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

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** 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.

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?

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.

1. 20 pages problem. For the younger kids I'll have 4 pages stolen and probably sheets of papers or a magasine to explore. For the older kids there could be something related to 3 instead of 2. Still thinking about it. 1.1. Triangle with incircles. Have no idea so far - the problem seems to be so complicated that I don't see the essense that can be reframed in an easier way. 2. Pins - a lot of play with matches and LEGO heroes. The photo with kids on the grass looks inspiring - will definetly try it too! 2.1. 2x2x2... will become 2+2+2... 97 times for younger kids. Not 100 because some of them might know how to multiplicate by 100. With older we can look and see what happens with the last digit of powers of 3 or 4 or.... ;-) 2.2. modules. Don't know yet. 3. Comparing powers seems suitable for 10-years old and up, if we discuss powers before. For younger kids would try comparing other expressions without actually evaluating them. Like 1354+235 <> 1357+200 or 21*2 <> 17*3 I like the idea of comparing without evaluating.

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.]

I plan to work on the young adult problems with my teen and friends. We'll print out the problems, and separate them from the hints. We'll read the problems together, then we will use [GeoGebra][1] and [Wolfram|Alpha][2] to model them. I will save the W|A input and the GeoGebra applets to tell the stories next week. Last year, I made [a 3D paper model of the AT-AT walker legs and Luke's speeder][3] for my 15 seconds in the crowd-sourced remake of "The Empire Strikes Back." I have an idea of steampunk-style movable papercrafts for the young kid level. "Paper gears" web search has quite a few good ideas, for example. ![Gear Art][4] Artwork by [Studio L3][5] [1]: http://www.geogebra.org/cms/en/ [2]: http://www.wolframalpha.com/ [3]: http://www.starwarsuncut.com/scene/1085 [4]: /storage/temp/73-studiol3gearscogs.jpg [5]: http://studiol3.blogspot.com/2011/07/gingersnaps-get-all-mechanical.html

For those without gears sets.... one of my best investments has been an old-fashioned egg beater. It not only demonstrates gears, but we have used it in a number of other science and math demonstrations or activities. PLUS, I have my son or my students beat whipped cream the old way before adding it to a treat, which helps them to work off some of the calories they are about to consume. It is cheap, readily available, distills the gear action to something even a really young child can see (and move backwards and forwards), and it is useful in real life as well. Carol

Finally Problem #3: I chose to think about the alternate problem, and I think I have a great way to work with my 7yo, and we'll see what my 4yo can do, but I'll be introducing multiplication to him. I may just do it with addition for him. With my 7yo, I'm going to work with multiplication, rather than exponentiation. She has been working a little on multiplication, but its still a new thing for her. I'm going to ask if she can figure out which is larger, 10x5, 12x3, or 8x7. Note that in these three problem I altered the numbers by increasing one by two from the first problem and lowering the other by two. So if this were a sum, they would all three be the same (maybe we'll even do that first!) We could do each of these problems, but given that multiplication is still new to Harriet, she has to work them out by hand and should be interested in thinking about it in a different way. 10x5 (10 five times) should be easy since counting by 10s is easy. So we can change the problem and figure out which is bigger of 10x5, 12x5, or 8x5 easily. Then we could think about how 12x3 is related to 12x5 and how 8x7 is related to 8x5. We'll use pennies to show them all as arrays I think. For my 4yo, I think I'll see what he can do with very small numbers, focusing on the same question for addition, and then multiplication if he handles addition well. He hasn't seen multiplication before, so I'd be curious what he'd do with it!

Now for problem #2: We'll use matches for our manipulative, since I think they will be easiest for my 4yo to handle. I'm going to introduce the problem to both of them having them each build a circle of 5 and then asking if there are spots where two heads meet, where two tails meet, and then where a head & tail meet. Then I'll start giving them some challenges, perhaps just centering first on those head/head meeting places. Can you make different number of head/head meetings (1, 2, 3, 4, etc) with different numbers of matches. For my 7yo, I'd love to get her started with record-keeping about solutions, so I might have her record all of the different ways you could arrange, say 5 matches. I imagine that she'll notice relationships between the H/H and T/T and H/T vertices, but we might explicitly explore this. For my 4yo, I'll probably keep it more loose, looking for whether he can make circles that have criteria I set out, and letting him give me some challenges too. I also love the Showing the idea of looking at the sequence 1, 2, 2x2, 2x2x2 etc with my 7yo. She understands a little about multiplication, and I'm going to use some manupulatives to try and build this up, and we may then look at writing down all these numbers and see if she can figure out patterns in their unit digit.

