Comments and answers for "ASSIGNMENT 2: Share your stories about problem groups 1, 2 and 3"
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The latest comments and answers for the question "ASSIGNMENT 2: Share your stories about problem groups 1, 2 and 3"Comment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/706/view.html
Ali, that excited comment you have in bold, about a line as a triangle with one 180-degree angle... What a good start of an ongoing activity! It's for the part of the book about the Math Circle infrastructure, such as warm-ups.
You can have a standing board, a poster paper, or a picture frame as an ongoing collection of funny, wise things kids say or draw. Just write down that saying about the line and the triangle on a card, and put it up on the board with other such sayings. Kids love their collections. They spend good time reading past ideas, too!Sun, 04 Aug 2013 06:35:15 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/705/view.html
Marianna, I've been thinking about your comment, "I guess they liked it because they knew what to do immediately." For this problem, I gather you meant that they knew particular ways of solving it. But I want to take this idea in a more general, even generic directions.
The general problem-solving techniques such as "draw a picture" is also SOMETHING TO DO - and it can be done for ANY problem! In the book, we need to explain how motivating and empowering it feels to always have something you can do. And that it's one of the reasons the techniques work.Sun, 04 Aug 2013 06:24:52 GMTMaria DroujkovaComment by ccross on ccross's answer
https://naturalmath.com/community/comments/684/view.html
I don't know--no one commented particularly on the writing aspect. I suppose if they didn't like it they wouldn't have done it, since this crowd is not an incredibly cooperative one.Thu, 01 Aug 2013 11:07:58 GMTccrossComment by ccross on ccross's answer
https://naturalmath.com/community/comments/683/view.html
I've been using different groups of students with all the activities, so each time it is a different dynamic. I don't have a class or group that I'm doing this with--I just do it with my son and whatever of his friends we have at the time.Thu, 01 Aug 2013 11:06:58 GMTccrossAnswer by Rodi.Steinig
https://naturalmath.com/community/answers/666/view.html
**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.Tue, 30 Jul 2013 16:39:11 GMTRodi.SteinigComment by Rodi.Steinig on Rodi.Steinig's answer
https://naturalmath.com/community/comments/663/view.html
To internalize such messages, I think it takes a lot of modelling and repetition. We as adults need to verbalize our "self-talk" in front of students. They need to know that we have emotional reactions, etc., see it repeatedly, and be guided through strategies again and again.
I think that students who are confident in math could be engaged by visible discouragement (see it as a challenge), but the typical student lacks the confidence, desire, or strategies to rise to the occasion. If we coach them over time, however, on how to rise to the occasion, they may.Tue, 30 Jul 2013 16:12:51 GMTRodi.SteinigAnswer by Rodi.Steinig
https://naturalmath.com/community/answers/662/view.html
**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.Tue, 30 Jul 2013 16:08:39 GMTRodi.SteinigComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/660/view.html
Nikki, the power of words is amazing! "Play math" vs. "do math" makes all the difference. Or the words your niece used for direction, "up" and "down." We need to help kids use their own words and their own voice!Tue, 30 Jul 2013 08:06:32 GMTMaria DroujkovaAnswer by Maria Droujkova
https://naturalmath.com/community/answers/648/view.html
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.jpgWed, 24 Jul 2013 18:55:43 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/639/view.html
How can we remember to come back to a problem that was a success? I like to put up pictures on my bulletin board in the kitchen. Say, if you have exponential kittens up, you can later pose other puzzles and problems about them. What generation are 8 kittens? Can there be 25? What is the last digit of the 100th generation?
Sometimes we have stacks of sketches and computer printouts growing and branching on the bulletin board, like weird math bushes.
I also keep activity boxes or baskets for experiments like Problem 3.Wed, 24 Jul 2013 07:39:49 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/638/view.html
Kids who live pretend play may love math stories! Like the one about the king and the rice on the chessboard, or the one about Hotel Infinity. Living Math site lists "math readers": http://www.livingmath.net/ReaderLists/tabid/268/language/en-US/Default.aspx
You can also play your own story from any problem!
