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

For Olga 6) We had a stuffed snowman and a bear, sitting against each other. I drew one cat on one side of the piece of paper, and two on the other. We showed the paper to our animals and they had to decide how many mats the cats on the paper would need. Olga attached correct number of mats (sticky notes) every time, even when I made different pages with different number of cats. ![alt text][1] [1]: /storage/temp/146-6.jpg

For my 22 months old daughter. 4) I made a hanger balance. I made a few bags with 5 counting bears in them, adding some loose bears to the equation. It turned out that my daughter didn't understand the idea of the balance scale :o) We covered it by putting a bag into each bucket, one bag she stuffed with bears by herself (she absolutely loved the part of stuffing the bag), the other bag was empty in the beginning, and we balanced the scale by putting bears one by one into that empty bag. At some point the scale was balanced, we took out both bags and compared the quantity in each bag by taking bears out and lining them up on the floor, making two lines. The lines were the same length! ![alt text][1] 5) Olga didn’t agree that her solution with four pencils were not correct. She insisted that all the pencils touch each other through the pencils between them. I guess it was obvious for her that she could walk from one side to the other, so they were touching. When I showed her the correct solution, she looked at me like I wasn’t getting it :) ![alt text][2] [1]: /storage/temp/144-4.jpg [2]: /storage/temp/145-5.jpg

![Math in the Air-Conditioning][1]![K-1st grade group][2] K-1 group: This week was much less successful with my group, perhaps a combination of tired kids and hot summer days, and the fact that a basement full of toys was just around the corner... Manipulatives would definitely have helped with the algebra puzzles---or else we should have spent more time to play with and internalize the notion of balancing weights. After the success of the gear pictures last week, I thought the kids would take to the balance pictures, but they were just confused. Next week: More hands-on stuff! They played with the craft sticks for awhile, and two of the kids were able to get 5 sticks all touching each other. The other boy just wanted to make a pattern (which would be a great activity for exploration, but which wasn't our goal this time). They had fun laughing at the various combinations of animal bodies and heads, but they were not at all interested in counting the possibilities. One mother came up with a good way of counting: she arranged the bodies in a circle and then moved the heads around the circle so each head had a chance to visit each body. When they got back to the original configuration, she knew she had counted them all. The kids enjoyed the 3 bears logic puzzles for a little while, but by then they were just tired of the whole thing, so they did more wild-guessing than actual reasoning. The parents, however, really thought hard about the puzzles. My daughter said, "It's funny how our meeting turned into math club for the parents by the end!" [1]: /storage/temp/122-edit1.jpg [2]: /storage/temp/123-edit2.jpg

