Paul described the idea of casting out nines. He reminded us of the rule that the sum of a number’s digits equals the remainder when that number is divided by nine. It was here that he introduced to us the concept of modular arithmetic. He then used this to explain why the sums work, and followed this by showing us a trick. That is, if you write a number (the more random and complicated, the better), then reverse its digits and subtract the two numbers, you can figure out any missing digit in the answer by omitting zeros, thinking in terms of mod 9, and hearing the rest of the digits. He modeled this trick by standing with his back to the board and showing us how he could figure out the number.
This activity not only showed us the fun of clever math tricks to play on our students, but also gave a brief but thorough explanation of the relationship between any number and nine. By knowing this trick we can “Make it easier” to solve various problems or tricks involving these relationships.
The downfall of this activity was that it happened very quickly. There was a lot of explanation to happen in a very short time. I think many of us who had never seen modular arithmetic still find the details quite vague and even those of us who are familiar with modular arithmetic find much of the casting out nines concept to be fuzzy. Also, although we had several opportunities to experience modular arithmetic in later problems, we never again encountered a problem that would use casting out nines, so I think it would be helpful for us to know what other problems this strategy would be used for. I think it would also be good for us to discuss or at least reflect on our own how we could explain this to our students without using ‘mod’ language. More specifically, I think we could talk in terms of remainders to use language our students are familiar with to encourage their understanding of nines. It would also be helpful for us to practice these tricks with each other so that we feel comfortable and can be more fluid with them in front of our students.