bracelets with fifth graders.

2009-09-29T04:29:00.000-07:00

Taking an idea from Ariane, I modified the activity to multiplying beads rather than adding beads. I did this simply to relate to what we are doing in class. So most of the class was working with multiplication. However, I did have a group that did the addition as written up. The questions were posted and I added a few questions:

What patterns do you notice?

If you used only odd (or only even numbers) could you make a prediction about the ones digit in the next answer?

If you used odd and even numbers could you make a prediction about the ones digit in the next answer?

All of the students enjoyed the activity. The higher students, who tend to be bored during regular math instruction and need to be challenged, enjoyed looking for patterns and relationships. The students who tend to struggle enjoyed that they had an entry point.

The answers to the questions were not as thoughtful and deep as one would hope. That was to be expected due to the age of the students and the lack of exposure of problem solving and justifying or proving their reasoning.

I also did a quick poll as to whether the students would like to do this activity again, never again, or if I made them. The class was pretty much divided.

Bead Activity

2009-09-28T12:06:00.000-07:00

My students used the bead activity as an introduction to patterns/algorithms;i.e. order of operations. The students so enjoyed the activity. I had them write down their two numbers on a sheet of paper and do the algorithm to find when the two numbers repeat. After completing the pencil/paper task the students actually made their necklace/bracelet from beads and pipe cleaners. The task became quite competitive and students worked diligently!

It gave me chill bumps to see the interaction and chat regarding algorithms.

This also led into the "Math in Action" photo for the new concept. The students were to take a picture of an algorithm and use pencil/paper to describe the algorithm. . . rec'd some real unique ideas!

Math Bracelets

2009-09-23T12:39:00.000-07:00

Wow! My students were hooked on the math bracelet activity! They wanted to try multiplying the numbers after they were successful with adding the numbers. Two students now have a contest going to see who can make the longest bracelet. The numbers were great to set the pattern and practice some basic computation. My enrichment class expanded the activity by creating a color code for the beads. One student suggested using to beads when we get to double digit answers. I was able to get some digital photos of the colored bracelets. Hopefully, I will be able to share them next time we meet.
One of my goals was to share this activity with our special education teacher. I was also able to share it with my team. They thought it was great.

Divide and Conquer

2009-09-21T15:49:00.000-07:00

Divide and Conquer is a game played by subtracting proper divisors from the initial number. One is always a proper divisor and no number can be divided by itself. The game is won when a player presents his/her opponent with one.
As we worked with 100, which has several proper divisors, we tried strategies that included making a list and drawing a diagram on graph paper. In the list we found a proper divisor of 100, and then subtracted the divisor from 100. We repeated the process with each new difference until one player could present his opponent with one. Similarly, we drew a 10 x 10 grid on graph paper and took turns crossing off proper divisors until one square remained.
We discussed that many middle school students will learn how to play this game with the initial number of 30, They will memorized the specific steps for 30 and attempt to generalize those exact steps to other numbers. The conclusion was that in order for students to truly demonstrate their understanding of factors in this game, they need to play it with other numbers.
I shared this game with my summer school students. They were thrilled to be able to beat the teacher.

Finding the last digits of 7 to the 9999th power

2009-09-16T16:50:00.000-07:00

Now here is a question. Suppose you took 7 to the 9999^th power. That would be a number with 8450 digits. In 12 point type, that is a number about 70 feet long. What are the last three digits of that number? Why would we ask such a question? No reason, that’s just what we do.

One problem solving strategy is to create an organized list and look for patterns. Using Excel, here are the first 10 powers of 7 (formulas are listed in column C).

	A	B
1	7	7
2	49	=A1*7
3	343	=A2*7
4	2401	=A3*7
5	16807	=A4*7
6	117649	=A5*7
7	823543	=A6*7
8	5764801	=A7*7
9	40353607	=A8*7
10	282475249	=A9*7

No obvious patterns seem to jump out and the numbers are already getting pretty big. Is there a better way? Asking for the last three digits is the same as asking for the remainder when you divide by 1000. Excel has a modulo function we can use to find the last three digits of the powers of 7. We only need to multiply these values by 7 to find the last the last three digits of the next power of 7 and the mod function returns only the last three digits.

As a shot in the dark, what is the value of the first 25 values for the powers of 7 mod 1000? It would be very cool is some relatively small power of 7 ended in 001 (or written in mod notation: 7^x mod 1000 ≡ 1) since powers of that number would also end in 001.

