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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jamestantongears.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 Denise Gaskins
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Jul 14, 2013 at 05:27 PM

Since my teen group is meeting a week late, I'm still working on this assignment. I decided to print out the problems on full-sheet sticker paper, so I can cut them up for the kids to stick in their notebooks. The triangle problem and the absolute value equation problem are both beyond what my students have studied, so I am giving them the basic situations without a question, just to see what questions they can find to ask. Only on the exponents problem do I expect to get solutions (I'll save it for last, so we can end the day with some sort of resolution.) I modified Carol's Problem Solving Techniques file (from the Assignment #3 page) and combined it with excerpts from [Math Forum's Notice & Wonder worksheet][1] to make a couple of problem solving stickers, too. I'll make new problem-solving stickers for each week, as we go through the techniques. [Problem Solving Stickers, week 1][2] [1]:
http://mathforum.org/workshops/nctm2010/handouts/MathForumNoticeWonderRecordSheet.pdf [2]: /storage/temp/95-math+group+problem+stickers,+week+
1.pdf

math group problem stickers, week 1.pdf
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**Answer** by ali_qasimpouri
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Jul 14, 2013 at 02:32 PM

Our strategy is based on collaboration between two brothers Parsa and Amir. We play Gears and Pins for warm-up! Then they face problems and try to start from what they know! They should be critical students from start to end of class from different points of view. Parsa has important role to simplify what Amir does not know.

**Answer** by Lobr23
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Jul 10, 2013 at 09:30 PM

1. Gears: 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. 2. 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. 3. 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.,

**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 mirandamiranda
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Jul 09, 2013 at 11:33 PM

I just wanted to share a conversation I had with my husband about the problems last night. We were talking about the the pin problem and he wanted to know why the pins had to be in a loop. He tried to put them (well, we used matches) together in a star shape and we started discussing how many head to head connections we might be able to get if we omitted the loop requirement. We tried to connect as many heads to as many other heads as possible, although I queried how we would count such connections. Match heads are small so I thought only 4 could connect all touching one another, but then he put a match on top of them and made another! He claimed we could connect infinitely many, but I am not so sure... How many dimensions are we allowed to use?!
We didn't get as far as recording our observations (it was past bedtime but you know how it is when you start talking about maths...) but I really appreciated the reminder to let go of the rules and see where it takes you.

**Answer** by nikkilinn
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Jul 09, 2013 at 03:50 PM

Problem 1 - We have the Gears!Gears! set, but haven't opened it yet, so I will let the girls (5 and 2 1/2) experiment with them and soo how quickly they figure them out. Problem 2 - We will use some magnet rods and connectors, matchsticks and dolls to work this one out. I loved the idea of "stinky feet" that another participant had, and I will most likely use that as well! Problem 3 - For the 5 year old, we will use the iPod app, Flow. It uses the same concept and gets more difficult with each level. I may let the 2 1/2 year old try her hand at the beginning levels too. I will also use some animal families and fences we have with our barn set, and let her help the "mommies" find their babies.

**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 rachaeljanae
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Jul 08, 2013 at 02:51 PM

I don't like to get to creative with presentation, unless it really facilitates the thinking process. Most of the time, I've found, the more creative I get, the more it interferes with the actual problem. So my plan seems overly simple, but I think my kids will bite. Here it is: Problem #1
http://www.youtube.com/watch?v=WRYZHWSSs_Y Ask questions: • How is it that movement in one direction can produce movement in another direction? • Can you think of other examples where moving an object in one direction, causes another object to move in another? List examples. • How can we demonstrate this principle to someone who doesn't understand? Are there household objects we could use? Problem #2 Give the kids each three pencils. Give an example of all three pointed the same way. How many other ways could we arrange the pencils in a loop? Add another pencil and do the same things. Conduct an experiment and collect data. Do 4, 5 and 6 pencils. Find a pattern. Can we predict how many combinations we would have with 7,8, 9 and 10? If we multiply two by itself, what do we get? Another time, and another. Use cuisinairre rods and make each number in terms of ones, tens and hundreds. Is there a pattern emerging? Can we use a pattern to predict what the ones place will be each time? How can we demonstrate and teach this to our friends? Problem #3 Print off a copy of the original problem and pose it to the students. Give them scissors and have them cut out the circles. Use a stapler and string to connect each of the circles to the correct match. • Can you arrange the circles back into order and not cross paths? • How would this problem be easier? • Can you manipulate things around so it works better? Is there a trick?

