This question was **closed** Dec 07, 2013 at 05:41 PM by Maria Droujkova for the following reason: The course is over.

This week, tell the stories of what your kids did with problems 1, 2 and 3. What questions did the kids ask? How did they change the problems? What was unexpected? What surprised you? What did you like about the experience? What would you change next time? [The problems 1-3 pdf file][1]. You can upload photos of your adventures, if you click the picture icon, and then "Choose file", "upload" and "accept": ![Upload Instructions 1][2] Then: ![Upload Instructions 2][3] [1]:
https://docs.google.com/file/d/0B6enMfoYXJb3UHlGODAwSFUwQlk/edit [2]: /storage/temp/75-uploadphotos.png [3]: /storage/temp/76-uploadphotos2.png

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Comment

**Answer** by Rodi.Steinig
·
Jul 30, 2013 at 04:39 PM

**COMMENT ABOUT MIRANDA'S REPORT**
For some reason, I can't get the comment field to work under Miranda's report. Hers is the 8th one, above the photo of the 2 girls. She wrote about handling distraction and physicality, and here's my comment about that:
I have this struggle too, especially when there is a wide age range (more than 2 years), or larger group. It sounds like you have some good ideas, though. I definitely agree that having more activities prepared is a good thing. For this age range, I also try to have at least one attention-focusing quick activity to bring out as needed, and also one narrative story related to the problem, such as a history vignette.
Another thing I think about in my groups is whether the peripheral activity is in fact distracting. Some kids need to move their bodies to think more clearly. My own daughter is one of these. I tend to let the physical activity go on if it does not impact anyone else's ability to enjoy working on the problem. Those physical kids get their energy discharged and return to the group typically, whereas had I coerced them into cooperating, their resentment might block their ability to contribute meaningfully to the math circle.
I have, at times, let this go too far, and have lost the group, and frustrated some kids, so it's a fine balance.

**Answer** by Rodi.Steinig
·
Jul 30, 2013 at 04:08 PM

**COMMENT ABOUT MIRANDA'S REPORT**
For some reason, I can't get the comment field to work under Miranda's report. Hers is the 8th one, above the photo of the 2 girls. She wrote about handling distraction and physicality, and here's my comment about that:
I have this struggle too, especially when there is a wide age range (more than 2 years), or larger group. It sounds like you have some good ideas, though. I definitely agree that having more activities prepared is a good thing. For this age range, I also try to have at least one attention-focusing quick activity to bring out as needed, and also one narrative story related to the problem, such as a history vignette.
Another thing I think about in my groups is whether the peripheral activity is in fact distracting. Some kids need to move their bodies to think more clearly. My own daughter is one of these. I tend to let the physical activity go on if it does not impact anyone else's ability to enjoy working on the problem. Those physical kids get their energy discharged and return to the group typically, whereas had I coerced them into cooperating, their resentment might block their ability to contribute meaningfully to the math circle.
I have, at times, let this go too far, and have lost the group, and frustrated some kids, so it's a fine balance.

**Answer** by Maria Droujkova , Make math your own, to make your own math
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Jul 24, 2013 at 06:55 PM

My kid M. and a friend C. solved the young adult version of problem 2. At the beginning, as James described, I was scared of the problem, especially when both M. and C. said they can't quite remember what that |x| sign means. They did recall it quickly. M. and C. bravely followed James' advice: "DO SOMETHING!" It amazes me how computer tools enable exploration of topics my teens barely know. M. and C. can successfully solve problems without knowing what I would consider prerequisites. They don't just have machines solve problems for them, but observe patterns, make hypotheses, test conclusions. It's really, really different from the methods I used when I was learning these topics! But I consider it a type of REAL problem solving. I have hard time keeping up. C. started by trying to find a pair of numbers such as |x+y|+|x-y|=2, and M. by trying to graph this with the 2D grapher [
https://www.desmos.com/calculator][1] - which did not work. Then M. opened a spreadsheet and programmed the third column to compute |x+y|+|x-y| using the first two columns. By then, C. found that (1, 1) works, and M. used this pair to test the formula. M. and C. tried multiple pairs of numbers and observed the results. Within 20 trials or so, they noticed that one of the numbers can be 1 or -1, and the other 1 or under. What about other numbers? M. then programmed a column (x) to go from 0 up by increments of .1, and the column next to it (y) to be 1. They discarded (by coloring purple) the values that did not produce |x+y|+|x-y|=2 Then they tested x and -x in the quadratic formula and found the maximum of 8. They came up with quite a few hypotheses and questions on the way, such as: - Is it all commutative for x and y? - Does the value |x+y|+|x-y| double when you double x? y? They checked all these numerically, using the spreadsheet. At some point, they plotted [|x+y|+|x-y|=2][2] and [the quadratic][3] using Wolfram|Alpha. I also showed them my favorite 3d grapher [
http://www.math.uri.edu/~bkaskosz/flashmo/graph3d2/][4] C. likes biology. He compared this numeric problem solving to the genetic algorithms he's been showing us, such as [BoxCar2D][5], where random racing cars evolve toward better and better models. Says C.: "We tried to do something, checked if it works, and did more of what worked." ![M. and C. solving Problem 2][6] [1]:
https://www.desmos.com/calculator [2]:
http://www.wolframalpha.com/input/?i=%7Cx%2By%7C%2B%7Cx-y%7C%3D2 [3]:
http://www.wolframalpha.com/input/?i=z%3Dx%5E2-6*x%2By%5E2 [4]:
http://www.math.uri.edu/~bkaskosz/flashmo/graph3d2/ [5]:
http://boxcar2d.com/ [6]: /storage/temp/132-problem2mandc.jpg

