Depth · one concept from first principles · step 2 of 4
Add each pair of neighbors in the bottom row.
For example (x+2y)+(x+y)=2x+3y: add the x's and the y's separately.
7.EE.A.1Draw A DiagramAdd neighbors again.
(2x+y)+(2x-y)=4x: the y's cancel.
7.EE.A.1Convert To AlgebraAdd the two pairs.
Like terms combine into one term.
6.EE.A.3Convert To AlgebraThe last sum: the +4y and -4y cancel.
Only x is left.
7.EE.A.1Convert To AlgebraEach bottom brick is counted 1, 4, 6, 4, 1 times on the way up: 1+4+6+4+1=16 copies of x, while the mirror-image y parts cancel.
Symmetric left and right halves cancel.
6.EE.A.3Look For A PatternAt the top, the bottom bricks are counted , , , , times, so the parts build and the parts cancel.
Working down from the top, a brick's count is the sum of the counts of the bricks it feeds, since its expression travels up through each of them separately: ; ; ; ; .
The parts give groups of , which is .
Pairing the ends, once against once and four times against four times, each pair adds something and then takes the same amount back, and the middle brick has no .
Add the x's with the x's and the y's with the y's, and watch symmetry do the cancelling.
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