Before introducing integers, we watched the movie Upside Down (trailer: http://www.youtube.com/watch?v=3veONCcRWbw). In it there’s an elevator with integers. Floor zero is in the middle of the two planets. The positive floors lead to the top planet, and the negative floors go to the planet below.

We make an Integer Elevator, and the students use it to do integer operations. It’s basically a vertical number line.

examples:
4-7 means you (starting at floor zero) go up 4 floors and then down 7 floors. You end on floor -3.
2(3) means you go up 3 floors twice.
2(-3) means you go down 3 floors twice.
(-2)(-3) means you go down 3 floors but twice in the opposite direction, which takes you to floor 6.

*The elevator thing doesn’t work for (-1/2)(-3.07), but I’m pretty satisfied that it helps the students understand (-2)(-3) first.

Is it a real world application? Yes, but from a different world, so I think it leans more to the perplexing side.

Mr. Meyer, any chance you attended an AP Institute in San Diego?

I think its important when students challenge the notion of (-1)*(-1)=1 we encourage this questioning and do not just shut them down. To appeal to both types of students and motivations I think I would answer that (-1)*(-1) doesn’t have to equal 1, but we defined it this way as a convention. It is not an arbitrary convention, it was done for many practical reasons for modeling real world behaviors described above. It was also done for abstract reasons such as preserving the distributive property also mentioned above. Deciding that (-1)*(-1)= -1 is within the powers of a mathematician, however, he or she has to be aware of the consequences that follow… such as no distributive property and many real world modeling problems that won’t make sense. Perhaps it would be good to address the problems with (-1)*(-1) = -1 to shine light on the virtues of (-1)*(-1)=1.

The zero power is tough. Intuitively the answer is zero, but that makes problems. Here’s an example, or train of thought.

Suppose we are talking about things that exist. Then an exponent of three has a lot of existence it represents a cube. Two is less (only length and width). One is less still. An exponent of zero still exists; it still has position. So, it is one for that reason. Making it zero would take it out of existence.

Higher exponents would mean more existence; we would add time, color, etc. Exponents that are less than zero imply less existence which is why they are fractions (and smaller the more negative they get.

Has anyone asked about zero factorial?

10^2 = 100
10^1 = 10
10^0 = 53

There is no *logical* reason for refusing to admit 53 in the “pattern” above. The fact that you cite a particular numerical pattern and a particular graphical pattern just means that we *really, really want* to define x^0 as 1, but that doesn’t cancel the fact that it’s a definition, in a technical sense.