Odd-one-out, Rolling Dice

Attending: G, S, N

 

Activity 1: Odd One Out

We started out by playing “Odd one out” with cards I made.  The cards have pictures of obtuse and acute triangles, various quadrilaterals, and higher polygons.  Some are convex, some not.  Some are simply connected, some not.  Some shapes are all curved, some have curves and “corners”, some have curves, corners, and straight sides too.  The game works as follows:  I lay down four cards and ask the boys to tell me which one doesn’t belong and why.

I start off easy:  three reds and a green.  N immediately got it, S held back a bit so that he could give a more subtle reason.  It turns out that the green shape wasn’t convex, so he says that is the real reason.  We go on to harder problems — hexagons vs. heptagons, convex vs. non-convex, corners vs. no corners, etc.  Following S’s lead, we start spending more time with each set and try to find as many solutions as possible.  N seems to follow, but doesn’t contribute secondary solutions.

Activity 1.5: Tower of Hanoi

G arrived, and S set up a massive Venn diagram classification problem for him.  N is bored, so I have him do Towers of Hanoi again.  He is very fluid at moving the top three blocks around, capable with the fourth, but still messes up a bit with block 5.  Eventually he gets it, so I give him 6.  Eventually he gets that too, but needs a little help in the depth of it.

Activity 2: Dice rolling

I set up a 20-by-12 grid with the numbers 0,1,2,3,…, 19 written below the 20 columns.  I explain to the boys that we are going to each pick a number then have a race.  We’ll roll two die, and every time the numbers on the die add up to your number, you move forward one space.  The first person to get to 12 wins.

I tell them that just to see how the other numbers do, I’ll mark how each one is doing (a weak explanation, I have a better variant in the next round).  I have the boys pick.  N picks 7.  S is upset, but picks 6.  G picks 5?!  I take 8.  Obviously this isn’t the first time they’ve played dice.  We play, get a nice distribution, and 6 wins.

The next time each of us picks 3 numbers (I get the hopeless number 1).  This made the game much more engaging for them:  they are constantly trying to figure out where they stand but the game keeps moving on so they need to find mental counting shortcuts.  They also pay much more attention to the tails of the distribution.  S won again, but nobody was too competitive despite all the cheering that was going on while they played.

We kept track of the two distributions on the same chart:  for the first game we placed Xs for each roll, for the second game we placed Os.  This made it easy to compare the distributions and discuss.  This activity went very well.

Afterwards, S went and drew the expected distribution, and figured out the expected distribution for 3 dice rolls too.

Pigeonhole Principle, “Probability”, and 5 choose 2

Attending: S, N

Working with stuff from Zvonkin’s book again.  I’ll start getting more creative one of these days.

Activity 1: Finding pairs of shoes

S was late to come down, so I started with N.  I started by telling him the story of a poor man who had all of his shoes in his basement, but the basement light was out and he could only fetch one at a time.  He had some red shoes and some blue shoes and needed to get a matching pair.  I used counting frogs for the “shoes” (N called them “frog slippers”) and used a winter hat for the “basement”.  To start, I put 2 blue frogs and 2 red frogs in the hat then had N draw one at a time.  The first round went like this:

He draws a blue. I ask  ”Do you have a pair yet?”

N (laughing): “Noooooo!”

He draws a red.  I ask “Do you have a pair yet?”  He says “No”.

I ask “How many more draws do you think it will take?”

He laughs and says he’ll get a pair no matter what next, because I didn’t put any other colors in.  He drew a blue and got a pair.

So the pigeonhole principle is intuitive to him, at least when it’s about to be binding.  We repeated it a few more times so he could “get lucky” and get a pair in just 2 draws.  We talked a bit about impossible (getting a pair in 1 draw), possible (getting a pair in 2 draws), and certain (getting a pair in 3 draws).

Then S joined us and I added a third color.  We went through it all again, and once again N knew when it was certain he’d get a pair.  We tried 5 colors and it was the same, but he couldn’t say how many draws would be certain to yield a pair until he had already started drawing.  S figured out the formula for the number of draws needed to be certain of a pair when you have colors.

