ASSIGNMENT 6: Share your stories about problem groups 7-10

**Problem 7**: I had laminated and cut-up shapes that the kids could draw lines on and erase. There were slightly older kids at this class and they breezed through this rather. We had a bit of discussion as to whether the two hidden shapes had to be exact copies or whether reflections were ok - there were some we couldn't solve if the former! We tried cutting out and making our own but this proved harder than people anticipated. We also veered more to making symmetrical shapes only as these were easiest to produce and I did not have a chance to sit down and demonstrate the technique. If I were to do this activity again, I might introduce it by creating a puzzle first then producing the pre-made ones to figure out. One child worked for the entire session trying to figure out on shape. Unfortunately (in my opinion) his friend knew how to do it and in the end showed him - but I was impressed by his determination! I was not really sure how to address this beyond expressing this feeling, but I felt that an opportunity was missed somehow. **Problem 8**: I had another laminated 'cake' but the kids were more interested in using the slices of bread I also provided! They figured out two, three and six but no-one managed four! We attempted larger numbers but the manipulatives kind of fell apart - in more than one sense! So there were claims of 48 pieces but it was hard to check accuracy through the crumbs... Also a lot of bread got eaten. The kids loved cutting up the bread though. I think with a smaller group or more structured environment we could have got even further. **Problem 9**: I drew maps on big sheets of paper and used small toy dogs. The kids could see the 'land' or 'water' pretty quickly. They were not vastly interested in figuring out patterns or exploring this any further really, but they did enjoy lining dogs up on all the islands!

My kids (ages 7, 6, and 4) tried problems 7, 8, and 9. Problem 7: I had worked on a shape (cut two identical shapes, then drew an outline around a shape I created by joining the 2, and cut the outlined shape out). My 7 year old took the shape I created and cut it so that the pieces joined together. Then all 3 set off to create their own shpaes. It ended up being an exercise on symmetry, but they enjoyed creating their own symmetrical shapes so I saw it as a success! Problem 8: we drew islands and water using chalk on the driveway. The kids had fun naming their islands and figured out when their stuffed animal was in the water vs. on the island. They had a great time and even exclaimed, "This is much more fun than I thought it would be!" If I were to do it over, I would have drawn the islands in concentric circles, instead they were more like globs in different places, which may not have illustrated the point as effectively. Problem 9: I had the kids pick out 3 of their favorite dolls / stuffed animals. Then we went outside to have a "birthday party." I made a cake with chalk in the shape of a short L and said it was "L" for "Lori" (my name) since it was my party. I asked them how we would divide the cake for the 3 of us (my 6 year old was still working on her island). They drew lines to divide the shape into three equal pieces. Then I drew that one square on each of their "plates." I asked them to invite one of their friends, then divide up the piece of cake between the two of them. Then invited another friend. Then another. What was interesting was with the 2nd friend (3 attendees total), my 7 year old ended up dividing it into 6 equal pieces. When I asked her why, she said she was saving 3 of the pieces "with icing, for later." I guess I hadn't thought about that possibility.

Abbie couldn't figure out how to split the cakes, but she had fun with it anyway. Any time you can do math with a bright pink marker, that's good, right? ![Pink power!][1] [1]: /storage/temp/156-too+cute.jpg

More Math in the Park pictures... ![Cutting up cakes][1] ![Does it make 10?][2] [1]: /storage/temp/154-cutting+up+cakes.jpg [2]: /storage/temp/155-checking+the+sum.jpg

