This question was **closed** Dec 07, 2013 at 05:43 PM by Maria Droujkova for the following reason: The course is over.

Here are problems that go with the first three solving techniques. The goal of this course is to adapt the same or similar problems for different levels and interests. ![Gear Problem Illustration][1] **1. Look at the PDF document [mpsMOOC13_Problems_1_2_3][2]** (updated links) **2. Reply here: how do you plan to adapt the problems for your kids?** There are many adaptations. You can use objects and characters to pretend-play a story about the problem, or model it in Minecraft. You can ask the kids to make the problem harder or easier mathematically. Your child's goal may be to create a beautiful and meaningful illustration for the problem, or to make a short video about a solution. Maybe you will go on a scavenger hunt for a math idea from a problem - in the park, at the supermarket, around the house. ![alt text][3] **Reply here by July 7. Do problems 1, 2 and 3 with the kids by July 14.** Remember your math dreams as you plan! [1]: /storage/temp/57-jamestantongears.png [2]:
https://docs.google.com/file/d/0B6enMfoYXJb3UHlGODAwSFUwQlk/edit [3]: /storage/temp/58-zomebubblemodeling.png

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Comment

Maria Droujkova

Am I missing something else though - is there something to read for the older kids.

If you follow the links to James Tanton's MAA Inspiration (for example, the one for Problem 1 is
http://www.jamestanton.com/wp-content/uploads/2012/09/MAA_AMC_Inspiration_Letter-1_2012.pdf), he has related math problems for middle school and high school students. Carol

1. For my 2 y.o. and 3 y.o. I could not imagine something else than a ladder and a rope with toys to raise up and down. 2. I like the idea with dolls which are going to sleep on the common pillow or animals to eat from the bowl. 3. I'm going to try play dough and modeling of it. May be abstract figures to connect them colour to colour or some animals and their meal. My babies enjoy modeling of the sausages and balls:)

**Answer** by ccross
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Jul 01, 2013 at 02:18 PM

Here is a good resource to go along with the second version of Problem #2 (2 to the 100th power):
http://freemars.org/jeff/2exp100/question.htm I plan to use that after we work out the last digit part of the problem. Carol

**Answer** by ccross
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Jul 01, 2013 at 03:11 PM

Just some quick feedback. I tried the gear problem with my 14 year old son and his 13 year old friend. I just gave them the graphic without explanation. My son started immediately rotating his finger for each gear and figured it out within 20 seconds (although he got one wrong, but that was speed and carelessness, not lack of understanding). His friend took longer--maybe five minutes--before getting them all right. She said she had been trying to figure out a pattern, but ended up getting too confused and so drew the direction on each gear. So we talked some more about possible patterns. My son kind of figured it out but couldn't articulate it. I had them count up the gears on each path--what did they notice? Then it became clear. Why would you have to do that? For my son, his way was just about as quick...but not as accurate. Their way was fine for that problem, but what if there were 20+ gears? 50+ gears? But imagining the same problem on a much larger scale, they could see that establishing a pattern of odd one way, even the other would be much quicker and more accurate for larger gear configurations.
I'm going to try the same thing this evening with her younger brother, who is only 5 but more mechanically inclined than either of the teenagers. We'll see what he comes up with....
Carol

Carol, can you please share your thoughts on how you plan(ned) to try the problems, with older kids and the young one? This thread is for sharing the plans and ideas before you run activities. I am sure people will find your quick feedback helpful as well! We are trying to help people with the preparation PROCESS.

Well, my planning process is kind of just what I indicated above. I read the problem, work at it until I understand it myself, find any other resources that might help me explain it, print out the problem without the explanations, and then do it with people. I don't know--there isn't a whole lot of planning process behind it all.

Is there a place we are supposed to be putting what we are finding? I've done two other sets of problems as well now. For me it is better to do the problems right away before I've forgotten, plus with the holiday coming up, it is harder to get with other people in a non-party atmosphere as the week goes on.

We'll have a place for reports on problems 1, 2 and 3 next Monday. Just save your file locally for now. I am trying to keep us synchronized.
Before July 7 - plan problems 1-3
July 7-14 - report on problems 1-3, plan problems 4-6
...