[link text][1]OUR STORY ABOUT SOLVING STANTON MAA PROBLEM 5 (aka, placing 3 items on 25 place grid) I did this problem with my 14-year-old son, who considers math to be his worst subject, and his 17-year-old friend, who is very strong in math. I created a sheet with the problem and the steps that Stanton uses (Have an emotional reaction, What is the problem asking?, Can you draw a picture? Can you solve a smaller version of the problem?, etc.) with blank spaces inbetween for them to work out their solutions. My son reads the problem and has his emotional reaction: “I have no idea. I don’t even understand what the problem is saying.” Meanwhile, his friend starts working out the entire problem in his head. “Hmmmm....that means there would be 25 options for the first piece, 16 options for the second piece, and 9 options for the third piece....” Me (thinking): “Man, this is incredible. He is working it out just as Stanton eventually walked us through in his paper. Except it took me going over it about 3 times to understand what he was doing WHILE he was explaining it...” Back to friend: “So that’s easy if the question is asking about permutations. Is it asking about permutations or instances?” Me: “Uhhhhhhhh......” (thinking: PERMUTATIONS? INSTANCES? I don’t think I have those words in my Stanton cheat sheet.....) Friend (clearly noting I have no idea how to answer his earlier question): “Does it matter in what order they are? Or it is just how many separate position combinations are possible?” Me: “Uh, well, no, it’s not different orders of the same pattern. Each pattern is just counted once.” Friend: “OK, well, that makes it a bit more difficult. Then I guess we have to divide by 3 factorial.” Me (thinking): “THREE FACTORIAL! I KNOW that’s not in my cheat sheet. What is a factorial again......?” Me (aloud, knowing that he is supposed to divide by 6 from said cheat sheet): “So what is 3 factorial?” Friend: “6.” Son (piping in for the first time): “Don’t you know that? EVERYBODY knows that 3 factorial is 6.” ME (thinking): “Gee, thanks, son. At least he refrained from say ‘DUH’.” Son continues his emotional response by ranting on again about one of his ongoing math complaints that he doesn't think that 0 factorial should be 1. He brings this issue up on a recurring basis, and so is glad for yet another opportunity to insist that it shouldn’t be that way. Meanwhile, friend announces: “OK, so that would be 600.” Which is the answer. But so far, not one thing has gone down on the paper. And, of course, the problem-solving technique is supposed to be solving a smaller version of the problem first, which obviously was not necessary for 17-year-old friend. So, Me thinking: “NO! You did it wrong! It was too easy! What about a smaller version?” But Me saying: “That’s fantastic! Great job, Friend. Son, did you understand what Friend did with this problem.” Son, of course, did not, having been distracted during the calculation phase by his 0 factorial rant. Me: “Friend, can you explain it to Son.” Friend patiently tries to explain to Son, but Son is not understanding the basic concept of the whole problem. Friend: “So, how many options are there to place the first circle?” Son: “Uh, 25?” Friend: “Right. Then how many options are there to place the second circle?” Son: “25.” Friend: “No, no...” Son: “Oh yeah, I see. 24.” Friend: “No, because it can’t be in the same row or column as the first one....” Son: “Huh?” Repeat several times. Now for my only substantive contribution to the problem: “OK, Son, think of it this way. Imagine you have a chess board with 25 squares....” Son: “Chess boards have 64 squares...” Me: “Fine, imagine a MINI chess board with 25 squares.” Son: “OK.” Me: “So now imagine that you have three rooks. What you are trying to do is figure out how many separate ways you could place those three rooks on the board so that none of them could capture the other ones.” Son, brightening: “OK.” Friend, meanwhile, has been reading the piece of paper for the first time. Now he is scratching his head. “Solve a smaller version of the problem? Huh? I don’t know... I’m just going to solve the problem my way.” Me: “That’s fine. Go ahead.” Friend (now starting to draw the problem): “So say we put the first one here.” (Draws on the grid and colors out the squares in that row and column.) “How many squares does that eliminate?” Son: “10” Friend: “No.” Son: “5 in the row and 5 in the column.” Friend: “No, no.” Me: “Honey, you are counting the same square twice. You can’t count the square both on the row and on the column.” Son: “Oh, right. 9.” Friend: “Right, 9. So how many squares does that leave as options for the second piece?” Son: “16.” Friend: “Right. That means there are 25 options to place the first piece, and 16 options for the second piece for each of those options.” Son: “Okay.” Friend: “So that is 25 x 16. But now we have to figure out about the third piece.” (Draws a second piece and colors off that row and column.) “How many places does that leave for the third piece?” Son: “9.” Friend: “Right, for each of all those options there are 9 options for the third piece. So that is 25 x 16 x 9.” Son: “Okay.” Me: “Do you want to use the calculator on my iPhone?” Son: “Nah.” Me (resisting rolling my eyes and thinking): “Yeah, like my son is going to be able to figure that out in his head....” Friend: “But that equation gives all the permutations of the three pieces. But we don’t care about the order--we only want to count the same position once.” (Draws three circles and labels them A B C.) “So, this counts ABC and BCA and CAB and CBA and...and BAC...and ACB. But because they are all the same position, we only want to count them once.” Son: “Yeah.” Friend: “So that means we have to divide by 6.” Son: “Yeah.” Friend: “So we have 25 x 16 x 9, but 6 doesn’t go into any of those evenly. But if we divide 9 by 3 we get 3, and if we divide 16 by 2 we get 8, so that’s 8 times 3, which is 24. So then we have 25 x 24. But if we split 24 into 4 x 6, then we can multiply 25 by 4 to get 100, then 100 times 6 is 600.” Son: “YEAH!” Me: Speechless and astounded. Me: “So what did you get on your SAT scores?” (Just kidding! I didn’t really say that.) I did praise both the boys about working through the problem so well, and also told Friend what a great job he did explaining the problem. I told him, truthfully, I couldn’t have worked it out nearly as well as he did. So there you have it--the Good, the Bad, and the Ugly (the Ugly mostly being Me.) Lessons learned? Well.... The way the boys did it, we never got to the problem-solving technique we were supposed to be doing, which was #5: Solve a Smaller Version of the Same Problem. But they did use #4: The Power of Drawing a Picture, and #9: Avoid Hard Work. This isn’t one of the official Problem-Solving Techniques, although Maria talks about it a lot herself. But making a story out of the problem, or relating it to real-world items with which he was familiar was necessary for my son to make sense of the problem. Friend is pretty awesome at mathematical problem solving. Son is easily intimidated, but stuck with it and got the problem once his friend explained it. So to some extent, that demonstrated #7: Perserverance is Key. It was really fun to see the boys doing math together. And how come I’m the only one who doesn’t know that 3 factorial is 6? Does everyone else walking around out there just automatically know that? Cheat sheet needs some additional vocabulary for math-impaired moms to use when working with students proficient in math. [1]: /storage/temp/137-problem+5+ sheet.pdf

