
Traffic Jam Activity


Teacher Lesson Plan
This activity is aligned to NCTM Standards  Grades 68: Algebra, Problem Solving, Reasoning and Proof, and Communication and to California Mathematics Standards Grade 7:
Algebra and Functions #1 and Mathematical Reasoning #1.1, 1.2, 2.2, 2.4, 2.5, 2.6, 3.2
Glencoe's Interactive Mathematics text provides an activity (Units
712, p. 5) called Hop, Skip, Jump. I first encountered this
activity during the Math Forum's 1996 Summer Institute, where it was called
Traffic
Jam.
Here's the problem:
There are seven stepping stones and six people. On the three lefthand
stones, facing the center, stand three of the people. The other three
people stand on the three righthand stones, also facing the center. The
center stone is not occupied.
The challenge: exchanging places
Everyone must move so that the people originally standing on the righthand
stepping stones are on the lefthand stones, and those originally standing
on the lefthand stepping stones are on the righthand stones, with the
center stone again unoccupied.
The rules:
 After each move, each person must be standing on a stepping stone.
 If you start on the left, you may only move to the right. If you start
on the right, you may only move to the left.
 You may "jump" another person if there is an empty stone on the other
side. You may not "jump" more than one person.
 Only one person can move at a time.
Large movement experience:
Each group of 6 students is given 7 sheets of paper to use as stepping
stones. Areas of the room are assigned to each group and the activity
begins.
Allow enough time for groups to try to find the minimum number of moves
necessary to complete the task.
Simulating the activity:
Once the activity has been experienced as large movement, students use what
has been learned to try it on a smaller scale.
Each group is given 6 small plastic figures or other objects with which to
simulate the activity while looking at page 5 of Glencoe's Interactive
Mathematics text, where there is a diagram of the bases. As groups try
to find the fewest number of moves necessary to complete the exchange of
places, the teacher circulates among them to monitor the activity.
Interactive Web activity:
To simulate the Traffic Jam activity, Mike Morton wrote a Java applet for
the Math Forum. Students can work individually, in pairs, in groups, or
with one classroom display to further investigate the problem.
Traffic
Jam  Java applet
Be sure to manipulate the various options that Mike has made available,
including:
 background color
 foreground color
 level of difficulty
 show history and redraw history
After the students have tried the easy, medium and hard levels, encourage
them to look for a pattern.
At this point, students have investigated the problem using large
manipulatives (their bodies), small manipulatives (plastic figures), and
technology (the Java applet). Some students will discover the minimum
number of moves for 6 people because they successfully complete the
activity using the Java applet, and the computer tells them they are
correct! Other students will not master the activity, but may have a better
understanding of the task.
Once more have the students assemble in their groups of six and repeat the
activity using their bodies and the paper stepping stones. As they repeat
the activity, observe groups that are successful and ask them to think of
some "rules" to account for their success.
Extending the activity  looking for patterns


Have the students sit down and think in terms of a pattern:
 What if there are only 2 people and 3 spaces?
How many moves does it take for the two people to exchange positions?
 What if there are 4 people and 5 spaces?
How many moves does it take for 4 people to exchange positions?
 What about 6?
 What about 8?
 What about 10?
.....
 Can you find a pattern for any number of people?
Writing the answer algebraically


Students can first make a data table using the information gathered so far.
There might just be columns for the number of pairs, the number
of people, and the first 3 entries for the minimum number of
moves.
number of pairs
1
2
3
4
5
6

number of people
2
4
6
8
10
12

minimum number of moves
3
8
15
...
...
...

Ask students: What patterns do you see? Are there any relationships
among the numbers in any of the three columns? Consider just the first and
third column. What if we let n equal the number of pairs? Can we
generate any of the numbers in the minimum number of moves column?
Does 1^2 + 2(1) = 3?
Does 2^2 + 2(2) = 8?
Does 3^2 + 2(3) = 15?
The completed table might look like this:
number of pairs
1
2
3
4
5
6
...
n

number of people
2
4
6
8
10
12
...
2n

min. number of moves
3
8
15
24
35
48
...
n^2 + 2(n)

another view
1^2 + 2(1) = 3
2^2 + 2(2) = 8
3^2 + 2(3) = 15
4^2 + 2(4) = 24
5^2 + 2(5) = 35
6^2 + 2(6) = 48
...
n^2 + 2(n) = n(n + 2)

