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

For my 22 months old: 4) I will make a hanger balance and weight our counting bears. I will make bags with 3 or 4 of them, and hope she can make a bag of her own, when she finds out how many bears are in each bag. 5) I will play around with pencils, may be she can make 4 of them touch each other. 6) I will draw one cat on one side, and two on the other. then I will tell her a story, and ask her to give me right number of mats (sticky notes) for the cats.

My students strategy: start from smaller version and from what they know!

For my 11 year old we will: 4. do a compute the sum problem 5. do a young adult level problem 6. solve the horse problems with gears instead of horses

Problem 4: I agree with one of the other participants, I plan to change the balance problem to one without fractions: one apple on one side, 5 apples on the other. Though we may not use apples, I'll use something of standard measure that I have in the house --- like pennies. If my kids can't see the answer after drawing the picture, we may try it with the hanger and items in a bag. Problem 5: We will try the pencil problem as written. Problem 6: We will try the picture problem as written, using the 3 and 4 horses. I am very interested to see how the kids work through this problem initially, as my 6 year old is quite literal.

We had a short planning meeting with the two people who will work with me on these problems, my kid M. and a friend C. There were some concerns from C. about the level of math required, and if he's good enough at math. So my preparation had to do with explaining that the goal is to adapt problems, not some sort of test of math prowess. Again, I plan to use computer tools a lot. Last time, one of the better activities was inviting students to explain their reasoning to one another. They both liked it (in either role). I plan on promoting this even more. I am concerned about problem #4, because it only discusses heterosexual families. I am pretty sure the issue will come up in some form, because it's a hot topic for young adults. So much so I am afraid it is a distraction from math, much like other people said toys are too much of a distraction. We'll see.

**Problem 4** I think I am going to make a treasure theme for this and at least the next problem. So I will tell a story, something like a little girl finds a big pot of gold coins and bags of coins at the end of the rainbow, and then she has to figure out how many coins in each bag. I am planning to draw some balance scales on pieces of card and glue on sequins and little felt bag shapes in several different puzzle equations of varying difficulties. Then I will give the kids blank scales outlines and sequins and bags to rearrange and find answers. And maybe create their own problems too. If there are some littler ones who struggle with this I might just get them arranging coins and bags and trying to get them to balance. **Problem 5** I am painting some craft sticks gold and am going to use these. The story will be that the Goblin King is upset his treasure has gone missing and he has cast a spell on his remaining gold - it is safe as long as it is touching all the other bars. Then the kids can try and save as many gold bars as possible. I tried this with pencils this morning and we had trouble with the pencils rolling about. Craft sticks feel more stable and I have more of them! Although I think we will have to be careful about what counts as touching as it is not quite as clear as with larger objects. I also love the idea of adapting the combinations puzzle in various ways - maybe I will come up with something here. **Problem 6** I am not sure we will reach this, but if we do I think I will just go for simple two and three person truth telling puzzles. Maybe the guards of the treasure have to be bested before you can get into the treasure room...

Problem #5 We will work fairly with the question as stated. But we will use a variety of modeling media with different properties that bear on the solution. For example, we will use dry spaghetti noodles, but we will also use... cooked spaghettis. For the squeamish at heart, though, we will have socks. The idea is that hard and soft materials differ in their malleability. A sock can be oriented up to touch another sock but then gently fold back to touch other socks beyond, so that it touches a greater overall total of socks. That was discovered by Sockrates while taking a bath with Archimedes. Problem #6. Again, we will work with the problem as writ. We will provide paper and crayons and ask the kids to (attempt to) enact the tiny drama. At one point we might use a page with different pictures on each side and ask half of the kids to close their eyes while the others look up at the page... switch groups while flipping the page... different number. How can that be? We're thinking of introducing a whiteboard and markers this time, rather than use lots of paper when it is uncalled for.

Problem #4 As it so happens... my student Kiera Chase and I have been creating a digital learning environment for algebra, Giant Steps for Algebra (GS4A), that is built on a number-line model of algebraic propositions (borrowed from Dickinson & Eade (2004 in For the Learning of Mathematics). We've so far had a [modest paper][1] in the proceedings of Interaction Design and Children 2013 that took place in NYC a few weeks ago, but Kiera is developing this into a dissertation project (yoo-hoo!). So for this problem we will have a lo-tech treasure hunt. For example, for 3x+2=4x-1 is a story about a giant. "The giant has stolen the elves' treasure. He escaped their land and voyaged to a desert island. After mooring, he set off walking along a path. You are an elf positioned on this island, and you are spying on the giant. The giant walked 3 giant steps and then another 2 meters, buried some the treasure, covered it up real well, and then went back to the ship. On the second day, the giant wanted to bury more treasure in exactly the same place. Setting off along the same path, he walked 4 steps and then, feeling he'd gone too far, he walked back one meter and buried the treasure in the same spot. The giant then covered up the treasure as well as all his tracks. He returned to the ship and sailed off. Your job is to tell your fellow elves exactly where the treasure is." So I'm thinking we'll have several stories of this ilk, beginning with very simple ones and then gradually getting into more complex numbers. We will make paper and crayons available, but we will enact the stories (with volunteer parent-giant and kid-elves) and leave traces on the floor. Perhaps we will use string or such to mark the concatenated intervals. Problems might be: x + 3 = 2x , 2x + 4 = 3x + 2, etc. [1]: http://edrl.berkeley.edu/sites/default/files/ChaseAbrahamson-IDC2013.pdf