![Land or water?][1] **K-1st Grade Group** Kiara's mom mentioned that as they were driving to the park, Kia said, "I like Fridays. I like math!" **[1]** The kids enjoyed playing with the pond mazes (the dinosaurs on the outside need to decide if their friend is in the land or the water). They would take a wild guess, and then move step by step into the maze: "Splash!" That's water. "Thump!" That's land. Even more, they liked drawing the mazes. We started by drawing a big circlish shape for the outside boundary, and then erasing a section and turning the inside into a squiggle. The girls, especially, didn't care about solving each other's mazes, but just wanted to get to their own turn of drawing. And they discovered that if the squiggle criss-crossed itself, they still had a workable "land or water" puzzle, but the land was not all connected---now they had islands in the lake. **[2]** The cake-dividing puzzle went much as it was described in the "Technique 9" section, except that Micah tried to cut up the "each has three friends" cake pieces with two diagonals. He recognized that he had more than 3 pieces, so he said, "They each get two." When his mom pointed out that didn't quite add up, he was able to erase and fix it by using parallel lines. The boys didn't get to the "hundred friends" question, but the girls did. Kia said, "I'm going to need a bigger cake!" She drew a big rectangle on the white board and started dividing. Her lines criss-crossed, and some of them were a bit wiggly, but she got pretty close to 100 pieces. (Within experimental error?) ![Tens Memory game][2] **[3]** The Tens Memory game was a big hit. The idea of figuring out "how many more do I need" was new to both of the kids, but they caught on after a few turns. We let the kids take a free turn whenever they got a match (until we were down to fewer than ten cards remaining), but not the parents. I think Micah won, though we didn't bother to count our cards at the end. By that time, the kids were more interested in running over to the playground while the parents cleaned up. [1]: /storage/temp/152-land+or+water.jpg [2]: /storage/temp/153-matching+10.jpg

9) I cut L-shaped piece of paper, it was a piece of cheese for a dog, a cat and a bear. It turned out that Olga didn’t understand what divide in equal parts is. Her brother promptly explained it to her dividing small rectangular in two squares between two toys. Then it was fairly easy for her to see (or at least understand) how to divide the piece in three equal parts by sitting there her toys. Then the pieces turned out to be too big and the toys invited their friends over, then those friend invited more friends, totally she divided the cheese between 12 toys. ![alt text][1] 10) We went to the mall with an escalator, drove it 5 or 6 times. First I carried my daughter with a relaxed pace counting every time I stepped on a step. We counted steps we went, there were 12 of them. I asked her if we would count more or less steps, if we went faster. To my surprise she answered correctly, by insisting that there would be more steps. We tried and counted 20 steps. She guessed that there would be less steps if we went slower, which we checked too. When I asked her how many steps we would step to if we would freeze on the escalator, she giggled saying that there would be none. [1]: /storage/temp/149-9.jpg

For 22 months old Olga 7) We made our own puzzle. I cut two dogs out of paper and traced them touching each other. That was our mat for the dogs. Olga figured out how to put dogs to the mat, so they could sleep comfortably. Comfort of the living beings (or just the toys representing them) is the main concern of my toddler.![alt text][1] 8) We drew land-water-land closed curves on the pavement. In the middle we had water; we marked it blue and put a frog there. I marked Olga’s shoes with blue and black chalk. She started walking from the middle with her blue shoe on water, by checking what color her shoe was she could tell if she stepped on water or land. Then she could safely place animals on land and the gees and ducks on water.![alt text][2] [1]: /storage/temp/147-7.jpg [2]: /storage/temp/148-8.jpg

In terms of your overall question, the kids didn't have too many difficulties with the problems I gave them, so it didn't take a lot of perseverance or rethinking for them to do them. The MAA problems would have required them, but they intimidated me, so I didn't want to give those ones to the students (although, obviously, Friend could probably have figured them all out and explained them to me!) So I think this set of problems were the least interesting ones for the teenagers. And I didn't really see them using the earlier problem-solving techniques, like drawing a picture or solving a smaller version of the problem. But it was awesome watching Friend solve the first-100-odd-numbers problem! The whole thing has been interesting and valuable, though. I appreciate the effort that went into all of this. I don't mean to sound too whiny about supporting less-confident math adults. I just think, if you want this book to serve a broader audience than people who are already pretty proficient in doing and teaching math, that you need to add some material to support the rest of us. So all my comments requesting more information are just intended to make the product more useful for more math-intimidated folks like me. So thanks for having me!

