Before diving into this topic I asked students to share conjectures they had written down from last class. There wasn't much forthcoming in this front with one notable exception. One student talked about the idea that while prime numbers are infinite, the spacing between them continues to increase and that as you continue going further out on the number line that the spread between the prime numbers becomes infinite. It was a nice informal way of thinking of limiting processes and the idea that finding ever larger primes will become next to impossible as the computational time required will exceed fixed time. Interesting ideas that really go beyond the scope of a high school discrete math class.
To start off the investigation of relative prime numbers I posted pairs of numbers and asked students to consider how they could group these pairs based upon ideas and properties they associate with prime numbers. The goal was for students to consider the multiplicative properties and factors of these numbers.
Many students grabbed their divisibility test sheet and started to consider primes that divided each number. Some students considered differences. One group in particular was stuck on the additive relationship of the pairs. I mentioned to this group that prime numbers were based on multiplicative properties and they then started to consider factors.
Most of the groupings looked at values divisible by 3 or values divisible by 5, etc. One student noticed that there were two pairs of numbers that did not have any common factors. Eureka! I was able to then highlight the idea that many of the pairs had two or more common factors but that for these two particular pairs the only common factor was 1.
I then defined relatively prime composite numbers using this idea. I provided the example of 15 and 16 being composite numbers that are relatively prime. I asked students what an equivalent definition would be using the greatest common divisor. This seemed to stump the class. I asked students what the greatest common divisor of 15 and 16 was? Several students responded that it was 1. I asked again how the greatest common divisor could be used to define relative prime composite numbers.
One student tentatively said that two composite numbers would be relatively prime if their greatest common divisor was equal to 1. This was precisely what is needed.
Students then used their Sieve of Eratosthenes to look for patterns for relatively prime composite numbers. This was a struggle for many students but a couple of patterns were identified. First, students noted that they could not have two relatively prime even numbers. Next, students noted that the two composite numbers either needed to be both odd numbers or that one had to be even and one odd.
Most students were still not comfortable with the idea of integers being relatively prime, so I asked each student to identify 10 pairs of relatively prime composite numbers and to group their pairs. It was interesting to see the tactics that students used. Some picked a value like 15 and started to find multiple values that would be relatively prime to 15. Others were randomly picking values and checking.
I then had students write down their responses to the following questions:
- What strategies did you use to find relatively prime numbers?
- Which pair is the closest together?
- Which pair is the farthest apart?
- What made a pair easy or hard to find?
- Are some numbers used more than others?
As this was the end of the class I posed on last question: what is a composite number that is relatively prime to 30? It was interesting to hear students frantically trying values only to find they didn't work. I purposely picked 30 since the smallest relative prime composite number available would be 49. Finally, one student said that 49 would work.
For homework, I asked students
- to verify that 49 works,
- to determine if 49 is the smallest composite number that is relatively prime to 30, and
- to explain why 49 would have to be the smallest composite number, if it was the smallest
We'll start by looking at the responses to these questions and then generalize the idea of two values being relatively prime.
Visit the class summary for a student's perspective and to view the lesson slides.
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