**Answer** by Denise Gaskins
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Jul 02, 2013 at 07:06 PM

Still working on plans, but here are some thoughts for my K-1st grade group:
**Problem #1:** I don't like the term "Successful Flailing" for the first technique. When you're flailing, the whole point is that there's no guarantee you'll be successful. How about "Organized Flailing" (like on James Tanton's original AMC puzzle page) or "Creative Flailing" or simply "Experimentation"?
For the Gears puzzle, I want the kids to notice that there really are two different directions to turn. I thought we could start by spinning around, just for fun, and then maybe have a couple of adults demonstrate spinning in the same direction and then spinning opposite, so the kids can see the difference. I have some Gears that I'll let the kids play with before they try the paper puzzles.
But I'm not sure all this scaffolding is a good idea, because it eliminates the "flailing" factor. So perhaps I need to just hand out the worksheet first and see what they make of it, and then add the activities if needed?
**Problem #2:** I like the action figures and Post-It pillows (or colorful index cards) approach, so I hope I can find some of my kids' old Barbies and GI Joe guys around here. Star Wars figures would be even better (our group is mostly boys), but I think all of those are lost or broken. :(
After working the puzzle with 6 dolls, I think I'll extend it to 7: "Maybe Penny didn't count her dolls right. If she had 7 dolls, could she use four pillows?" It shouldn't take them too long to figure out that Penny needs 8 to make the loop she described, with 4 pairs of heads. If the kids are in a talkative mood, perhaps we'll see how many things they can think of that come in pairs, or in other sized sets...
Then I think I'll send the kids out into the grass to make body-shapes. That should provide a nice break and burn off some energy. It would be nice to have a snack when they come back to the picnic table. Hm, I wonder if I can talk the parents into taking turns bringing snacks. Though now I'm second-guessing the plan: it might be better to do the body shapes in the grass first, as a break after the Gears puzzle. Then a snack after the loop puzzle will make a break between that and problem #3.
**Problem #3:** I think we will have three families, which means I can grab pictures from Facebook and put the kids and their parents on the worksheet. Since this is the third puzzle, "It's time to go home" makes a fitting story for it, and we don't want them to run into each other, because somebody might get hurt, so that's why the paths can't cross. I think yarn will work well for trying out different paths (fastened at the kid pictures, with the other end loose to make a path to the parent).
I think I'll make one laminated "manipulative" picture where the parents aren't pasted down but instead are attached to the yarn, like the "move it around" pictures in the handout. That might help them get the idea of trying more complicated paths than just straight lines.

One of the issues I have with the gears activity and my kids who are young is that without actually having gears to play with I don't think they'll make much out of it. The idea of gears doesn't make a lot of sense if you haven't seen gears. So I do need to find a way to get a set of gears. Do the kids you are working with have gear experience? Is there anything else that has that same property that they could manipulate physically?

Denise, your idea of introducing gears to your kids so very naturally is really amazing. MashAllah!

**Problem #3 reconsidered:** I think yarn will be too hard to work with in the park, if there's any breeze---and I sure hope there is! So I will try to find enough rope to act the problem out on the grass, with the kids trailing rope behind them to mark their paths. Then we'll see if they can translate their method to paper (which I'll laminate, so they can use dry-erase markers).
**Problem #4?:** If we have time, I'll adapt the exponents problem to addition (biggest, middle, smallest: 3 Bears story?), and then have them each make up a put-the-in-order puzzle to share.

**Answer** by faroop
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Jul 03, 2013 at 02:22 PM

I am thinking about the second version of problem #1. I have worked with my older daughter (recently turned 7) on even and odd, which she learned in school. I'm thinking about how I bring these ideas to my son who recently turned 4 and involve my daughter at the same time. I am thinking of using pennies (or another counter, but pennies are always available) and presenting even numbers as "friendly" numbers because of the way they pair up and odd numbers as "lonely" numbers because of the one left out. We could talk about which numbers are friendly and which are lonely (I'm open to other language here if anyone has some, I'd even be open to just using even/odd, but I want to be emphasizing the pairing). Then we could get to talking about what happens when a number meets (gets added to) the next number up. What happens when 2 (friendly) meets 3 (lonely)? How about when three numbers in a row meet up? For my daughter, who has already explored some of this, I could start her thinking on longer strings of consecutive numbers.

