"Let's do it."

Crafted cubes were ripped apart. The X² term was a square of nine blocks, the X term was a bar of 3. "What have you got and what is left?" Several different organizations of the pieces emerged. Two people recombined their blocks to produce a paired structure. "How is this structure related to X³-1?" I asked.

Two teachers struggled with their cubes to produce a segment of the original that had some of the cubic shape. The remains of the cube minus one unit were slowly reassembled into something similar to the first piece.

"How are these two pieces related to each other, and to X³-1?"

Someone's face gleamed when the two pieces fit like parts of a Soma cube. When flipped, the second piece fit right on top of the first one. Factoring was now a puzzle solved in a different way. The -1 was a hole in the middle. Other representations showed the hole in the side or on the edge. Three teachers grouped their pieces horizontally. Two others produced another format to visualize X³-1. This view was layered. There were two X² pieces on the bottom. The two X terms were on the top along with the two units.

"What would you like to view next?" I asked. Pairs formed again to explore different values of X. More models were built and ripped apart into their factors.

"Where did the hole show up?" was a common question. A layer cake model generalized well for X=6. Different grouping strategies invented by several pairs were shared. A new way to see what indeed the factors of X³-1 emerged.

"What was the central insight that aided you in seeing a new way?"

"Seeing the (X-1) in (X-1)(X² + X + 1) as one quantity multiplying another. It is more than just a symbol."

After fiddling with the cube puzzles someone else entered the discussion. "X³-1 is (X-1)(X² + X + 1) because it simply couldn' t be anything else." He demonstrated by taking apart pieces of the X=3 and X=4 representations. "If you used (X² + X + 2) or (X² + X - 1) the amounts would not be correct."

We ended the inquiry with more conjectures: What do prime polynomials look like quantitatively? Do prime polynomials evaluate to prime numbers? What do other factorable polynomials look like when viewed as quantitative representations? Are there general patterns? We've demonstrated this factoring for a few integers, but can we show it's true for all integers without using the notation of synthetic multiplication?

We agreed to continue our discussions online using the INTEC discussion server. The polynomial graveyard was not so dead after all, when viewed in light of the quantities it contained.