ASSIGNMENT 1: How do you plan to adapt problem groups 1, 2 and 3?

Since my teen group is meeting a week late, I'm still working on this assignment. I decided to print out the problems on full-sheet sticker paper, so I can cut them up for the kids to stick in their notebooks. The triangle problem and the absolute value equation problem are both beyond what my students have studied, so I am giving them the basic situations without a question, just to see what questions they can find to ask. Only on the exponents problem do I expect to get solutions (I'll save it for last, so we can end the day with some sort of resolution.) I modified Carol's Problem Solving Techniques file (from the Assignment #3 page) and combined it with excerpts from [Math Forum's Notice & Wonder worksheet][1] to make a couple of problem solving stickers, too. I'll make new problem-solving stickers for each week, as we go through the techniques. [Problem Solving Stickers, week 1][2] [1]: http://mathforum.org/workshops/nctm2010/handouts/MathForumNoticeWonderRecordSheet.pdf [2]: /storage/temp/95-math+group+problem+stickers,+week+ 1.pdf

Our strategy is based on collaboration between two brothers Parsa and Amir. We play Gears and Pins for warm-up! Then they face problems and try to start from what they know! They should be critical students from start to end of class from different points of view. Parsa has important role to simplify what Amir does not know.

1. Gears: I plan to have the kids work out the solution using the gear set we have. They can also experiment with the raising/lowering of the toys as shown in the picture. For the page number problem: my 7 and 6 year olds understand even/odd but not my 4 year old. I thought we would try to make a game of it using a small number of pages, with the single digit page numbers, out of one of their books and we would see who could calculate the sum and whether it's even and odd. We'll try this a few times and then see if we can see the pattern. 2. Pin positions: We'll use their Lego people to do this problem. First, I'll give them time to just play with the different combinations --- head-head, feet-feet, head-feet. Then, I'll present the problem using the story and pillows made out of the Post It notes in the example. For the multiplication series problem, I was thinking of making a story for them and drawing a picture. My kids like cats so the story would be something like: "A little girl named Eva discovered two cats in her yard. She named them Jill and Jillian, brought them home and gave them food, water, and a warm blanket. The next day, Eva awoke to find 4 kittens --- Jill had 2 kittens, and Jillian had 2 kittens. The next day, each of the kittens had had 2 kittens --- so 8 more kittens." We could then look at the last digit pattern and I would ask them, what would be the last digit after 100 days? We can test it out on the calculator or computer. 3. I'll use the cardboard cutouts and a white board to have the kids draw lines, beginning with the "wishful thinking" placement, then moving them around to find the solution. For the exponent problem, I thought I would instead use an addition problem with multiple addends, more than they are comfortable with. We would then use the wishful thinking method to see if they could order them.,

I just wanted to share a conversation I had with my husband about the problems last night. We were talking about the the pin problem and he wanted to know why the pins had to be in a loop. He tried to put them (well, we used matches) together in a star shape and we started discussing how many head to head connections we might be able to get if we omitted the loop requirement. We tried to connect as many heads to as many other heads as possible, although I queried how we would count such connections. Match heads are small so I thought only 4 could connect all touching one another, but then he put a match on top of them and made another! He claimed we could connect infinitely many, but I am not so sure... How many dimensions are we allowed to use?! We didn't get as far as recording our observations (it was past bedtime but you know how it is when you start talking about maths...) but I really appreciated the reminder to let go of the rules and see where it takes you.

Problem 1 - We have the Gears!Gears! set, but haven't opened it yet, so I will let the girls (5 and 2 1/2) experiment with them and soo how quickly they figure them out. Problem 2 - We will use some magnet rods and connectors, matchsticks and dolls to work this one out. I loved the idea of "stinky feet" that another participant had, and I will most likely use that as well! Problem 3 - For the 5 year old, we will use the iPod app, Flow. It uses the same concept and gets more difficult with each level. I may let the 2 1/2 year old try her hand at the beginning levels too. I will also use some animal families and fences we have with our barn set, and let her help the "mommies" find their babies.

