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.


  1. OmG! I love T Math works!!! I'm impressed!! Excelent work!!! Congratulations!

  2. Awesome, awesome work!

    I love that the children can show US how to work with something when they have been properly prepared. :)

    1. I second this one for sure!! I could see so many past works tying in to this particular work. It is amazing to see how the lesson sequence builds, expands, and flows deeper and deeper. Thank You Jessica, for being my guide, and enabling me to guide my little ones!

  3. I get so excited when someone posts about this "later" math works. It's so rare to see them in action online and having seen them somewhere else can only help me to feel calmer when we do them. Plus, I can call T if I get stuck. After getting hopelessly stuck on geometric multiplication this week it will be nice to have T on speed dial from now on. (btw, after two days of fail in a row, I wound up letting the kids watch a video of it online and they said "OH! Why didn't you just show us that way?"

    1. Awww!! Too funny. Maybe next time T can just face-time with them and walk them through it. Or, I wonder if that would work before they got side tracked with another topic. Well, it would be worth giving it a try I suppose. Or, maybe T could make a demonstration video and send THAT along. And maybe Me-Too could give S a racks and tubes lesson!!