ASSIGNMENT 2: Share your stories about problem groups 1, 2 and 3

**(1) Warm-up** Gears was easy for both of them and tried to increase speed! For mix-pins Parsa tried to approach algorithmically and find a solution which works for making mix for random number of pins! Amir tried to approach visually. But he started from Ten-Pins! Parsa has simplifier role for Amir. He tried to show Amir that starting from smaller number of pins works better. Amir reached the right solution which Parsa did. but in a wrong way! This was good thing for both of them. Amir revised his steps and found his mistake. **Problem1(Area of Triangle):** Parsa knows Thales Theorem. So he solved it easily! Amir knows how to solve areas of circles and right triangles. Amir tried to start from what he know and redrew it. But he stuck! **Parsa drew a line and asked: "Amir what is it? What are possibilities?" Then Parsa told that it can be a triangle! a triangle with one 180 degree angle and two zero angles!** He could not continue exactly from what he thought. But finding heights from shadows with a ruler could be a great experience for them! **Problem2 (finding max value):** This problem was challenge for both of them! Amir only knows that we can use variable x for representing different values but he does not know about algebraic operations or equations and Abs function. But Parsa know Abs function and quadratic equation but he had no visual idea about equation of circle or square. We tried to remove what we don't know and turned it to simpler form: x+y + x-y = 2 and x - 6x + y Amir learned simple equations from Parsa and they could struggle with solution. **Problem3(comparison):** Parsa had no problem to solve problem. But his challenge was to teach square root to Amir! His strategy was similar to folding and unfolding! Amir learned and solved it!

Kids liked the problem with gears very much. Older kids also. I guess they liked it because they knew what to do immediately. And it is funny to look at their rotation, too. We've also played the problem of 20 (or 4) pages, matches and triangles. I'd love to try it with integer powers also but we haven't had enough time for them this week. One of the kids has had a moment of enlightenment with 20 pages, and it was great to see his face and hear him cry "Aha!!" For others it was difficult to get the idea - why are we questioning about odd or even sum at all? They were staring at the book without any idea, and after some time of talking about sums the've got bored. Pins or matches were also difficult and that amased me a lot because kids of 9-10 years were not able to solve the problem for younger kids about 6 matches and 4 pilows. They were trying to put 6 or 10 matches in the right way and then they told me it was not possible, but they didn't explain why. With the triangles I've drawn a picture and ask my son what similar figures did he see there. He've found a lot, but not the pair of triangles that helps to solve the bigger problem. The good thing is that I saw him thinking about the problem without looking at the picture and he confirmed that he can see it in his mind. Next time I'll talk to them more before giving a problem - about what they know, their ideas, what they want etc.

Oh, I forgot this part of problem 1, the one about the book pages. I made a sheet and had them write about their thinking process in solving the problem. Out of five people, two adults, two teens, and one child, the adults and one teen figured it out, and the others gave up. Here are how the ones who completed the problem explained their thinking: **Adult 1**: There are 20 pages. On each page, there is one odd number and one even number. I separate the odds and the evens. If you add evens to evens, they will be even. If you add odds to odds an even number of times, the result will be even. Therefore, the answer has to be even. **Adult 2**: I visualize the numbers as blocks. Even numbers create whole squares or rectangles. Odd numbers have a missing piece. But if you can match two odd numbers together, they can connect to make a whole square or rectangle--in other words, they make an even number. As long as every odd number has a mate, there are no missing pieces. Adding in the even numbers doesn't change that, so I visualize all of the numbers coming together to make a big, even rectangle. No matter what pages are ripped out, there is an even and an odd number on each. Since 20 is an even number, we know each odd number has a mate. I tested it out on 1-20 and 2-21. **Teen**: Even. There are two pages on each paper. One is odd and one is even, so it is like 1 + 2 = 3. 20 x 3 = 60. So, I got three very different successful approaches to the problem. Carol

