As I walked around the room today and watched students engage in the learning activity I had a realization. "We're pretty smart!" The students were actively trying to discover how changing each coefficient, a, b, and c, in the quadratic model, y = ax^2 + bx + c, affects how the parabola looks. As I listened and watched I heard some beautiful thinking...
- "As the b value gets more positive the parabola slides in the southwest direction." (Armando)
- "When b is negative the parabola moves into the fourth quadrant and when b is positive the parabola moves into the third quadrant." (Torian)
- "I don't see a pattern when I'm changing the b value, the shift in the vertex when I change b seems somewhat random. How can I find the pattern?" (Iaisha)
- "If a is large and positive the parabola is skinny and if a is small and positive then the parabola is wide." (Jessica's group)
- "c just shifts the graph up and down. If c is positive the curve just moves up and if c is negative than the curve just moves down." (Saul's group)
Testing a variety of equations in the graphing calculator and exploring how manipulating the number changes the look of the curve, the kids actively pushed to discover mathematics. They still need guidance, support and modeling. They still get stuck from time to time and don't quite know how to get themselves unstuck. But, they're working hard to learn and when I step back for a moment and observe rather than engage I realize that we're actually pretty smart.
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