Nikolai: Problem 7. I just started listing the square numbers out and found a pattern without an 8. The number 8 is not a square number, and you cannot make a number less than 10 out of tens. So the only part we are interested in will be the last digit. If we look at the square of some number, we can write down (10a+b)^2=100a^2 +20ab+b^2. We are interested only in the final digit, in this case b^2. b can only be equal [0,9], and none of them gives 8 as its square. We are interested only in the last digit because the tens and hundreds always end in zeros, thus not effecting the last digit which cannot be 8. Problem 8. I took 65, and broke it up into prime factors of 13 and 5. Since 5 has only one digit, we cannot reverse it, which means that we can only reverse 13. So the answer is 31*5=155 Problem 9. I started by looking at the series 1+3+5+7+….+n, finding the last number of the list which is (2n-1) here n is the number of the elements in the series of all odd numbers, in our case n=100. The sum of the first and the last elements is equal 1+(2n-1)=2n. If n is an even number we will have n/2 pairs each equals 2n. So if we want to get the answer the formula will be 2n*(n/2)=n*n=n^2. For n=100 the answer is 10000. Problem 10. I made a list starting from 10-49 and matching 10-49 with 90-51. 51-90 are the banned numbers because 10 and 90, 11+89 and so on equal 100. Then we have 91-99, they work, but all together we have just 40+9=49 numbers. Two numbers are missing, so the answer is no. The most surprising and unexpected part was how much harder it was to write the problems down comparing to solving them. It took like 80% of all time and effort to write them down.

But for Problem 9--OH MY GOODNESS!!!! This one I gave to my son and his math-brilliant friend that I discussed in my write up about Problem 5. I gave them the problem on paper of "What is the sum of the first 100 odd numbers?" In the course, the authors suggest it is related to squares. But as soon as he read the problem, Friend started talking about Pascal being a child prodigy and figuring out an easy way to add up the first 100 numbers. He figured out that 1 plus 100 would be 101, as would be 2 plus 99, 3 plus 98, etc. So the answer was 50 times 101, or 5050. So then Friend started to figure out how to adapt that to odd numbers. His monologue (since neither my son nor I had anything to contribute to this whole thing) was something along the lines of: "OK, so we know the first odd number is 1, but we have to figure out the 100th odd number. So the formula for odd numbers is 2n - 1, like 2*1-1 is 1, 2*2-1 is 3, etc. 2*100-1 is 199, so that is the 100th odd number. 199 plus 1 is 200, multiple that by 50, and the answer is 10,000." It was just like that. It was amazing! Friend figured the whole thing out in like 10 seconds. He had to go over it all again more slowly for my son, but even my son understood his explanation pretty quickly. So besides being blown away, again, by what a math phenom Friend is, I just thought his approach was so much better than the "official" we had in the text. I realized that while I was going to suggest doing it like it was in the text, so that they would notice the square thing, I didn't have a clue WHY that worked. Which I felt like I should have if I were presenting this to students, or at least to students who aren't as math brilliant as Friend. But that didn't occur to me until Friend demonstrated his approach that did make sense and that I could explain (once he had explained it to me). So that made me feel bad about myself, because I realized I was just passing on my childhood experience with math, where I just did things the way the teachers/books told me to do them without understanding why they worked. That is why I would prefer a place to look for answers and how they are worked out completely and why that approach works, so that I really know that I understand the solution, rather than just the hints that are given in the text as it is. But I would add Friend's approach to answering the question as well. It was beautiful. It was like watching a Maestro at work!

For Problem 8, again, I put the problems (including the MAA one) on paper and did it with two teenagers. They worked them through pretty systematically. It seemed pretty simple and straight-forward for them, so not much to report. After they got the "correct" answers, they did play around a little bit about if there was another solution involving fractions or negative numbers or something, but that petered out fairly quickly.

For Problem 7, I printed out the shapes on paper and gave them to three teenagers. I brought scissors so they could cut them out, but they didn't need to; they could just draw the separation line. I gave them the square number digit 8 problem on paper as well. We had to look up definition of square number and integer (so I would add those terms to the vocabulary list I would add for those of us with less math proficiency). Everyone worked out the problem as suggested. Not a lot to report about this problem-solving session, other than after it was done, we looked at the pattern of how 1 and 9, 2 and 8, 3 and 7, and 4 and 6 had the same squared digits, which I had never noticed before. We mused about why that was, but I had no clue, so I don't know if there is an explanation for that or not.