**PROBLEM 5 (Solve a smaller version of the problem)** I brought in a menu from our favorite pizza shop, Fino’s. The cover has a map of the 20 regions of Italy. I told the kids that they were going on a pizza-recipe research trip, and had to choose 3 regions to visit. They debated, chose, and drew a dot on the board to represent each region. The kids chose Sicily, and island first. I told them that they have a choice of boat or plane to reach their second region, Abruzzo, and then a choice of three highways to get to their final stop, Emilio-Romagna. (The kids named the highways after themselves.) *How many different routes from S to A to E-S are there?* At this point, I must confess that I’m writing this 2 weeks after teaching it, so my recollections of the kids’ responses and reactions are fuzzy. I do know that I had originally planned to give a harder problem, have them get stumped, and then use the pizza-map question above to demonstrate the strategy of solving a smaller version. But the kids worked so hard on problem 4 that the younger kids’ brains were fried; I was afraid I’d lose them with a hard problem. So I started with the easy and decided to build up to hard, and hoped to somehow still get that strategy across. J, one of the younger kids, leaped to the board, excitedly. (She’s the one who loved the Gas Water Electricity problem, and saw similarities here.) She started mapping out and counting, then everyone else joined in. Conjectures arose: 5? 6? The older kids easily saw that 6 routes were possible. The younger kids understood that with some explanation by the older ones. R noticed that 2*3=6, but did not make the leap to a generalization. I said nothing. Then I asked them to add another city (City T – was is Tuscany or Trentino Alto Adize?), with four highways leading to it from City E-S. Now how many routes are possible? Y and K started sketching them out and counting them. J and D tried to, but got a bit overwhelmed. Fortunately, because I was allowing kids to write on the board too, they stayed involved. R started looking for a shortcut involving multiplication. Conjectures were 9, 10, 14, and 24. After some tedium, some agreed on 24, while others took a break. I suggested coming back to the problem later. During the break, R, I allowed R to explore this at her level. I asked her how many routes are possible if you went to all 20 regions in the numerical order the pizza menu ascribed? I told her that the number of highways between regions alternated between 3 and 4. She struggled with diagramming this for a bit, then the exciting part began. She started to experiment with developing her own algorithm. This is where I introduced the idea of using the smaller problem as a strategy. We ran out of time, but she definitely wants to come back to this later. I think I could develop a whole 6-week math circle on combinatorics, using the premise of opening a pizza shop. I never even got to the questions I had originally planned involving pizza topping combinations. NOTE: I’m not including the photo of the collaborative boardwork for this problem since the kids’ names are on it. But, if the photo is needed for reference (and not publication) I can provide it. I also have a photo of R’s work on the hard problem.

