Pascal's Triangles and Distribution

Distribution of a perfect square often confuses students. When given (x+3)^2 their first impulse is to square x and square 3. It take a little work for them to see that it is instead (x+3)(x+3). After they get to this point they are typically pretty good at distribution using the FOIL method or the area model. My favorite is obviously the area model because it can be extended to other problems. I've posted about this method earlier.

Here's an extension for students to play with: Pascal's Triangle
Have students identify as many patterns as they can. Ask them to add a new row. Can they make their own triangle with a different rule?  Every time I do this, I learn something more myself about this cool triangle.

          1             (x+y)^0    1
        1  1           (x+y)^1   x+y
      1  2  1         (x+y)^2   x^2 + 2xy + y^2
    1  3  3  1       (x+y)^3   x^3 + 3x^2y + 3xy^2 + y^3
  1  4  6  4  1     (x+y)^4   x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
1 5 10 10  5 1   (x+y)^5  . . .