This activity is aligned to NCTM Standards - Grades 6-8: 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 7-12, 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:

1. After each move, each person must be standing on a stepping stone.
2. 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.
3. You may "jump" another person if there is an empty stone on the other side. You may not "jump" more than one person.
4. Only one person can move at a time.

 

Introducing the activity

    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.

 

Using Manipulatives

    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.

 

Using Technology

    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.

Be sure to manipulate the various options that Mike has made available, including:

1. background color
2. foreground color
3. level of difficulty
4. show history and redraw history

After the students have tried the easy, medium and hard levels, encourage them to look for a pattern.

 

Revisiting the Activity

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:

1. What if there are only 2 people and 3 spaces?
   How many moves does it take for the two people to exchange positions?
2. What if there are 4 people and 5 spaces?
   How many moves does it take for 4 people to exchange positions?
3. What about 6?
4. What about 8?
5. What about 10?
   .....
   
6. 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.

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:

 

Extensions/Resources

Solutions and Reflections - from participants in Math Forum Institute

Purpose of Algebra - from the Dr. Math Archives

Algebra - Fun with Calendars by Cynthia Lanius

The Million $ Mission by Cynthia Lanius

"Middle School Mathematics Curriculum" - Suzanne Alejandre


Developing a mathematics curriculum that integrates activities, exercises, manipulatives and technology: a position paper for the NCTM's Standards 2000 and Technology Conference held in Washington, DC (June 5-6, 1998).