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?
[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."
This idea should be taken with a grain of salt. I'm not sure if the problem actually is teaching students long division without an application. Also, long division is not just about finding the quotient and remainder of any division problem between positive integers. It's also about using the quotient and remainder to compute the answer of the division problem in mixed fraction notation. If the problem is just the lack of an application, here's one possible application for long division. It's computing the speed of a car when you know how long it takes for it to get somewhere and how far it needs to travel to get there. If that's not the problem, maybe the problem is that some students can't remember how to do long division because they were just simply taught how to do it without an understanding. Maybe they're less likely to forget it if they're guided to teach themselves how to do it. First, they could be taught that in order to compute the answer to the division problem as a mixed fraction, you first have to compute the quotient and the remainder of the division problem. Then the teacher could teach the students that for any decimal notation of a number $\geq$ 10, the number it represents is the number the string of all digits but the last one represents multiplied by 10 plus the number the last digit represents. Then the students might teach themselves that if you take off the last digit of the dividend and then compute the quotient and remainder of that division problem, you can use that to compute the quotient and remainder of the original division problem.