7A had a lively discussion that helped to expand our understanding of exponents. We began with what we knew about positive whole number exponents with base 10 (blue ink). When we arrived at 10 to the zero power, the class had to figure what made sense. They proposed that the answer could be 0, 1, or 10. We then took arguments for each possible solution. It was rewarding to see their pattern recognition skills come into focus. The two clarifying explanations are in green ink. NSK noticed that the difference in the standard forms was a 9000, 900, 90, and 9 and rationalized that 10^0 had to be 1 for the pattern to continue. EH noticed that the pattern between values was to divide by 10 so the next number in the pattern was 1. This convinced us that 10^0 power was 1. Then they tackled 10 to the power of negative 1. There was much debate about whether the negative exponent would make the number negative. Logic prevailed and we applied our patterns from before to extrapolate that negative exponents made them fractions. It was invigorating to hear the healthy debate and hypotheses being shared.
I was excited to hear the new questions that were then generated by these new discoveries:
"Can you have two to the power of 4 to the power of 2? Can powers have power?"
"Can exponents be fractions?"
The doors have been opened to a whole new world. Check back for other musings and discoveries.