Thursday, March 31, 2016

Moving Forward

T really likes this cubing material. It is definitely one of the most expensive Montessori materials we own (bells and tone-bars aside.) But the rate at which he is going to "grow-out" of this material, has me wondering if it was worth the money. Maybe the fact that he loves these puzzle pieces so much makes it worth the cost. Also, maybe the fact that this material made this set of mathematical concepts so completely and simply apparent to T makes it worth the cost.

I also wonder how the other two will fare through these lessons. I suspect they will take more time working with this material.
More cubing. But not a whole lot more cubing somehow. (Our previous cubing experience can be found here.) A note to myself and others: although these presentations reflect the KotU albums T decided to not follow the presentation. So, if I get confused about what step came next, or you can't follow my post with your album next to you, it is because T left some steps out. 

In the shot above, T is making the 8-cube with the cube of 3 and the cube of 5. He also made the 8 cube - Ninja Turtle style (purple and green) and Christmas style (white and red.)
I selected the boy and girl cube (blue and pink) cube and T set to work naming each term algebraically. Here, I pointed out that the length of the side of the first term (in this case the pink-3) is the height of the second term (in this case the blue 6 squares.)
T then labeled each term. From left to right: a2, a2b, a2b, ab2, ab2, ab2, a2b, b3.

T then organized and combined all like terms: a3, 3a2b, 3ab2, b3.
T first figured out each term on a separate blank paper ticket. Then he wrote down the equation in it's entirety.
Then T figured out how to algebraically describe the whole cube: (a+b)3.

Then we dug out our primary sensorial binomial cube material. I seriously don't know what the album introduction was, since T didn't pay any attention to what I was starting to say. Instead, he said something like, "I already KNOW MOM."

Edit: I finally really read the album page and found that the first presentation focuses on how to create the cube of (a+b) starting with the square of (a+b). It is the same production as the bead-bar exercise we did here. The point of the second part of this binomial cube lesson is to know the algebraic "names" of each piece by sight and to extrapolate a "rule" for binomials. As you can see below, T assigned algebraic "names" to each piece and now we just need to review the "rule."

After this cubing lesson we can either double back around and finish out the squaring sequence or continue on to cubing a trinomial.
Now if this were pink....
D did a little sand tray writing. He is going through our Dwyer reading folders slowly exploring the different spellings for these long-vowel sounds.
This is print not cursive. Interesting.

The other night, I got up out of bed to see why T was still awake at 11pm. He said that he had finished reading this entire book in one evening and was still sad about Charlotte. I told him to think happy thoughts and to go to sleep already. Secretly I was glad that he stayed up reading E.B. White and not something less-than-ideal like Captain Underpants.
S finished the Daily Math Word Problems Grade 2 book, and is on to the grade 3 book.
S helped D read these first readers. They do not follow the Montessori phonogram sequence so he can't read them independently quite yet. 
T got in on the Daily Math Word Problems too.
D started the addition snake game this week! I wrote more about this work here.

D was super excited to be doing a big-boy work.

Stay tuned, there is more to come soon!

Tuesday, March 29, 2016

Spring Has Sprung

Major changes are afoot. I think I caught the spring cleaning bug. (And am managing to muster a tiny bit more energy expenditure these days.) Our classroom is due for a re-arrange. We can do that mid-year as homeschoolers, no? Actually there will not be too much upheaval. I only need to build one shelving unit. But the best part about this is that the kids are growing vertically and I think we can finally convert all of our shelving units to the 40" elementary height. More on this transformation later.
D is moving forward!! He decided that dynamic subtraction on the stamp game was HISTORY, and here he has begun static multiplication. The rulers are his own addition. Here he is multiplying 2,311 by 3. 
Here he is showing me his last subtraction stamp game problem.  

A note to self about D's writing: I noticed, only after looking at this shot on the computer, that one of his "5's" is backward. D rarely reverses his numbers. And he rarely reverses his letters. Additionally, before yesterday, he had never written from right to left. S did this ALL the time. But that time, D reversed his writing and wrote his name, in cursive, from right to left. Each letter was correctly orientated. I don't know why he decided to write from right to left that one time. He was writing on a laminated cover and he was holding the booklet upside down, so the coil binding was on the right. Maybe he thought that he should start writing next to the binding, which would have been correct if the binding had be on the left hand side. This handwriting thing has always been interesting to me. I wonder what was going on in his-I'm-a-Little-Lefty-brain.

