The CME curriculum has a really clever way to help kids gain another perspective on rational exponents. What’s so confusing about rational exponents — I think — is that it’s very difficult to make sense of them using the model of exponentiation as repeated multiplication. How do you multiply something half a time?
CME subverts this by actually talking about multiplication of something a fractional number of times.
What follows is a direct quote from the CME textbook:
You can use a geometric sequence to list the integer powers of any number by starting with 1 and repeatedly multiplying by that number. Here are the powers of 27: 1; 27, 729; 19683…
If you multiply 1 by 27 four times, or multiply 4 factors of 27, you get 531,441. So But what is ? How can you multiply 1 by something of one time?
One way is to insert extra terms in the geometric sequence between 1 and 27:
1, __, __, 27, __, __, 729, ___, …
Assume it remains a geometric sequence. What goes in the blanks?
As far as turning this into a problem that kids could solve, I’m imagining something like giving them a set of problems that includes both the sort of incomplete sequences that were shown above, and also a request to imagine what it might mean to multiply 1 by 16 a half of one time. But if you’ve got a different idea on how to turn this into a good problem, drop into the comments.