I don't like to get to creative with presentation, unless it really facilitates the thinking process. Most of the time, I've found, the more creative I get, the more it interferes with the actual problem. So my plan seems overly simple, but I think my kids will bite. Here it is: Problem #1 http://www.youtube.com/watch?v=WRYZHWSSs_Y Ask questions: • How is it that movement in one direction can produce movement in another direction? • Can you think of other examples where moving an object in one direction, causes another object to move in another? List examples. • How can we demonstrate this principle to someone who doesn't understand? Are there household objects we could use? Problem #2 Give the kids each three pencils. Give an example of all three pointed the same way. How many other ways could we arrange the pencils in a loop? Add another pencil and do the same things. Conduct an experiment and collect data. Do 4, 5 and 6 pencils. Find a pattern. Can we predict how many combinations we would have with 7,8, 9 and 10? If we multiply two by itself, what do we get? Another time, and another. Use cuisinairre rods and make each number in terms of ones, tens and hundreds. Is there a pattern emerging? Can we use a pattern to predict what the ones place will be each time? How can we demonstrate and teach this to our friends? Problem #3 Print off a copy of the original problem and pose it to the students. Give them scissors and have them cut out the circles. Use a stapler and string to connect each of the circles to the correct match. • Can you arrange the circles back into order and not cross paths? • How would this problem be easier? • Can you manipulate things around so it works better? Is there a trick?

Vacation is over and time to get into play. I almost brought these problems up to the myriad number of cousins this week, but they were so busy creating their own fun I didn't want to interrupt. To introduce the gears I thought I would cut some basic gears from paper plates and have the kids play with them. They can even draw arrows on the plates to signify movement. The pin problem would have been great to try the children turning around in the circle, but I won't have that many kids now that we are home. I think I will try any pin like object. Something with a head and foot. I've never really understood why we want to find the last digit in a large number though so it is difficult to determine why we want to learn this. For problem 3 we will try string and people. Try to figure out a method of connecting three people to three chairs or something without crossing paths. We can scaffold like they did with the pictures, but moving the chairs from directly across to the final position.

Currently my plan is to lobby the principal of the elementary school where I work, to allow me to set up a 'problem bank' for the teachers. I think I'll try to set up the problems pretty close to how its laid out now, and adjusting as I get more feedback. I think the important part will be to try to emphasize problem solving means NOT giving away the answers. I know that initially you were looking for us to adapt these problems, but I find it hard to do so, until actually trying them out. I really like that problem of connecting the lines in Technique 3. That one works for all ages, and I can't wait to spring it on everyone I meet.

Problem 1: For the 6 and 7 year old, I will let them play with real gears and see what happens. We plan on having them try to make the last gear in a sequence turn a certain way. Problem 2: We will definitely go the action figure route (and maybe include our 2-year old). The oldest boy did the extension problem (2^100) already. For that, I first had him try it on the calculator. It returned roughly 1.26e30. We had a discussion about why the letter "e" shows up and about scientific notation. I then had him compute (by head) the first 10 powers of 2. He quickly saw the pattern of the last digit. He ruled out 2 and 8 as possible ending because these correspond to odd powers. He didn't quite make the connection that he needed to divide 100 by 4 and check the remainder, so instead he used a laminated number chart to see if the sequence 2, 6, 10,... contains 100. When it didn't, he tried 4, 8, 12, ... and saw that it did. This allowed him to conclude that the answer is 6. Problem 3: For both of the older boys, we will place the areas taped on a white sheet, and place it in a transparent folder. We will let them draw on the folder with white-board markers and see if they make any observations. As we need, we can remove the paper and move around the different areas.

Nikolai: I will be working on my S.D.F (self flying device). I will: 1. use gears to transmit power and change rotation between motors and see how many I need to make all of them to rotate the same direction. 2. use wires to transmit power between motors and lights air conditioners etc. in some tricky situations. 3. have a limited number of rods for holding up my machine and I have to have it not fall apart. I will practice. Sofia: with my little one I will: 1. play with ropes and stuffed animals, trying to predict the direction. 2. make roads between animals and their food storage, so they don't intersect. 3. play with dolls and the pillows, predicting how many pillows we will need.