Here are the "Truth or Liar?" puzzles I am going to try with my two math circles, in case you want to use them, too. In the first two, Mama and Papa are setting a puzzle for Baby Bear to solve. On the second page, Baby joins in the puzzle-making. Because the Three Bears picture is copyrighted, please don't share the worksheet beyond this course (unless you make your own image). [Three Bears Liar Puzzles (pdf)][1] [1]: /storage/temp/107-three+bears+liar+puzzles+ temp.pdf

My thoughts so far... Problem 4 – We will use our balance scale, various household objects – right now I am thinking marbles, counting bears, perhaps candy, and I am sure the girls will throw in a few ideas of their own, and a few ziploc bags to group objects in. I will let the girls experiment and compare weights and see where this leads them. Problem 5 - We will use pencils and Cuisenaire rods to play with this one. I will also see if using the game “Battleship” will work to adapt the young adult problem in a way that my 5 year old can envision it. I will have her see how many ships she can fit on the board without any of them using the same row or column, trying different sizes and directions. Problem 6 - Still working on this one. It is a complicated concept but i am confident that a fun, clear way to do this will come to me if I keep considering it. :)

I have been thinking about how to do these problems, but I'm hampered by not having met with my groups yet. The K-1st grade group doesn't meet until Friday, and the teen group will not have their first meeting until next week, due to scheduling conflicts. I'll prepare 3-4 problems for each group, but I have no idea whether we'll actually get to more than one of them in the time we have. **K-1st grade:** **Visual algebra --** I want to make a worksheet for them with a couple of problems (NOT starting with one that comes out a fraction!), and then a blank balance for them to make up a problem of their own for the group to solve. Maria at Math Mammoth has a sample worksheet that might be useful to some of you, but it's too busy for this age: [ http://www.mathmammoth.com/preview/balance_problems.pdf][1] **Arrangement puzzle --** The family that's bringing action figures said we could use them for two weeks. We'll make this an extension of the "Planet PeeYu" puzzle from last time. Dr. Sploosh invented a formula that eliminates foot odor, so the PeeYuvians are experimenting with new arrangements. What shapes can they make now? Captain Mack said he figured out how four PeeYuvians can arrange themselves so that each one of them touches all three of the others. Do you believe him? **False sentences --** I don't like the picture puzzle because the answer seems too much like a trick. We'll try the false sentences instead, but make it more concrete by changing it to lying bears (since I used a three-bears theme in one of the first week puzzles). We'll do a couple of simpler 2-person "who's telling the truth" puzzles and work up to the three part puzzle given in the handout. This sounds like a huge mental workout, so I won't even bother planning a fourth question. If we get done early, we'll take a hike in the woods. **Teen group** They have done some [knights and knaves puzzles][2] in the past, so the false sentences puzzle should make a nice warm-up. I like all the problems in the MAA-AMC newsletters, so we'll probably go with those pretty much as written. I especially like the extension idea in the graphing problem that we allow teleporting and figure out ways to explain each of the graphs. [1]: http://www.mathmammoth.com/preview/balance_problems.pdf [2]: http://letsplaymath.net/2012/04/02/skit-knights-and-knaves-logic-puzzles/

Problem #4 I think I might try the sum problem with both my 4-year-old and 6-year-old. I think I will start with some beads on hand that we can arrange however they see fit. I will tell them the story about Gauss adding all the numbers from 1 to 100 and work on that related problem first. I don't want to over-script what will happen next, but try to react to their ideas about the story/problem. If all goes well, we will attempt the sum described in Dr. T's problem. Problem #5 I'm going to try modifying the AMC problem. Instead of an array, I'm going to cut up squares of colored paper (5 colors) and put different animal stickers on them (5 kinds). The story will be that we are going to choose 3 animals to add to a zoo. We want to have variety, so we need to choose animals that are all on different-colored backgrounds and we need to choose 3 different kinds of animals. How many different possible combinations are there? This is the "full" problem, but I will try to adapt further by starting with a couple simpler versions. What if we just look at the animals? How many different ways could you choose 2 out of five possible kinds? 3? 4? 1? 5? What if they come in different colors? (start with two different colors, then work up to five if possible.) Problem #6 This will be for the 6-year-old. He is interested in things like: "Will it ever be tomorrow?" So I will piggy-back on that and ask about: "This sentence is false." We will try some knights and knaves puzzles before we attempt this one. I use problems like this with my geometry (high school students) when I teach indirect proof, so I have some idea of how I would organize it for them, but I want to see what my kids do with these and react according to whatever they bring to the puzzle.

