How to teach the division algorithm? 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?
 A: 
[Cue sci-fi sequence.]
A number of clones in an underground cloning lab escape from their enclosure. Most of the clones are from the same batch, and they all weigh 150 pounds. There is also a clone from a later batch--not as fully developed, only weighing 50 pounds. They manage to reach the elevator out of the facility, and through [plot contrivance] are able to make it operational for one trip up, only. The elevator has a 2000 pound carrying capacity. How many of the grown clones will escape the facility (assuming that the volume of the elevator is not a restriction)? Will the elevator be able to carry the youngster?
Tune in next time, for Division Algorithm!
[Cut to black.]

Okay, that isn't a real life example, but it certainly isn't dull.

You have a pocket full of quarters (39 in all) and a serious gumball craving. The supermarket gumball machine charges a dollar per gumball. (The money-grubbers!) How many gumballs can you get, and how many quarters will you have left?

That isn't quite as exciting, but it's more "real."
A: I found Cameron Blue's answer confusing so I wrote my own answer. How do you compute a quotient and a remainder of a division problem? It's by computing the quotient and remainder on division of the original dividend by the divisor, in terms of the quotient and remainder of the division of a new divison problem with the original divisor without the last digit as a divisor and the same dividend, and recognizing that you can iterate this process of changing the division problem. Now the quetion is "How do you do a single iteration of that process?" Let's say you have a really large number and you want to find its quotient and remainder on division by 257. Once you find the quotient and remainder of the original dividend without the last digit on division by 257, you can find the quotient and remainder of the original division problem as follows. What does it mean to say the quotient and remainder on the original dividend without the last digit are? It means it can be expressed as 257 $\times$ a natural number + a natural number from 0 to 256. So if you multiply that by 10 than add a number between 0 and 9, then this new number can be expressed as 257 $\times$ 10 times the quotient + 10 times the remainder plus that number from 0 to 9. Now this number is small enough that we can find its quotient and remainder on division by 257 in our head. Its remainder will now become the remainder of the original problem and once you add this quotient onto the number that was used in the sentence "So if you multiply that by 10 than add a number between 0 and 9, then this new number can be expressed as 257 $\times$ 10 times the quotient + 10 times the remainder plus that number from 0 to 9," you get the quotient of the original division problem.
