Numeric and Geometric Patterns

Today, I learned that group worthy tasks have multiple entry points for every problem. When students solve math problems together, share their ideas, and when multiple solutions from the group are valued, learning takes place at an accelerated rate. Math is no longer obsessed with finding only the right answer but by finding a host of solutions. Indeed, as human beings we naturally see patterns in the world. In the “Sneaky Snake” math problem below, I stumbled upon a realization. First, I saw a geometric pattern where a central block increased by one tall and by one wide. I wrote a numeric equation for each consecutive group. When comparing this new group, I saw yet another pattern emerge. 5 = (6 x 4) + 2,   6 = (7 x 5) + 2,   7 = (8 x 6) + 2.  I could see that the next number in line would be one greater and one less than the number “n”, where 2 is constant. This lead to the algebraic equation of n = (n - 1)(n + 1) + 2.  The remaining question I have is how many ways can the formula be solved?  I welcome pattern recognition in a linear numeric manner and in a geometric visual approach, and I can see the benefits of encouraging both in the classroom. While there is always a definitive right or wrong answer, there is never a single solution. How we get there is up for grabs. It all depends on how we look at it.