We went on to the story of the three-legged man getting shoes from the patient (it was fun acting it out).  We started with 2 colors and worked up to 5.  It took some effort but S found the formula for the number needed to be certain of a pair.  We also noticed that while it was possible to get a pair in 2 draws, it was pretty unlikely.  I didn’t say any more about probability.

Activity 2: 5 choose 2

S disappeared a bit and I asked N to arrange 3 blue frogs and 2 red frogs in a row.  Then I asked him if he could make a different arrangement.  N problem.  Then I asked him to make as many different arrangements as he could.  He started off surprisingly systematically:  he put together all patterns with the two reds next to each other:  RRBBB, BRRBB, BBRRB, BBBRR.  After that he appeared to be going randomly, but from the way he was staring at it it seemed like he wanted some structure.

S came down and saw how N started.  I told him what we were doing and he structured the solutions in a really interesting way.  He said that you could think of the row of frogs as making a loop, so the two reds would always be either touching each other or have just one blue between them.  So to get all 10 of the solutions you just put down those two examples and look at the 5 rotations of each one:

Reds touching:  RRBBB  BRRBB BBRRB BBBRR RBBBR

Reds separated by one: RBRBB BRBRB BBRBR RBBRB BRBBR

Even better, he ran out of red and blue and had no problem representing the rows as “3 of one color, 2 of another”.  I showed him my decomposition:  4 with Red in the first position, 3 with the first red in the 2nd position, 2 with the first red in the 3rd position, 1 with the first red in the 4th position and noted that my decomposition made it natural to see that the answer would be a triangle number.

Eventually N got all 10, but didn’t know if there were more.  Later in the day he wanted to do it again.  I also had G work on this in the afternoon (he was studying Tagalog when his little brothers were with me).  I think that making structured lists like this is probably good for him too, so I’ll try to schedule so he can join us.  Funny, it was supposed to be just me and N.

Permutations, Towers of Hanoi

Attending: S, N

Activity 1: Permutations

I got this idea from Zvonkin’s book.  I set out 4 “armies” of 12 counting frogs: one red, one blue, one green, one orange.  I  also set out 12 wood blocks I called “tables” and explained

The armies are tired of fighting each other and want to have a great feast.  You need to make table arrangements for them, but here are the rules:  every table must have one member from each army, and the seating arrangements must be different at each table.  If you break these rules, there will be no peace in the land.

I set up two tables to demonstrate, “accidentally” setting the second one identical to the first, then correcting it by rotating.  This was intended to give a hint about how to make new arrangements and to show them that the rules allowed rotations.

I let N get started and he put a couple of tables together, making a few repeats that I pointed out and he corrected.  Then he asked S to take over.  S methodically set all 6 tables that had a blue frog in the North seat.  N didn’t quite catch on, but he did realize that as long as he didn’t put a blue frog in the North seat his chances of success were decent.  Again, with a few corrections he got it.

While N was working I asked S:  what if I said that rotated seating arrangements don’t count as different?  How many tables could we set?  He answered correctly.  Then I asked: what if reflections don’t count either?  He immediately said there would be 3, but came to doubt a bit later.

Activity 2: Towers of Hanoi

Yesterday I taught N the Towers of Hanoi puzzle, and he sort of got it, but stumbled a bit on larger towers.  Today I gave him a 5-item tower and asked him to solve it.  Again, he had some trouble at the end.  While doing that, S and I discussed the number of moves needed to shift towers of different sizes.  He saw the algorithm to use, and pretty much deduced the recursion.  He also deduced the right rule from observing the first few numbers in the sequence.  But he couldn’t deduce the rule from the recursion until we had a discussion about induction over lunch.  I also told him the legend of the monks in  Hanoi moving 64 disks around and asked him to estimate how long it would take before the world ended.  I let him use a calculator and he answered at lunch.

Wrap-up

While N and I worked a bit on the towers, I asked S to produce the three unique seating arrangements for the frog armies.  This is when he had a bit of doubt:  did I mean reflection along the North-South axis or East-West (I guess NE-SW didn’t occur to him).  I told him that they were the same since we could rotate, but should have said less and said it differently. He got three distinct representatives fairly directly, but it required some thought.

I’m back

I’m going to start blogging again, but I’m too lazy to come up with a new blog name. So here I am.

The new material is going to be different. I planning to keep a journal to document the Math-related activities I’m doing with my three elementary-school-aged boys. This means it will mostly be about ideas that kind-of worked, with the occasional resounding success or abject failure.