Not a lot of insight to report about the first part of problem 3, which I again did with about 10 people from 5 to 65. Some got it, some didn't, at least within the time frame. My husband and my son, both of whom have good 3-D visual minds, figured it out first. Then, once some people had figured it out, younger children just got discouraged and gave up (so I did the "what would be the easiest way to place them for this problem" thing, which they enjoyed much more). Eventually one other adult and one other teen figured it out, but after that point even the older students gave up and just wanted to see the answer from the ones who figured it out. So sustaining interest in figuring it out yourself after some peers have done it was an issue for me. We also did the MAA-AMC #3 problem with a different group of two moms and four children. One family considered themselves good at math; the other family didn't. I gave them the problem "What is the correct ordering of the three numbers 8^10 , 12^5 and 2^24 ? I first asked them what they just intuitively thought was the answer, and almost everyone chose D giving highest exponent highest value. Then the math family started just powering through the calculations (each working separately without using calculators or anything) while the non-math family just kind of floundered around. So I started to work with them about writing out the equations and could they do anything to simplify or make the equations more comparable. They wrote them out OK, but they still didn't get anything they could do. Eventually all three of the math family finished their calculations, apparently correctly, because each one handed in a written answer that was the correct choice. Then they started fiddling around with the re-formulating the equation to make them more comparable. With a few hints or prompts from me, they figured out the alternative approach as well. I think the non-math trio understood what we did once we showed them and explained it to them, but I don't know that they would have ever figured it out themselves. So, in general, I'm having a problem supporting or encouraging the people who can't figure it out fairly easily to stick with it or to be more creative in their thinking about how to figure it out.

**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.

PROBLEM 1 ![alt text][1] I did a simplified version of the AMC problem from Essay 1: “A farmer wants to maximize the amount of grass she can plant in her sheep pen with a limited amount of fencing material. How should she build her fence?” My students were R, age 13, and J, age 9, both homeschoolers with strong conceptual knowledge but few algorithms under their belts. R is eager and confident in math, J is the opposite. To make it even more challenging, they are sisters. From the start, R attempted to distill the essence of question while J tried to change it. This battle played out over the entire hour. J didn’t like any of the premises, including the fact that I called it a “problem.” I read her the explanation of the word “problem” from the prelude to assignment 1 (who wrote this?). That calmed her a bit. Throughout, each girl alternated between enthusiasm and discouragement. As we worked, I talked about “flailing” as a problem-solving technique. J said “I don’t want to flail.” I suggested we do “organized flailing,” and we sallied forth with that as our approach. Each time discouragement nearly prevailed, I read one of Dr. Tanton’s steps to problem solving from Essay 1. Each of these steps rejuvenated our problem solvers. I read the whole paragraph that goes along with Step 1 (“Take a deep breath and relax”) and both girls breathed a sigh of relief. At one point later J stormed out of the room, but was listening when I read Step 3. She bounced back in saying, “I think I have an idea!” R was ready to give up near the end but was energized by the un-numbered step “identify the penultimate step.” We attacked the problem by using props (2 pieces of yarn), asking questions, making assumptions, changing assumptions, forming conjectures, and rejecting conjectures. By the end, we hadn’t a definitive answer, but our flailing had been successful, albeit exhausting, so far. Both girls want to finish the problem at a future date. Some of the **questions** that arose from this problem (few were answered today): • Which piece of yarn is bigger? • What is the question? • Can grass grow everywhere in the pen? • How can we know for sure which shape has the biggest area? • What is area? (“How much space something takes up in 2 dimensions,” says R.) • Do we want grass coverage in 2 dimensions or 3? • What is a dimension? (“Counting how big things get,” says J.) • How many dimensions does a point have? • Can we trust our eyes if we make the shapes with yarn and eyeball them? **Conjectures**: • Shape doesn’t matter • Shape does matter • A square fence would be best • A circle would be best (This became the girls’ working conjecture before problem-solving began.) • The farmer could replant seeds indefinitely • The farmer could create a system to replenish the grass (“A circle might be bigger but replenishing is better,” argued J, since the pen could then be built more comfortably/humanely for our sheep Justin when maximum area was not the only consideration.) • The fence could allow the sheep to get grass from outside the pen • We could use various methods to answer this (eyeballing, area formulas, measuring, guessing, doodling…) **Assumptions**: • The farmer can replant/replenish (We did the question with this assumption, with J inventing numerous engineering solutions.) • The farmer cannot replant/replenish (We spent most time operating under this assumption.) • The farmer is using the most efficient planting methods At this point, the girls worked together with a ruler to make measurements of their yarn shapes. R began applying area formulas, while J began to draw shapes to make sense of it. She had finally let go of her desire to change the problem from a math question to an engineering challenge. “I can think much better when I’m doodling,” she explained. R calculated areas for a rectangle, square, and circle. The circle was the biggest so far. But more **questions** arose: • Are our results significant, since we just measured rough yarn shapes? • How could we calculate this without tedious arithmetic? (R did not want to use a calculator, but the fractions in the measurements were ugly – so common when using the English system!) • How do you multiply fractions on a simple calculator? • What other shapes should we try? While this problem was very challenging to address with such different ages, I loved that they each benefited from the other’s contribution. The older child gained from the “thinking outside the box.” The younger child heard a lot of math terms for the first time, and is very very curious to learn what they mean. I am building her math studies for 4th grade based upon her questions from these problems. I was surprised that no one thought of super-imposing the shapes, or of getting some actual grass seed and counting how many fill the space. I also was surprised by the pretty interesting tangential discussions of dimensions and pi. I enjoyed seeing the power of verbal acknowledgements of the emotional side of mathematics. I wish we had time for R to do this problem in an abstract manner (no props, no numbers); I asked her to think about this on her own later. I also would have like to have a few more kids (we will next week). Then kids can have closer-aged peers to bounce ideas around. I’ll send separate reports about problems 2 and 3. I had hoped, unrealistically, to do all 3 in one sitting. [1]: /storage/temp/90-img_0849+(640x480)+(2).jpg