So after thinking some more and talking with a friend, I've decided: (1) I'm going to start my 7yo with basically the second version, slightly smaller numbers ("there are four pages missing from my book, what's the sum of the page numbers"?)

(2)For my 4yo, we actually thought of a completely different related problem that I think would interest him visually a little more. I'm going to first lay out for him using pennies (or another counter) all the numbers 1-10 on a long table so he has room. I'll arrange them paired up so he can easily see the even/odd thing, so it may come up if he notices that. Then I'll ask him to rearrange the 3 pennies so they make a triangle, showing him if he's not sure. Then we'll go onto rearranging 6 into a triangle and see if he can do that.

Depending on where he goes with that I might have him arrange 10 as a triangle, or have him try to make other shapes out of other numbers. If he struggles with making meaningful arrangements I'm going to have dot paper available for him.

**Answer** by ccross
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Jul 05, 2013 at 10:25 PM

For those without gears sets.... one of my best investments has been an old-fashioned egg beater. It not only demonstrates gears, but we have used it in a number of other science and math demonstrations or activities. PLUS, I have my son or my students beat whipped cream the old way before adding it to a treat, which helps them to work off some of the calories they are about to consume. It is cheap, readily available, distills the gear action to something even a really young child can see (and move backwards and forwards), and it is useful in real life as well.
Carol

What a great idea, my kids love to use the hand mixer at our house!!

