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Celeste
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| Posted: Sept 05 2007 at 1:37pm | IP Logged
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Bead bar decanomial question: we've laid out the bars, step one. Now what? The next step is a little fuzzy--replacing the bars with other bars, before building the squares.
When I replace bars with bars, is it supposed to look like this? (I don't have enough bead bars. . . .)
I didn't do it that way; I only replaced the ones in the left column.
If we only replace the 1-bars, then when it comes time to replace the bars with squares, we're replacing combinations of 2- and 3-bars with 3-squares, for example. It's fun, makes you think harder, but is it correct?
Then when we replace the bars with squares, is there a specific way to lay them on the mat?
Thank you so much! Looking forward to advice from the experts!
Celeste
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montessori_lori
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| Posted: Sept 05 2007 at 4:43pm | IP Logged
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Here's what I got from looking through my album:
The first Decanomial presentation consists only of laying out the bars. No substitutions are done at this time. There are two ways of doing it: vertical and horizontal.
In the vertical presentation, the multiplicand is fixed (the first number); the multiplier changes. For example, 1 x 1, 2 x 1, 3 x 1, 4 x 1. You would be making rows of beads starting at the left and going to the bottom before starting the next row.
In the horizontal presentation, you would construct the rows across the top to the far right, then move to the next row. The multiplicand changes and the multiplier is fixed. Example: 1 x 1, 1 x 2, 1 x 3, 1 x 4, etc.
Again, these first two presentations are for the memorization of the multiplication tables. No substitutions (squares, cubes) are done at this time.
As an extension for a child who has mastered the first two layouts, you may begin to construct the squares and cubes.
Replace the bars on both sides, saying the equations as you go (3 x 2 = 6, then replace the 3 two-bars with a 6. On the other side of the square, say 2 x 3 = 6, then replace those bars with a 6).
Then, bring the like bars together - the number of bars should be the same as the number itself (for instance, you should be left with 5 five-bars, 6 six-bars, 7 seven-bars, etc.) Those will become your squares.
After exchanging all the bars for squares, the number of squares should also equal the number itself (2 two-squares, 3 three-squares, 4 four-squares, etc.) Those should be stacked (in order from 1-10) and replaced with the appropriate cube.
After that, the cubes should be stacked starting with 10 and going up to 1.
It's a spectacular presentation if done correctly - but I don't recommend doing it unless you have enough beads, squares, and cubes to do it right. My album says that the final layout is for kids 8+. Hope that helps!
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Meredith
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| Posted: Sept 05 2007 at 8:24pm | IP Logged
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This helps ALOT Lori, thank you so much for going through the motions here  I can't wait to do this one with my olders!!
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Meredith
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Celeste
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| Posted: Sept 05 2007 at 8:43pm | IP Logged
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Thanks, Lori. Got the horizontal, vertical, and angular layouts; no problem. Still not sure about the "first passage" from the beads to the tower.
According to RD Math III: "Replace 2 single red beads with a green bar of 2. Replace 3 single red beads with a pink bar of 3. Replace 3 green bars of 2 with 2 pink bars of 3. Replace 4 red beads with a yellow bar of 4.
Replace 4 green bars of 2 with 2 yellow bars of 4. Continue in this manner until all loose bars below (to the left of) the diagonal have been "operated on"; that side of the diagonal has changed color."
So. I started to do it. It's very cool. But the traditional decanomial box, with 55 of each color, DOES NOT HAVE ENOUGH BEADS.
They're very tricksy, they are. They casually mention the need for "colored bead bars," but they don't say how many. But now I know. One hundred ten bars, 81 nines, 64 eights . . . .
How does this work in a "real" Montessori classroom--i.e., whence do they derive the extra beads?
Celeste
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montessori_lori
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| Posted: Sept 05 2007 at 8:53pm | IP Logged
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Most Montessori classrooms have more than one bead bar box - and if need be, we go borrow one from another classroom!
I found that a great mat for this work is a large square of black felt. You can buy it, uncut, at Michael's or Hobby Lobby. The kids like to roll it up like a rug when they're finished.
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Celeste
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| Posted: Sept 05 2007 at 9:56pm | IP Logged
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Meredith, I'm coming over to borrow beads from your classroom. 
Thanks for the felt mat tip, Lori; our regular mats are wobbly--and I'll bet the bead colors look great on the black!
Celeste
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montessori_lori
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| Posted: Sept 05 2007 at 10:45pm | IP Logged
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The beads do look beautiful on black - I always took a picture whenever a child finished the layout. It's such a big task!
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Celeste
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| Posted: Sept 09 2007 at 10:00pm | IP Logged
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Another decanomial question for you, Lori, I hope the last for a while! I am making the geometric decanomial with graph paper, as described in RD Math III, pp. 32. It says to make all the squares blue and place in an envelope. (I get that part.) The other rectangles to be made are well described--but no color is given for them. Are they supposed to be colorless, as a step toward abstraction?
It seems an interesting exercise--geometrically exploring the commutative property of multiplication.
TIA. You're so kind to share your formation and expertise with us!
Celeste
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montessori_lori
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| Posted: Sept 10 2007 at 8:09am | IP Logged
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Sorry, I don't have that album and I've never heard of that extension. I don't think they'd be colorless - perhaps they are also meant to be blue?
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4Real Forums : Nurturing the Years of Wonder
Topic: Decanomial (elementary) question