What about math? Roleplaying kids do care. But they establish the world, the characters, and the story first. These kids' math is APPLIED: making animals comfortable, or saving them from predators. Maybe your girl will grow up to save the oceans, or to eradicate world hunger?Wed, 24 Jul 2013 07:32:06 GMTMaria DroujkovaAnswer by Silina
https://naturalmath.com/community/answers/636/view.html
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).jpgWed, 24 Jul 2013 03:34:14 GMTSilinaAnswer by Silina
https://naturalmath.com/community/answers/635/view.html
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).jpgWed, 24 Jul 2013 03:27:36 GMTSilinaAnswer by Silina
https://naturalmath.com/community/answers/634/view.html
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).jpgWed, 24 Jul 2013 02:48:48 GMTSilinaAnswer by Lobr23
https://naturalmath.com/community/answers/625/view.html
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.Mon, 22 Jul 2013 18:57:13 GMTLobr23Answer by Denise Gaskins
https://naturalmath.com/community/answers/618/view.html
![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.htmlMon, 22 Jul 2013 15:09:10 GMTDenise GaskinsComment by MobySnoodles on MobySnoodles's answer
https://naturalmath.com/community/comments/616/view.html
It's interesting how computer games (Dragon Box, Free Flow) provide support for the kids. And comfort, and confidence!
Problem 6 is an "insight" problem. These problems have unique "tricks" to them, but you can discover the tricks by observation. If your son liked that one, you can search for more "insight problems" for him.Mon, 22 Jul 2013 09:19:36 GMTMobySnoodlesAnswer by cakeroberts
https://naturalmath.com/community/answers/609/view.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.zipSat, 20 Jul 2013 14:30:20 GMTcakerobertsComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/605/view.html
Rodi, you said James' messages make enthusiasm prevail over discouragement. What do you think would it take to internalize such messages? Or is visible discouragement a way to engage you, emotionally and intellectually?
Love all the conjectures!Fri, 19 Jul 2013 22:08:29 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/603/view.html
Carol, did the process of writing help them? Did it feel good to them?Fri, 19 Jul 2013 19:59:01 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/602/view.html
I wonder what's the boundary between the joy and the frustration here. Does it have to do with words/symbols (frustration) vs. objects (joy)? Or is it about algebra vs. spatial reasoning? Or is it about the difficulty level?Fri, 19 Jul 2013 19:53:47 GMTMaria DroujkovaAnswer by Maria Droujkova
https://naturalmath.com/community/answers/600/view.html
**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.jpgFri, 19 Jul 2013 10:30:25 GMTMaria DroujkovaAnswer by Maria Droujkova
https://naturalmath.com/community/answers/599/view.html
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.jpgFri, 19 Jul 2013 08:30:07 GMTMaria DroujkovaComment by andyklee on andyklee's answer
https://naturalmath.com/community/comments/597/view.html
We built the 'jammed wheel' example with Lego gears. But it wasn't possible to turn one gear and see the others turn. So our conclusion was based on visualization and thinking rather than physical reality.Thu, 18 Jul 2013 13:27:38 GMTandykleeComment by yelenam on yelenam's answer
https://naturalmath.com/community/comments/590/view.html
Miranda, it is something I too struggle with - maintaining the balance between letting the children explore and keeping them focused. You was thinking about your comments on Problem #3 - would it work out if, instead of splitting the kids into groups, some kids could be given a task of documenting the solutions (photos or drawings). Also, maybe once the kids played the problem once, they could break into teams and create challenges for each other to solve? I'm very interested to see what solution you have for the next meeting.Wed, 17 Jul 2013 22:59:38 GMTyelenamComment by yelenam on yelenam's answer
https://naturalmath.com/community/comments/589/view.html
Carol, will you be working with the same diverse group on other problems in the course? What would you do to sustain group members interest? Denise suggested asking questions. I've tried that (in a different course) and ended up with a couple of very vocal group members who would pretty much out-shout others. An even bigger problem was that some kids needed more time to come up with an answer (or with how to explain the answer) and taking turns didn't work out very well. So I guess my question is how to balance this issue with keeping it a social activity?Wed, 17 Jul 2013 22:49:10 GMTyelenamComment by yelenam on yelenam's answer
https://naturalmath.com/community/comments/588/view.html