**PROBLEM 7 (Perseverance)** “Is it possible to find a weaving pattern other than ‘over 1 under 1 (O1U1)’ to make a design of non-adjacent squares?” I asked the students (J and D, age 9, and K and N, age 11). The problem was written on the whiteboard; the O1U1 pattern was drawn on the board, and demonstrated with paper strips. We acknowledged our emotional reactions. The kids asked some clarifying questions (Adjacent? Weaving pattern? Square?) and then sat quietly with no idea what to do. I asked them to think back on problem-solving strategies they already knew. “Let’s dance,” said J. The others agreed. “Why dance?” J said that it would be good to take a break from the problem, then to refocus. Y said we could square-dance to get our bodies involved in problem solving, since the problem is about squares. Two excellent reasons, I concurred, but we really didn’t have time to dance since we had 4 problems and 90 minutes. Silence again. I read to them Tanton’s Essay 9 anecdote about the mathematicians working (one typing furiously vs. the other staring at the ceiling), and dramatized it with puppets. The kids debated which one was working harder, then agreed that the staring mathematician was probably doing more thinking than the typing. With this in mind, they decided to take a break from the weaving question and come back to it. I introduced the word “perseverance,” explained it, and agreed that this was a good spot to perservere by taking a break and then returning to the problem in just a bit. We set aside Problem 7 to work on Problem 8, and then returned to the weaving problem after snack break. When we returned, K suggested that the strategy “Do Something” could mean to pick up some paper strips, weave them, and see what happens. So the kids did, and rich math unfolded. They saw that they could get bigger non-adjacent squares by increasing the scale of the U1O1 pattern to U2O2 and U3O3. K posited that with bigger paper, we could probably demonstrate that U404 would work. (Had we more time, I would have asked them to prove that without using paper.) Then the kids tried making patterns with different numbers of “over (O)” and “under (U)” moves. D’s work made it obvious that more than one naming/counting convention for the pattern exists: she was counting vertically while we had been counting horizontally. She then switched to the horizontal counting convention for consistency’s sake. K decided to try a basket-weaving pattern she knew. With some careful counting, she formed the conjecture that the basket-weaving pattern was the same as U=O. She felt pretty certain that no pattern exists with unequal U and O. Y, however, was determined to use her strips to discover a pattern with unequal U and O. Both took strips home to persevere. The younger girls, J and D, worked more with the U=O pattern, and then began folding their strips into a stair pattern. **PROBLEM 8 (Second Guess the Author)** I used puppets to introduce the game Knights and Liars, a Raymond Smullyan classic. On the Island of Knights and Liars, every person is either a Knight, who always tells the truth, or a Liar, who always lies. The kids chose puppets and I gave the puppets statements. The students had to decide who was a Knight and who was a Liar. After a few rounds of this, I introduced the idea that they could use the strategy of second guessing the author to help answer some harder questions. One puppet said “2 consecutive numbers added together equals 12.” The other puppet said “2 consecutive numbers added together equals 13.” If one is a Knight and the other a Liar, who is which? The older kids started trying numbers immediately. I asked them to keep their conjectures to themselves. The younger kids questioned the terms in the problem. I asked the kids to see whether the given numbers could help them figure out what was going on. D posited that the author may have been using an alternative definition of “add” since she noticed that in one statement the digits of the problem were consecutive and in the other they were not. We discussed digits briefly, and I specified that the author did in fact intend the traditional definition. After much discussion, everyone understood the problem, and three of the four kids agreed on an answer. “But why did the author give you those numbers?” I asked. “Surely there’s some mathematical wisdom to be gleaned from this question, other than rote addition practice.” They looked more at the numbers (12 and 13) and K suggested it could be about “even” and “odd.” We discussed these terms and their characteristics. (“Even means you can smash it in half and get whole numbers,” said J.) We experimented with sums of consecutive numbers. Everyone seemed to have a firm grasp of the problem and all concepts involved. I was ready to put the problem to bed with this question: “So, we all agree that you can’t have 2 consecutive even numbers, right?” “No,” argued J, throwing an exciting wrench into the problem. “What if your consecutive numbers are multiples of 10?” Everyone agreed that she had a good point, and that we needed to further refine the question. I asked the kids to reword the original question, replacing the vague word “numbers” with something more specific. I expected to hear something like “whole numbers” or “integers,” but J bested my expectations with the incredibly specific phrase “multiples of one.” Then everyone was happy to put that problem to bed. I mentioned to the older kids that I had a harder problem that we could do later if we had time. Then we took a quick break