Friday, February 24, 2017

More Math: Group Division and Dividing Decimal Fractions

More math. Yes, more math. T is pretty close to the end of the elementary Math album. But there are times when we need to go back and review something that, well, just isn't there in the brain anymore. This happened recently with fractions and finding least common multiples. And this happened with S, when we had to back and do the golden-bead change game before continuing work with the Large Bead Frame. She is quite far behind in the math album, but who's counting? Me? Yeah, probably. But after a circle back around, and a rather quick-ish review, the kids take up their work where they left off and keep on going.

Here, T was finishing up group division. Yes, you go back to the stamp game! This comes after distributive division in the form of racks and tubes. I had to really try to wrap my head around this lesson to try to figure out how this is actually different than distributive division.

In distributive division, you say, okay, I have two friends, and including me, there are three people who get to share the cookies. Okay, I've got a basket of cookies here, and I am going to distribute them, "one for you, one for you, one for me. Now, one more for you, one more for you...." You get the idea. If there are twelve cookies, each person ends up with four distributions.

In group division, we say, we have 12 cookies in the basket and two fuzzy blue friends with big bulgy eyes that don't use articles in their sentences. We also say that you are also blue and fuzzy and you want cookies too. Your first blue fuzzy friend says, "Me hungry, give me cookies?" You take the basket and start making groups of three cookies and tell the other monsters no one can sample till you are done counting. This first group of three cookies contains a cookie for each friend and for you. You can make four groups of three cookies using all 12 cookies. So, that means each monster can get a cookie 4 times.

And then I said, isn't that mostly the same idea?

Well, it gets better as your divisor increases!
Before the divisor increases, we'll take the example above.

T made piles of seven of each category. (It would be like he gave six friends and himself each hundreds first, and then tens, and then units.) There are two piles of (7) hundreds, six piles of (7) tens, and three piles of (7) units with one left over. Each of six friends and T would get a hundred stamp two times, a tens stamp six times, and a unit stamp three times. The quotient is two hundreds, six tens and three units, or 263.
Here, T used a two digit divisor. In the top row he made (32) hundreds. In the second and third rows, he made (32) tens, and in the fourth, fifth, and six rows he made (32) units. He made one group of hundreds, two groups of tens, and three groups of units, so his quotient was 123. Am I losing you yet?
Here, he made one group of (48) hundreds. He made no groups of (48) tens, and one group of (48) units which you can't see in its entirety. This meant that his quotient was (1) hundred and (1) unit or 101. Now I am positive that you are turning blue and fuzzy and thinking about cookies out of sheer "what???" slack-jawed confusion. This is group division. It seem mind-bending to me, but T seemed to absorb it like a-no-big-deal-sponge. And then we'll see if that no-big-deal-brain retains any of this.

T had a tendency to stack his groups into piles. In this way, you can't see a multi-category group as easily. If the tiles are laid out all next to each other it is easier to see that you have (48) hundreds, rather than a stack of some number of green and red tiles.
T also finished up the decimal fraction part of the album. Here he is dividing decimal fractions by decimal fractions. Here we were using distributive division.

Above, his problem is 8.6 / 4.3 =. He put out 4 green unit skittles and 3 blue skittles to represent tenths. (The skittles were borrowed from the stamp game.) He then counted out 8 green unit cubes and 6 light blue tenth cubes from our decimal fraction materials. And here he distributed the 8.6 dividend amount among his 4.3 divisor amount to find that each unit skittle received (2) units.  This was his quotient.
In this example, T is dividing 0.4 / .25 =. He has placed two blue skittles to represent the 0.2 in his divisor, and five red skittles to represent the 0.05 in the divisor. The green disk (also from the stamp game) is to remind us that the unit category would go here. He also has counted out four light blue cubes to represent his 0.4 dividend.
He distributed two light blue tenth cubes to each blue skittle, and then realized that he needed to get some hundredths to distribute to his red skittles. He exchanged each light blue tenth cube for (10) orange hundreth cubes, and distributed five of these to each red skittle. The blue skittles represent 0.1, or tenths, and are worth ten times what each red skittle is worth. (Red skittles represent hundreths or 0.01.) If you give blue skittles units, the red skittles would get tenths. If you give the blue skittles tenths, the red skittles will get hundreths. Blue skittles receive then times what red skittles receive.