the kids I worked with (K-1) knew left and right, so that's the terms we used. As with Carol's situation, the kids figured out the alternating pattern quickly, but did not make the odd/even connection. I felt there was no way to guide them without pretty much giving out the answer. Interestingly, when we played #2 problem with one of the kids a few days later, he did figure out the relationship between "stinky feet" and odd/even number of action figures. Now I'd love to return to the gears activity and see if he will make the connection!Wed, 17 Jul 2013 22:42:25 GMTyelenamComment by yelenam on yelenam's answer
https://naturalmath.com/community/comments/587/view.html
Lizza, the pictures are so colorful! Reminds me of the Free Flow app and gumballs :) I found it interesting that, as you described it "the idea of non-crossing was totally not clear" to your daughter. When I played this game with my son, this was also our stumbling block. What are some real-life situations to which we can link the "no-crossing" rule, preferably something that kids have witnessed or participated in? Dog walking, perhaps?Wed, 17 Jul 2013 22:36:43 GMTyelenamComment by yelenam on yelenam's answer
https://naturalmath.com/community/comments/586/view.html
Andy, you mention that you tried the gears problem on paper, with actual gears and also online. Could you tell a bit more about what you and the kids did in each medium? Did you build the "jammed wheel" model? I was thinking about doing it as a "next step" with my child and thought that maybe now that he'd discovered the pattern, it'd be more interesting to do it on paper first and then model it with Legos.Wed, 17 Jul 2013 22:27:01 GMTyelenamAnswer by Denise Gaskins
https://naturalmath.com/community/answers/581/view.html
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+k-1.pdfWed, 17 Jul 2013 14:12:33 GMTDenise GaskinsComment by dendari on dendari's answer
https://naturalmath.com/community/comments/580/view.html
The movie Hugo has a lot of cool gears in it.
My then 5 year-old spent a lot of time on Flow Free last year also.Tue, 16 Jul 2013 08:57:31 GMTdendariAnswer by nikkilinn
https://naturalmath.com/community/answers/579/view.html
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.Tue, 16 Jul 2013 08:28:58 GMTnikkilinnComment by MobySnoodles on MobySnoodles's answer
https://naturalmath.com/community/comments/578/view.html
Rodi, can you draw your vision? I can't see what you see!Tue, 16 Jul 2013 07:13:27 GMTMobySnoodlesComment by RosieL52 on RosieL52's answer
https://naturalmath.com/community/comments/576/view.html
What if you drew the arrows in different colors? One color for CW and one color for CCW. This doesn't alleviate the problem of juggling two writing utensils (unless you have those double-sided crayons you sometimes get at restaurants - yellow on one side, red on the other side).
This would keep the direction concrete but also make the pattern more visual.Mon, 15 Jul 2013 22:04:22 GMTRosieL52Answer by abrador
https://naturalmath.com/community/answers/573/view.html
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.jpgMon, 15 Jul 2013 16:55:50 GMTabradorAnswer by nikkilineham
https://naturalmath.com/community/answers/572/view.html
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.Mon, 15 Jul 2013 12:47:03 GMTnikkilinehamAnswer by dendari
https://naturalmath.com/community/answers/570/view.html
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.Mon, 15 Jul 2013 12:03:12 GMTdendariAnswer by RosieL52
https://naturalmath.com/community/answers/568/view.html
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."Mon, 15 Jul 2013 10:08:58 GMTRosieL52Answer by Rodi.Steinig
https://naturalmath.com/community/answers/565/view.html
![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.
[1]: /storage/temp/102-img_0861+(800x600).jpgSun, 14 Jul 2013 22:44:01 GMTRodi.SteinigAnswer by Rodi.Steinig
https://naturalmath.com/community/answers/564/view.html
![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.
[1]: /storage/temp/100-img_0855+(300x225).jpg
[2]: /storage/temp/101-img_0856+(300x225).jpgSun, 14 Jul 2013 22:27:17 GMTRodi.SteinigAnswer by Viktor Freiman
https://naturalmath.com/community/answers/561/view.html
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/).Sun, 14 Jul 2013 20:23:59 GMTViktor FreimanAnswer by Lizza-veta
https://naturalmath.com/community/answers/559/view.html
We were working today with Problem #1 with a ladder, a rope and a bucket. Throwing the rope over an odd crossbar we have the bucket moves up while the rope is pull down. And when the rope goes over an even crossbar than the bucket moves up pulling the rope up too. Children didn't ask anything, just enjoed the pulling the rope. I tried to pay their attention to the fact of different ways of the rope to rise up the bucket but they were busy enough just pulling the rope up, the higher is the better.