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**Answer** by Silina
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Jul 24, 2013 at 03:34 AM

2. We played with stuffed animals and pillows. I told her a story about an enchanted meadow where the animals could have the same dream, but only if they laid in the circle touching each other. They were allowed pillows only if two slept head-to-head. Played with my daughter arranging pillows and animals, guessing how to put them. For example, if you have four animals and one pillow, or four toys and 2 pillows. My toddler was mostly concern about toys’ comfort :o)![alt text][1] 3. Connected a mouse, a wolf and a bunny with their food. Made paths out of yarn and after the hint Olga got the idea of pushing the other paths around to get the food to the designated spot. She tried to feed the wolf other food so he wouldn't be such a thread to the bunny and the mouse. ![alt text][2] The most unexpected and surprising part was that my daughter was not like my son used to be. She loved role playing part the most (took forever to be done with), and didn’t care about math underneath it. I really loved doing something “scientific” with her, next time I will let more time for the role playing. [1]: /storage/temp/131-5+(640x478).jpg [2]: /storage/temp/130-1+(640x427).jpg

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**Answer** by Silina
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Jul 24, 2013 at 03:27 AM

From Sofia and Olga (22 months old) 1. Chairs didn't spark my toddler’s attention. Tried Lego rods and a car on a rope. She pulled the rope but refused to look into relationship between the numbers of rods and the direction :o) ![alt text][1] Then I tried the cart and the poles where I could pull her, she loved it. She noticed the direction of the pulling, but didn't care about prediction. ![alt text][2] [1]: /storage/temp/128-3+(640x427).jpg [2]: /storage/temp/129-4+(640x478).jpg

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**Answer** by Silina
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Jul 24, 2013 at 02:48 AM

Nikolai's part: 1. I wanted to transmit the rotation from one motor to the four propellers by the gears. It turned out it was very easy, and I already knew that it would take odd number of gears to transfer the spin in the same direction, but I also got the idea that I could also have three or four motors turning the same motor this would have tons of power which I could make faster by the loss of some power with a few more gears. ![alt text][1] 2. The pins (matches) problem: I started by placing all ten matches without a single mix connection. then i saw that you can only have an even number of mix connections for an even number of matches. The page problem: Since there are 20 pages ripped out there are 40 numbers in all, 20 of them are even and twenty are odd two even numbers make an even number and two odd numbers also make an even number so you have 10 pairs of each. 2^100 last digit is going to be 2 I just looked at 2^0=1 2^1=2 2^2=4 2^3=8 2^4=16 2^5=32 2^6=64 2^7=128 2^8=256 And saw that 2,4,8,6 as the last digits kept repeating themselves every four numbers, and since 100 is divisible by 4, the last digit is two. 3. My model or actual flying thing can use this in reverse with one motor spinning four or three propellers but this may not work because the power will only be of one motor. So I’ll probably stick to plan A. ![alt text][2] The problem of connecting A’s with A’s, B’s with B’s, and C’s with C’s was fairly easy and interesting. I thought this would be hard but when I saw The Art and Craft of Problem Solving I remembered I did this problem already. My mom decided to have my little sister do this as well so she got out some Lego’s and dolls for my little sister. [1]: /storage/temp/126-2+(640x478).jpg [2]: /storage/temp/127-1+(277x640).jpg