**Answer** by RosieL52
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Jul 08, 2013 at 12:01 AM

**Problem #1** John exclaims “Twenty pages have been ripped from my textbook.” Hilde replied, not so hopefully: “I bet the sum of the missing page numbers is even.” Is Hilde right? I’m adapting this problem for my 6-year old son. First I will need to see if he reliably understands the concept of odd vs. even numbers. Since the solution relies on understanding when the sum of numbers is odd vs. even I will begin with a game. I’ll ask him to pick a playing card out of a deck (he’s really interested in playing cards right now). We will look at the card together and notice whether the number is odd or even. The playing cards have a nice visual built in if I need to reference it: there are “pairs” of hearts/spades/diamonds/clubs on the even cards. After I know he can distinguish between odd/even I’ll change the game. He will pick a card. I will pick a card. I will show my card and ask him if the sum of his card and my card is odd or even. I will then ask him if the product is odd or even. Based on his answers I will then “guess” whether his “mystery card” is odd or even. We’ll do this for a few rounds and also switch and see if he can guess my card. We will keep track of the responses for each round of the game in a table (parity of mystery card/parity of “other card”/ parity of sum/ parity of product). I’m putting both sum and product information in to begin with, but as we play I imagine we will try some rounds that include only a sum, and some that include only a product. This may end up being too complicated - I may need to pare it down to just look at sums. Hopefully we can keep both elements in the game and we can talk about whether or not we would need to ask both the product and sum questions and which one gives the most useful information (and in which situations). If we discover the “rule” I’d like to look at why the sum of two odd numbers is even, etc. using stones or some other small manipulative. I think this will be a pretty substantial amount of puzzling for one session, so I’m not sure whether we’ll get to the original framework of the problem with the page numbers. If we do, it will probably not be on the same day. In addition to understanding how odd/even numbers “work,” the solution to the problem also relies on understanding the way pages are numbered in a book. My next step would be to make sure he understands those conventions. (My son has experience writing his own “books” and numbering his own pages - not necessarily with the same conventions typically used in a printed book.) I will begin with a simpler question. Holding onto one of his books, I will say that I have chosen a page and that the sum of the numbers on that piece of paper is 19. Can you find the page? If he is successful with this question, I’ll try another similar question. (The sum of the numbers on a different page is 51. Can you find the page?) I’ll continue with these types of questions until we can successfully find the pages, then I’ll ask about one that has no solution: The sum of the numbers is 20. Can you find the page? What if the sum of the numbers is 8? Can you find the page? Why don’t these examples work? The goal here is to notice that on any given page there is always one odd and one even number so the sum of the numbers will always be odd (building on the work from the card game). Next, we will look at what happens if there are two paper pages involved. What will the parity of the sum of the four numbers on two pages be? How about the six numbers on three pages? How about the eight numbers on four pages? We will make a table recording the results and see if there are any patterns. We will guess about the ten numbers on five pages, then about the 30 numbers on 15 pages, and about the 40 numbers on 20 pages. **Problem #2** I’m going to adapt the pin problem for my 4-year-old son. With what I have available, we will make pipe cleaner people and arrange them in loops. We will color code the different possible combinations. BIG yellow (post-it-note)pillows to share when heads touch. small blue pillows for the single heads (head-foot combinations). No pillows when two pairs of feet touch. We will start with a small number of “people” (three, then four, then five,...). How many small blue pillows are necessary? What is the biggest number of small blue pillows that could be needed? What is the smallest number of small blue pillows that could be needed? On a sheet of poster board we can keep track of how many pillows were used for each number of people by sticking the pillows to the board. We will look for patterns and see if we can guess what will happen for a larger number. **Problem #2b** I’m going to adapt the powers of 2 problem for my 6-year-old son. My son dislikes the arithmetic speed tests he has at school. If he senses that the problem I’m going to give him has any sort of this flavor, he will abandon it (and me) as quickly as possible. I’ll need a sneak-attack approach to this problem. If I can find it at the library I would like to start with the book “One Grain of Rice” by Demi (
http://www.amazon.com/s/ref=nb_sb_ss_c_0_15?url=search-alias%3Dstripbooks&field-keywords=one+grain+of+rice&sprefix=one+grain+of+ri%2Caps%2C216). We can start to think about just how BIG 2^64 is. (How much bigger would 2^100 be?) I think I can then bring him around to considering what digit this number might end with. I will have a hundred board available and also four colors of crayons/markers. We will color in the powers of two, one at a time with me handing him a crayon for each. Once we’ve used each color once we’ll begin the same cycle of colors again with the first color. Hopefully this will make the pattern “pop.” A single hundred board doesn’t take us very far into the pattern, so I’m thinking we will write each subsequent power of two underneath the appropriate column until we have enough data to consider. I think he will be able to see the pattern and make a conjecture about the 64th and 100th powers of two, but I _really_ want him to think about _why_ the pattern holds. He is accustomed to working with golden bead materials in his Montessori classroom. The final digit corresponds to the units, so if we look at doubling the number of golden beads maybe I can get him to think (in the abstract) about doubling each group of beads (hundreds, tens, ones/units) and realize that for this puzzle he really only has to think about what happens to the units. We don’t have any golden bead materials outside of school, but pebbles should be a fine replacement for the purposes of this problem. **Problem #3** I’m going to try the A-B-C problem wih my 4-year-old and 6-year-old (separately). We’ll do this in the sand. The A, B, and C will be squirrels (red, gray, and black) represented by rocks. The a, b, and c will be their favorite foods (acorn, pinecone, and maple seed). We will try different arrangements of squirrels-foods and see if we can connect them. I suspect we will find solutions fairly quickly... So we will also try some variations. 1) Add a sunflower-seed-loving chipmunk to the mix. Does the addition of a fourth animal and food make it any harder to connect the animals to their favorite foods without crossing paths? 2) What if the three squirrels each want access to two foods. Can we still make paths that don’t cross? 3) Each animal wants access to all three foods. Can we still make paths that don’t cross? (Now it’s really the classic gas-water-electricity puzzle in disguise.) I’m not sure if the A-B-C problem was supposed to be adapted, but I have no great ideas about how to make the exponent problem into something meaningful for the 4-6 crowd. Instead of: What is the correct ordering of the three numbers 10^8, 5^12, and 2^24? I could imagine asking: Which is bigger, 8 groups of 10 or 12 groups of 5? For my 6-year-old it would be “Which is a better prize: 8 $10 bills, 12 $5 bills, or 24 $2 bills? (He is very interested in money... and the idea of a $2 bill - which he has never seen - might be a bit of an extra hook I could use.) This does not involve the exponents at all, but does get at the idea of having different small-big number combinations and how that impacts the size of the result of performing a particular mathematical operation. It also gets at the idea of breaking down numbers into more manageable parts (factoring) that is useful in the solution of the original problem involving exponents. I could also imagine asking... if the right moment presented itself. Anything that smells remotely like “arithmetic speed test” to my 6-year-old puts him in shut-down-mode. Which is the biggest: 10*10*10*10, 5*5*5*5*5*5, or 2*2*2*2*2*2*2*2*2*2*2*2?