4. For weighting we'll make a hanger balance. We'll also draw a picture of a balance and we've got some equal small LEGO pieces which we'll put into it. For the some we've got blocks and tins to play with 5+4+3+2+1 before getting to the square. Men and woman will become male ducks, female ducks and offspring. Each female has 1 duckling. We've got 5 adult ducks of which 2 are males. How many duckling have we got? For bigger kids we'll introduce the idea 'of each 5 adults 2 are male' and we'll see what happens. 5. Pensils are prepared, and carrots too. 3 blocks in the square 5x5 will become a square 3x3, a frog and a mosquitoe. We have to place the in the square avoiding the frog seeing the mosquitoe (otherwise it will eat it!). Then we can enlarge the square and introduce a dog (dogs eat frogs and mosquitoes bite dogs). 6. We decided to draw balls instead of horses because drawing horses would drive the kids too far from math ;-) The FALSE statements will become 2 persons saying: - One of us lies. - We both lie. It will be played with figures. Then we might add more persons saying more lies.

PROBLEM 4 (Drawing a Picture) I will give a problem about eating trail mix in the car. Something like "Your mom says you have to eat 5 nuts for every 2 M&Ms in the mix. If you want to eat 1 of every color of M&Ms, and save them for last, how many nuts do you have to eat?" I expect to have more kids, ages 7-11, next time, so will prepare a few problems like this of varying difficulty, and have paper and markers available. PROBLEM 5 (Solve a smaller version of the problem) I hope to do a problem where the kids need to come up with combinations of pizza toppings to make various named pizzas. Most likely I'll tell them they're designing the menu for the pizza shop, and need to know how many different named pizzas will be on there. Since these are younger kids, though, I suspect that this may be too complicated and will need to do something involving arrangements/permutations instead - like designing wedding cakes with action figures of the bridal party on top of it. Or a ranking of dogs in a dog show. PROBLEM 6 (Eliminate incorrect answer choices) I like Dr. T's problem with the statements, and will probably use that, unless I find something like it in one of my favorite books, "What is the Name of this Book?" by Raymond Smullyan.

When I was working with children from pre-school and elementary levels in 1995-2003, I was collecting a lot of evidence of their ingeniosity while solving challenging tasks like ones listed as 4-5-6. Namely, the power of pictures (schems, drawing) and solving with smaller number of objects was described (along with many other examples) in my article I published in 2006: http://www.math.umt.edu/tmme/vol3no1/TMMEv3n1a3.pdf As example of the riddle that can get students used to some situations which look 'impossible' to handle and perhaps get more insight into the beauty of mathematics by seeing something very unexpected : like 'we have 5 apples in a box which we divide among 5 people and one apple will have to remain in the box'. How do we do this? So actually we learn to think 'outside of the box' - very important aspect of mathematics learning and teaching in more creative way, I think. As for the puzzle with true-false choices, I remember another task well analyzed in the literature : http://en.wikipedia.org/wiki/Wason_selection_task I find that doing such task with young students (and everyone else) we may contribute to the development of the deductive reasoning.

As I said in the other forum, I had issues with some of the weaker math students not being able to figure anything out on their own, and/or in just giving up once someone else had figured out the problem. So something I'm doing with the next set is that I've created two sheets to hand out to everyone with the problems (I'm working with teens, so we're mostly just working on paper). One is the list of problem-solving techniques from this MOOC (the ones we have so far, at least), and the other is my adaptation of Tanton's problem-solving process from his Curriculum Inspiration essays. I'm hoping this will help some of the less-confident students to get beyond their "I don't know how to do this" blockage. I've written them both a bit generically in terms of subject (that is, I want to use them with my history and literature classes as well), but also added some of the other things I use in my classes and teaching (such as Thinking Maps). So they may not be appropriate to other people besides me. However, I have attached them to this post just in case they would be helpful to others. Carol [link text][1] [link text][2] [1]: /storage/temp/86-10+problem+solving+ techniques.pdf [2]: /storage/temp/87-steps+in+problem+ solving.pdf

This may seem semantic, but I was bothered by the way that Question 6 is written. How could a teacher answer anything except "Correct" to a question that asks how many items a child sees? The teacher can't say, "No, you don't see three items, you see four." The child sees what s/he sees. But what a child (or person) sees may have no correlation to the number of items there actually are. Lashana could see three, Juan could see four, but there could be twelve of them there, but the children can't see most of them because they are hidden, or it is an Escher-like pattern and they are only "seeing" the dark ones and not the "empty space," or whatever. I'm thinking particularly of those puzzles that have triangles within triangle, where many people only see the individual triangles and miss the larger triangles that are made of sets of the individual triangles put together (I hope people know those kinds of puzzles...sort of like this![alt text][1] [1]: /storage/temp/83-triangles-1.gif But I'm not sure how to rewrite it more precisely without giving it away. If the teacher asked "How many horses are on this page?," both three and four would still be correct answers because there ARE three and four horses on the page--there is just more than that. But if you ask, "What is the total number of horses on the page?" then three and four are both wrong answers, because there are seven horses on the page. But if you specific "face" or "side" or something, you are kind of giving it away. So I don't know how to express this problem in a logically correct way. Carol

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