Discussion:  Roundtable 
Topic:  Traffic Jam Activity 
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Subject:  RE: Traffic Jam: number of moves 
Author:  SteveW 
Date:  Aug 3 2004 
2. If you know how many people there are on each side, can you tell me the
minimum number of moves it will take to complete the exchange?
First we wanted to get the "idea down." and then we wanted to find the minimum
number of moves. We defined "n".
Question: Did you decide initially what "n" was? (Should I use a variable?) As
a high school teacher and working towards the quadratics, you would have to
define the two variables in some way.
We had a communication problem to start with  are we talking about the number
of people on each side or the total number of people?
Once you have the number of minimum moves for specific cases, describe it
algebraically, and then go on to write the description of that.
Many of us arrived at some form of n^2 +2n. And, connecting this expression to
the patterns noticed in the moves, we noticed that the number of jumps is always
n^2. When you count the number of steps, it's 2n.
The number of jumps could be calculated making use of a variation of Gauss'
formula  recognizing if you add the number of jumps that you go through (e.g.
1+2+3+4+3+2+1 when there are four pieces/people on each side), you are twice
adding all of the integers, 1 to n1, plus the "nth term" in the middle. 2*
[n(n1)/2] + n, which simplifies to n^2.
If you use n as odd, you get a different quadratic? (an odd number of n's) If
you don't define the rules as clearly, you get different results. Let the
students define their own rules and then discover the pattern.
What about putting it on a square grid rather than just linear. For example a 6
by 6 or ? (Are there lots of empty spaces?)  Cut The Knot
http://www.cuttheknot.org/SimpleGames/FrogsAndToads.shtml
The "kind of move" came into play.
We moved to the level of abstraction almost immediately so that we didn't
really learn to "play the game" until we backed up a little. What in the
patterns of the movement made the abstraction come out.
The lesson plan with the tool is important. There are key moments where the
teacher can help with a welltimed question, for instance about connecting
mathematical expressions to the pattern of moves.
Manipulatives vs virtual manipulatives vs paper/pencil vs (body)
kinesthetic
Part of what you want them to do is use mathematical language to describe the
activity. At the least I want the user to define the action. (Might you restrict
it to letters?) No restriction would be better. You want the development of the
language.
Could you use music? numeric, alphabetical (Some folks respond better to sound
patterns.)
Ways to extend the problem out further  "What if you didn't have an even
number of figures on each side?"
A good feature of this problem is its richness. It entertains from cradle to
grave.
Even in presenting this to teachers, using the kinesthetic first is a good
idea.
Have students try the activity with their bodies WITHOUT talking.
Conway's Game of Life  connection (point to some online instances) Used a
binary code to view the idea.
 
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