Problem 6. Horse problem 6. Nikolai as always had tons of ideas right away. Some of them are: 1. One kid had some kind of vision impairment, like double vision, then he realized that it would work for 4 and 2 horses, not 4 and 3. 2. Lashana couldn't see the whole page and missed one horse. 3. Lashana had to look through the mirror and one horse was cut out. 4. The teacher blended the page, wrinkling one horse. 5. Juan sits at the desk where he can see the OTHER side of the page through the mirror. There is one more horse! He ends up with 4 horses on the page. 6. Finally, Nikolai figured out that there were two sides with 3 and 4 horses :) Sofia

Nikolai: Problem 4. I simplified the problem first to 1+2+3+4+5+6+7+6+5+4+3+2+1. I looked at the picture as if it was a triangle where one got one segment, two got two, and so on all the way to seven. Then I cut that triangle down the middle, then put two halves together and got the answer 7x7=49. You can see that for the sum 1+2+3+…+n+(n-1)+…..+2+1 the result is n^2.![alt text][1]![alt text][2] [1]: /storage/temp/140-img_20130730_112224+(320x166).jpg [2]: /storage/temp/141-img_20130730_112336+(302x320).jpg

Nikolai's Problem 5. ![alt text][1] I simplified the problem by looking at three blocks on the 3x3 board. Found 6 ways to put the blocks on the board. Then I saw that for the first block I had 9 ways to place it on the board, for the second one – 4 ways and for the third one – 1 way. So it would give me 9x4x1 combinations, but I had just 6. I cut tons of 3x3 paper squares and started writing down all the combinations, making a tree of decisions. I realized that we didn’t care what number the block was, just its position. I eliminated all the same combinations, and was left with just 6. Then I looked at the case with 2 blocks on the 2x2 board. My decision tree was even more obvious for this case.![alt text][2] I saw that I counted 9x4x1 and 4x1 in the case where our blocks are numbered. It’s as if we were pulling the numbers from the bag, so we found the combinations of the numbered blocks. So the bag part is as if we are pulling the blocks from the bag one by one. Three possibilities for block #1, two - for block #2, and one - for block #3. 3x2x1 is 3!, which means that we have to divide 36 by 6 or 3! To get the answer. For the general case where M equals the lengths of one side of the board, N equals the amount of the blocks you have to place on it. The result is ((M*M)*((M-1)(M-1)…((M-N)(M-N)))/N! [1]: /storage/temp/143-img_20130806_232642+(305x320).jpg [2]: /storage/temp/142-img_20130731_155028+(320x240).jpg

Problem 4: We used pennies instead of the apples and I changed the numbers to avoid fractions. I presented a problem and when all three kids seemed confused, I drew a picture. It still seemed difficult --- the kids were just throwing out guesses, so as I went to find cups and pennies, my 7 year old answered correctly. She then explained why it was right. She deduced it by first thinking of a higher number, then when realizing that wasn't correct, she reduced it by 1, "checked" the answer, and found it was correct. I also showed her another way to solve it by eliminating the same amounts on both sides. Problem 5: This time my 6 year old solved the problem first. She's my visual learner. She took the 4 pencils and immediately put them in a triangle shape with one in the middle. Problem 6: Both my 6 and 7 year olds were able to answer the horse problem correctly, without much effort. Their answer when I asked how it could be possible was simple: the paper has two sides. It's been amazing for me to see how kids naturally think "out of the box" when rules have not yet been applied. It's as if their natural penchant for imaginary thinking, the unreal being real --- or at least possible ---, allows them to come at problems with a completely open mind. Through both sets of problems, my 4 year old son has observed passively. He does not seem interested in the problems or their solutions. Both my 6 and 7 year old enjoyed this set more than last week's, and have been asking for more problems.

