Discrete Math - Day 23

Today we continued to explore prime numbers and their properties. The class first verified that they had all the prime numbers less than 500 captured. We next discussed patterns that students found. Students talked about whole columns that could be eliminated and one student mentioned that the values along a diagonal were being eliminated, although she couldn't remember for which value this occurred.

The discussion then centered around how many prime numbers needed to be checked. Some students didn't stop until they were up in the seventies. One student said she stopped when the product of the next two primes exceeded 500. This meant that when she saw 23 x 29 > 500 she stopped. No one else had any thoughts about how far they had to go, so I asked the class how this student knew that she could stop looking for primes at this point.

There was a painfully long wait (over 2 minutes) of utter silence. I re-phrased the question and asked what was happening as you moved through the list of primes, crossing out multiples, that indicated you could stop at 23. Again there was no response.

I am not sure what is at the root of this issue. Is this a problem with student number sense? Is it an issue that students are not used to being active learners and just want an answer? Is it too early in the morning for them to be awake and thinking? It is something that I will have to monitor. I will say that students' natural abilities to be inquisitive and to reason are inhibited at this point.

I discussed what is happening as we proceed through the list. Each time, multiples of that prime are eliminated. When we get through 19, we have covered all numbers and their multiples that can be formed with primes valued at 19 and less. For 23, we already have 19 x 23 = 437. There are no other multiples of 23 that are not already crossed out, so there is no need to check further. A number like 31 has all of the lower multiples crossed off, such as 62, 93, 155, etc. Thus, one all the prime numbers less than square root of 500 have been checked there is no need to continue.

We then looked at some divisibility rules. Students by-and-large were not familiar with divisibility rules. For the most part students aren't interested in these rules because the can just pull out a calculator and check. Given that I have rearranged the course material, it may have been better to introduce these rules later when looking at modular arithmetic. I will move this piece for the next course.

As it was, most students did attempt to make use of the rules and work through trying to determine from the rules whether or not different values were prime. I discussed the divisibility rule for 7 as an illustration of how things work.

Finally we looked at the question of how many prime numbers exist. This was at the end of class but there was some interest in the philosophical nature of the question and the idea of the infinite. One student said they would go on forever because numbers themselves go on for ever. Another thought there might a fixed number. The question came up of couldn't there be some formula or equation we could use to determine the answer. This was a perfect set up for proving the infinitude of prime numbers.

There wasn't enough time in class to go through the proof but I was able to wrap up with the ideas that the students presented and that we would use these to prove the conjecture that there are infinitely many primes.

Visit the class summary for a student's perspective of the class and to view the lesson slides.