to snack on some rice crackers. **PROBLEM 10 (Go to Extremes)** Because J and D had been folding the weaving paper into stairs, we skipped problem 9 and jumped into question 10, Dr. T’s escalator problem, which was greeted with confusion. What was it asking? I reread the problem. More confusion. The kids suggested that I use the paper stairs to demonstrate, so I picked up one of our snacks. Now a rice cracker was Dr. T going up the paper escalator, counting his steps. This clarified everything, but the kids still didn’t have a clue about the answer. They had some guesses, though. I introduced the strategy of going to extremes, and asked for the most extreme number we could try. K suggested trying 1, instead of 20, as the number of stairs climbed before the escalator reached the top. With demonstration and thinking, the kids figured out the problem. I then asked whether 1 had really been the most extreme number to try. The kids laughed as we discussed using zero and negative numbers (walking backwards down the escalator). **PROBLEM 8 (revisited)** We had time for one more problem before parents arrived for pick-up. K requested the “harder version” of problem 8 because “I love hard math problems.” I was concerned that the younger kids wouldn't understand it at all, but when I mentioned that this problem involved the puppets and Knights and Liars, D begged for it. So I put J and D to work operating the puppets. I whispered the puppet lines into their ears, and they participated in the math parts as they were able. First J and D did a demonstration to make clear which puppet was the Knight and which was the Liar. Then the Liar said “2 numbers multiplied by each other equal 39.” The Knight contradicted, saying “She is mistaken. She reversed the digits of the 2-digit factor. The 2 numbers do not multiply to 39.” I put these statements on the board with slots drawn instead of variables. The kids used the second-guess-the-author strategy to realize that the (unstated) problem was about finding 2 numbers that multiply to 39. Unfortunately, no one knew any factors besides 39 and 1. I suggested just trying the problem using 39 and 1 for now. We did, and discovered that when 39 is reversed to 93, the correct product is 93, and the initial conjecture about who was the Knight and who was the Liar was confirmed. Then I broke the news that 39 is actually not prime, and while the product 39*1 worked to answer the question, it is not the number the author intended. What number might that have been? Now the younger kids got more involved. They worked together to come up with 13*3 as another possible pair. Interestingly, the kids saw that the “correct” answer to the multiplication with this new set of factors is the same as before: 93. They found that to be cool, and wondered why. Unfortunately, we were out of time and didn't have time to explore why the answer was 93 again. We did, though, talk about how this problem might have been asked differently. What if no specific number had been given as the supposed initial answer? What if the original answer had been a number with more sets of factors, such as 36? What if the question had been phrased using variables (a*b=39) instead of words? We talked about how there are different approaches to solving math problems – with words, numbers, pictures, props, variables, and even, as the kids were folding paper all over the room, origami. We had no time to get to **problem and strategy #9 (“Avoid hard work”)**. Unfortunately, all the kids are now off to camps and vacations, so we will not be gathering as a group again until the fall. Fortunately, we did cover 9 out of 10 strategies, and I think everyone was able to get enough out of the strategies to incorporate them into their future problem-solving endeavors. **What problems worked well? What kept the kids going when they had difficulties? What could be done better?** In this set, all of the problems worked well. Perserverence, changing strategies, and puppets kept the kids going. I asked the kids at the end for their feedback. They said that they really enjoyed using props this week (to foster understanding, and to have fun). They also enjoyed using the Knights and Liars framework to do arithmetic because of the added thinking required. (My observation here, and over the years, is that kids like to think hard.) And they enjoyed using puppets. The paper weaving strips I had were not all of identical width, which cause a bit of difficulty in assessing whether shapes were squares or rectangles, although interesting learning emerged from that seeming challenge. The bigger difficulty, again this week, was the wide age spread of kids. The older kids had to hold back just a bit so the younger kids could understand. And at least one of the younger kids felt intimidated by the math abilities of the older kids. I suspect that with problems involving less arithmetic and more of anything else, this would be less of a problem. **Personal Comments** Many thanks go to D for bringing her camera and photographing the session while she participated. (These photos will be posted here shortly.) Also thanks to the parents for bringing their kids. K’s mother emailed afterwards “K really enjoyed herself and has been enthusiastically telling me about her experiences. I have been trying to convince her for years that she does like Math, although she may not enjoy computation activities. Tonight she told me: I really do like Math!” D’s and Y’s mother was impressed with the strategies the kids learned, especially Perseverance.