This also reminds me to note here, that D can now, write his full name in cursive, correctly. I can't show a pic here on the blog, but I wanted to note it here for my future self that at 4 years, 10 months, D has arrived at this milestone. And now, everything he owns has his name on it. Okay, not everything, but a lot of things that don't need his name on them. 
This is how D feels about moving onward and upward.
S is so very far behind in math in my mind. She is more than 8 and has yet to really delve into the math album. She is just more interested in other areas. Here is her first introduction to the racks and tubes. I think that the album recommends that the child be DONE with the racks and tubes at age 8 because they are moving into the I-am-becoming-unappreciative-of-small-bead-manipulation by this time in their life. S has continually proved to be a bit "behind" the scope and sequence age recommendations so we'll see how she does. For now, she doesn't mind at all moving the beads. This is the first post I did that mentioned racks and tubes!
T is reading our Sir Cumference books. Hopefully we'll get to some more traditional geometry soon. I had a dream last night about tutoring high school trig.
But for now, T has started some logic training. Can I call it that? He generally has a pretty logical mind, so for him, at this age, Sudoku puzzles and grid logic games are "relaxing" to him. While I know that some might find this anxiety inducing, I find it exciting that I now get to "share" this kind of fun activity with my son.

I've read that math logic can be considered an entirely separate topic. I don't know if I agree with that statement completely, but I do know that I've needed strong logic skills in many of my math-related classes; honors calculus in undergrad and in graduate-level statistics, operations, and economics/trade. Having strong logic skills can be helpful for many kinds of analysis, but it can also make you feel a bit itchy like, why does the rest of the world think that is true, when clearly, correlation doesn't mean causation? Hopefully, T doesn't get itchy as he continues to improve his logical reasoning skills. But I do plan to incorporate logic exercises into our math work.

For now, I am helping him out when he gets stuck with these beginner puzzles, but I am also realizing that I have a bit to brush up on too.
Stay tuned to see what other math logic activities we have going on. 
After reading about that ancient Chinese Shang Dynasty's use of pictographs in the Stories of the World, S set to work using our Egyptian hieroglyphs stencil.
Spring has sprung was seriously evident this weekend when S harvested some March. This wasn't the entire plant. There is still more growing outside in the ground. There is a 18" colander under the parsley. Chimichurri, charmoula, gremalata anyone? Humn. Those would all be great with some grilled meats. Well, it IS warm enough to grill...It is continually amazing to this New Englander that it is this warm around here in March.

More to come soon!

Friday, March 25, 2016

Cubing, Mario Style

We've been at the cubing these days. Again, I am having trouble gauging how much is really sinking to permanent memory and how much repetition T really needs. He just "gets" it instantly and can independently repeat all steps in the "problem" sequence fiddling with the manipulatives and then on paper. So we'll see how this sticks.

Above, is a picture from the KotU album passing from one cube to another presentation sequence. This week we started with exercise 2, passing from a cube to a non-successive cube. First, we reviewed this presentation: passing from a cube to its successive cube, found here.  T took the 6-cube and puzzled together the 7-cube. (Okay, that post was in September. I suppose that this lesson, for whatever, reason has really sunk into permanent memory, if he is still able to figure this out 6 months later.) Here, 6 months later, T is taking the 4-cube and puzzling together the 6-cube.
He figured out that 43 + 2(42) + 2(42) + 2(42) + 4(22) + 4(22) + 4(22) + 23 = 63.
He also was able to describe his cube numerically on paper. He decided to be cute and write his version of the equation super small and combine like-terms in his head, leaving those in his math-wake, wondering how to follow what he did. Then he mathematically proved that cube he constructed was equivalent to the 6-cube.
And then we went on a small tangent making statues that look suspiciously Minecraft-like.

Then we moved on to the cubing a binomial presentation. (NOTE to me and others: this photo shot sequence doesn't reflect all of the presentation steps in the KotU album. Halfway through T decided that he didn't need to use bead bars any more, so he skipped a few steps.) T was surprised at how many different materials we needed to pull out for this lesson. (The album says you need, blank paper tickets, paper parentheses, colored number tiles, cut out wooden symbols, bead bars, and the squaring and cubing material. It doesn't, however, suggest getting a bigger table.)  I was surprised at how T was able to ignore most of the materials and focus on the math sequences in his head. 