**Answer** by ccross
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Jul 01, 2013 at 02:50 PM

I am reproducing the graphics for all of these problems, blown up, without the explanations so that my students can try them first without hints. Is there a way to isolate just the problem graphics from the text? I could do the first one, because the gear jpg or png is on the website, but the other ones I've either had to cut out using Jing or find a similar graphic on the web. I'm making multiple copies so I don't want to just draw them.
Thanks for any assistance with this-
Carol

There is no easier way than what you describe, because the pictures are placeholders. It is an excellent idea to create printouts, once we have final pictures - thanks for the recommendation! Meanwhile, send me the handouts you made, or put them up on line for everyone, please.

Hello - I would love to print out the gear graphics too - have you had any luck on resources to share? I have no idea what Jing is...

The gear graphic was attached to the top of this page. It is jamestantongears.png (24.5 kB) . I was also able to just use Google Image search and found a graphic for the third? problem of connecting the boxes without crossing the lines from another site. Jing is software that lets you capture items you see on the screen as image files. You can download it for free here:
http://www.techsmith.com/jing.html . It may be that will be the only way we can print out the images from future problems--by capturing them as pictures in Jing and then making our own document to print out. HTH, Carol

Thanks! I was looking for images of the other gear problems, so perhaps I'll try Jing.

**Answer** by adamglesser
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Jul 01, 2013 at 05:49 PM

My oldest son (7) looked at the 2^100 problem over my shoulder and wanted to know how big it is. Although this problem is not about that, it is an excellent opportunity to get in one of my favorite estimation tricks: 2^10 = 1024 ~ 1000 = 10^3. Thus, 2^100 = (2^10)^10 ~ (10^3)^10 = 10^30.

**Answer** by ccross
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Jul 02, 2013 at 09:39 AM

For the 20-missing-pages problem, I wrote out the problem and created a small worksheet for people to record their thinking process about how to solve the issue. I think people kind of intuitively know the answer, but have different ways to "prove" it. I'm more interested in the thinking involved than in the answer. And sure enough, when I tried it with people last night, people approached the problem in completely different ways.[link text][1] I'll attach the sheet here--I have the problem copied twice on the same page just to save paper. [1]: /storage/temp/
68-20bookpages.pdf

20bookpages.pdf
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**Answer** by Denise Gaskins
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Jul 02, 2013 at 08:03 PM

Tentative plans for my young- to mid-teen group (average about eighth grade):
**Problem #1:** I think they will find the page number puzzle an easy warm up, and a chance to practice justification. "How do you *know*?"
The original AMC puzzle will be too hard for my group, but I might just give them the diagram and see what questions they can think of to ask, without the pressure of answering any of them. Though I'm pretty sure they can find the height of the triangle. [Incidentally, the most natural way for me to think about the height was with an infinite series of circles. Perhaps I've been watching too many Vi Hart videos?]
**Problem #2:** I think they will find the 2^100 problem interesting. Again, I'll ask what extensions they can think of. I like the link that Carol gave. Perhaps I'll ask them the folded-paper question as a take-home to ponder...
I'd like to have them try the absolute value problem, but in a simplified version: I'll ask how many points they can find that fit the first equation (and ignore the quadratic part entirely). If they find the square, that should give them a "That's cool!" feeling.
***Problem #3:** I think my kids will be able to think their way through the AMC8 problem as it stands. Then they can probably make up some problems of their own. If they aren't too tired (summer afternoons in the park can get HOT!), I may split them into two groups and have them make up a problem for the other group to solve.
It would be interesting to see what they think about the different ways to explain exponents, but I think the kids will be worn out on thinking by this time.

