Fraction Bars to Understand Rational Exponents

September 27, 2016

I have been using number lines to teach negative exponents (I will write another post about this soon!) and the built in conceptual understanding explains to my students WHY a zero exponent yields 1 and WHY negative exponents are not necessarily negative numbers and they never reach zero. 

So WHAT could I do to teach rational exponents?

 I have used this activity before with my Integrated Math 2 students and it yielded a deep understanding of rational exponents. They understood what the fraction MEANT and we were able to step off from there to the more symbolic representations and procedures.

**I've always taught these with blank diagrams because I didn't have fraction bars. Future me will have them draw the diagrams or buy fraction bars.**

Here's how I bait the hook...

"Start with 64. 

What two numbers multiply to 64? 8*8.

Okay, so HALF the perfect square factors of 64 is 8 and two 8's multiplied make 64?

Interesting....

Can you make 64 another way?"

Some questions as I walk around: "How can we make 8? How is THAT related to 64?"

Then it gets fun because you give them specific fraction bars (or question them to get here) to get the square root of 8 with the 1/4 sizes (and later 1/8). This is where they understand that 1/2 power is a square root (if they haven't already). 

Here's the dialogue:

"So if two 8's multiply to 64 and they create the 1/2 pieces, what is the value for the 1/4 pieces?" We want them to realize that two 1/4 pieces multiply to 8, so it must be square root of 8. So half of the perfect square factors of 8 is square root of 8. Half factors must be roots. 

This is when I let a student explain (usually by the aid of a calculator discovery) that an exponent of 1/2= a square root. We then test our theory and understand cube roots and 1/3 etc. 

You can even extend their thinking to double roots really being 1/2*1/2 or 1/4. 

The concern

We are using lengths to explain products. Not the best plan so I front load and tell them that these lengths are representing multiplication but not modeling multiplication. Once we realize the rational exponent connection to radicals, I would explain that the lengths are ACTUALLY representing the rules of the exponents. 

An example:

64^1=64^(1/2)*64^(1/2)    We can use the fraction bars to see that the exponents are added in multiplication.  

Thoughts?

Thanks for reading!!