# MATH with my KIDS

## Kid Snippets: Math Class

If you’re a math teacher, is this ever what class feels like?

## Latin Squares, Squared Squares, and Legoed Squares

I introduced my kids to Latin Squares the other day. If you know Sudoku then you have seen examples of Latin squares. The idea is to fill in a grid of squares with colors or numbers or some other symbols, such that each symbol appears exactly once in each row and each column. (see also the Wikipedia Latin square entry). I handed them graph paper and let them loose. This is what my eight-year-old produced:
Note that she produced some Latin squares, but also explored other ideas.

I wasn’t sure initially how long my kids would be content to explore Latin squares, so I also planned to tell them about squared squares. A squared square is just a square made up of smaller squares (this is easy to accomplish). A perfect squared square is much more difficult, consisting of squares each of different sizes. For a long time it was thought that perfect squared squares didn’t exist, but they do! Here is the smallest (in some sense) one:
We got excited about it and decided to make one for our wall:

Next, the ten-year-old wanted to produce some squared squares of his own, but found the perfect ones difficult (they were thought to be completely non-existent for a long time after all). He settled for producing imperfect ones. After making an imperfect squared square on graph paper, he decided to reproduce it with Legos. Here is the result: This led us to considered perfect Legoed squares. That is to say, squares made up of some number of Lego pieces, each piece having a unique size. Here is one of our first examples: It’s a 3×3 made up of a 1×1, a 1×2, and a 2×3. Here’s another, somewhat larger: We found a bunch more (maybe ten or so total). I’m not sure that we got all of them. How many can you find?

Look for up-coming posts on: 1) further progress in building our marble computer, and 2) teaching my five-year-old to count (which required some deep thought about just what counting is).

## Sequences and Creative Math for Kindergartners

I visited my kids’ classes for career day in May. I have a son in Kindergarten and a daughter in pre-K. I also visited two other kindergarten classes. I told the kids that I am a research mathematician: mathematician means I do math. Research means I do math that nobody has ever seen before, math that I make up.

I mentioned that math is lots of things: It’s thinking about shapes and numbers and patterns, and other stuff. This day we were going to do numbers and patterns.

I showed them a board covered with three lines of (blank) sticky notes. Under each note was a number.The point is to try to guess which number is coming next.

We started with the blue line of numbers.

Once I had uncovered the 1 and 2 almost everyone expected the next number to be 3.

When I showed them that it was a 1, they caught on pretty quickly.

They had fun shouting out the 1-2-1-2-1-2 pattern as I frantically tried to keep up with them on the board.

Next, the pink ones.

I started by showing the 1-2-1 with a warning that I was being tricky. A 2 should come next, right? Wrong! But I didn’t tell them what it was yet. I put the sticky notes back on and explained that I was going to pull them off in a different order to see if it gave them a hint. (At this, some kids suggested I pull them off starting at the right instead of the left.)

I pulled off every other sticky: 1-skip a sticky-1-skip a sticky-1-skip… Okay, done with the ones. Next we have:

2-skip a sticky-2-skip a sticky-2-skip… (pulling off every other sticky that is left after the first pass). Okay, done with the twos. So the next one should be (the one that everyone thought should be a 2):

It’s a 3! And most of the students guessed that it was a 3!

And so it goes: 3-skip a sticky-3-skip a sticky-3 skip…

Next we have the fours: 4-skip a sticky (at this point we hit the end of the board, but if the board were longer…we would continue 4-skip a sticky-4-skip a sticky-4-skip…

Only one pink number left…

Everyone knows it’s a 5 by now! If we had a long enough board we would have 5-skip a sticky-5-skip a sticky-5-skip…

The pink numbers are:

1-2-1-3-1-2-1-4-1-2-1-3-1-2-1-5-1

Some interesting things (I think) about these pink numbers: Once you’ve done the ones and twos the row looks exactly like it would if you had pulled off the same patterns on the blue row! Nevertheless, most kids pick up on the fact that 3 should come next (well…I did give a big clue “done with the ones! . . . Done with the twos!”)

These pink numbers are called the ruler sequence, which comes up in the towers of Hanoi problem.

Now for the greens (the stickies look yellow in these pictures, but they are really green (yellowish-green)).

This row is really sneaky (but usually the kids understand it before their teachers! A testament to the freedom that young minds who haven’t had mathphobia drilled into them are capable of!), so I give a big hint. What do we have so far (on the green row)? One. How many ones? One one.

So that is what comes next. You start with a 1 then you say “look, it’s 1 one,” and that’s what you put next 1-1.

Now what do you have? 1-1-1. You have 3 ones. So next comes:

3-1.

Now you have 1-1-1-3-1. Four ones and one three. So next is:

4-1-1-3.

Now you have:

1-1-1-3-1-4-1-1-3. That is, six ones, two threes, and one four. So next will be:

6-1-2-3-1-4.

And now you have:

1-1-1-3-1-4-1-1-3-6-1-2-3-1-4. That is eight ones, one two, three threes, two fours and one six.

There is only room on the board to put 8-1. But if we had more room we would keep going. If we had paper going all the way around the school… I asked the kids if they thought we would ever get a 5. We haven’t so far! (At this someone always pointed out that we got a 5 on the pink line. A good observation! But we are just looking at the green line now.) What about 10? If we kept going would we get a 10? Would we ever get a 20? Or 100? Would we ever get a thousand or a million or a billion?

I’m pretty sure that you do get a 5 and a 10. I don’t know about 20, 100, 1,000, 1,000,000 or 1,000,000,000. I’m guessing that you do get them… Why? well, my reasoning is this: in some sense you have infinitely many chances to hit those numbers. Maybe at one point you will have 20 ones. If not, maybe you will get 20 twos. Of course maybe you will skip right over 20 twos. So maybe you will get 20 threes or 20 fours or 20 fives or…, you get the idea.

I told the kids that I didn’t know if you ever get 20 or 100 or 1,000. All I know is that you keep getting bigger and bigger numbers (can you see why?). I pointed out to the kids that this is a math question that I don’t know the answer to, and (as far as I know) neither does anybody else. This is math research, and it’s math research that they can do! They can figure out if we ever get a 5 or a 10. A million…probably they won’t figure that out until they learn some computer programming…

The kids were perfectly willing to accept that this is a math question that has an answer, but nobody knows that answer yet.

As for me, I’m conjecturing (conjecture=educated guess) that if you continue with the blue numbers, you will eventually hit any positive, whole number that you might care about. I already told you my reasons for making this conjecture. Can you prove it? Can you prove me wrong? If you’re a teacher, give the problem to your students. Maybe they’ll have an idea about it. Then let me know.

Finally, I wanted to let the kids make up their own number sequences. I handed out sheets of paper with five slots for numbers and five sticky notes. I told them that they could write any numbers they want in the blanks. They could be tricky, or they could be easy (like 1-1-1-1-1). Then, cover them up, and see if someone can guess the numbers.

Do they have to be able to articulate some reasoning for the patterns behind their numbers? No. But if they can, that’s great. There doesn’t have to be any reasoning at all. The point of this exercise was for the kids to do some free-form creative math. Just release your inhibitions (actually the kindergartners don’t have many inhibitions. The grown-ups on the other hand…) and write down some silly or crazy or boring or weird or hard-to-guess or happy, or sad pattern. The kids loved this part. They especially loved seeing if I could guess their numbers.

Let me know what you think about these sequences and this activity!