How best to teach the division algorithm?

[matqkks]

What is the best way to introduce the division algorithm? Are there real life examples of an application of this algorithm. At present I state and prove the division algorithm and then do some numerical examples but most of the students find this approach pretty dry and boring. I would like to bring this topic to live but how?

 

[SteamKing]

Give a test with only long division problems and tell your students if they fail the test, they fail the course. That should keep them focused and alert.

 

[Mark44]

Give a test with only long division problems and tell your students if they fail the test, they fail the course. That should keep them focused and alert.

With no calculators allowed...

 

[dkotschessaa]

I know this isn't too exciting if your audience doesn't already appreciate number theory or abstract math, but perhaps if you started to prove a result and then "realized" that you didn't have the tools you needed. This tool turns out to be the division algorithm, which you then introduce. For example, I found this homework question for (I think) a number theory class:

[STRIKE]Use the division algorithm to[/STRIKE] prove that every odd integer is either of the form
4k + 1 or of the form 4k + 3 for some integer k.

Perhaps just posing that question to the class (without mentioning the division algorithm) and seeing if it sparks some discussion. Just a thought.

-Dave K

 

[micromass]

The division algorithm for the integers is pretty boring in itself. If you say that it proves that long division actually works, then people might be more interested.
In my opinion, the "coolness" of the division algorithm is in its applications. For example, the Euclidean algorithm for calculating greatest common divisors is very nice. Bezout's theorem is just the reverse of the Euclidean algorithm, so that's pretty cool too. Explaining how to write a number in another base using the division algorithm is also fun.

Furthermore, you should absolutely mention that the division algorithm generalizes to other "structures" as well. For example, polynomials satisfy it. And Gaussian integers too. So they also satisfy the Euclidean algorithm and Bezout's theorem.

 

[lurflurf]

With many calculators allowed...

What level are we talking about? Third grade, Third year graduate students, or somewhere in–between? By hand or with a computer. If you are talking about integers by hand I do not like the usual guess and check method double division and its variants eliminate multiplication and guessing, there is the chunking method, if you like guessing an extension that allows negative and multi-digit guesses can eliminate the tedium.

Edit:I just remembered you are probably talking about your number theory class. So you probably want to talk about modular arithmetic (and partition functions) and generalizations. You might also like to bring up clocks and periodic functions like cosine. A related example that might amuse them is

$$\lim_{n \rightarrow \infty} \sin ( 2 \pi e n! )=2 \pi $$
It uses the useful ideal that separating a real number into an integer and fraction is the same as taking the remainder upon division by 1.
If you use complex number it is a good time to mention C~R~R[x]/(x^2+1)
Related to that if you have been talking about integrals or pi
$$\int_0^1 \! \frac{x^4(1-x)^4}{1+x^2} \, \mathop{dx}=\frac{22}{7}-\pi$$
Which can be done by division algorithm on the polynomial

 

[verty]

My teacher showed us the short method first, calling it "division", then only later said, actually there is another way called long division, ...

This worked because long division was easier knowing the short method and one could check one's answers. We didn't learn synthetic division although it was in the book, and I never used it. I would leave that or do it with some time between, like months later.