**Answer** by sutton_c
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Jul 07, 2013 at 12:55 PM

**Problem #1** For my 5 year old and 3 year old girls, I am going to create a circus story about clowns on spinning merry-go-rounds. They connect the merry-go-rounds to make a huge spinning clown show. We have a motorized gear toy to play with. It is a little bit limited but will get the basic idea. And since they are actually spinning, they can see the directions easily. My mother-in-law also has some plastic gears that are like spinning lego blocks that you can spin with a handle . I can extend the circus story to acrobats using ropes and pulleys for up and down patterns. Actually this might be a better starting point since clockwise and counter-clockwise rotations don't have a lot of meaning to young kids in a digital world. For my grade 8 math class, I would stick with the gears and extend the problem to explore ratios and rates. Ratios of gear teeth determines speeds of gears. The story could be a factory trying to conenct a power source to a machine requiring a certain speed of rotation. The challenge would be to use as few or as many or a limited number of gears. **Problem #2** My initial thought here is to use bar magnets so that the connection and repulsion make the pattern obvious. Many bar magnets are coloured red for North pole and blue for South pole so colour matching works too. As an alternative we could decorate some popsicle sticks with our two favourite colours and then build shapes with matching (or non-matching) corners. As an extension, we could mix up all our sticks and try to make a shape with all the colours in the rainbow with no matching corners. **Problem #3** I think it is important here to have physical tools that work. I don't think drawing the connections will work for my young girls (and most of my struggling grade 8's). They need something physically connected that they can move around to help them visualize the paths. I am going to have my girls create a city by drawing roads on a big piece of paper. Then we will create buildings and cars out of cardboard and colour them matching colours. We will connect the pairs using thread or yarn of the same colour. We will glue down the buildings and play a game to try get the cars to move to different parts of the city without crossing the other cars path.

**Answer** by jessecarrell
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Jul 07, 2013 at 01:53 AM

We are creating a storyline for the problems; 1. We have at one end of the platform a gear driven treadmill, at the other end of the platform is the power source driven by a crank that moves only in one direction and all we have to connect them, to save us or reward us, is more gears. We then let the kids experiment with configurations to push away the undesirable stuff or pull in the reward. Then we change the platform (length and shape) and let them do it again. Finally trying to educe from them the pattern to produce the desired effect. 2. We have discovered life on planet PeeYu, they communicate by creating figures lying on the ground; the problem is that everyone’s feet stink horribly. Sending a message with lots of peoples head and feet near each other is offensive. We need to find out which shapes we can use to make sure that no one’s head is next to another’s feet, so as not to offend them and start an intergalactic war. We will use Lego figures and baby dolls to work with the various age groups. 3. We created a cooperative challenge. We have 6 kids, we will pair them up and place them on one side of the room. We will have 3 stations on a table parallel or spread across the other side of the room. Each station will have a desirable treat and a number. Each team will have an anchor and a runner. The anchor stays in place holding one end of the string. The Runner draws a card with a number for a station that they need to reach. The challenge is to reach all three treats without crossing the strings in order for them to get to eat the treats.

I love the "planet PeeYu" idea! I bet the boys in my group will enjoy that.

**Answer** by dendari
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Jul 08, 2013 at 11:54 AM

Vacation is over and time to get into play.
I almost brought these problems up to the myriad number of cousins this week, but they were so busy creating their own fun I didn't want to interrupt.
To introduce the gears I thought I would cut some basic gears from paper plates and have the kids play with them. They can even draw arrows on the plates to signify movement.
The pin problem would have been great to try the children turning around in the circle, but I won't have that many kids now that we are home. I think I will try any pin like object. Something with a head and foot. I've never really understood why we want to find the last digit in a large number though so it is difficult to determine why we want to learn this.
For problem 3 we will try string and people. Try to figure out a method of connecting three people to three chairs or something without crossing paths. We can scaffold like they did with the pictures, but moving the chairs from directly across to the final position.

**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 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 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 ChrisYu
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Jul 08, 2013 at 10:11 AM

Currently my plan is to lobby the principal of the elementary school where I work, to allow me to set up a 'problem bank' for the teachers. I think I'll try to set up the problems pretty close to how its laid out now, and adjusting as I get more feedback. I think the important part will be to try to emphasize problem solving means NOT giving away the answers. I know that initially you were looking for us to adapt these problems, but I find it hard to do so, until actually trying them out. I really like that problem of connecting the lines in Technique 3. That one works for all ages, and I can't wait to spring it on everyone I meet.

**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 adamglesser
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Jul 08, 2013 at 02:52 AM

Problem 1: For the 6 and 7 year old, I will let them play with real gears and see what happens. We plan on having them try to make the last gear in a sequence turn a certain way. Problem 2: We will definitely go the action figure route (and maybe include our 2-year old). The oldest boy did the extension problem (2^100) already. For that, I first had him try it on the calculator. It returned roughly 1.26e30. We had a discussion about why the letter "e" shows up and about scientific notation. I then had him compute (by head) the first 10 powers of 2. He quickly saw the pattern of the last digit. He ruled out 2 and 8 as possible ending because these correspond to odd powers. He didn't quite make the connection that he needed to divide 100 by 4 and check the remainder, so instead he used a laminated number chart to see if the sequence 2, 6, 10,... contains 100. When it didn't, he tried 4, 8, 12, ... and saw that it did. This allowed him to conclude that the answer is 6. Problem 3: For both of the older boys, we will place the areas taped on a white sheet, and place it in a transparent folder. We will let them draw on the folder with white-board markers and see if they make any observations. As we need, we can remove the paper and move around the different areas.

**Answer** by abrador
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Jul 06, 2013 at 02:52 PM

Problem #3 (Connecting three pairs on a surface). I remember this from my childhood. We used to speak of three houses that each should connect to three utility providers (water, electricity, gas) without the lines crossing. Yeah, ... Flatland civil engineering?... I think a nice variation on this for summer is to play it on the beach, where we connect lines in the sand. The last impossible connections demand digging tunnels! Perhaps we could bring some color yarn strings, which only add to the fun of digging tunnels. And, thinking back to my childhood version, we could even pour some water through the tunnel...

**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....

**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.

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