# Infinitely Many Bounces

We recently finished up our series chapter.  One of the last questions we talked about was about a bouncing ball.  We just touched the surface of infinite geometric series, so I didn’t want to start out the question with infinitely many bounces.  I wanted to scaffold the question so it was easier for them, then hit them with the zinger!

First of all, their illustrations were awesome (and varied).  Almost every student found the total distance after five bounces.  Once they had confidence, I threw down the challenge, “What if I asked you what the total distance would be in infinitely many bounces?”  Since they all summed 5 bounces painfully by hand, they looked at me like I was crazy.  There were even some moans (infinitely many bounces, ALL that math)?  But after I gave them “the look”, and they knew there had to be a better way.  I told them they could work on the problem by themselves or in groups.  Several kids wanted some help, or even a hint.  But since I had just decided to ask them about infinitely many bounces, I didn’t have the answer yet.  I truly had nothing to give them, and I purposefully tried NOT to think of the formula or answer so I couldn’t give it away.  I try not to help students too much, but helping is in my teaching nature so sometimes I just can’t help myself!

Many groups of students came up with the infinite sum, and developed explicit formulas that seemed to work (but didn’t recreate the sequence of bounces).  With five minutes left, one student actually started screaming in class, “I got it!  I think I got it!”  And she did.

I didn’t tell her she was correct until the next day (I didn’t want the answer to get out before I gave all classes a chance to solve it).  When she found out she was correct, she literally went screaming down the hall.  The history teacher walking down the hall at that moment thought something was actually wrong with her.

THAT is the joy of mathematics, and I wish every student could feel it just once.  But more than once, I wish they could feel it everyday.  I also wish I could do this everyday.

I just discovered that we have motion sensors that will record bouncing data for us!  We could collect our own data.

I need to get better.

# Exploring Convergent and Divergent Geometric Series with Desmos

I could not find a Desmos teacher activity exploration for series, so I made my own.  Everything is better with Desmos!

This activity works best if students are already familiar with geometric sequences and series.  They are really just exploring convergent and divergent.  I instruct them to look up the words convergent and divergent in the dictionary.  I thought knowing these definitions would help it make sense.  I also included a geometric sequence and series link from Math Is Fun at the end.

Enjoy!