**Problem #1** John exclaims “Twenty pages have been ripped from my textbook.” Hilde replied, not so hopefully: “I bet the sum of the missing page numbers is even.” Is Hilde right? I’m adapting this problem for my 6-year old son. First I will need to see if he reliably understands the concept of odd vs. even numbers. Since the solution relies on understanding when the sum of numbers is odd vs. even I will begin with a game. I’ll ask him to pick a playing card out of a deck (he’s really interested in playing cards right now). We will look at the card together and notice whether the number is odd or even. The playing cards have a nice visual built in if I need to reference it: there are “pairs” of hearts/spades/diamonds/clubs on the even cards. After I know he can distinguish between odd/even I’ll change the game. He will pick a card. I will pick a card. I will show my card and ask him if the sum of his card and my card is odd or even. I will then ask him if the product is odd or even. Based on his answers I will then “guess” whether his “mystery card” is odd or even. We’ll do this for a few rounds and also switch and see if he can guess my card. We will keep track of the responses for each round of the game in a table (parity of mystery card/parity of “other card”/ parity of sum/ parity of product). I’m putting both sum and product information in to begin with, but as we play I imagine we will try some rounds that include only a sum, and some that include only a product. This may end up being too complicated - I may need to pare it down to just look at sums. Hopefully we can keep both elements in the game and we can talk about whether or not we would need to ask both the product and sum questions and which one gives the most useful information (and in which situations). If we discover the “rule” I’d like to look at why the sum of two odd numbers is even, etc. using stones or some other small manipulative. I think this will be a pretty substantial amount of puzzling for one session, so I’m not sure whether we’ll get to the original framework of the problem with the page numbers. If we do, it will probably not be on the same day. In addition to understanding how odd/even numbers “work,” the solution to the problem also relies on understanding the way pages are numbered in a book. My next step would be to make sure he understands those conventions. (My son has experience writing his own “books” and numbering his own pages - not necessarily with the same conventions typically used in a printed book.) I will begin with a simpler question. Holding onto one of his books, I will say that I have chosen a page and that the sum of the numbers on that piece of paper is 19. Can you find the page? If he is successful with this question, I’ll try another similar question. (The sum of the numbers on a different page is 51. Can you find the page?) I’ll continue with these types of questions until we can successfully find the pages, then I’ll ask about one that has no solution: The sum of the numbers is 20. Can you find the page? What if the sum of the numbers is 8? Can you find the page? Why don’t these examples work? The goal here is to notice that on any given page there is always one odd and one even number so the sum of the numbers will always be odd (building on the work from the card game). Next, we will look at what happens if there are two paper pages involved. What will the parity of the sum of the four numbers on two pages be? How about the six numbers on three pages? How about the eight numbers on four pages? We will make a table recording the results and see if there are any patterns. We will guess about the ten numbers on five pages, then about the 30 numbers on 15 pages, and about the 40 numbers on 20 pages. **Problem #2** I’m going to adapt the pin problem for my 4-year-old son. With what I have available, we will make pipe cleaner people and arrange them in loops. We will color code the different possible combinations. BIG yellow (post-it-note)pillows to share when heads touch. small blue pillows for the single heads (head-foot combinations). No pillows when two pairs of feet touch. We will start with a small number of “people” (three, then four, then five,...). How many small blue pillows are necessary? What is the biggest number of small blue pillows that could be needed? What is the smallest number of small blue pillows that could be needed? On a sheet of poster board we can keep track of how many pillows were used for each number of people by sticking the pillows to the board. We will look for patterns and see if we can guess what will happen for a larger number. **Problem #2b** I’m going to adapt the powers of 2 problem for my 6-year-old son. My son dislikes the arithmetic speed tests he has at school. If he senses that the problem I’m going to give him has any sort of this flavor, he will abandon it (and me) as quickly as possible. I’ll need a sneak-attack approach to this problem. If I can find it at the library I would like to start with the book “One Grain of Rice” by Demi ( http://www.amazon.com/s/ref=nb_sb_ss_c_0_15?url=search-alias%3Dstripbooks&field-keywords=one+grain+of+rice&sprefix=one+grain+of+ri%2Caps%2C216). We can start to think about just how BIG 2^64 is. (How much bigger would 2^100 be?) I think I can then bring him around to considering what digit this number might end with. I will have a hundred board available and also four colors of crayons/markers. We will color in the powers of two, one at a time with me handing him a crayon for each. Once we’ve used each color once we’ll begin the same cycle of colors again with the first color. Hopefully this will make the pattern “pop.” A single hundred board doesn’t take us very far into the pattern, so I’m thinking we will write each subsequent power of two underneath the appropriate column until we have enough data to consider. I think he will be able to see the pattern and make a conjecture about the 64th and 100th powers of two, but I _really_ want him to think about _why_ the pattern holds. He is accustomed to working with golden bead materials in his Montessori classroom. The final digit corresponds to the units, so if we look at doubling the number of golden beads maybe I can get him to think (in the abstract) about doubling each group of beads (hundreds, tens, ones/units) and realize that for this puzzle he really only has to think about what happens to the units. We don’t have any golden bead materials outside of school, but pebbles should be a fine replacement for the purposes of this problem. **Problem #3** I’m going to try the A-B-C problem wih my 4-year-old and 6-year-old (separately). We’ll do this in the sand. The A, B, and C will be squirrels (red, gray, and black) represented by rocks. The a, b, and c will be their favorite foods (acorn, pinecone, and maple seed). We will try different arrangements of squirrels-foods and see if we can connect them. I suspect we will find solutions fairly quickly... So we will also try some variations. 1) Add a sunflower-seed-loving chipmunk to the mix. Does the addition of a fourth animal and food make it any harder to connect the animals to their favorite foods without crossing paths? 2) What if the three squirrels each want access to two foods. Can we still make paths that don’t cross? 3) Each animal wants access to all three foods. Can we still make paths that don’t cross? (Now it’s really the classic gas-water-electricity puzzle in disguise.) I’m not sure if the A-B-C problem was supposed to be adapted, but I have no great ideas about how to make the exponent problem into something meaningful for the 4-6 crowd. Instead of: What is the correct ordering of the three numbers 10^8, 5^12, and 2^24? I could imagine asking: Which is bigger, 8 groups of 10 or 12 groups of 5? For my 6-year-old it would be “Which is a better prize: 8 $10 bills, 12 $5 bills, or 24 $2 bills? (He is very interested in money... and the idea of a $2 bill - which he has never seen - might be a bit of an extra hook I could use.) This does not involve the exponents at all, but does get at the idea of having different small-big number combinations and how that impacts the size of the result of performing a particular mathematical operation. It also gets at the idea of breaking down numbers into more manageable parts (factoring) that is useful in the solution of the original problem involving exponents. I could also imagine asking... if the right moment presented itself. Anything that smells remotely like “arithmetic speed test” to my 6-year-old puts him in shut-down-mode. Which is the biggest: 10*10*10*10, 5*5*5*5*5*5, or 2*2*2*2*2*2*2*2*2*2*2*2?