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**Answer** by Lobr23
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Jul 22, 2013 at 06:57 PM

Problem 1 Plan: 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. Problem 1 Actual: With the first picture, we worked through the gear problem quite easily using the gear set that we have first. Once my 6 year old understood the pattern, she could work the other pictures out without using the actual gears. For the page numbers, my kids seemed to lose interest (maybe relevance?) quite quickly and had moved on to other activities. If I were to do it differently, I might try to make it into a game or mystery to keep their interest. Problem 2 Plan: 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. Problem 2 Actual: The kids liked this problem the best. First, they were unable to arrange the people with 4 pillows. So my oldest said, Let's add 2 more people. And they solved the problem. The kittens story was a great success. My 7 year old recognized the pattern and could guess the next sets of numbers. However, because we have not covered division and I thought her attention span was too low at this point to try to explain it, I did not have her try to determine what the last digit of the 100th iteration would be. Problem 3 plan: 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., Problem 3 actual: Unfortunately, we did not solve this problem. I had done the cutouts and attached using yarn, and placed it on the kitchen table to work on like a puzzle. But the setup was too difficult for them --- the yarn kept getting tied up. If I were to do it again, I might use a dry erase board or even some type of computer game to be able for them to better attempt a solution.

**Answer** by Denise Gaskins
·
Jul 22, 2013 at 03:09 PM

![Math in the Park, teen group][1]![thinking about circles][2] I can tell I'm never going to get around to writing up detailed reports, so here is the gist of our experience: The problems were much more difficult for my teen group than I had expected. I brought five problems, but we only got to two of them. That makes sense, I guess, since each of the MAA-AMC problems are meant to take a whole class period (or longer) to explore. Way too much for my one-hour-a-week group! Next time, I'll only bring 3 problems, and I won't be surprised if we don't get beyond the first one. My (14yo algebra student) daughter spent a couple more days during the week on the inscribed circles and absolute value equation problems. She immediately saw the infinite series for the height of the triangle (based on [Don Cohen's "Infinite Cake" activity][3] that we did a couple years ago), but I challenged her to prove that the pattern of each radius being half the length of the one below it would continue, and after a couple of false starts (which may have been my not understanding what she meant) she succeeded in explaining that the triangle showed fractal-like self-similarity. I gave her the challenge of finding all the values of x and y that satisfied the absolute value equation. She worked at it for a couple days until she was satisfied that she'd found all possible integer values, but she didn't think to graph them until I suggested it. She recognized that she'd made a big assumption at the beginning (in limiting herself to integers), but she was afraid that allowing fractions would lead to infinite possibilities. I told her that it was her choice whether she pursued it further or not. I think she may, someday, but a stomach flu has put a crimp in her math for now... [1]: /storage/temp/120-gedc0927.jpg [2]: /storage/temp/121-gedc0926.jpg [3]:
http://www.mathman.biz/html/annaL.html

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**Answer** by cakeroberts
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Jul 20, 2013 at 02:30 PM

Technique 2 (young kids) I forgot to include this photo in my original post. My son loved working with Lego minifigs: [link text][1] Technique 3 (young kids to young teen) I wanted to use something tangible for this problem because I knew my seven year-old would get frustrated quickly if he had to erase a lot. When I presented the problem, he immediately ran for his large erasable grid. He drew the letters and I cut 3 very long pieces of string and taped one to each letter at the top of the board. We talked about straight lines and curved lines, and he stumbled on the idea of going out of bounds in order to go around the B. When I told him that he couldn’t go out of bounds but he should run with his “around” idea, a lightbulb went off. We had A and C tackled very quickly and B a short moment later. I think blowing up this problem and dealing with it in a large space was really helpful. Technique 4 (young kids) I found with the first two techniques that the young kid level was relatively easy and intellectually exciting for my 7 year-old, but the kid to young teen level quickly led to tears. For the moment I will pursue only the young kid level in order to try to build some confidence; I am hoping to come back to the kid to teen level problems. I was completely prepared to build a scale for this one and do some brute reckoning. As I talked about the problem with my seven year-old, he quickly determined that it is like his algebra video game Dragon Box. I helped set up the algebraic equation and he quickly solved the problem. When we reached the answer he smiled and shook his head, clearly surprised that the answer was less than 1. Technique 5 (young kids to teens) I gave my seven year-old a pile of pens to work with. He quickly solved it. As I am keen to build his confidence, I don't want to push the harder problems. That said, some of these are quickly parsed. Technique 6 (young kid level) My son wanted to explore the possible ways that the teacher could manipulate the paper (cutting, folding, covering). When I told him that there was no manipulation of the paper, but that he should consider the properties of a piece of paper he immediately had the solution. He looked genuinely surprised and happy by the trick that solved the problem. [1]: /storage/temp/114-technique+2.jpg+2.zip