**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.

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.

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

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.

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

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

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

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.

![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

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

Technique 1 Gears • My son drew arrows on gears to indicate direction. • As problems got more complicated, we suggested he could substitute symbols for arrows; he chose to use 1 and 2. • When the most complicated problem was solved, he re-substituted incorrectly and started to get very confused and frustrated. Book pages • When we moved on to the kid/young teen level, my son became very frustrated talking about the problem. My husband made a book with him and they numbered pages and then talked about tearing out different pages. • We found it useful to let my son discover the alternating pattern: o If you tear out one (an ODD number) page, the sum of page numbers is ODD o If you tear out 2(EVEN) pages, the sum is EVEN • After doing a few more, my son noticed the pattern and he extrapolated to: 20 is EVEN therefore the sum is EVEN Technique 2 Pins • We gathered up 10 Lego minifigures and made 10 pillows out of paper. My husband said that 10 people were going camping but they all forgot their pillows. For safety from bears, they have to sleep in a circle. What is the minimum number of pillows that they need to make if they each share a pillow with one other person(H:H)? What is the maximum number of pillows that they could make (H: P)? Is it possible to arrange the sleepers such that they require any number of pillows between 5 and 10? Once he understood the upper and lower limits, he loved the challenge of manipulating the figures to see if all outcomes between 5 and 10 were possible.

**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

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) [1]: /storage/temp/110-problem3spreadsheetbinarytr.jpg

I'm not sure what the problem numbers are, but we worked on the Wheels with Cogs problem, and the 2^100 problem, yesterday. It was really fun! I worked with Alyssa (going into 8th grade) and Taryn (going into 6th grade). Both attended the Rio Rancho Math Camp last month. **What we learned about Wheels with Cogs** If one wheel is going clockwise, any wheel that touches it is going counterclockwise. If the wheels are different sizes, then they go around at different speeds. For example, if a wheel with 5 cogs is the 'driver', and it is touching a wheel with 9 cogs, then for every revolution of the 5 cogged wheel, the 9 cogged wheel goes around 5/9ths. It would be interesting to do some measurements and do the math to figure out if the distance travelled along the circumference of both circles is the same or not. If you make a circle of cogged wheels, and you have an odd number, then you end up with a conflict of which way one of them should turn—counting one way from the start, the "last" one should be clockwise, but counting the other way, the "last" one should be counterclockwise. If you actually try to build this, when you put the last wheel in, all the wheels will stop turning. We did most of this exercise on pencil and paper, but we also used some lego pieces that were cogged wheels for the circle example. We also looked at some cool animations of cogged wheels spinning online. **What we learned about 2^100** Both Alyssa and Taryn started writing down all the powers of two starting with 4, 8, 16, 32, etc. They thought the brute force method would work. They started slowing down at about 1024, and then they decided to go as far as 2^20. When they got there, Taryn thought he could multiply that answer by 5 to get 2^100. I tried to convince him that was nothing close to doubling 2^20 another 80 times. I think he understood that. Then Alyssa started noticing the pattern of 2, 4, 8, 6 in the last digit. She then wrote down 2, 4, 6, 8 as a group of four numbers, 25 times. Checked by counting the digits up to 100, and said that 6 was the last digit. I tried to point out that she could have divided 100 by 4 and figured out that since the remainder was zero, the answer was the last digit in the group of four. It might have been better to draw a circle with the numbers 2, 4, 8, 6 around the outside of the circle, and then count to 100, and see if she noticed that this has something to do with division and remainders. We then checked the answer on the internet and found that 6 was the correct answer--if you can believe anything you read on the internet. From a teaching standpoint, I noticed that it is tricky to offer the kids guidance and support without leading them directly to the answer I wanted them to find. Asking open ended questions is not as easy as it sounds! These were both great exercises--lots of fun to watch the kids think. They were both really engaged.