I am adapting for ages four and six. Problem 1: First, I'm going to ensure that they understand the idea of turning in different directions. I'll point out the clock on the wall and how the hands go around one way. Always one way? Do they ever go the other way? Then let's call this way that clocks go "clockwise." What if they went the other way? That would be different. We'll call that "counterclockwise." Then we'll stand up and spin around both ways. This should be fun! Then I'll bring out the first picture, printed to fill the sheet of paper. I'll see what they make of it. If they get it right away, I'll leave it at that, but if they don't, I'll facilitate by bringing out their Gears toys, and we can build the diagram with those. (If we did not have the Gears, I would have them try being the gears themselves, "spinning" one another by touching hands as they turn around.) Then we'll add a new gear. Now what? Then another. Now what? Another. Now what? Is there a pattern? If they don't see the pattern, we'll make a chart together showing the number of gears and whether the last gear turns the same or opposite way as the first. After the chart is made, the pattern should become obvious. Then we'll look at the big diagram. What are the answers? Can we tell from the pattern we found? We'll test our answers by building it with the gears. Problem 2: My children like to draw, and they like little figures. I'll tell them the story of Penny and ask them if they'd like the draw loops or make them with figures. Each child can do whatever he prefers. We'll see about doing a loop of ten with five mixes. If that's daunting, we'll make smaller loops. We'll try making of loops of, say, five many different ways. (They can pick the quantity of objects in their loops.) I will act as their record keeper, writing down the number of each type of joint. From this, we'll see what they find out. For my six year old, I will extend this to the powers of two version. He loves patterns. I'll say something like, "Hey, what pattern do you get when you multiply two by itself over and over?" I've seen him write this pattern down many times while playing on his own, so I'm fairly certain he'll excitedly write it down right away. Then I'll ask him what the final digit would be if he did it 100 times. Perhaps he will see the pattern. If not, I'll ask him what the final digits are of the ones he has written. Is there a pattern? Problem 3: First I will see what they make of the problem as written. If that is frustrating, I'll ask, "What if the lower and A and C were switched? Would it be easier?" Then I'll give them each 3 pairs of large card stock dots connected by very long pieces of yarn. What can we do with these? Let's make our own diagrams. Let's try to make strange paths now. What is the strangest path you can make? What happens with two pairs? Can we make the paths not cross if the matches are on opposite sides? How about with three pairs? Can we build this problem?