I’m doing this for my own benefit. I want to have a record of what we’ve done so I can review it and learn how to do things better. So it might be boring. You’ve been warned.

Changes

Obviously posting has been slow lately.  It’s about to get slower.

I’m retiring this blog.

The kids are doing well, learning a lot, and all of that.  We are traveling, trying to sell our house,and dealing with other tedium.  I’m appalled at the way our government is handling the economic situation.  And so on and  yadda yadda.

I’ve just gotten tired of writing about it.

I’d like to spend my time writing better researched posts, polishing them more carefully, and tying them together more coherently.  I also want to spend some time thinking about how we can use technology to improve the education process, and start building tools and content directly targeted at students and teachers.

Of course I can do much of this by just making this a better blog, but I’m choosing not to do that.  “Coherent” and “polished” would ruin the spirit of this blog anyway.  This blog is about picking fights over homeschooling, calling people on their hypocrisy, and cute little “kids say the darndest things” posts.

It is also about learning — both me and the boys.  The Mathematics posts are still my favorites, especially the ones collected here and the little series about the Riemann zeta function (which I really should continue).

So what is in the future for the Schmidt family?

We’re moving to Seattle where I will start working on a Ph.D. in Computer Science at UW.  I am excited about getting started, and once I get established there I’m sure I’ll have plenty to say about it.  In the meantime, we’ll be on the road most of the next four months, we’ll have spotty internet access, and I will squander much of that access on facebook and runsaturday.  I’ll still pester some of you with my comments too, and do my best to keep up with your posts in my reader.

I do want to thank all of you who read this blog and all of you who left a comment or two or twenty.  I had a lot of fun with some of the discussions and was genuinely surprised at the response I got.

BTW, if you want to buy a house in Texas, I’ll give you a good deal

Tschüss!

Time for substance, not rhetoric

I’m busy and I don’t have time to fisk Obama’s whole speech.  That would just leave me irritable anyway.  But there was one little section that jumped out at me:

The weight of this crisis will not determine the destiny of this nation. The answers to our problems don’t lie beyond our reach. They exist in our laboratories and universities; in our fields and our factories; in the imaginations of our entrepreneurs and the pride of the hardest-working people on Earth. Those qualities that have made America the greatest force of progress and prosperity in human history we still possess in ample measure.

Oh really?  So that’s why we are giving trillions to bankers and letting the laboratories and universities, fields and factories, entrepreneurs and hardworking people rejoice over a few crumbs from the ‘stimulus’ package?  Just imagine the good jobs that would be created, the innovation we would see, the wealth that would be created if we actually funneled all these banker giveaways into science, the arts, and education.

But no, it is more important to preserve the status quo, preserve income inequality, and do what we can to make debt slaves out of our populace.

He goes on to say

What is required now is for this country to pull together, confront boldly the challenges we face, and take responsibility for our future once more.

I’ll pull together when you start putting your money where your mouth is.

By the way, our health insurance premiums went up 44% this year.  I feel lucky that I even have my lousy coverage.  I’m still waiting for that ‘change’.

Introducing binary numbers: part I.I

In my first post about binary numbers, I mentioned that I wanted to teach my kids how 0′s and 1′s could be used to represent colors.  Instead of talking about it too much, I just built a little web page so they could explore it on their own.

Try it here.

It is pretty bare-bones, but seems to get the point across pretty well.  My kids have already given me a long list of requested improvements: sliders, preset colors, mixing arbitrary colors, and more.  Please let me know if you have any suggestions!

This little web page did lead to some interesting discussions.  It let us start talking about how a computer would store a picture, one block of color at a time.  Then we noticed that if we just changed one “unimportant” 0 to a 1 or 1 to a 0, we couldn’t notice the change in colors.  This means that we an send secret messages in pictures:

1. Take a picture
2. Write your message in binary (using a code like the one we made up in my last post)
3. Change every fourth bit in the picture to match the next bit in your message
4. Now you have a picture that looks just like the original, but contains a secret message.

Gunnar really liked this idea. Of course we didn’t talk about file formats or any of that stuff yet, so he doesn’t really understand how to do it.  But we’ll get there.  And we’ll send secret messages.