Rational exponents is a topic that is traditionally "how-to" lesson. However, the notation has meaning. I'll explain in greater detail below.
This is a two day lesson that should have students able to convert between radical notation and rational exponents, able to simplify rational exponential expressions and higher order roots. Each day has a quality assignment with it.
There are a few pages of notes for teachers that guide you and the slides have notes to help you pace appropriately and to let you know where you should probably guide more and where you should allow the students to grapple with understanding.
Here's the idea behind the language:
Multiplication is repeated addition.
Division is the inverse of multiplication, yet it is not repeated subtraction. If it were, 15/0 = infinity. That’s an inconsistent way to consider division.
Division is better thought of as asking a question about multiplication relationships. For example, 24÷8 asks what plus itself 8 times is 24? Then with a fraction 24÷(2/3) asks, what plus itself three times is twenty four? What’s the second step (the numerator).
Similarly, rational exponents do the same, except exponents are repeated multiplication. That’s why the division symbol is used to introduce the ideas. We quickly discard it and adopt the traditional notation.
For more lessons and mathy-type things, please visit my website: http://thebeardedmathman.com