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**Answer** by Maria Droujkova , Make math your own, to make your own math
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Jul 19, 2013 at 10:30 AM

**Problem 1 (young adult level)** This proved to be a more difficult problem. All three of us tackled it by brute force at first. M. used GeoGebra to draw (NOT construct) a sketch of the problem. GeoGebra computed the area of the triangle, as drawn. But it also supported visual examination, which produced useful insights. For example, if you draw a horizontal line between the two circles, the top triangle is exactly 1/2 as tall as the bottom triangle. Having *seen* that using the pretty exact GeoGebra prototype, you can then *prove* it rigorously. P. drew a lot of lines and did a lot of trig with them. He got far enough to convince himself he could finish the calculations, then said it's just arithmetic from there. I wrote a system of four quadratic equations with four variables, and solved them by substitution. Then we went to search for beauty and order, since none of our solutions had any! M. and me drew multiple circles getting smaller and smaller. P. started to compute their diameters: 4, 2, 1, 1/2, 1/4... I helped P. to find the sum of the infinite series - it was a fun aside, using halves of halves of halves etc. within a square. Once we had the height of the triangle, we used similar triangles to find the rest of the info needed for the area. P. and M. thought younger kids could do the part of exploration about the infinite series, maybe watching Vi Hart's video "Infinity Elephants." [
http://www.youtube.com/watch?v=DK5Z709J2eo][1] We then spent some good time cutting paper and working with GeoGebra to make the construction more visual and more accessible. It's a work in progress. ![Problem 1 Sketches][3] [GeoGebra app][2] [1]:
http://www.youtube.com/watch?v=DK5Z709J2eo [2]: /storage/temp/112-problem1model_2.ggb [3]: /storage/temp/111-problem1handsketches.jpg

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**Answer** by Maria Droujkova , Make math your own, to make your own math
·
Jul 19, 2013 at 08:30 AM

We did AMC problems 3 and 1 (in that order) with two young adults, my kid M. and a friend P. Both of them unschooled until their teens, and now work, as a game designer and a software developer. M. requested a computer (by pushing my chair to roll away from it and taking over the keyboard) and P. used pen and paper.
Most curious: how M. and P. used estimation or computer calculation as a tool, in both problems. First, they got the answers from estimation or from computers. Second, they spent a long time making the solutions more elegant, more logical, and more accessible.
The first step is a quick and dirty way to arrive at answers. The second step is about following mathematical values; it is harder, takes longer, and is more creative. This reminded me of the archetypal engineers (first step) and mathematicians (second step) in popular jokes. M. and P. played both roles, sequentially!
**Problem 3**
P. quickly wrote *10^8=100000000*. Then P. said there is "something" you can do with powers to reduce them. He decided to estimate. He instantly knew 2^10=1024 (being a programmer), so 2^12=4096. The square of that is *about 16 with six zeroes*.
5^12=25^6=625^3, which is about 600^3 or *216 with six zeroes*. The estimates were enough to order the three numbers. Meanwhile, M. programmed a spreadsheet to make the power series, and obtained the answer to the problem by reading its last lines. I asked why M. would not simply enter =2^24, but M. said the growth patterns are interesting to see.
Since M. was done solving the problem, I asked how would you make this problem accessible to young kids? M. was sketching some binary trees.
P. explained to M. why it was nice to have 4^12 (so that it could be compared with 5^12 easily), and where it came from. P. did some algebra with fractional powers, which came to: 10^8=[10^(2/3)]^12 He looked at 100^(1/3). "Is it about 5? No, we get 125 from raising 5 to the third power. And 4 is too small."
I asked, "So it is between 4 and 5?" and P. saw that it helps us to solve the problem! Since we have 4^12, 5^12, and (something in between)^12
We had a long conversation of adapting the problem for younger kids. A simpler version is something like, "compare 2^6 and 5^3." Five-year-olds could switch from 2^6 to 4^3 using the binary tree (see the picture below - going from black to orange). P. tried to work out how to make fractional power accessible to young ones, but it's still work in progress.
![Problem 3][1]
(to be continued)
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**Answer** by Denise Gaskins
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Jul 17, 2013 at 02:12 PM