My girls became too distracted with the balance weight to complete problem 4. They did weigh just about everything within reach, and figured out what objects weigh the same, so perhaps they came close to the problem in their own way? We used our three bears play set for problem 6, and ended up with a true +true and false +true style game. From Homeschooling 2013 Here is a video of what we came up with. (I can't make it embed for some reason.) http://youtu.be/Tn6beLM_I4Y

In my case, it has also been taking longer than expected. Some of that is that I am trying to work with the teen problems (having a teenager with teen friends), and since my high school math skills are rusty, it is taking a lot of time for me to even figure out the problems myself before I can present them to the students. And some of them I can't figure out, even some of the MAA problems where Stanton has worked them out. So I'm only giving them the ones that I can figure out myself, or can understand Stanton's explanation of how to work them out. So that reduces the number that I can give them. Still, it is taking a significant amount of time. It's been worthwhile, but more work than I expected. I notice that there seems to have been a big drop off among the participants who are just "parents," rather than people who are running math clubs and such. I assume that many are like me and found the level of mathematics intimidating, but haven't had the time or support to push through working through the problems.

**PROBLEM 6 (Eliminate incorrect answer choices)** After writing up my plan to adapt the sample problem, I came up with a better idea and went with it. I made up a multiple choice word problem with bar graphs as answer choices (see attachment, which my computer refuses to rotate 180 degrees). I drew the bar graphs on the board, and read the following to the kids: *“Which could be a bar graph of how many jelly beans Jean ate during the week if she ate more than her parents allowed on only one day, and on the next day ate exactly the maximum allowed.”* I wish the photo of these graphs on the board had turned out because for answer choices A, B, and C, I filled in numbers on the y-axis, but did not for choices D and E. I let the kids provide those numbers and they had great fun doing so. The image attached is my rough sketch. FYI, the top right graph is intended to show the same number on Monday and Friday. Everyone, no matter what age, loved this problem. I asked the younger kids to explain the graphs to the older ones to make sure everyone understood. They did. I asked each younger child to eliminate one choice and give a reason. They did. Soon, we only had choices D and E left, and everyone was scratching their heads. A discussion of “must be true” vs. “could be true” emerged. Finally the group decided to choose D because it must be true. **CITIZEN SCIENCE WITH THE KIDS** [link text][1] After each of my sessions, I asked the kids what worked, what didn’t, and if I were to repeat these problems with other groups, how I should change them. In today’s session (problems 4, 5, and 6) everyone said that they loved problem 6. It was fun, it was challenging, and it was accessible for a wide range of ages. J and R enjoyed problem 5 (J because it was visual vs. numeric, and R because it went into a very challenging math topic). K, D, and N didn’t enjoy that as much, but I’m not sure why. It’s possible that I let J and R, in their enthusiasm, dominate the group. D and N reported that they both really enjoyed problem 4 with the ratios. I think they liked that it was a fun problem (the M&M scenario) and also that it was a new but accessible math concept. K said she liked everything. [1]: /storage/temp/139-jelly+bean+bar+graph+ problem001.pdf

[link text][1] I did all the number 6 problems--the paper, the True/False statements, and the graphs--with a number of students 10-14, including my son. I didn't like the way the problem was originally written, so I rewrote it as: A teacher draws a number of circles on a piece of paper. The first class comes in. The teacher holds up the piece of paper to the class and asks, “Exactly how many circles can you see on this paper?” “THREE!” answers the class. The second class comes in. The teacher holds up the same piece of paper and asks, “Exactly how many circles can you see on this paper?” “FOUR!” answers the class. If both classes were right about the number of circles they saw, then how many circles are on the paper? Explain your reasoning. No one figured out the paper problem without some pretty substantial prompting. I had written out all the problems and printed the first two on the front part of the piece of paper and the graph problem on the back to give them a clue. So when they got stuck, I told them to examine their paper and see what they noticed about it. However, most of them thought first that there were three items on the front and one additional item on the back that could be seen by the light in one class but not the other (and so they answered 4). It was interesting to me that they got the concept of different things on the front and back, but chose what to me is the less-obvious answer of being able to see an additional circle THROUGH the paper than just having 3 on one side and 4 on the other. So maybe the rewrite still needs work.... On the logic problem, everyone chose the right answer, although my son was the only one who could articulate the way he figured it out. On the graph problem, again my son was the only one who could articulate his reasoning. It was interesting to me that instead of tracing around the rectangle and trying to translate that to directions in the graphs, which is how I did it, he thought of the direction she would be jogging, based on the graph, and decided that the correct answer was the only one that could represent a circuit around a course. For the others, they began saying they didn't know and just wanted to give up. So I encouraged them to eliminate the ones they thought it COULDN'T be and see where that got them. Eventually, everyone ended up with the correct answer. But they couldn't explain why they eliminated the ones they did--they just said it "didn't seem right" or something like that (all of these students did the problem separately, so they weren't influencing each other). But I told them all they had a great math intuition, and so just to trust that, and that they would pick up the vocabulary to explain it later on. So everyone ended up feeling empowered by these problems, even if they couldn't explain their reasoning. [1]: /storage/temp/138-problem+6+ sheet.pdf