![![alt text][1]][1]
[1]: /storage/temp/99-14072013-sg208523-2.jpgSun, 14 Jul 2013 17:25:14 GMTLizza-vetaAnswer by ali_qasimpouri
https://naturalmath.com/community/answers/558/view.html
**(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!Sun, 14 Jul 2013 16:04:00 GMTali_qasimpouriAnswer by Marianna
https://naturalmath.com/community/answers/557/view.html
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.Sun, 14 Jul 2013 15:35:06 GMTMariannaComment by ccross on ccross's answer
https://naturalmath.com/community/comments/554/view.html
I had the same experience with introducing physical gears. All of my students already had the concept of the gears moving in different directions, so when I gave them physical gears to work with, they got more wrapped up in either playing with them (not necessarily towards the direction issue--like rolling them across the table) or getting frustrated if they didn't work perfectly that they didn't want to do the problems. They were much more focused on working on the problems/looking at the directions of the gears when I removed the physical gears from the workspace.Sat, 13 Jul 2013 14:10:56 GMTccrossComment by Denise Gaskins on Denise Gaskins's answer
https://naturalmath.com/community/comments/553/view.html
We ended up using arrows after all, because the puzzles were hard enough without adding an extra layer of abstraction. Arrows connected more closely to the actual movement than a color-code, and the kids didn't have to juggle two markers. Our kids made their arrows with nice, big swoops of motion that made the direction of rotation clear, but they did have trouble with fine-motor-muscle control when drawing the arrow tips, which caused some confusion on the longer chains of gears.Sat, 13 Jul 2013 13:57:00 GMTDenise GaskinsAnswer by Denise Gaskins
https://naturalmath.com/community/answers/552/view.html
I'm working on a report about the K-1st grade group, but I need to wait for permission from one of the parents for photos. Meanwhile, here are some highlights:
We had five children playing with the math problems and five adults (counting my teen daughter, who helped when she wasn’t taking pictures), so after I introduced a puzzle, we were usually able to work one-on-one with the kids: to respond directly to what each child was doing, reflect their thinking, and offer little suggestions to nudge them along. We adults probably talked more than we should have—listening without offering advice has always been hard for me.
To my surprise, the children found the gear pictures easier to work with than the physical gear set, perhaps because the lack of motion let them focus on the teeth where two gears met: this gear is turning this way, so which direction will it push the next gear? All five children worked intently for several minutes.
The three older children successfully solved even the challenge puzzle (the long, branching line of gears)—well, sort of: I think I saw final answers that went both directions, so one of them must have been wrong, but the kids clearly understood how each gear turned the next one, even if they lost track of which way the arrows were pointing as they went along. The younger siblings had the idea of spinning, but drawing arrows to keep track of direction was beyond them: one girl drew circles inside most of the gears, though she skipped some of the smaller ones, and a boy made several long chains of circles on his paper.
I adapted the exponents problem as an addition puzzle, to put in order the sums 10+8, 5+12, and 2+24. I used a three-bears coloring picture (Papa Bear likes big numbers, etc.) for sorting out the answers. The kids immediately said that 24 was a Papa Bear number and 2 belonged to Baby Bear. Two boys insisted that 1 should be on the page somewhere because it was wrong to let 2 be the smallest number, and then one of them pointed out that zero would be even smaller.
None of the children recognized the sums as representing individual things to be sorted. In their opinion, “10+8” wasn’t a number.
The kids went to the playground while the parents helped pick stuff up. One girl told my daughter, who was acting as playground guard: “I’m done with math. It was really fun!”Sat, 13 Jul 2013 13:51:46 GMTDenise GaskinsAnswer by cakeroberts
https://naturalmath.com/community/answers/551/view.html
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.Sat, 13 Jul 2013 00:41:36 GMTcakerobertsAnswer by cakeroberts
https://naturalmath.com/community/answers/550/view.html
Technique 1
Gears
• First my son drew arrows on gears to indicate direction
• As the problems got more complicated, we suggested he could substitute symbols for arrows. He chose to substitute 1 and 2.
• When the most complicated problem was solved, he re-substituted incorrectly and started to get really confused and frustrated.
Book pages
• My husband and son made a book and they numbered the pages and then started tearing out different pages to see the result.