**Problem #1** I'd like to take the idea of the hard-to-fathom interplay between length and area from the AMC triangle problem to flail about with these questions: what is length, what is area, how can they be measured, and how can their relationships be generalized? I'll use a sheep stuffed animal and lengths of yarn to explain that a farmer has unlimited grass seed and plenty of land, but a limited amount of fencing (the yarn). How should the farmer place the fencing to maximize the amount of grass grown in the enclosure for the sheep to eat? **Problem #2** I'd like to look for patterns (and possibly generalizations) in exponential growth using stuffed animals and rubber bands. The problem will be this: "You live in a world run by animals. They have hired you, a lowly human, to do the bookkeeping for their circus. They have an act where the animals hang from a crane - first one animal hangs, then another animal's head connects to the first animal's foot, etc, etc. How many animals must the boss dog hire?" The question is intentionally vague and will require students to do something, and do something else, and something else, etc. I'm really curious to see whether the kids limit the number of legs each animal can have/use. **Problem #3** I'd like to explore the classic math problem called "gas water electricity," which looks a lot like the Paul Zeitz problem for technique 3 in the pdf. I did this problem in my math circle once as an example of a problem that can't be solved. I'd like to revisit it now with wishful thinking and find a solution based upon shaking up assumptions. NOTE: I may have to further adapt the problems based upon the ages of kids that show up. So far I just have 2 kids, ages 8 and 13.

Technique 1 – as I don’t have cogs, I’ll invent a game for the kids – each kid will have to turn around for 3 times (not to get dizzy) as cogs would do. With 3 kids in a line – first turns to the left, second to the right, third to the left. I’ll work with all my neighbors’ kids as I need 16 kids. With my granddaughter (2 years old), I’ll use a rope and some toys – as described in the experiment. I’m still thinking at the triangle & 2 circles….. what do you mean by ‘younger kids’ (“Help us to adapt it for younger kids”)? Technique 2- For the 2 years old, I’ll use her dolls; for the 10 years old I’ll use pens, discuss about patterns and finally, let’s hope that they’ll discover the last digit of the 2 multiply with itself one-hundred times. Don’t know what to say, yet, about the ‘maximum problem’ …. Techniques 3 - for the 2 years old – we’ll dig in sand or we’ll use ropes and toys. For the question about numbers raised to integer powers – don’t know yet.

Prob. 1: All 3 kids (ages 3,5,7) will be participating together, so I’ll try to start with the toddler version for the 3-year old and then work up to the gears problem for the older two. For the toddler version, I plan to use a long string and various stuffed animals, as suggested. Build on the story line of will a specific animal go up or down? For example, how do we let the pteranadon down to get some water? Set up golf clubs parallel to ground at different heights to experiment with a bowl of water on the ground. Let children explore and demonstrate. Maybe try to make a little video. For 5-7 year olds, take out gears from the lego box and have them explore setting up a few in a row and predict what happens at the end, starting small. Show the diagrams from the problem and have markers and paper available to create/draw their own diagrams. Suggest creating a system that will perform some silly or simple task at the end as a possible extension/application. Prob. 2: We’ll try using Straws and Connectors for this problem on the floor, with different colors for straws and connectors (have a connector on one end only). We’ll start with small numbers and I’ll try to get my 5 and 7 year olds to collaboratively on making a chart to log the results. Prob. 3: We’ll try this in the sand at the beach with largeish objects and foot paths, and have the kids help make a story out of it. If that doesn’t work so well I have some long strings so we can explore the problem by using some Little People toys at home.