A	B	C
1	7
2	49	=MOD(B1*7,1000)
3	343	=MOD(B2*7,1000)
4	401	=MOD(B3*7,1000)
5	807	=MOD(B4*7,1000)
6	649	=MOD(B5*7,1000)
7	543	=MOD(B6*7,1000)
8	801	=MOD(B7*7,1000)
9	607	=MOD(B8*7,1000)
10	249	=MOD(B9*7,1000)
11	743	=MOD(B10*7,1000)
12	201	=MOD(B11*7,1000)
13	407	=MOD(B12*7,1000)
14	849	=MOD(B13*7,1000)
15	943	=MOD(B14*7,1000)
16	601	=MOD(B15*7,1000)
17	207	=MOD(B16*7,1000)
18	449	=MOD(B17*7,1000)
19	143	=MOD(B18*7,1000)
20	1	=MOD(B19*7,1000)
21	7	=MOD(B20*7,1000)
22	49	=MOD(B21*7,1000)
23	343	=MOD(B22*7,1000)
24	401	=MOD(B23*7,1000)
25	807	=MOD(B24*7,1000)

Holy Math Smack! 7²⁰ends in 001 or 7²⁰≡ 1mod 1000, then the pattern begins to repeat. That means that 7²⁰≡ (7²⁰)ⁿ ≡ 1mod 1000. So, think of 7⁹⁹⁹⁹ as 7²⁰x7²⁰x7²⁰…7¹⁹ . so 7⁹⁹⁹⁹ ends on the 19^th step of a repeating 20 step cycle. That value, according to the Excel sheet, is 143. QED. (that is Latin abbreviation you put at the end of a proof. I think it means “Thank God its over” but you should google it to be sure).

So even though 7⁹⁹⁹⁹is impossibly large on the scale of 8^th grade math, we can make some statements about it, such as the last three digits have to be 143. Now, can you find the first three digits?

Clock Arithmetic

The problem above makes use of a repeating pattern in the last three digits of the powers of 7.

Repeating patterns such at this come up often enough to be interesting. Here is a simple case that would be accessible to 4^th or 4^th graders.

My friend, Mary O’Dell works in a candy shop. Part of her job is to bag the candy when a fresh batch comes out of the kitchen. As one of the benefits of working at the shop, she gets to keep any candies left over after bagging – that is, if there are some candies left but not enough to fill a whole bag, she gets to keep them. Today she is bagging candies 9 to a bag and the cooks made 665 in this batch. How many candies does Mary get to keep? What are some numbers that would leave Mary several candies? What are some numbers that would leave her with no leftovers at all?

Clock arithmetic is a common example. If it is 8:00 am now, what time will it be 133 hours later?

A mathematician would ask, “what are the patterns in these kinds of problems?” The first thing we’d need is a notation – a way of writing these ideas down. In this case us a mod notation. If we want to look at Mary O’Dell bagging 9 candies per bag then we can start with the number of candies which was 665 in the example above, then designate the number is a bag (or the number in any group) by using the notation “mod 9”. The notation would be 665 mod 9. We use a congruence notation rather than an equal sign because we are trying to say these are numbers with the same remainder when divided by 9 and these numbers are not equal, their remainder is when they are divided by 9. We use the notation ≡ and we read it as “is congruent to” so we would write 665 mod 9 ≡ 8 and we read it as “666 mod 9 is congruent to 8”

2009-09-14T18:13:00.000-07:00

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:

is relevant to experiences of target population;
can unfold over time and is not sequential and closed;
is cooperative, drawing on current knowledge;
has outcome accessible to self-verification;
has accessible extensions;
has a problem solving trajectory that can be shared;
has a solution involving fundamental math;
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

2009-07-20T12:10:00.000-07:00

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

2009-07-13T12:45:00.000-07:00

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

2009-07-10T09:42:00.000-07:00

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

2009-07-09T21:36:00.000-07:00

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

2009-07-09T14:27:00.000-07:00

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

2009-07-09T09:05:00.000-07:00

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.

Puppies and Kittens Game

2009-07-09T05:42:00.000-07:00

The Puppies and Kittens Game

DIRECTIONS

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.

2009-07-08T13:41:00.000-07:00

Proposed Fall Class Schedule
Mondays 4:30-6:30
August 31
Sept 14
Sept 28
Oct 12
Oct 26
Nov 9
Nov 23
Dec 7

2009-07-08T13:27:00.000-07:00

Wyoming Math Circle Implementation Plan

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

Individual plans
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.

Mathematical Game - Put Down

2009-07-08T09:57:00.000-07:00

Mathematical Game – Put Down
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.

2009-07-08T08:26:00.001-07:00

Wow we really like the ropes

2009-07-08T06:22:00.000-07:00

Welcome!

2009-07-07T17:18:00.001-07:00

Welcome to the blog for the Wyoming Math Teachers' Circle.