My students tried to implement their strategy(smaller version and from what they know) by drawing. Their drawing was cool for problem 4. Parsa's drawing was more abstract than Amir. But Amir tried to tell story... This drawing became a challenge. Because after solving the problem, they tried to make simple model that express the problem. Problem 5 was very interesting. Amir tried to start from smaller size of problem. for 3x3, he found all of the possible states, but he could not prove that there is no more! Instead of problem 6. We tried one of Edward De Buno's problem (we have four forks and three bottle. we want to locate these bottles with distance more than length of fork and try to form a surface by forks above the bottle and locate glass of water top of them. Parsa tried with 3 forks. But Amir tried with 4 forks. I thought that Parsa will came with solution. But Amir did!

We did problems 4 and 5 (young adult level) with my kid M. and a friend C. They did have a bit too much fun with the wording of Problem 4, for example: "Oh, I know, there are no men there, because all husbands can be female." Denise's wording about nobles and body-guards was popular. This problem was also very quick and easy. They spent more time drawing 1.5 married women per each single woman than actually solving it. C. initially mixed up the numbers two and three, and we had a nice chat of why it's better to solve problems with other people (it's easier to catch typos). ![alt text][1] With Problem 5, drawing helped to solve it, and its "native" advice (use smaller numbers) helped to check the solution. M. and C. started with the strategy of situating one peg (25 positions) and then the second peg (16 position). At first, C. thought that produces 25*16=400 combinations. I asked if they were all different, and both C. and M. quickly said you must divide the number by two to count unique combinations. With the third peg, they divided 200*9 possible combinations by three, by analogy, but weren't sure of that decision. They asked how to check, and I reminded of the advice to use smaller numbers. So they tried smaller squares (3x3) and the same strategy worked. They still weren't quite sure, and (for the first time in our problem-solving together) checked the answer on the problem sheet. Oh, and we had a funny moment when M. was reading the problem: "No two pegs are in the same row or column... What about the third one? (pause) Oh yeah! I feel silly. For a moment, it was a genuine question!" ![alt text][2] [1]: /storage/temp/134-problem4.jpg [2]: /storage/temp/135-problem5.jpg

**Problems 4-6, comment about "adapting by time,"** in response to the instruction "Just write something about the pace, because it will help the study of adaptations." In week 1, I did each problem in a separate math session on a different day. In week 2, I did all 3 problems in one session. In week 1, the math went deeper. In week 2, we didn't go as deep, but more people's interests were served since each child had something she really loved. I will write up our week 2 experience in the next few days. The pace of this class is a bit different from what I expected. From an initial email or web post, I had the impression that the time commitment was about 2 hours per week. I should have realized that for me, 2 hours means 4-6. (Have you heard that general advice in the world of home repair to set your timeline and budget, then multiply them by three?) Reading the assignments and essays and doing the problems in them, then planning the new lesson, writing about the plan, doing it with the kids, and then writing the follow-up really does add up to a lot of time for me. Then again, I may be doing more than expected because I am hoping to generate new ideas for my future math circles, as well as learn more math.

