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!

Thursday, February 23, 2017

Rocks and Fossils, and Grammar and Other Projects

Old things that are hard: these two explorations weren't all that close to each other, but I'll try my best to remember my information sources.  If my mind seems a bit sprout-y it is because it is spring and I am presently wondering, HOW are the boys destroying my measuring tape outside? The SOUNDS are tipping me off that there may be some impermissible tape-measure behavior going on. Hold, on....

Okay, I'm back. First, different types of rock. I think that this is sedimentary, or layered rock. I adapted this set of instructions from Homeschool Den. I just used what I had on hand.
Here, we have layers people. Color-full-sugar-filled ones, which incidentally, the kids didn't eat because we are on an elimination diet. Amazing.

This was pre-heat metamorphic rock/sugar.
After we stuck the metamorphic specimen in the oven and added heat, it looked like the mess on the right. The specimen on the left, is igneous: one piece that had cooled from a very high temperature into a more homogeneous mass.

Then we sang the three types of rock song to the tune of "row, row, row your boat."
Sedimentary rock,
Has been formed in layers,
Often found near water sources,
With fossils from decayers.

Then there's igneous rock,
Here since earth was born,
Molten lava, cooled and hardened,
That's how it is formed.

These two types of rock ,
Can also be transformed,
With pressure, heat and chemicals,
Metamorphic they'll become.

Kids loved singing that one.
And we explored some of our personal rock collection specimens. These specimens are large enough to be held in the palm of your hand. They are from and I purchased the Premium Rock Collection. It comes with a handy booklet for those who don't have a strong geology background. I'd say that the booklet is more for adult useage, or middle-schooler/high-school usage, or for the young extreme enthusiast. My kids weren't interested in the more detailed information. This collection was a bit more pricy, but I really wanted the kids to have large-enough samples to hold, feel, and identify. Most smaller sized sets I found on other websites, featured quarter sized rocks, and I felt these samples were too small. I was quite pleased with this sample set.

D figured out that pumice floats.
I think that these are two selenite crystals, purchased on a trip to some underground caverns.
Some of these other specimens we picked up on that same trip south where we went to visit the caverns and "pan for gold." This is S's personal collection. She is very proud of her collection. Her rock collection is second only to her chicken bone collection. (I am not kidding, in her room she has a box of chicken bones collected from the dinner table.) S was very excited we went to the museum to look at bones!
Recently we went to a museum that had a lot of dinosaur fossils found in Texas.

T, S and D all got to converse with the paleontologist who was working at the museum that day. The tooth she is holding in her hand is from a woolly mammoth and the kids got to touch it and hold it.
This is the top part of a donkey skull and S is holding a horse's tooth. 

We visited early on a Saturday and there weren't many families at the museum yet. The exhibits were pretty small and old-looking. But it was wonderful to be able to ask the staff person any question we liked about bones and get some up-close, hands-on, personalized lessons about fossils. This first-hand interaction isn't something we've ever gotten at a larger museum and this conversation made the kids' day.
S sure looks serious doing her sentence diagramming. She and T really like these exercises.
This work was D. This work also reminded me that I have to remind him what the different parts of speech are, again. Though he has some parts down pat, others are still, lost in that goes-in-and-never-comes-out-again bin. All my children need this constant grammar review over 3-4 years and the older ones still can't identify a conjunction.
D has also started our word study cards from ETC. Here he is alphabetizing this card deck.

These are a couple of T's sentence diagrams. I guess this lesson was about prepositional phrases.
I mentioned this book in a previous post.  It is Drawing Sentences by Eugene Moutoux. The kids like the exercises and I love the instruction and descriptions. His website is here and you can now buy it on Amazon.
Over the past few months we've been doing a lot of Catechesis of the Good Shepherd materials making. For those familiar with level I materials, this one T is working on could seem a lot like the Empty Tomb work. This is because it was patterned after that material. I've cut and glued two of these "tombs" so far and I still can't seem to get the outside dimensions quite right. There are instructions, but they are vague in parts and you really need to have it in your head, what your cut list is supposed to be, and how to scale-down this ginormous material to "fit" into your atrium space. Maybe by number 3, I'll produce something satisfactory.
Here T is taking the wooden outer and making his own Minecraft "land" with styrofoam, wood glue, and joint compound.
This is what our school table looks like sometimes.
D was helping me paint the, now-that-I-think-about-it-it's-too-short-walls for the city of Jerusalem. This one is staying at home. The other one went to the atrium all ready.
He's a precise little boy.
I haven't a clue when these were shot, maybe in early February? But it was 90 degrees here today. The weather man said we were in early spring and S planted some Hollyhocks before the boys complained it was too hot and wanted to go inside. We live south I guess.