After our first meetings, I have a better idea what levels my groups are working at: **K-1st group =** lots of intuition, counting, minimal addition **teen group =** rules-oriented, some intuition, mostly at pre- or early-algebra level (ie, not yet comfortable with exponents) I wrote a thorough report of our K-1 meeting: [Math in the Park K-1][1] The teen group only got through a couple of the puzzles. They are not going to be able to work through the MAA AMC puzzles as written, so I either need to focus on one problem per session or choose easier versions for them. And I need more practice at scaffolding while "being invisible." That's hard! We started with the isosceles triangle with inscribed circles, to see what they would notice about the drawing. They had memorized (or partially memorized) some area formulas, which they tried to apply (using a ruler) with varying success. One of the boys came up with this question: "Can circles really be perfectly round?" Then we moved to the 2^100 problem, which I thought would be a relatively quick one. Not so---it took up the rest of our hour. I gave them the stack of paper (cut in half 100 times) question to take home and wonder about. [1]: /storage/temp/106-math+in+the+park+
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**Answer** by nikkilinn
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Jul 16, 2013 at 08:28 AM

Problem 1 – I introduced a set of gears to my daughters, 2 ½ & 5, along with the help of my husband, who demonstrated some ways to make them work together. As Rodi mentioned above, the movement of the gears was better visualized as a wave, rather than gears moving in opposite directions. The girls immediately caught on to how the gears fit together to move, and enjoyed building different shapes and “machines”. My 5 yr old and I discussed what kinds of things are powered by gears...clocks, bikes, etc. and did a mini-scavenger hunt to find some around the house. We also spoke about what could stop a gear from working properly – misalignment, something caught in the gears, etc. She then built a “clock” and played a game with her sister in which the gears broke due to one of these and they had to fix it. It was great to see the interaction between the two as they looked for the “broken” gear and either taught the other how to fix it, or had each other find the problem on their own. Problem 2 – I used counting bears to illustrate the concept of 2^100, borrowing another member's idea of cats and kittens. We began with a mama bear who had 2 baby bears, who then each had 2 baby bears of their own. My 5 yr old continued up to 32 bears, and then we examined the patterns together. Problem 3 – For this problem, we utilized the iPhone app, Flow Free. The game encourages the girls to find the most efficient path to the color's mate without crossing any lines. The difficulty increases with each level. Both girls caught on right away, and their problem solving skills improved as they continued. My 5 yr old is currently on level 28, and the 2 ½ year old made it to level 5.

**Answer** by abrador
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Jul 15, 2013 at 04:55 PM

Math Circles 2013-07-13 We were 7 kids and 8 adults (!), we worked about 80 min. on a Saturday morning at Dor's place. We got to work on two of the three problems, with the first completed satisfactorily and the second partially completed. We ended off with a hand game that Dor had learnt in a German biergarten many years ago. All parents were thoroughly pleased as, I think, were the kids. The atmosphere was relaxed and playful. Dor did most of the facilitation, with sporadic contributions from other parents. Then almost everyone stayed on for another couple of hours for games and lunch. And iPad. And the parents who were not making lunch sat around to discuss subversive mathematics pedagogy... We made new friends. There was a sense of something very positive, and there was general interest in doing this again next week, hosted by Silvia & Russ. For Problem 1, a useful model was a sequence of domino blocks that were either face-up for clockwise or face-down for counterclockwise. ![alt text][1] For Problem 2, kids enjoyed working with matches, ![alt text][2] and some clued in that they could go 3D... Here are some implementation issues that came up in our postmortem, I mean postvivo: Parents' role during the session. Are they spectators or facilitators? If they are to help, how might they do so? Do all parents need to be on board with the lesson plan and goals? For the most during our session, the parents sat back and watched. We would like to change that. One of the motivations to involve more parents is to create opportunities for these parents to practice, and possibly get feedback, on how to work with children on these problems. Forms of representation, and in particular symbolizing. We recognize that some forms of representation are uniquely powerful models, for example algebraic symbolic notation supports reasoning, inference, and generalization in ways that transcend figural and diagrammatic models. In that sense, moving toward symbols appears to be a positive decision. And yet we noted that the concrete models children built with the substantive materials were often adequate for thinking through the (gears) problem. The models appeared to bear all the information relevant to solving the problem - there was no need for another level or phase of representing. Other times, though, multiple media can be very useful for problem solving, such as in the case of the Pins problem, where it might have helped to keep a record of aggregated findings from the group and then detect patterns within that aggregation. The question here is how to introduce paper and pencils (or markers) in a way that does not appear contrived. Embracing group variability. Our kids were K-3, so that they had had quite a variable exposure to mathematics content. Also, we had one ADHD child, and one child on the Autistic spectrum, so that we witnessed a variety of engagement forms. Finally, a couple of kids (boys) tended to dominate group conversations. We thought it might be useful to develop strategies for embracing this diversity. We would rather not divide and conquer, and yet sometimes that seemed to be the only way of maximizing engagement. Keeping kids on the theme. The kids were creative in taking the problems in innovative directions. Sometimes this was an opportunity to make explicit the problem's premises, such as when kids started building vertically the pins (matches), holding them up with clay. That is when we realized we had never stated that we are working only flat on the surface. Other times, we were not quite sure how or even whether to get kids back on track. [1]: /storage/temp/104-mathcircle2013-07-13-berkeleyproblem1d.jpg [2]: /storage/temp/105-mathcircle2013-07-13-berkeleyproblem2a.jpg