![alt text][1] I had a group of 5 girls, two age 9 (J and D), two age 11 (K and Y), and one 13 (R). D, Y, and K have not been in a math circle before, but R and J have. Since R is so much older, I asked her to contribute questions but not conjectures or solution methods unless I directed. **Problem 4** *“If your mom says you have to eat your trail mix in a nut-to-candy ratio of 5:2, and the candies are all M&Ms, and you want to eat one of each color of candy, how many nuts do you have to eat?*” After K immediately named the 6 colors of M&Ms, the kids came up with 4 approaches to solving the problem: arithmetic, using actual M&Ms, using physical objects to represent M&Ms, and guesstimating. Since the object of this lesson was to introduce the strategy of “the power of drawing a picture,” I prompted them a bit for the drawing approach. Before drawing, the group had 2 conjectures as to the answer: 30 (the quick reply) and 15 (after more thought). Then J helped me draw it, with input from the rest of the group. The younger girls were uninvolved in forming the conjectures. D came into the class having no idea what a ratio is, but understood once we began drawing. J had experienced the trail-mix ratio scenario in our car numerous times, but wasn’t sure how to project a ratio onto bigger numbers. I asked these 2 younger girls what to draw first, and D instructed J to draw 6 M&Ms. Then J suggested drawing 5 nuts connected to each M&M. We did, and counted 30 nuts. Everyone realized that this wasn’t the answer, since we had just drawn a 5:1 ratio. “The problem would be so much easier if it did specify a 5:1 ratio,” I agreed. “Could we use this wishful thinking to move toward our answer. R disagreed that using the 5:1 ration constituted “wishful thinking.” To her and K, it was just common sense. Someone suggested cutting the nut number in half since the candy part of the ratio was actually doubled. “How could you check if your reasoning is correct?” I asked, and the group suggested drawing. So we did, with one leading question from me. Then the answer was obvious to everyone. I reiterated the strategy of drawing a picture, and someone asked “Does that always work?” “I don’t know, what do you think?” I replied. “It works most of the time,” said N. I wondered aloud whether we could think of any problems where drawing would not work. R suggested maybe some algebra problems, but I reminded her of the usefulness of the Babylonian box. I asked the group if they wanted to see another M&M ratio problem to test the usefulness of the box strategy, and they all said yes. *“New rule: for every candy you eat, you must eat a raisin. If you eat a raisin, you do not have to eat a nut, but you can if you want. At the end, what would you have eaten more of?”* Almost immediately, J said, “it depends,” but everyone else was silent. We ended up drawing a chart full of pictures with different scenarios. The students solved the problem with the pictures, and agreed that the picture was helpful, but in this case, not necessary. In the prior problem, drawing the picture had been necessary for the younger kids. In both problems, the kids came up with great assumptions, including (1) you like M&Ms or are at least willing to eat them, (2) you have enough food to use the given ratios, and (3, giggle, giggle) you obey the rule and don’t sneak extra M&Ms. ![alt text][2] [1]: /storage/temp/119-img_0892+(640x480).jpg [2]: /storage/temp/117-img_0890+(480x640).jpg

My teen group started by discussing area (leftover from the inscribed-circles problem last week) and headed off on a few rabbit trails, then came back and attempted to develop a justification for the formula for area of a triangle. We never got to the puzzles from this week's list, and since I'd like to keep following area until I'm sure it makes sense to them, it may be awhile before that group gets back to these puzzles. We won't be meeting again until August due to schedule conflicts. How long are you willing to wait for our report?

Another fun day. For Problem #4, recall that we changed the twin-pan metaphor of algebraic equivalence into a number-line model. Here's a picture of Neomi's (9 year old) solution to 3x+2= 4x-1. First, here is the story: "The giant has stolen the elves' treasure. He escaped their land and voyaged to a desert island. After mooring, he set off walking along a path. You are an elf positioned on this island, and you are spying on the giant to find out what he does with the treasure. Starting from the port and walking along the only path there, the giant walked 3 giant steps and then another 2 meters. He buried some of the treasure, covered it up real well, and then went back to the ship, covering up his tracks. On the second day, the giant wanted to bury more treasure in exactly the same place, but he was not quite sure where exactly that place was... Silly giant! So, setting off along the same path, he walked 4 steps and then, feeling he'd gone too far, he walked back one meter. Yes! He found the treasure from the day before. So, now he buried the rest of treasure in exactly the same spot. The giant then covered up the treasure as well as all his tracks, so that nobody will know where the treasure is. He returned to the ship and sailed off. Your job is to tell your fellow elves exactly where the treasure is. You need to tell them how many meters they need to walk from the docks to the treasure place." ![alt text][1] Notice in her drawing, on the right, half-way up, it says "1GS = 3m" (One giant step equals three meters). Here's another solution to the same problem: ![alt text][2] Again, you can see "Day 1" on the right and "Day 2" on the left. The white clay indicates that the meter is going backward, and the little blue lego piece is the treasure. For Problem #5, we used pencils but then also socks, which was fun. I think today we struck a better tone. Kids were more relaxed, and we distributed the work over groups. We also had two new parents, which seems to be a good trend. [1]: /storage/temp/115-mathcircle-2013-07-20-giantsteps2.jpg [2]: /storage/temp/116-mathcircle-2013-07-20-giantsteps4.jpg

