Discrete Math - Day 47

First, a little mea culpa. In my last entry, I indicated to students that we could use congruence property 1 to show that having 3k₁ ≡ 6 (mod 5) means that k₁ ≡ 2 (mod 5). In fact, while we proved that if a ≡ b (mod m) then for any integer k, ka ≡ kb (mod m). We actually are using the converse of this, i.e. if ka ≡ kb (mod m) then a ≡ b (mod m) for any integer k such that gcd(k, m) = 1, which we did not prove. At this juncture, I wanted to convey the sense of what is happening. The class will see the need for gcd(k, m) = 1 shortly, in fact they already saw this in one of the encoding schemes where different values were mapped to the same letter.

I started by asking students how things went with trying to solve the Fred's Baseball Card problem using the algorithm from yesterday's class. As expected, many didn't try or simply got stuck. I had decided last night that I wasn't going to press the use of this algorithm. Instead, I used it to point out some key aspects that I felt they should know or be able to do.

With that, I wrote out the congruence statements for the problem and told students I expected them to be able to do this. 

1. x ≡ 1 (mod 2)
2. x ≡ 1 (mod 3)
3. x ≡ 1 (mod 4) 

Students then wondered about the last condition of even piles when divided by 7. Some wondered if this meant the value was congruent to zero, which is precisely what is happening.

• x ≡ 0 (mod 7) 

I told them they should be able to find the linear equation that describes the solution set and to find the slope of this equation using the least common multiple.

I also said they should be able to perform some of the algebraic substitutions, although I am not expecting them at this point to work completely through an entire system.

Everyone seemed to agree that they could do this. I then told them I expected them to be able to take a congruence statement and write the equivalent equality expression using the definition of congruence.

With that said, I focused on modular arithmetic. My goal was to continue pursuing this topic from a different perspective. Students generally find this topic interesting, as was the case today.

First, I revisited the equivalence relationship properties we had explored a couple of classes ago. It appears that congruence behaves much like equality from an arithmetic perspective. Can this be the case?

The answer is yes. In fact, modular arithmetic possesses the same properties as integer arithmetic. Modular arithmetic has associative properties of addition and multiplication, a distributive property of multiplication over addition, and commutative properties of addition and multiplication.

I did not ask students to prove these results here but indicated that a question on the final could ask them to prove one or more of these properties. I did ask them to try these with some values so they could see that the properties held.

I like to point out that algebra is really the study of structures and trying to understand what entities possess these structures. For example, polynomials and matrices possess many of the same structures as integers. With matrices, multiplication is generally non-commutative. This is another aspect that I like to point out. The commutative property is special.

I then showed students addition and multiplication tables for arithmetic mod 3.


+ 0 1 2          x 0 1 2
0 0 1 2          0 0 0 01 1 2 0          1 0 1 22 2 0 1          2 0 2 1
This took a little bit of time for students to absorb. There were a lot of good questions about the workings of these tables. I pointed out that arithmetic mod 3 actually is more complete than arithmetic with integers because mod 3 arithmetic actually provides multiplicative inverses whereas integer arithmetic does not.

As we talked through how the tables worked students were able to verify through values that the tables worked. At this point, one student wondered if this idea would work with other values for m. I love when a student anticipates what is coming next.

I produced the next slide that asked students to create addition and multiplication tables for mod 5 arithmetic. The one student apologized to the class profusely for coming up with the idea. The class started to tackle these tables. There were a few questions and points of hesitation but as the proceeded they started to understand how the tables worked. Some students wanted to just write down the values regardless of how large they were. I would ask what happens when they divided those values by 5. The students would realize they could reduce the result to a value that was less than 5.

Unfortunately class ended at this point. The class found this an interesting study. We'll continue to look at this idea and what happens if m is not a prime.

I will be at the NCTM National Conference in Denver for the next two classes. I have students write a research paper on cryptology and will use my time away to have the class work with the school librarian to start researching their topic and putting together their resource materials.

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