• 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? My son loved thinking about these imaginary campers and manipulating the pillows and minifigs.Sat, 13 Jul 2013 00:35:42 GMTcakerobertsAnswer by Rodi.Steinig
https://naturalmath.com/community/answers/549/view.html
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).jpgThu, 11 Jul 2013 23:51:05 GMTRodi.SteinigAnswer by Rodi.Steinig
https://naturalmath.com/community/answers/548/view.html
QUESTION/COMMENT ABOUT THE GEAR PROBLEM:
When I read and did the gear question myself, I never thought about "changing direction." Most writing I have seen about this problem talks about the change of direction in terms of the gears rotating either clockwise or counterclockwise - total reversal of direction. When I did the problem, OTOH, I visualized, and even drew, a flow of motion that never reversed itself; it was basically a curve or wave that pushed the next gear along in that same direction (north/south/east/west, perhaps). In this model, nothing alternates. I'd like to find out from people doing this problem with kids how children conceptualize the movement, and would like to ask people to mention it in their write-ups if anything notable comes up about this.
Thanks!Thu, 11 Jul 2013 23:00:26 GMTRodi.SteinigAnswer by mirandamiranda
https://naturalmath.com/community/answers/540/view.html
I attempted the problems with a group of kids aged roughly 5-9, weighted to the younger end. There was a couple of younger ones in the mix but I think they were a bit shy and didn't manage to really get involved.
{I have not yet really worked on Problem 1 but hope to do so with my own kids}
**Problem 3** We did this first. The two older kids (8 and 9) worked on the diagram on a whiteboard and solved it pretty quickly. Other younger kids managed it afterward on the board too - one good strategy was to join one diagonal and then it clicked that you had to go *round* the other 'homes' to match the pairs.
I also set up chairs with baby and adult stuffed animals to join up. I attached the baby animals with yarn to their chairs so the yarn would trace the paths. I was a little worried they'd get very tangled but this wasn't too much of a problem. The younger kids really enjoyed moving the animals, they quickly caught on to the idea of changing the arrangement of the parent animals. In fact they were more interested in finding different arrangements than in solving the problem! We did manage it eventually although I think the fact that they couldn't see the arrangement from above hampered understanding to some extent. There was also some disagreement about who would get to move each animal...
It definitely piqued the kids interest, but I had one group who got it very quickly and another who wanted to play for a while. This meant I lost some of my students early on as they went off to play outside (the timing of this first meeting is not ideal as it is right after another class for most of the kids).
**Problem 2** My eldest then wanted to move on the 'paper dolls' she had made. I explained the principle and the kids were very excited by the idea of getting into loops themselves. They had a bit of trouble arranging themselves but quickly caught onto the idea of one 'pillow' per 2 heads. One child very quickly said that there would be half the number of pillows as children for an even number of children. We began to test the theory but got rather distracted by tangential explorations into different body arrangements. They tried linking different parts of their bodies and making strange kinds of loops.
I think given some more time and focus we might have tried it again with the paper dolls and noticed some more patterns. In fact I would like to try this again with my own children. Unfortunately in our math circle the kids then got involved in playing musical chairs and that was kind of the end...
I think what I found hardest was keeping the attention of more kids than about 4 at a time. I think a couple of the older ones might have benefitted from more involved problems as well, which I had not prepared.
The strategy of getting the kids physically involved in acting out the problems worked great in terms of engaging them - they loved it. But it also gave them ample opportunity for distraction and play - they were not vastly invested in solving the problem, and needed frequent nudges from me to get very far in reaching an answer. I think it worked for open ended problems, but not so much for getting from a to b. A bit like trying to walk somewhere with young children in fact! It would be interesting to see how far they would have got without my prodding, although I suspect it would have ended up in piles of giggling children!
I also feel like a larger group need time to cohere. This was the first time we had done this, and the kids were already a bit stir-crazy from their day. After the novelty of play dynamics with a given group wears off I think (hope?) I might have more luck engaging them in specific problems and their possible solutions.
Next time I think I will be more clear about starting and explaining the problem; there was a fair bit of peripheral activity which i think was distracting. It might also be worth splitting the kids into groups to attack the problems. Better get planning!Thu, 11 Jul 2013 01:28:49 GMTmirandamirandaComment by Denise Gaskins on Denise Gaskins's answer
https://naturalmath.com/community/comments/539/view.html
"..So sustaining interest in figuring it out yourself after some peers have done it was an issue for me."