**Answer** by nikkilineham
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Jul 15, 2013 at 12:47 PM

I worked with my five year old niece, Jayda, for the first time. She is about to enter kindergarten and so I really had no idea what her skill level was, although we've done lots of counting together. The first thing she said to me was that she was excited to "play math" with me, which is now a phrase I will use rather than "do math". I made cardboard cut-outs of the cogs and she quickly found the pattern and we worked through those problems quite quickly. I scribed for her and she showed me with her fingers the way the arrows would go. Interestingly, she used the words "up" and "down" to indicate their direction. Next, we used some of her toys to look at the head-to-head problem and she played around for a bit, not really understanding what head-to-head meant at first but then figured it out. Her six dolls didn't join in a circle like the picture, so she found a seventh and that completed the circle. I found this interesting as she could have just adjusted the six dolls into more a circle. Finally, I made cut-outs of baby/mommy animals and placed them on a big piece of paper and told her that the mommies needed to rescue their babies but couldn't cross paths. She initially rearranged them a bit, but not at all in the way I'd guess and then drew lines. We turned the paper over and this time I told her she couldn't move them and she drew lines with no problems. She seemed to have fun and I think the skill level of the problems were right on for her. I also worked with an adult friend, Vanessa, who is totally math phobic. She has been wanting to go to college to re-train but hasn't due to her fear of being incapable of doing math. I asked her to help me with this project so that she could try math, as an adult, and basically dip her toes into the idea of returning to school. I had no idea what her skill level was either but was very careful about not scaring her and making her too anxious. She confided that she had had nightmares the previous night about not being able to do the problems. So, I sat with her and certainly helped, but basically tried to let her do what she thought and only jumped in when she was completely stuck. We did the pages ripped from the book problem and she tried different start points and saw the pattern quickly. She did bring up points like what if the pages ripped were from the front of the book where there are no page numbers and other good questions that would certainly affect the solution. Next we moved onto the 2^100 question. Her approach was to actually do 2^100. I let her carry on for a while before I suggested we look for patterns and then we discovered the solution. She said that she wouldn't have thought to use patterns had I not suggested it. She had no problems at all doing the ABC problem once she knew you could draw lines outside the box. Finally we looked at the exponents problem and she really wasn't sure what to do with that one at all. I learned that she actually doesn't know her times tables all that well, furthermore, she had no idea what multiplication meant and the different strategies you could use.

**Answer** by dendari
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Jul 15, 2013 at 12:03 PM

We have played with the first two problems so far. My older boy is definitely more confident with the math and the patterns. At least he has the language to recognize and explain what he is seeing.
http://youtu.be/b0Jub6-Bcms Both boys loved creating the gears and could tell easily which direction each should spin. I had just started the pin problem and you can see my old boy caught the even odd question right away, but then a reminder went off on my phone and I had to leave for a meeting. We did not get a chance to return to the math.
http://youtu.be/dM4YnCDI2CQ Edit Finally got around to problem 3. I laid out three strings and crossed them as in the problem. I told the boys that they couldn't move the ends, but had to make the lines so they didn't cross. At first the 8 year old picked up a string and said it was too hard. He then dropped the string but it landed behind one of the ends. I could see understanding click and he quickly picked up strings and rearranged them in a correct order. It took about 10 seconds. The younger boy had been distrcted and came back to see the problem solved and was disappointed. I quickly made the problem a bit more difficult by saying they could not go behind the middle end point. The problem was still dolved in seconds.

