Every so often, I’m inserting some posts on pedagogy, especially for those of us who aren’t as math-inclined. If you find this helpful, just let me know in the comments.

Today, I’m differentiating between the old way I used to prepare the students for a lesson on exponents and the way I do it now.

**Old Way**

1. Ask “What’s an exponent?”

2. Here are the five mail rules you need to know.

3. Apply them.

**New Way**

1. Ask “What’s an exponent?”

2. Use their definitions with a bit of refinement.

3. Get their thinking about a few cases involving exponential, expanded, and standard forms.

4. Have them deduce what 3¹ and 3° power would look like.

5. If they can see the operation of division, then prompt them to ask if this works for every case in general.

6. Use the same thinking from 3-4 to get the values for 3¹ and 3°.

7. Let them establish the (real) first two laws of exponents.

8. Use similar tabular thinking to deduce product, quotient, and power of a power laws.

Usually, when students are just given the laws, there’s rarely reason why we give it to them. It gets shoved on their laps, especially x to the zero power. Some of it might be because of a lack of understanding on the part of the teacher, but I also suspect it’s because we don’t see the importance of taking them through similar steps that an actual mathematician would go through.

But I changed that last year, and into this year.

My main objective was to make sure to establish the foundation for every argument we will make about exponents from here until October.

Here’s a sample of what I mean:

Exponential | Expanded | Standard |

x^{3} |
x · x · x | n/a |

x^{2} |
x · x | n/a |

x^{1} |
x | n/a |

x^{0} |
1 | 1 |

I’m hoping for a couple of things here. First, I hope they see that, as the exponent decreases, the amount of x’s in the expanded column decreases, just to solidify our definition of “exponent.” Also, I hope they see some sort of division (making the connection between multiplication and division matters here). I use a specific case first, and it’s usually a positive base, just so they’re familiar with the numbers, then work my way towards this.

By the way, once they see the 1 at the end, they should also be able to see that Icould have had a multiplication of 1 all along since 1 times any number equals that number.

Hope that helps. Best,

**Mr. Vilson, who will have a post like this about once a week …**

## Comments 2

Of course, it works if you extend it into negative exponents as well! I was an engineering major before I switched into education. I remember explaining zero exponents to the rest of my elementary math education class. No one had ever bothered to show them!

Author

Exactly right, Anne. You took the words out of my mouth (and the post I was going to write in a few weeks). I’m hoping that they can see the relationship between the negative exponent and the fractional ways I write it. Let’s see how craftily I explain it. -rubs chin-

Thank you.