He ran out of light blue tenth cubes to give out, and instead gave out orange hundreth cubes to each blue skittle. Then he needed to get some thousanth cubes to give out to each red skittle, so he exchanged a single orange hundreth cube for (10) green thousandth cubes and distributed one to each red skittle. He continued distributing and exchanging until he ran out of cubes.

Then we asked, "what would one unit have received?" There wasn't a unit in our divisor, but we put the green circle tile as a place holder. And what one unit would receive is our quotient. A unit skittle would receive ten times what a blue tenth skittle would receive. In this case, a blue tenth skittle received 0.16, so a unit skittle would receive 1.6. Our quotient is 1.6. And now, I am quite sure I've lost everyone.

Somehow a minecraft guy got in this shot and he is doing some exchanging.
This is the same deal, but the divisor doesn't contain any tenths and I believe the quotient is a repeating decimal.

The problem was 0.36 / 0.027=.

The divisor is represented in skittles and the missing place values, units and tenths, are the blue and green circles. T started out with three light blue tenth cubes and six orange hundreth cubes. He distributed a tenth cube under each of the red skittles. Then each green thousanth skittle received a hundredths cube. T didn't have enough blue tenth cubes to give each red skittle another distribution so he exchanged his remaining light blue tenth cube for (10) organge hundreth cubes and gave one of these to each of the red skittles. The green thousandth skittles then needed to received green thousandth cubes, so T exchanged an orange hundreth cube for (10) light green thousanth cubes and distributed these. T continued distributing and exchanging to find that this problem never ends because it is a repeating decimal. He also figured out that the unit skittle would have received 13.3333.

Interesting stuff humn? I am glad that T understood it all because I barely understood what was going on. I feel calculus is easier than this.

One of the very cool things about Montessori math is that the child builds upon the concepts they've already absorbed. Color coding is reinforced again and again. The child circles back around to reuse essential materials and in doing so, is invited to dig deeper into more complex topics. To the outsider, who learned math entirely differently, it is like learning a new foreign language to figure out how to present these lessons. But to the child, who has been absorbing and using these coding systems and apparatus for 6 years running now, the mystery and complexity of the mathematical operation is completely eliminated and the genius of mathematical illustration is allowed to shine through. This is what continually amazes me about Montessori's presentations.

Ah, a weekend is upon us and the weather will be a bit cooler! (I know, you are probably like, for real? It is February!! REALLY! It's been near 90 a few days running now and we are having to water grass already! 80s will be nice.)  I think we'll be barbecuing some tandoori-style lamb kabobs, baking gluten-free-egg-and-dairy-free pitas, planting hollyhocks, phlox and daylillies, doing a ton of other lawn and garden work, and cheering on our first lacrosse game of the season. Okay, well, S and I won't be there at the game. We'll be at a flute lesson trying to figure out dynamics and other fun stuff. Hope everyone has a wonderful weekend and week to come!

1 comment:

  1. I had a similar adult experience with the racks and tubes for division.

    Back up to division with the bows:
    During training, I was sitting there like "What on earth is going on? I don't get it, this is crazy. I know something good is going on so I TRUST it, but HUH!?"

    Most people were nodding like they got it, a few slightly glazed looks.

    And one woman burst into tears.

    Then we moved into racks and tubes. She kept crying but kept telling the trainer she was fine, go on, continue.

    SHE understood long division for the first time in her entire life. She'd never understood WHAT she was doing before.

    I was at the other end of the spectrum. I understood SO well, the materials were a hindrance for me.

    Everyone else was somewhere in between.

    We ALL had to go back and work with this material intensely - either from just now "getting" it or from the need to figure out what is going on with this material.

    And honestly? I'm glad I didn't draw it for my math exam - I would have failed. It took presenting it to my son and his friends in co-op a few years later, before I truly-truly-truly understood the racks and tubes.

    It took me a short time (longer than I would like to admit) to understand the group division too. I really had to work with it before I saw the columns and the groups.

    So - you're in good company!