## Developing rational exponents through geometric sequences

Soon I’ll be teaching rational exponents in algebra 2. I’ve never found an intuitive way of teaching it…until now. Thanks Bowen.

The approach leverages geometric sequences. I’d love to regurgitate it, but this tweet from Bowen is my source and sums it up:

The unit prior focused on sequences with a heavy emphasis on geometric sequences, so this is the perfect bridge to developing this idea that most students find confusing. It all comes back to repeated multiplication, as it should.

In the past, I’ve usually had students enter various expressions (e.g. 100^(1/2)) into their calculators to stumble upon the relationship between rational exponents and radicals:

But this painfully ignores the mathematics behind exponentiation and instead lures them into believing that these two concepts are magically connected through a few keystrokes on their calculator. It treats rational exponents as an isolated concept and unrelated to repeated multiplication.

After discovering my new strategy for teaching rational exponents, I found this video from Vi Hart on logarithms. The similarities run deep.

Now I can’t wait to teach logarithms.

bp

## Entry ticket

Yesterday my students were introduced to logarithms. Today, to follow up, I used an “entry ticket” requiring students to evaluate basic logarithms. It’s pretty simple and requires no prep. It’s one of the few times during the year it can work because of the minimal computation involved (another one is the unit circle).

Here’s the deal. I stood outside my classroom before class and had students line up single file along the wall. I waited for the late bell to ring and asked the first person in line to evaluate a given logarithm. For example:

If he answers correctly, he’s allowed to enter the classroom. If he doesn’t, then to the back of the line he goes! The person next in line steps up and the process continues until everyone has gained entry. For 27 kids, it took about 20 minutes.

I’ve done this several times and I’m still surprised on how wildly successful, and effective, it is. The kids love it! They always embrace the challenge with smiles and good energy. Even random kids passing by my classroom and other staff members root them on or cackle when students are sent to the back.

Other details:

• I made up the problems on the spot, so planning was minimal.
• I used a whiteboard to present the problems.
• Kids that were absent yesterday had to learn from someone else in line that was present for the lesson. Interdependency!
• After students were in the classroom, they played log war.

bp