**Answer** by mirandamiranda
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Jul 03, 2013 at 12:21 AM

This is challenging for me! Part of the problem is that I am not sure how many kids or what ages will be coming along, which complicates things. I am also a bit nervous about leading a group beyond my own children! But here are my thoughts so far.
**Problem 1** The resource centre where I will likely be running the group has a set of gears that I will hopefully be able to use. The kids can put them on baseboards and explore the problem that way. I also thought that younger kids especially might enjoy pretending to be gears - we could line up with our arms out and spin each other around to see how the gears interact. Although perhaps I should try to show them a real gear first so they have something to work with mentally. A friend also has magnetic gears that might work.
I like the idea of printing out the gear problems too so that older kids might be able to work independently on them.
**Problem 2** I can see a lot of potential for kinaesthetic learning here too! I was thinking of getting the kids to lie on the floor in a circle and see if they could fit 4 (or however many) 'pillows' into the circle under double heads. Or maybe using paper dolls - my oldest is very keen on those, she could make some for me! And I'd like to prepare sheets to fill in any patterns we see, just a basic table with number of pins/dolls, number of meeting points etc.
**Problem 3** The idea of using pictures with yarn is great. I was also thinking dry erase boards, or playground chalk outside so we can try various paths. Maybe putting something up on the wall so more people can see and try out paths?
I would like to try and present each problem in the form of a story, which they all seem pretty amenable to. This feels like it would be a good way to engage the kids initially and get them motivated to solving the problem.
I'm looking forward to reading more ideas here!

**Answer** by mgrunk
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Jul 03, 2013 at 02:45 PM

Problem 1 - we have lego mindcraft gears -lots of them, so we'll play with those and them move onto paper - my son in looking at the paper already intuited how to determine which direction the last gear would move - quicker than I did. Now to get him to verbalize it. I also have a knex gear set. I thought we could try and build structures, 3 dimensional. Problem 2 - my son that likes Minecraft said that in creative mode, you can place a piston that has a brown end and a gray end and then place a certain number of pistons until you have the number you want, you can try it again in another area with the same number of pistons but using different orders. You could use a half block to mark each green to green match. I also have 50 pegs that I'm painting into people. We can play with those. 3. I was thinking of printing out matching pictures and stapling yarn onto them and letting them play w/ moving them around w/. We can also bring up paint on the computer and create the objects and draw lines - it's easy to erase them too.

**Answer** by yelenam
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Jul 04, 2013 at 12:53 PM

1. I have a Gears!Gears! set, so I'm just going to let the kids play with it for a while before showing a diagram. The Gears! set comes with 6 interlocking plates on which gears can be assembled. I thought it'd be fun to give each kid a plate and ask to create gears puzzles for each other. Then we can connect the plates into one big puzzle and try to figure out which way will a certain gear rotate. 2. I think I'll use our vast collection of Star Wars toys to pretend play through this problem. Also, we might venture into the 2^100 problem. I am thinking the kids can fold large sheets of thin paper and punch holes in them using a hole punch. With 3 kids and 1 adult taking turns folding, the pattern might become even more obvious. 3. I was thinking about playing out a story (maybe Moon exploration where astronauts need to move from their rovers to craters to collect rocks), but maybe using sand for tracing the paths.

**Answer** by Marianna
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Jul 05, 2013 at 09:18 AM

1. 20 pages problem. For the younger kids I'll have 4 pages stolen and probably sheets of papers or a magasine to explore. For the older kids there could be something related to 3 instead of 2. Still thinking about it. 1.1. Triangle with incircles. Have no idea so far - the problem seems to be so complicated that I don't see the essense that can be reframed in an easier way. 2. Pins - a lot of play with matches and LEGO heroes. The photo with kids on the grass looks inspiring - will definetly try it too! 2.1. 2x2x2... will become 2+2+2... 97 times for younger kids. Not 100 because some of them might know how to multiplicate by 100. With older we can look and see what happens with the last digit of powers of 3 or 4 or.... ;-) 2.2. modules. Don't know yet. 3. Comparing powers seems suitable for 10-years old and up, if we discuss powers before. For younger kids would try comparing other expressions without actually evaluating them. Like 1354+235 <> 1357+200 or 21*2 <> 17*3 I like the idea of comparing without evaluating.