Stamp Game Division, a Dog House, Flute, Flowers, and Lacrosse

Where does the time go? It has been over a month since I last updated this space. Hopefully these next three rather brief posts will highlight the major recent events.
D is done with stamp game division. I think that this happened about a month ago. For the moment we are in a math holding pattern and he is working hard in other areas (like using up my dry erase markers to make stick figure stories on the white board we use for All About Spelling tiles.)

I made problem cards through triple digit divisors. Here he divided 4,680 stamps among 312 skittles. The red skittles stand as centurion soldiers and represent 100. The blue skittle is a decurion and represents 10 and the green skittle is a unit foot soldier and represents only himself, one. He gave out his thousand stamps first to his centurions and the hundreds to the decurion and tens to his unit guys. When he ran out of thousands to give each centurions one he exchanged what he had left and have each centurion a hundred stamp, each decurion a tens stamp and each foot soldier a unit stamp. Then he counted up how much one foot soldier received, and it received 15.
Sometimes working through all the problem tickets was heavy stuff.
But he was happy that he completed this work and now we go on to elementary math. Okay, not really, I think that we have a bit to do in the way of fractions first. 
This is Chase. He is a wolf stuffed animal we purchased at the Native American History Museum in Washington D.C. about 4 years go. Recently I did a bit of surgery on Chase and re-stuffed his neck and his legs. Chase, unfortunately, has developed cataracts from being dropped on the wood flooring and his tail has needed some reattaching after he was pulled out from under someone suddenly. But he is very much still a favorite.
Chase recently got a new dog house.
D drew up the design, I did the cutting and D did the gluing, and the painting. That is "blankie" on top of his house in case he needs some extra insulation, since it is 90 degrees outside here today.
S is now about 1 1/2 years in flute age. She is working on vibrato and dynamics. I am working to find a longer flute for a still very small person. Her music has outgrown this beginner flute, but her body hasn't yet grown into a regular length flute. Somewhere there must be a somewhat affordable intermediate option right?
And it is that season again. Juggling flute lesson and lacrosse game times, and my husband's travel schedule, but that is another story. T is a growing sprout and this year pretty much needed entirely new gear: new helmet, new gloves, and new elbow pads, and new expensive. We are still hanging on to the shoulder pads.
This his him "fake" throwing at me behind the camera. His throwing and catching is better, but not great. He says he likes it, and doesn't complain about team practice, or practicing in the back yard. But his enthusiasm and gusto on the field leaves a lot to be desired. Hopefully once his team gets out there and starts to play a bit in a game situation, he'll buck up and help out his guys.
Meanwhile, you can see what is happening here in the background.
This little guys is pretty ambidextrous. We'll see if he even favors a side. He is wearing gloves because you always practice with gloves on. It is easier to handle the stick without gloves, but you'll never ever play without gloves on, so you better know how to do that fancy stick work with gloves on. Little D is also wearing a helmet to protect his big noggin because he is pretty accurate with the bounce back. That is that black vertical trampoline there on the right. You throw the ball to hit it, it will bounce back, and you catch it. It is like playing catch with yourself. Except little D can't really catch yet. So, when he throws the ball, it comes right straight back at him and hits him in the noggin. Native Americans started playing this sport with rocks and nets on sticks. Now, they use hard rubber balls with no give. It hurts when that thing hits you. So, D wears a helmet.

We ask little D if he'll play lacrosse like his hyung-a. He says, "no, I play soccer." I don't think that he really wants to get out there on the soccer field anytime soon, but we'll see.
And it is early spring here in the south. I've been hauling mulch, peat moss, and compost these past weeks and taking a lot of showers. S found this at Costco the other day. There are 24 bulbs in that bag, and she is beyond thrilled that she has to dig 24 holes out back.

We already potted up her amaryllis plants this year. She was gifted 4 bulbs over the last 4 years and a few made babies this year. So all together she potted 9. She is fortunate that her bedroom window faces south and has great exposure.
She was wondering where to put additional pots if another bulb were to make babies this year. She is certainly a proud gardener.