In the shot above, we are multiplying (4+6)2 x (4+6). (This shot was actually taken after the shot below...but it shows you the full equation, before we multiply through. Just ignore the bead bars. They aren't there yet.)
NOW look at the bead bars. And look at the equation again. T multiplied the first term, (4+6)2, through and got: (4+6)2 = 42+ (4x6) + (6x4) + 62. Then he set out his bead bars...and squished them all together so you can't see the separate terms in this shot. Top left, there should be a group of 4 4-bead bars that represents 42. At the top right, there should be a group of 4 purple 6-bars that represents (6x4). At the bottom left there should be a rectangle group of 6 4-bead bars that represents (4x6) and at the lower right, there should be a square of 6 6-bead bars that represents 62These beads aren't actually included in the final cube. I suppose that this representation is just a "reminder" of what is in that set of parentheses, and the distributive and commutative properties we cover in the numeration section of the math album. This part also look similar in theme to the decanomial square too. 
Then he multiplied each of these terms by (4+6). Below you can see his first term, 42, multiplied by 4, the grey number tile. He laid out 42 four times in bead bars.
Then he realized that this was equal to 43 and he made the physical exchange for a bead cube.
Here he is multiplying the (6x4) terms by 4, the grey number tile and laid out four groups of 4 6-bead bars.
Then he figured out that 4(4X6) is the same as 6(42). So he replaced the bead bars with 6 four squares.
He also pulled out 4 6-squares for his 4(62) term. I noted since he was all done multiplying the entire first term by 4, that this would be the first layer.
Then he set about multiplying the first term through by 6 and pulling out the correct squares and cubes. Here he is making 6 groups of 4 6-bead bars. I think. They look a little mashed up.
Here, he has a group of 6 4-squares, equivalent to 4taken 6 times, and two groups of 4 6-squares, equivalent to 6(4x6) or 4(62).
and finally, Yoshi is holding 63, or 62X6.
When you put all those terms together, (all those prisms and cubes together)...
you get the first layer...
and a second layer...
and finally the cube of 10, as presented by Yoshi.
To tell the truth, I was a bit intimidated by the presentation instructions. I had trouble picturing the substitutions, bead bars for wooden squares. But T pulled through and made the substitutions easily and smoothly and was able to show me how it all was done.
The next day we did exercise 2: cubing the binomial starting with the cube of the first term. Again, the instructions were confusing to me, so I just asked T to figure out how to make the cube of 9, using the cube of 3. These were the different prisms and cubes he and Yoshi came up with...(or the different terms)...
And with a little help from our friends Yoshi and Toad, this is his constructed cube of 9. 
Are you feeling a little throw back to primary sensorial lessons here? Binomial cube anyone? D saw this representation and immediately wanted to use the binomial cube.
Then T wrote down his equation, using the prisms and cubes as references. Next up, we will be isolating the point that the length of one side of the square face of the rectangular prisms is the height of the second term before moving on to algebraic notation. 
Someone else got in on the squaring and cubing action as well. 
Did we figure out that 16 2-squares aren't equivalent to 1 9-square?
I think we did figure that out! D was delighted with his pyramid.
I figured out from our testing this year that the two younger ones could use some money experience. So, S is back at our money cards from ETC. 

D is exploring our money boards from Hello Wood. This one is the penny dollar board.

D is showing a photo of his dollar board to T.
Here S is comparing our Andy Warhol art folder to this drink coaster I picked up at a special art museum. (I can't remember if I made this or we purchased this one from Montessori Print Shop. I think it was the latter, because I don't remember choosing any of these less-than-iconic pieces. The file I found today on the MPS site is similar to mine, but it has an artist description card that my set doesn't have. Oh, and the real clue is that I included the date of the piece on all of the cards I created. These don't have dates which means I picked them up from MPS.) Anyway, the black drink coaster is one of a set that I picked up at an art museum and S wanted to understand why it was a play on Warhol's style.
I've decided to start the Writing with Ease curriculum with my kids. I think T would benefit from it the most, but S and D could definitely use this too. I'll write more about this topic later.
And we are continuing with the Stories of the World. This first time through, I am reading the stories in sequence, with a focus on narration responses, and then I intend to double back around and explore some of the activities and suggested readings for the chapters we found most interesting. Or, at least this is my plan for now.
This is one of the coloring pages for the chapter about ancient Indian civilizations. Can you tell who colored which one?

We'll be back next week! Happy Easter.