**Answer** by RosieL52
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Jul 15, 2013 at 10:08 AM

The biggest take-away for me from the first set of problems is to not over-plan with my kids. The 6-year-old can smell "official math problem" a mile away and becomes immediately resistant. The key for me to engage him is to be playful. Problem #1 For the book-page numbers, he shut down quickly when we started adding page numbers together. Later, I tried the card game on him and it was a hit. He decided quickly that knowing the sum of the numbers on the two cards was sufficient for determining the parity and became uninterested in the product of the numbers. When I asked him how he was able to determine the parity of the mystery card, he said (something like): "When you add two evens together, the answer is even because there are no extras to start with so there are no extras when you add them. When you add an odd and an even, the odd has an extra but the even doesn't. So the answer will have an extra and will be odd. When you add an odd and an odd there are two extras that go together. So the answer is even because there aren't any extras left." Problem #2 My 6-year-old was not all that interested in the powers of two until I found an online hundred board where he could click to color them. He was able to guess at a pattern with the last digits, but at that point he was "done" and did not want to pursue any further questions on the matter. Problem #3 I tried the A-B-C problem with my 4-year-old. I started with "oh, look at this cool rock. Let's pretend it's a gray squirrel. Can you help me find ones to be a red squirrel and a black squirrel?" We also easily found their favorite foods (acorns, pinecones, maple seeds). I ended up using string for the paths and he had no trouble finding ways to connect the critters to their foods without crossing paths. At this point I asked him if two squirrels could share each food - could we add a second path for each squirrel so that still no paths would cross? He said "No, I think we should find more acorns and pinecones." Shortly thereafter we had another place to get to and I was unable to get the same momentum going for the problem. In the future I will need to make sure that there is adequate time to "play" and also "play math."

**Answer** by Rodi.Steinig
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Jul 14, 2013 at 10:44 PM

![alt text][1]
**PROBLEM 3**
I had chosen the classic math problem “Gas Water Electricity” to practice the strategy “Engage in wishful thinking.” I had done this in a math circle several years ago with the instruction “How can this be done?” It turns out that in the classic version, it can’t be done. One child in that circle had gotten very frustrated, nearly in tears. I was hoping that by rephrasing the instruction to “Can this be done?” and using the Wishful Thinking strategy, the girls would reach this conclusion in a less painful way. Immediately J said, “I like this problem.” She is a visual/spatial learner, a style that this problem as great for. Also, there is no arithmetic. As is her style, she immediately started changing the problem by moving the houses and declaring that lines can overlap without crossing and still be legal. In other words, she was engaging in wishful thinking on her own. I named this strategy, and identified it each time she used it to worked the problem.
Unfortunately, R had seen this problem before and knew that there was no solution in the classic sense. I tried to get her to engage in wishful thinking anyway to notice and attack assumptions, and just see if there’s a solution if you change the problem. “I don’t like this strategy of wishful thinking,” she pouted. She just couldn’t get past her preconceived notions of the question. She also stated “I don’t want to cheat.” To her, changing the problem is cheating. To J, changing the problem is opportunity.
There are a lot of things you can change and test in this problem. But my girls didn’t do them. R wasn’t willing to try. J was stuck on her overlapping idea – she still thinks it will work but is having trouble figuring out how to draw it. I hope to return to it with both of them in the future.
If I were to do this problem again, I’d do it the same way, but make sure that no one had seen it before.
One other observation I’d like to share about these 3 problems is the idea of sharing the board. When I lead math circles, I do it in a room with multiple boards and children are welcome to come up and take over the class at any time. At home, we only have a teeny whiteboard, making sharing virtually impossible. For the third problem I did let them use the board since we didn’t have long lists of numbers that couldn’t be erased. I think this contributed to J’s enjoyment. I hope to get a bigger board for home, or even just use sidewalk chalk outside to get everyone more involved.
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**Answer** by Rodi.Steinig
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Jul 14, 2013 at 10:27 PM

