Must it Always be True?

This morning on twitter, there was a problem that I just had to solve before going out the door. It is safe to say that these types of problems are my vice. Number Theory has always held a special interest to me despite, according to G.H, Hardy, having “absolutely no practical use.” (A Mathematician’s Apology, 2001). This has all changed with the inception of encryption.

I wish just to present the problem and then muse on its educational significance both for my personal learning of mathematics, and for that of my students.

N is the 4-digit integer 6_9_. If these two digits are reversed, explain why the resulting number must be 2970 more?
(posted by @dmarain to @cuttheknotmath)
I immediately look toward the base system when digits are switching around. When a digit moves from one place to another, it takes on a new meaning. I expect this new meaning will give me the difference I desire.
First number can be represented as (with a, and b in the set of base 10 digits):
6X1000 + aX100 + 9X10 + bx1
When we switch the digits, the new number becomes:
9X1000 + ax100 + 6×10 + bx1
The second is always larger (which is an interesting discussion to have with students) so we subtract to keep difference positive. We se that because the a and b did not switch orientation in the place value system, their value remains constant. Therefore, they will cancel out upon subtraction and have no bearing on the final solution. This is why it is a constant regardless of the two digits.
9000 +60 – (6000 + 90)
9060 – (6090)
In this way, we show that for the general case (there are 100 cases in total–2 spots, 10 digits) this fact is always true.
The idea of proof to students is very elitist. In high school, a list of examples where it holds is often sufficient. Once the list gets long enough, the proof is concluded. In this case, it would be easy enough to show the students there are 100 cases; this may discourage a plug-and-chug method. Instead, number tricks like this help students realize two things:
1. The basic qualities of our base 10 number system
2. The many interesting patterns that numbers create
In my curriculum, both of these topics are mandated. Posing this problem to a class of grade 10s gives them opportunity to create hypotheses, test them out, and then dive deeper into the number system to look around. I would imagine that such an activity would pair nicely with one on scientific notation or binary numbers. If they are really keen, a proof like the one above may be deciphered. Then we drag algebra into the mix as well.
There are many areas of useful mathematics that are left out of textbooks. As teachers, our pursuit of learning can greatly effect our teaching.

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