**Problem #1** For my 5 year old and 3 year old girls, I am going to create a circus story about clowns on spinning merry-go-rounds. They connect the merry-go-rounds to make a huge spinning clown show. We have a motorized gear toy to play with. It is a little bit limited but will get the basic idea. And since they are actually spinning, they can see the directions easily. My mother-in-law also has some plastic gears that are like spinning lego blocks that you can spin with a handle . I can extend the circus story to acrobats using ropes and pulleys for up and down patterns. Actually this might be a better starting point since clockwise and counter-clockwise rotations don't have a lot of meaning to young kids in a digital world. For my grade 8 math class, I would stick with the gears and extend the problem to explore ratios and rates. Ratios of gear teeth determines speeds of gears. The story could be a factory trying to conenct a power source to a machine requiring a certain speed of rotation. The challenge would be to use as few or as many or a limited number of gears. **Problem #2** My initial thought here is to use bar magnets so that the connection and repulsion make the pattern obvious. Many bar magnets are coloured red for North pole and blue for South pole so colour matching works too. As an alternative we could decorate some popsicle sticks with our two favourite colours and then build shapes with matching (or non-matching) corners. As an extension, we could mix up all our sticks and try to make a shape with all the colours in the rainbow with no matching corners. **Problem #3** I think it is important here to have physical tools that work. I don't think drawing the connections will work for my young girls (and most of my struggling grade 8's). They need something physically connected that they can move around to help them visualize the paths. I am going to have my girls create a city by drawing roads on a big piece of paper. Then we will create buildings and cars out of cardboard and colour them matching colours. We will connect the pairs using thread or yarn of the same colour. We will glue down the buildings and play a game to try get the cars to move to different parts of the city without crossing the other cars path.

We are creating a storyline for the problems; 1. We have at one end of the platform a gear driven treadmill, at the other end of the platform is the power source driven by a crank that moves only in one direction and all we have to connect them, to save us or reward us, is more gears. We then let the kids experiment with configurations to push away the undesirable stuff or pull in the reward. Then we change the platform (length and shape) and let them do it again. Finally trying to educe from them the pattern to produce the desired effect. 2. We have discovered life on planet PeeYu, they communicate by creating figures lying on the ground; the problem is that everyone’s feet stink horribly. Sending a message with lots of peoples head and feet near each other is offensive. We need to find out which shapes we can use to make sure that no one’s head is next to another’s feet, so as not to offend them and start an intergalactic war. We will use Lego figures and baby dolls to work with the various age groups. 3. We created a cooperative challenge. We have 6 kids, we will pair them up and place them on one side of the room. We will have 3 stations on a table parallel or spread across the other side of the room. Each station will have a desirable treat and a number. Each team will have an anchor and a runner. The anchor stays in place holding one end of the string. The Runner draws a card with a number for a station that they need to reach. The challenge is to reach all three treats without crossing the strings in order for them to get to eat the treats.

Problem #3 (Connecting three pairs on a surface). I remember this from my childhood. We used to speak of three houses that each should connect to three utility providers (water, electricity, gas) without the lines crossing. Yeah, ... Flatland civil engineering?... I think a nice variation on this for summer is to play it on the beach, where we connect lines in the sand. The last impossible connections demand digging tunnels! Perhaps we could bring some color yarn strings, which only add to the fun of digging tunnels. And, thinking back to my childhood version, we could even pour some water through the tunnel...

Problem #2 (head/point). Because my youngster is a very avid builder, I would look for objects that can connect to each other in different ways depending on whether they are head/head or head/point, and I would think about qualities of the various structures emerging from this, such as their engineering qualities (how strong is this structure? -- what can it support or pull?). And i would want these to be household objects. So for example take belts, the sort we wear. I am guessing we should collectively have about 7 of these at home. They best interconnect as head/point. It's tough connecting belts together as head/head or as point/point, so there is an intrinsic bias toward preferring head/point, and there might be a humorous challenge for connecting like sides (h/h, p/p); we can think of these as snakes dancing in a circle... Which reminds me of my variation on Problem #1 (rotating cogs. Hmm....

Problem #1 (rotating cogs). A variation for young children. We all stand in a circle, holding hands, and sing a dorky math song. Ok. Then, in a ripple effect that begins from the designated starter-person (e.g., the youngest child), and spreading to the right (anti-clockwise as viewed from above, every other person turns around facing out of the circle, but holding hands again. This continues around the circle until the ripple reaches all the way back to the first child. But wait... did the pattern work there, or were there two kids facing in-circle holding hands? Oh oh.. what do we do? Either we bring in another person (or cat, or stuffy, or action hero), or one person has to volunteer to stand out for a moment. Who should that be? And what if we were 13 people in our group? Would that work? 22? 99? [As an aside: As I was imagining this in my head, I got into topological confusions over how the hands would hold -- literally how the palms would clasp, thumbs and all. Is it like shaking hands?... Yes, shaking hands is like a degenerate form of a math dance :) . Then I remembered from dancing salsa that this works very well, angular momentum and such.... Actually, talking about folk dances, instead of holding hands one could lock arms at the elbows, which somehow makes for a more cohesive circle of people. In any case, there should be more mathy dances, just like those Hungarian CS dances.]