I am thinking about the second version of problem #1. I have worked with my older daughter (recently turned 7) on even and odd, which she learned in school. I'm thinking about how I bring these ideas to my son who recently turned 4 and involve my daughter at the same time. I am thinking of using pennies (or another counter, but pennies are always available) and presenting even numbers as "friendly" numbers because of the way they pair up and odd numbers as "lonely" numbers because of the one left out. We could talk about which numbers are friendly and which are lonely (I'm open to other language here if anyone has some, I'd even be open to just using even/odd, but I want to be emphasizing the pairing). Then we could get to talking about what happens when a number meets (gets added to) the next number up. What happens when 2 (friendly) meets 3 (lonely)? How about when three numbers in a row meet up? For my daughter, who has already explored some of this, I could start her thinking on longer strings of consecutive numbers.

1. I have a Gears!Gears! set, so I'm just going to let the kids play with it for a while before showing a diagram. The Gears! set comes with 6 interlocking plates on which gears can be assembled. I thought it'd be fun to give each kid a plate and ask to create gears puzzles for each other. Then we can connect the plates into one big puzzle and try to figure out which way will a certain gear rotate. 2. I think I'll use our vast collection of Star Wars toys to pretend play through this problem. Also, we might venture into the 2^100 problem. I am thinking the kids can fold large sheets of thin paper and punch holes in them using a hole punch. With 3 kids and 1 adult taking turns folding, the pattern might become even more obvious. 3. I was thinking about playing out a story (maybe Moon exploration where astronauts need to move from their rovers to craters to collect rocks), but maybe using sand for tracing the paths.

Just some quick feedback. I tried the gear problem with my 14 year old son and his 13 year old friend. I just gave them the graphic without explanation. My son started immediately rotating his finger for each gear and figured it out within 20 seconds (although he got one wrong, but that was speed and carelessness, not lack of understanding). His friend took longer--maybe five minutes--before getting them all right. She said she had been trying to figure out a pattern, but ended up getting too confused and so drew the direction on each gear. So we talked some more about possible patterns. My son kind of figured it out but couldn't articulate it. I had them count up the gears on each path--what did they notice? Then it became clear. Why would you have to do that? For my son, his way was just about as quick...but not as accurate. Their way was fine for that problem, but what if there were 20+ gears? 50+ gears? But imagining the same problem on a much larger scale, they could see that establishing a pattern of odd one way, even the other would be much quicker and more accurate for larger gear configurations. I'm going to try the same thing this evening with her younger brother, who is only 5 but more mechanically inclined than either of the teenagers. We'll see what he comes up with.... Carol

This is challenging for me! Part of the problem is that I am not sure how many kids or what ages will be coming along, which complicates things. I am also a bit nervous about leading a group beyond my own children! But here are my thoughts so far. **Problem 1** The resource centre where I will likely be running the group has a set of gears that I will hopefully be able to use. The kids can put them on baseboards and explore the problem that way. I also thought that younger kids especially might enjoy pretending to be gears - we could line up with our arms out and spin each other around to see how the gears interact. Although perhaps I should try to show them a real gear first so they have something to work with mentally. A friend also has magnetic gears that might work. I like the idea of printing out the gear problems too so that older kids might be able to work independently on them. **Problem 2** I can see a lot of potential for kinaesthetic learning here too! I was thinking of getting the kids to lie on the floor in a circle and see if they could fit 4 (or however many) 'pillows' into the circle under double heads. Or maybe using paper dolls - my oldest is very keen on those, she could make some for me! And I'd like to prepare sheets to fill in any patterns we see, just a basic table with number of pins/dolls, number of meeting points etc. **Problem 3** The idea of using pictures with yarn is great. I was also thinking dry erase boards, or playground chalk outside so we can try various paths. Maybe putting something up on the wall so more people can see and try out paths? I would like to try and present each problem in the form of a story, which they all seem pretty amenable to. This feels like it would be a good way to engage the kids initially and get them motivated to solving the problem. I'm looking forward to reading more ideas here!

For the 20-missing-pages problem, I wrote out the problem and created a small worksheet for people to record their thinking process about how to solve the issue. I think people kind of intuitively know the answer, but have different ways to "prove" it. I'm more interested in the thinking involved than in the answer. And sure enough, when I tried it with people last night, people approached the problem in completely different ways.[link text][1] I'll attach the sheet here--I have the problem copied twice on the same page just to save paper. [1]: /storage/temp/ 68-20bookpages.pdf

My oldest son (7) looked at the 2^100 problem over my shoulder and wanted to know how big it is. Although this problem is not about that, it is an excellent opportunity to get in one of my favorite estimation tricks: 2^10 = 1024 ~ 1000 = 10^3. Thus, 2^100 = (2^10)^10 ~ (10^3)^10 = 10^30.

Here is a good resource to go along with the second version of Problem #2 (2 to the 100th power): http://freemars.org/jeff/2exp100/question.htm I plan to use that after we work out the last digit part of the problem. Carol