Alex the Geat and Fractions

We've been focusing on history a lot this summer, trying to finish up the Ancients before fall. Not that you'd know it to read my blog! But we have. We've made it through Alexander the Great, who, it turns out, is really very interesting. So, we're reading along and learning about all the areas that he conquered (all of the known world, basically), and what he did in the areas that he conquered, and we learned that he had them pay tribute. The book estimated that it was about 10% of their production that the conquered areas had to pay. Now, we've talked some about fractions, but they're pretty tricky, so we stopped and talked about that. It went like this:

"The conquered areas had to pay one tenth of their production. Do you understand how much is one tenth?"

"No."

"Ok. So, if they have 10 cows [I draw 10 really ugly cow-like-objects on the markerboard], then one of them gets sent to Alexander. [I color one of the "cows" purple.] How would you write one tenth?"

We figure that out.

"If Alexander was really nice to the people, then they might only pay one ninth for tribute."

I think this sounds like a good time to talk a little more about fractions, so I go grab several pages of paper, which we pretend into chocolate bars, and some scissors for dividing them.

"OK. Which do you think is more, 1/2 or 1/3?"

"1/3."

"OK. Let's see if you're right."

I cut the first "chocolate bar" into 1/2s, and the second one into 1/3.

"Which one is bigger?"

He points to the 1/3; I set it on top of the 1/2.

"You'd rather have 1/3 of a bar, instead of a 1/2 of a chocolate bar?"

"NO!!" [giggle]

"OK. I'm going to divide the next one into quarters. Are they going to be bigger or smaller than the 1/3s?"

"Bigger."

"OK." [cut cut cut] "Here it is. What do you think? Is it bigger?"

"No."

"What if we make fifths? Will those be bigger?"

"No."

"OK." [cut cut cut] "Fifths are tricky to cut. These aren't perfect. But they're pretty close. What do you think, are they bigger or smaller than the fourths?"

"Smaller."

"Why?"

He had to think about that one for a while. If the numbers we were writing on the bottom were getting bigger, why weren't the pieces bigger and bigger? That was a tricky question. We talked about how that bottom number was talking about how many pieces we are making, and so what if we had 1/100? That's dividing our chocolate with 100 people! My goodness, that piece is going to be the size of an M&M! He thought that was pretty funny, and was now quite confident that he'd rather have 1/2, versus 1/100 of our "chocolate bar." So we went back to the original question: Would it really be nicer if Alexander was to ask the people for 1/9, rather than 1/10? Now, he was able to say he'd rather pay 1/10, and understand why that's actually a smaller number. I was pleased. And we continued to read about this interesting guy that took over the world 300 years before Christ.

Fractals

After Hero caught me reading this article, and asked what's that picture, I had to go ask one of my expert: my BIL. He informed me that it's a fractal, and my son immediately wanted to learn about them. Okaaay. I don't know a thing about fractals, so this is going to take some research. Turns out, fractals are pretty interesting. I found this site, which is supposed to be made for kids, though it still gets pretty technical from time to time. We read about them a little, and checked out some of the cool examples they have. There is some amazing variety in what a fractal might look like on this page!

We each made a Sierpinsky Triangle, using these instructions. We found some examples of kid-made fractals. And even a cool 3-D one made by preschoolers. That one I'd like to try to make, but I don't know how exactly. I don't have any triangle blocks. Where do you even get such a thing?

We haven't played with variables, as such, so he didn't understand this video, "Fractal Fractions," but he decided that he loved it! "I didn't understand anything about it. It's just so funny! I mean, I understood it, but it's soo funny!!"

And this one has some cool circle games that I'd like to play, but I'll have to see if Hero is up to playing with me, or if his interest is starting to wane, though, once again, he really liked the video. Especially the last couple seconds.

And, finally, here are some "fractal zoom" clips.

More Mathy Stuff

My 5yo thinks that negative numbers are cool and wants me to explain fractals. Fractals! I have no idea what a fractal is, beyond having seen a few pretty pictures. Needless to say, I spend a goodly amount of time trying to figure out mathy stuff so that I can help him satisfy his immense thirst. It's an interesting journey for me, since I left school convinced that I was math-impaired, and even went so far as to choose my major partly based on which of the options required zero math classes. (I studied Japanese. The math totally scared me off of astrophysics; I'm not so sure that I chose correctly.) Teaching this kid is fascinating because I'm learning so much. It has rocked my math world, and I'm discovering that this math stuff isn't so scary after all, and in fact, can be pretty interesting. Fun, even!