20 pages. Will discuss numbers other then 20 with older kids. Then maybe we'll come to a conclusion... hopefully. Triangle. Will draw a picture and discuss what similar figures we see and what are their proportions. Modules. We'll play with pairs of numbers and maximums, like - of all pairs of numbers (from natural to real depending) with the sum equal to 17 what is the greatest product? The smallest product?

- Of all pairs of LEGO pieces of the same color which pair has the greatest number of pins? And he smallest? - A farmer had hens and sheep. The count of all their legs is... What is the maximum possible number of animals? And the minimum?

**Answer** by faroop
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Jul 05, 2013 at 04:38 PM

Now for problem #2: We'll use matches for our manipulative, since I think they will be easiest for my 4yo to handle. I'm going to introduce the problem to both of them having them each build a circle of 5 and then asking if there are spots where two heads meet, where two tails meet, and then where a head & tail meet. Then I'll start giving them some challenges, perhaps just centering first on those head/head meeting places. Can you make different number of head/head meetings (1, 2, 3, 4, etc) with different numbers of matches. For my 7yo, I'd love to get her started with record-keeping about solutions, so I might have her record all of the different ways you could arrange, say 5 matches. I imagine that she'll notice relationships between the H/H and T/T and H/T vertices, but we might explicitly explore this. For my 4yo, I'll probably keep it more loose, looking for whether he can make circles that have criteria I set out, and letting him give me some challenges too. I also love the Showing the idea of looking at the sequence 1, 2, 2x2, 2x2x2 etc with my 7yo. She understands a little about multiplication, and I'm going to use some manupulatives to try and build this up, and we may then look at writing down all these numbers and see if she can figure out patterns in their unit digit.

**Answer** by faroop
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Jul 05, 2013 at 05:03 PM

Finally Problem #3: I chose to think about the alternate problem, and I think I have a great way to work with my 7yo, and we'll see what my 4yo can do, but I'll be introducing multiplication to him. I may just do it with addition for him. With my 7yo, I'm going to work with multiplication, rather than exponentiation. She has been working a little on multiplication, but its still a new thing for her. I'm going to ask if she can figure out which is larger, 10x5, 12x3, or 8x7. Note that in these three problem I altered the numbers by increasing one by two from the first problem and lowering the other by two. So if this were a sum, they would all three be the same (maybe we'll even do that first!) We could do each of these problems, but given that multiplication is still new to Harriet, she has to work them out by hand and should be interested in thinking about it in a different way. 10x5 (10 five times) should be easy since counting by 10s is easy. So we can change the problem and figure out which is bigger of 10x5, 12x5, or 8x5 easily. Then we could think about how 12x3 is related to 12x5 and how 8x7 is related to 8x5. We'll use pennies to show them all as arrays I think. For my 4yo, I think I'll see what he can do with very small numbers, focusing on the same question for addition, and then multiplication if he handles addition well. He hasn't seen multiplication before, so I'd be curious what he'd do with it!

**Answer** by yelenam
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Jul 05, 2013 at 10:45 PM

What to do when no gears are available... I think one of the solutions is to substitute gears with spools and yarn. Then the question changes to whether the yarn on the last spool would go over or under. An even more fun idea, especially for younger kids, might be to set up a slalom course like this one with some agility cones (or paper cups or other markers) ![alt text][1] And see if we can figure out whether the last cone/stick/paper cup will be passed on the left or right (and for kids who can't tell left from right yet, maybe mark each side with toys, like a dinosaur side or a teddy bear side). This can be done on a large scale or as a tabletop slalom course for a toy car. [1]:
http://library.thinkquest.org/11431/images/dribble.jpg

Hi - can you explain in greater detail what you mean by substituting the gears with yarn and spool? I am a visual person and I cannot see it! We do not have gears and I am working with a 3 and 6 year old so the simpler the better. Thanks.