I expect to have this sort of problem with my teen group, which includes students with widely varying backgrounds in math. Not as wide as your range, but enough that I'm concerned about discouraging the younger/weaker students. I want to focus on asking questions and extending the problems more than on getting answers, which I hope will give everyone a chance to contribute.Wed, 10 Jul 2013 22:44:04 GMTDenise GaskinsComment by Denise Gaskins on Denise Gaskins's answer
https://naturalmath.com/community/comments/538/view.html
Thank you for the tip on using color! I had already decided to avoid the words "clockwise" and "counterclockwise" with my K-1 group, but I hadn't thought about how easily they can mix up drawing arrows. Color-coding the "same" and "different" rotations will definitely bring out the pattern.Wed, 10 Jul 2013 22:22:42 GMTDenise GaskinsAnswer by ccross
https://naturalmath.com/community/answers/535/view.html
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.
CarolWed, 10 Jul 2013 21:26:50 GMTccrossAnswer by ccross
https://naturalmath.com/community/answers/534/view.html
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.Wed, 10 Jul 2013 21:12:12 GMTccrossAnswer by ccross
https://naturalmath.com/community/answers/532/view.html
Not a lot to say about the second problem, which I just did with my son, except that it linked up to some number pattern charts we had done earlier this year as part of our exploration of vedic math. Oh, and the resource I mentioned earlier, http://freemars.org/jeff/2exp100/question.htm , was a great addition to the 2^100 problem.
CarolWed, 10 Jul 2013 20:49:19 GMTccrossAnswer by ccross
https://naturalmath.com/community/answers/531/view.html
I did the first gear problem with about 10 people, ranging from 5 to 65 in age. Everyone I worked with had a pre-existing understanding of how gears worked, so everyone did the problem the exact same way. I gave them a sheet with the problem image on it and asked what way the gears with the arrows would turn. First, everyone just turned their fingers in the direction of the gear. But then it got too complicated (where the gears branch out), and so then they took a pencil and started drawing arrows on each gear.
Although they understood what they were doing, all the children made at least one error about which way the gears would turn. The biggest mistake, especially among younger children, was that they didn't always draw the arrow in the same place, and that led them astray. So, for example, if the gear was going clockwise, they drew an arrow at the top pointing right. Then, for the counter-clockwise gear, they would draw an arrow at the top pointing left. So they went on, alternating directions in which the arrow was pointing. However, at some points, for a clockwise gear, they would draw the arrow pointing right, but at the bottom. But that would actually move the gear in a counter-clockwise direction. The younger ones got frustrated when I pointed out their errors, because they thought they were right because the arrows were always alternating. I then asked them if they could figure out an easier way to do this than mechanically drawing an arrow on each gear. All the adults and a couple of the teens eventually figured it out they could just count and know odd is clockwise and even numbers were counterclockwise (or however it is in the puzzle), but some of the teens and none of the younger ones couldn't figure it out without some pretty direct instruction.
THEREFORE, if I were to do this again, what I would do, especially with younger students, is have them choose a color for clockwise and a color for counterclockwise and have them color the gears instead of drawing arrows. Not only would this eliminate errors, but it would be easier for them to see the pattern. The frustration level of being "wrong" when they knew they were "right" because the arrow was pointing in the opposite direction from the one before prevented a couple of the youngest students from completing the exercise while we were together, although the moms took some sheets home to try again when they were in a better frame of mind.Wed, 10 Jul 2013 20:46:13 GMTccrossAnswer by Lizza-veta
https://naturalmath.com/community/answers/526/view.html
Well we've tried yesterday with Problem #3. I was the best student among us )) My babies had a fun with modeling the balls and sausages, and enjoyed the story about Mother Ball, Father Ball and Baby Ball, which are needed to find each other. The abstract mind of my 3y.o. daughter (Maya) is enough to let her agree that Mother and Father could look like coloured balls, but 2y.o. son (Vic) was sceptical about this )))
So Maya tried to connect balls each other but she didn't keep in mind balls of similar coloures working with the nearest balls. And idea of non-crossing way seems totally not clear for her. And Vic had a fun copied our movement and activity with modelling, drawing the way between the balls but only as he understood it. So he draws something here and sticks something - he doesn't mind if it is fun.
alt textWed, 10 Jul 2013 07:10:22 GMTLizza-vetaAnswer by andyklee
https://naturalmath.com/community/answers/522/view.html
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.Tue, 09 Jul 2013 07:39:00 GMTandyklee