This edition of Simple But Evil, really is simple and evil. The prerequisite is that students should know at the least, multi-digit stacked addition and should have seen how roman numerals work. For example, they should be able to do this problem
and they should be able to convert, say
Here’s the challenge, suppose we never discovered our current place-value arithmetic, how would we do this addition?
One simple but evil rule is that we can’t convert this into
What’s the educational value of this?
Consider,
With near certainty, students will try to do this by inventing stacked arithmetic rules for the Roman numeral system. This won’t really work out and the eye-opener for us teaching is that most of our students have been following an algorithm.
In all the times I have given this problem, regardless of age group, only one student showed a method that was consistent, understandable, extendable, and rooted in mathematical thinking.
Of course, we can ask questions about subtraction. How would we do this subtraction without our place-value system?
The answer is
What I am looking for is reasoning within the Roman numeral system. This is difficult for students because they are conditioned to see the above problem as
There are probably many ways, but I like this one the best:
Thus,
This is the basic counting that students do when they first learn numbers. It’s also similar to methods of regrouping — i.e.,
Here’s another example with commentary.
First, we can establish a convention that we want our numbers written in a “counting form”.
That is, we note that the biggest unit is
And now, our Roman numeral subtraction is
Going back to our original addition problem
We can do the same thing by utilizing Roman numeral facts:
Let me know if you try this. I’ve found that I can have lots of conversations about arithmetic and can get students to see the standard arithmetic more clearly by stepping out into Roman numeral land.