![alt text][1] ![alt text][2]
**PROBLEM 2**
The strategies of “reread the question” and “do something” were of immense help in this problem. I asked R and J how many total animals would you need to hang them from a crane pyramid-style for a circus act, and let them define the terms of the problem. J really got into this information-gathering stage: she lined up and measured the heights of her stuffed animals and researched typical heights of cranes. She chose the Flat Top Tower Crane (344’ tall) for this problem, and determined at an average height of 1’ tall per animal that we would need 344 rows of animals. There were other questions, answers, and assumptions that went into clarifying the problem, prompting R to complain “I don’t like this stage. I want to get to the problem solving.” Her eyes quickly lit up, though, when she found a flaw in J’s calculations (they are sisters, after all). Since the girls had decided that the animals had to hang 1-animal length above the ground for dramatic effect, only 343 rows were needed. “Is it going to get harder?” she then asked.
“I’m confused,” said one girl and “I don’t understand this,” said the other, once we got into problem solving. We repeated the “reread” and “do something” again and again. I worked with each girl separately for parts, as J probed the problem with stuffed animals and rubber bands, and R did with a chart of data. They periodically helped each other. J got frustrated when she ran out of animals long before row #343. She doubted getting a solution. R got frustrated trying to generalize (Is this a function of number of animals, or number of legs? How do you express this with exponents?). Interestingly, they both got stuck at similar points. They both realized that this problem is tedious and boring without a generalization.
Once again, we reread and did something. J hung the animals from a hook on the ceiling and we tried to spread them out to see the row-to-row relationship better. I did ask some leading questions (something I don’t like to do, but will when a kid is getting too discouraged). Finally, she had an epiphany: “Each row is one Leg Number as big as the row before!” (The decided-upon Leg Number was 2, and since J hasn’t worked much with multiplication before, this was her conceptualization of that concept.)
“Yes, it is – you’re right!” I replied. She jumped up and down and whooped and hollered. This may have been her first math epiphany ever. In the meantime, R had written out her answer on the board as a sum of a list of exponential numbers. She was still discouraged that she didn’t have a way to simplify this. She was at a conceptual impasse, having never done much with exponents. I showed her how to represent this symbolically as a summation sign, but required her to provide the variables. That thrilled her. Using the sigma was powerful. What a relief to show something as a single instruction. Her task for the future will be to figure out a way to actually simplify this.
If I were to do this problem again, I would do it in a multi-week math circle. We could explore how the answer might (or might not) be different if we used 4-legged animals. Middle-school kids could learn so much about exponents from it. Younger kids could have more time to dramatize it, gain more conceptual understanding, and write out number sentences. I did like the idea of stating the steps (reread, do something) again and again as almost a mantra. This is something that we do in math, but don’t always remember to verbalize.
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**Answer** by Viktor Freiman
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Jul 14, 2013 at 08:23 PM

I tried first two activities with one member of my family (female, adult, not trained for maths except of regular high school courses). On the first problem (wit 20 pages out of the book), her initial question was if they took out 20 pages or 20 sheets of paper. After precision, she said that it must be even since 20 is an even number, but then she said we should start with page number 1, so the back will be 2, therefore, 1+2=3 and 3x20=60, so the sum should be even. And she added that this holds for any number of page we begin with. From my point of view, as educator, I see this problem as a very interesting to engage people, including very young one in a game of questioning – like what would be if 20 pages counted as 1 sheet=2 pages? What if we take out pages not as sequence but separately from different parts of the book? What would be with any n number taken out? What can we say about divisibility of the sum by a different number (like 3, 4, 5, etc.). Regarding the second task, my participant said first that it may be 0 as last digit since 100 finishes with 0 – interesting that for the second time she also thought that some hints are in a given number (like 20 in the first task and 100 in the second) but after she realized that it cannot be 0. Then she said it should be either one of 2, 4, 6 or 8 but could not figure out which one. It is interesting to see with this problem how it is important to develop (as early as possible) a culture of ‘seeing mathematics’ through different patterns (thus developing ‘mathematical cast of mind’ (term used by Krutetskii, 1976). The task can also be an excellent way to introduce math that is overlooked by curricula, like modular arithmetic, and do so using another context as well, like circular clock (see for example
http://www.shodor.org/interactivate/discussions/ClocksAndModular/).

ASSIGNMENT 1: How do you plan to adapt problem groups 1, 2 and 3? 38 Answers

ASSIGNMENT 3: How do you plan to adapt problem groups 4, 5 and 6? 18 Answers

CITIZEN SCIENCE 1: Ask about adapting problems 1 Answer

Lack of ease? 8 Answers

ASSIGNMENT 4: Share your stories about problem groups 4, 5 and 6 20 Answers

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