So, here are some of the things that I've been reading this week:

First of all, Let's Play Math is going into my feed so that I get to look at their stuff when they post it. That blog is fascinating.

Apparently, multiplication is NOT repeated addition. That idea (they say) works for integers, but no so much in other number sets. Like fractions. And mysterious things called "triangular numbers." And some other things that I don't believe that I've ever heard of. But, since Hero is starting to explore both multiplication and fractions a little, I'm trying to wrap my head around this idea at least a little bit before he decides that these topics are really interesting and inhales them. So here are a few of the articles I've been reading about this:

If It Ain't Repeated Addition, What Is It?
What's Wrong With Repeated Addition?
PUFM 1.5 Multiplication (This one has the pretty picture that spawned the, "What's a fractal?" question.)
It Ain't No Repeated Addition (This one's from a real live mathematician.)

We also watched some of Education Unboxed's videos. Education Unboxed is another awesome resource that I'll be visiting and revisiting. It's done by a homeschooling mom who started with Miquan, and has continued to collect teaching materials for doing math in that style, and made videos to share what she's learned. They're pretty cool. For instance, after watching this, Hero wants to build a pyramid with our rods tomorrow.

Cuisenaire Rods - Playing With Square Numbers from Education Unboxed on Vimeo.

And this one, also from Unboxed, is how I'll probably do multiplication with him the next time we play with that.


Introducing Multiplication and Division Concepts from Education Unboxed on Vimeo.

Though it may be a bit before we do that. He's enjoying doing "tricky math" which is 2 and 3 digit addition. He thinks giving me huge numbers, so I have to use our 1000 cube to answer his questions is pretty awesome. And he's getting pretty good with the addition, including carrying, though he hasn't seen it written out yet. We'll do that part soon.

Another thing I want to hit at some point with him, probably several points, is mental math. Only, I suck at mental math. Turns out, there are strategies for doing the stuff, and this post has a bunch of them.


Aaand, this one is not math, directly, but still related. It's a fun game to play with kids: How To Train Your Robot. Looks pretty awesome.

Oh, and those fractals he wants to know about? Turns out they're some kind of geometric shape. Like rectangles and pentagons, only different. And this site is for teaching kids about them. I'll definitely be going back there to check that out.

Money Games

I realized that Monkey is still a bit weak on money. Rather weak, in fact. We've done it before, but it's been quite some time and so we're practicing place value with coins this week. I got the rods out and made a "key" for him, so if he forgets he can look and see: a penny =1, a nickle=5, dimes=10, and quarters=25. I think I'm going to get some rubber bands out next time to bundle the rods into units of 25, rather than just individual rods adding up to 25. And then, although it's hard to see the movie I made, I also mirror what he's doing with coins in the rods. It seemed to work out pretty well.

On Learning to Count

Introducing Teen Numbers with Cuisenaire Rods from Education Unboxed on Vimeo.

We didn't use rods or Miquon when I was teaching Monkey how to count; I didn't know about them yet. But I love the way it's done  in this video (And what a cool table! How cool that the little girl gets to look at her exciting new discovery from all those angles!) One thing that I noticed as I was watching this clip is how nicely the Mom waits. I don't think I was that good at waiting through the faces and sillies that go with teaching a precocious child. I do think that my little Dragon will be ready for this sort of work early - he loves to "use" the rods while Monkey's working, and will often pile them up and then tell me "Eighteen!" Don't ask my why he loves that number; I have no idea. I must remember to keep my trap shut enough to let him actually figure it out, like this mom did. Hats off to her!

Classical Carnival: Call for Articles

It's been a while. Actually, quite a while. The carnival system was having issues, I got behind and overwhelmed, and the Carnival didn't happen. But I think I'm ready to try again with it, and I'm hoping that there's still interest. So click on over and share a post or two with us about your classical homeschooling efforts. Talk to us about what makes classical right for you. And, if you'd like to host a future edition on your blog, leave me a note in the comments and we'll talk.
 

Pianist

I was practicing this evening, but had to get up for a minute, and when I came back I found I'd lost my turn!

Owl Lapbook (Finally)

We weren't very efficient with the owl lapbook, but it's done and I'm finally putting it into his binder. It turned out nicely!