With that the course transitions to the second unit, which covers Number Theory and Cryptography. Last year I went a bit overboard on the number theory portion. Although some of it was interesting, at least to me, the discrete course almost turned into a number theory course. This year I've removed some of the prime number investigations and will bring them in at the end of the unit to fill in space, if needed.
As a means of introduction I briefly discuss that cryptography is used in everyday communication devices and on web sites. Cryptography helps to keep our personal information secure. However, cryptography is based on prime numbers, so we need to spend some time better understanding the properties and characteristics of prime numbers. And so begins our journey into number theory.
Many students haven't thought about integers and prime numbers for a good number of years. And, although students have had two years of algebra, they don't necessarily recognize the underlying algebraic structures that are at play. This provides the opportunity to cover familiar ground from a more detailed mathematical perspective. The importance of precise definitions, the use of algebraic structures, and the development of important mathematical relationships can be developed using ideas and concepts with which students can readily relate.
To start of, I ask students to define an integer. The purpose of this isn't to get to a formal definition but to have students agree as to what an integer looks like. Students discussed that integers can't have fractional parts or decimals; are values like 1, 2, 3; and do change values in increments of one. I wrote out the following:
...-2, -1, 0, 1, 2, ...
and repeated that these values have no fractional or decimal parts. I realize the latter is not technically true but for the purposes of understanding what an integer looks like it focuses students on the important idea.
Next I asked students to consider properties of integers that they know. Students talked about integers being positive and negative, that there was a zero value, and that they didn't have fractions.
I asked students about operations and one student said you could add two integers. The class giggled because they thought it was obvious, not realizing this was a key algebraic operation that can be performed over integers. I pressed students to consider other operations. They responded you could multiply but not divide, another key algebraic property. After a few more ideas we took a look at the properties of integers under addition and multiplication.
The properties looked at were:
Addition
- for any integers a and b, a + b is also an integer
- for any integer a and value zero, a + 0 = a
- for any integers a, b, and c, (a + b) + c = a + (b + c)
- for any integer a, there exists integer b such that a + b = 0
- for any integers a and b, a + b = b + a
Multiplication
- for any integers a and b, a x b is also an integer
- for any integer a and value one, a x 1 = a
- for any integers a, b, and c, (a x b) x c = a x (b x c)
- for any integers a and b, a x b = b x a
I present these in this manner because it sets up considering the similarity and differences of operations under addition and multiplication and mirrors how algebraic rings are defined.
This is an opportunity to discuss the ideas of closure, identities, associative property, commutative property, and the existence of an inverse (for integers there is an additive inverse but not a multiplicative inverse). I also relate this algebraic structure to other entities that possess the same basic properties: matrices and polynomials. Students have seen and worked with these but probably never connected the operations to a common algebraic structure, in this case what we call an algebraic ring.
I also point out that the commutative property is actually a special property that does not apply to all situations, for example matrix multiplication.
Students at this point realize there is a lot more going on in their known math world than they had previously considered.
I next asked students to define a composite number. Again, I wasn't looking for a formal definition at this point, only a working definition that we could use. Students came up with two ideas:
- A composite number is the product of two other integers other than one and itself.
- A composite number has factors other than 1 and itself.
The class was satisfied that these definitions fit the idea of composite number, so we moved to the next definition, that of a prime number. Here, students decided that a prime was only divisible by 1 or itself or that its only factors were 1 and itself.
This was perfect because it allowed me to bring up the question of whether or not 1 is a prime number. Students said that 1 was not. I told them that was correct and that 1 was the multiplicative unit. I asked students to consider the definition of a prime number and if 1 fit into the definition. They agreed that it did.
The question posed was how to define a prime number so that 1 would not be included in the definition. After some discussion, students decided that defining a prime as an integer with exactly two factors fit the bill.
This allowed us to now re-visit the definition of a composite number. Students concluded that a composite number would be a number with 3 or more factors. This left open the ability to characterize the multiplicative unit as a value with exactly 1 factor.
With these definitions out of the way it was now time to define what factor meant, since both the prime and composite number definitions required factors. Students decided that a factor b of a means that b divides a. I used this as an opportunity to introduce the symbol b | a to designate that b divides a.
I didn't emphasize that all of this is made possible through the properties of integers that we had discussed. This is something that I'll have to bring up tomorrow.
With these definitions in place we proceeded to look at a three questions about prime numbers:
- Is 221 prime? If not, what are its factors?
- What’s the largest divisor you need to check to be sure that 397 is prime? How do you know it’s the largest?
- Is 8171 prime, or not? How do you know for sure?
I allowed students to work on these on their own, no smartphone searches allowed, and then to discuss in their groups. Students wanted to know the easy way to get to these answers. I let them explore.
After trial and error, students found that 221 = 13 x 17. It's factors are 1, 13, 17, and 221. Students initially said the factors were just 13 and 17 but it's important to count all the factors.
Most students thought they would need to check values up to 200. We talked about this and I listed out the following primes: 2, 3, 5, 7, 11, 13, 17, 19, 23. After talking about this I claimed that once I that the first 8 primes did not divide into 397 that there was no need to check any further. We discussed this further and students were reluctant to accept that this would happen. I provided a simpler example, using 10 as the number to examine. This helped to illustrate what was happening.
For the last problem, we didn't verify if 8171 was prime or not but discussed how many primes would need to be checked. Students found they would need to check primes up to 90. I then asked students if they knew all the primes less than 100. Of course, they didn't and I didn't expect them to know.
This allowed me to introduce the Sieve of Eratosthenes. I provided a worksheet to use the sieve on the first 500 positive integers. I illustrated how the sieve works and asked students to work through the sieve as homework.
Tomorrow we'll use the results to examine the distribution of prime numbers.
Visit the class summary for a student's perspective of the class and to view the lesson slides.
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