6.7 Summary & Closing Puzzle

Summary

Key Points

  • The distributive property a(b+c) = ab + ac lets us “multiply by parts.” It holds for all numbers, positive, negative, fractional.
  • Extending it to two two-term brackets gives the master identity (a+b)(c+d) = ac + ad + bc + bd: multiply each term of the first by each term of the second.
  • Three special cases are worth memorising:
    • (a+b)^2 = a^2 + 2ab + b^2  (don’t forget the 2ab);
    • (a-b)^2 = a^2 - 2ab + b^2;
    • (a+b)(a-b) = a^2 - b^2.
  • Distributivity powers fast multiplication: write 11 = 10+1, 101 = 100+1, 99 = 100-1, 999 = 1000-1, etc.
  • Only like terms (same letters to the same powers) may be combined; exponents never change when you add.
  • One pattern can be described many ways, and expanding the expressions proves they are all equal. Finding multiple correct methods is a creative act.

A Closing Puzzle: Coin Conjoin

Try This Arrange 10 coins in a triangle (4 rows: 1+2+3+4). The challenge: turn the triangle upside down by sliding one coin at a time to a new position. A 3-coin triangle flips in 1 move; a 6-coin triangle in 2 moves; the 10-coin triangle in 3 moves. How many moves for the 15-coin triangle? Is there a simple formula for the minimum moves of any triangular arrangement? (Hint: count how many coins must change rows.)