**Problem 4** I changed the problem to be about gold coins and money bags instead of apples. The I created some problem sheets with sequins and paper cutouts (although everyone thought the bags were fish...). I also provided the kids with blank scales to copy / create problems. The manipulatives were popular and kids didn't get too distracted playing with them. I did have to have a good story as to why we couldn't just open the (imaginary) bags and look inside! The kids had a look at the problems but were vastly more interested in creating their own. Littler ones (5/6 ish) tended to just pile up bags and sequins, or created problems with infinitely many solutions (eg. x +2 = x + 2 where x are bags and constants are sequins). We talked a bit about how we couldn't solve them. Some children definitely felt that the bags did have fixed amounts inside, viz. the amounts they had decided would be in there. They also wanted to use bags of different colours together (I had stated that bags of the same colour had the same number of coins in them). This also meant their equations were not solveable! Older ones made some good problems, including one simultaneous equation! Although it was only partially solveable: x+y=4, x+y+z=4, ie z=0. I had in fact created one simultaneous equation problem and the same child (aged around 10) solved it, using the subtraction method. I was interested to see a couple of the kids came up with problems including empty bags, which had not occurred to me. I would have like to have been a bit more organised about sharing and solving kids' problems but the number of kids made it difficult, especially once some started to get distracted. Overall I thought it went well, the kids were really beginning to think algebraically even if they weren't all quite there. The scales model worked well to emphasise the need for both sides to be equal. **Problem 5** I used craft sticks (painted gold!) instead of pencils. The kids found them easy to manipulate and were invested in trying to solve the problem. Several managed to find a solution for 6 sticks. They tended to get the bottom three in a v-shape but needed a bit of prompting to get the top three in a similar shape so that all 6 were touching (I suggested they just needed to copy the shape they had already made underneath). We had attempts at 7, including changing the orientation of sticks, but no success that I saw at least. Actually I don't know how to do 7! Again younger children sometimes just piled sticks up but mostly understood why that wouldn't work. I would have liked to record the different solutions but again it was a bit chaotic so I was not really able to photograph or diagram. With older kids this might have been easier. **Problem 6** I hadn't really expected to get to this so only had a printout of a few 'Knights and Knaves' puzzles. However I adapted them to suit a game my children play called Opposite Land where you have to say the opposite of what you mean. So I talked about 'Opposites' who say the opposite and 'Normals' who say the truth. This ended up being problematic as we discovered the complexities of what opposite means! What is the 'opposite' of, for example, "Both of us are Opposites" (the first puzzle)? It does not appear to be the same as the negation of it, which could in fact be three things (both normal, A normal and B opposite, B normal and A opposite). So this got a bit in the way of solving the problem. Also the kids found it hard to manage a discussion without shouting over each other, storming off, etc etc... I think they were a bit tired by then to be honest. I feel that the group was quite large and this has presented extra problems than the mathematical ones. If anyone has any tips how to manage this it would be great! Ideally we would break up into smaller groups I think, and I'd love to be able to do the more complicated problems for interested kids. Overall, I am feeling like the sessions are fun and interesting, and feedback from parents and kids seemed positive. I am having trouble getting my own children to get as involved as I'd like though! They are definitely more distracted than when we work on problems at home.