This "slalom" approach is my natural way to solve a gear problem. I never remember the connection between even and odd, and I don't bother to re-think it through each time. I just run my finger along the edges of the gears, following the direction they will turn, until I get to the one I'm supposed to identify.

**Answer** by Christiane
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Jul 06, 2013 at 03:26 AM

Working with 4 and 6 year old. **Technique 1:** 1. Young kids level. We do not have gears, so to help them understand how these work, I will show them a youtube video of cogs in action. Then I will show the diagram of Dr T's problem as printed. See if they need manipulatives. Ask them for suggestions - do we have anything that moves like this? Can we make something similar? May have to make cardboard cutouts if they do not come up with other ideas that are workable. I will encourage them to guess an answer and have them explain it. Then we'll check our answer using the manipulatives we came up with. 2. Toddler time. We will do this with all sorts of dolls and furniture. Should be lots of fun. 3. Kid and young teen level. We will take their favorite book and look at the page numbers. Encourage them to find a pattern. Teach about odd and even numbers through use of tiles. Even - the stacks are the same height. Odd - not the same height etc. Then I will ask how will we approach this problem? How to find the answer? I'll ask - what happens when we add 2 even numbers? What happens when we add an odd an even number together? Is this always the case? How can we test to see if this is true? Hopefully they will come up with something otherwise I am stuck! 4. Young adult level - no idea how to do this. Need help please! **Technique 2:** Young kids level. Will use matchsticks as manipulatives. I will let them play with it, creating shapes. I will ask them if they notice anything interesting. Present the problem and see what ideas they come up with for getting to the solution. Kid and Young teen level. We haven't really done multiplication at this level so I don't know how to approach this without them getting overwhelmed. Might try with beans and they will be able to see how quickly the beans grow. Might introduce them to the calculator and play around with that. Young adult level. No idea.

**Answer** by abrador
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Jul 06, 2013 at 02:26 PM

Problem #1 (rotating cogs). A variation for young children. We all stand in a circle, holding hands, and sing a dorky math song. Ok. Then, in a ripple effect that begins from the designated starter-person (e.g., the youngest child), and spreading to the right (anti-clockwise as viewed from above, every other person turns around facing out of the circle, but holding hands again. This continues around the circle until the ripple reaches all the way back to the first child. But wait... did the pattern work there, or were there two kids facing in-circle holding hands? Oh oh.. what do we do? Either we bring in another person (or cat, or stuffy, or action hero), or one person has to volunteer to stand out for a moment. Who should that be? And what if we were 13 people in our group? Would that work? 22? 99? [As an aside: As I was imagining this in my head, I got into topological confusions over how the hands would hold -- literally how the palms would clasp, thumbs and all. Is it like shaking hands?... Yes, shaking hands is like a degenerate form of a math dance :) . Then I remembered from dancing salsa that this works very well, angular momentum and such.... Actually, talking about folk dances, instead of holding hands one could lock arms at the elbows, which somehow makes for a more cohesive circle of people. In any case, there should be more mathy dances, just like those Hungarian CS dances.]

**Answer** by abrador
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Jul 06, 2013 at 02:47 PM

Problem #2 (head/point). Because my youngster is a very avid builder, I would look for objects that can connect to each other in different ways depending on whether they are head/head or head/point, and I would think about qualities of the various structures emerging from this, such as their engineering qualities (how strong is this structure? -- what can it support or pull?). And i would want these to be household objects. So for example take belts, the sort we wear. I am guessing we should collectively have about 7 of these at home. They best interconnect as head/point. It's tough connecting belts together as head/head or as point/point, so there is an intrinsic bias toward preferring head/point, and there might be a humorous challenge for connecting like sides (h/h, p/p); we can think of these as snakes dancing in a circle... Which reminds me of my variation on Problem #1 (rotating cogs. Hmm....

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