So we left off with binomials, trinomials, quintonimials, and the decanomial.
From a small age, the Montessori child touches the beads and sees one is one thing and that three groups of three is a total of nine things. He can physically see that 1 x 2 has the same product as 2 x 1.
He can also see that when 10 groups of 10 beads are laid side by side they make a square and can count that it totals 100. They also see if you have 10 x 10 and 10 high that equals 1000. This cubing is not abstract it is really a cube. Really. Go get your legos and build it. I had a rocket scientist (literally) stand in awe as his 5 year old laid out the six chain and then laid down each square and then as his son brought the cube look at me and say that he never thought that a cube was a real thing. You are in good company if you never thought so.
When the 9 year old is ready, one of the last things he does in lower elementary is begin to lay out the decanomial. One red bead, next to it two red beads, next to that three red beads, etc. all the way across to ten. Then the next row 1 green (2) bar, next to it 2 green bars, and so on continuing across to 10 and down to 10. The tedium and need for perfectly aligned rows stressing the nine year old. Then come the first years. The little guys gather round ever so carefully not to upset the rows and rows of perfectly aligned beads. The six year old watches with rapt attention or asks to hand the builder his needed beads. The second years watch from a distance in their aloof state.
Then the exchange begins. 2 red beads equal the same as 1 green (2) bar with its two beads. “The red ones roll a bit. It might be more stable if we substituted the 2 bead bar. Let’s change it to a green bar.” We change them all out. Then the child discovers that we have the squares cutting right down the middle and flanking it if we regroup the beads they make squares that when gathered all together make a cube. The secrete of the decanomial is that it is really all the cubes of 1 to 10 put together.
I say all this to say that AV has done all this work. Now, he has laid out the decanomial once again.
This time with paper representations which we carefully looked at before placing letters over the multiplier and multiplicand. We then write it all down on one long piece of paper.
So 1 x 2 became a x b. We began the proof of equivalence between all the mathematics of the internal beads and the (a+b+c+d+e+f+g+h+i+j)squared.