Penny Jar Situations

In math, we've been studying constant rate of change (grown up language only) by thinking about penny jar situations.  A penny jar situation might sound like this: I start with 4 pennies in a jar.  Each round, I add 5 more pennies to the jar, so after round 1, there are 9 pennies in the jar, and after round 2, there are 14 pennies in the jar.  We've been talking about how you can figure out how many pennies are in the jar after any round without having to know how many pennies were in the jar in the previous rounds.  There are a bunch of ways to think about this.  For example, Charlie noticed, 

"If you know the 5th round and want to know what the 8th round is, you have to find out how much is in there, or how much rounds is in between the 5th and 8th rounds.  I add the number you're supposed to add (5, in this situation) three times.  I add three times because that's what's in between five and eight."

Here's Charlie's explanation of his work for the penny jar situation start with 4, add 5:

Sascha L., Braden and Frey thought about it another way.  They realized that as long as you know the situation, you can figure out the  number of pennies after any round:

Braden and Frey expressed this idea algebraically:

One of the ideas that came up as we were exploring ways to figure out how many pennies would be in the jar after any number of rounds was doubling.  It seems logical that you could double the 5th round to figure out the 10th round, for example, but we figured out that doubling doesn't work.  Why not, then?  Here's Jessie's work and explanation:

"The reason it doesn't work is you have a starting number.  If you did not have a starting number, the doubling would be normal.  In this case, our starting number is 4.  4 is throwing the doubling off.  The way you can use doubling to help you find the total number of pennies in a certain round is you double the (total number of pennies in the) round you are on, and then you subtract the number you started with.  But, to find it (the total number of pennies), you have to double the round too.  If you're on round 2 and you used the doubling method, you would find the answer to round 4:

(14 x 2) - 4 = 24 (the number of pennies in the 8th round)

We would love to answer your questions.  It would be very 2011 if you posted questions as comments, and students could respond.  :)

Have a great weekend!