What do Hercules, Kings, and Exponents all have in common?

Just introduced exponential functions in Algebra and I wanted to find an engaging way to get them hooked. I proposed this question to the students and asked them think & discuss with their groups:

A king has offered you a handsome reward for all your service to the kingdom. You may choose either:

- $1,000 on each square of a chess board until it is filled

- $0.01 on the first square, $0.02 on the second, $0.04 on the 3rd, until the board is filled

Which do you choose and why?

Naturally, most of them picked the $1000 deal- because obviously it will yield more money! Only one student bothered to actually do some calculating...like by the 17th square you'll start getting way more than $1000 per square. 

I had two students come to front of the room and handed them an envelope with the amount they would receive in it. They went nuts. Why would this happen? They thought & discussed. Geometric vs. Arithmetic sequences.... counter-intuitive but captivating!

I then showed them this clip of Hercules fighting the hydra. They then broke into 5 groups and each had a different type of hydra- one that loses one head and grows 3 back, one that loses one head and grows 4 back, etc. They had to make a table of the number of heads in each round, then write a function equation for it. This, of course, led easily into graphing and seeing the growth. Which hydra would you least like to face? The one with the biggest common ratio/rate of growth! Vi hart wrapped it up beautifully for the day with her binary trees video. Fun day!