For the Explore #MTBoS Week 1 Challenge, I am going to write about one of my favorite rich problems: The Traffic Jam Problem.

The problem is that there are six people and seven stepping stones. The three people on the left must switch places with the three people on the right. You can only jump one person at a time and you can only move in one direction. The Math Forum provides a great explanation of the challenge and rules.

I do this problem with my students in between our linear and quadratic functions units. I place colored paper on the floor in the front of the classroom to represent the stones and start with two student volunteers, usually one boy and one girl so it is easier to follow along as they switch places. Then we build up to two boys and two girls switching places. When the class is determined that we have done this in the fewest possible moves, I send everyone back to their seats and pass out these simple game boards and bags with 6 hershey kisses (3 each of two different colors) to each set of partners.

The hershey kisses could be substituted by any other manipulative, but it’s very important that the students can physically move around some pieces to try to solve this. This is the really fun part. There is usually a loud mix of excitement and frustration. Some students will begin to solve it and others will keep making the same wrong moves and get stuck. I ask all students to show me their solution, so I allow them to use their phones to film it so they don’t forget. If students finish quickly, I give them two more hershey kisses and let them try the next round.

On the whiteboard, we make a chart with columns for # of people, # of pairs of people, and # of moves. As a whole class we find a pattern and equation that represents the data. Most students try to make a linear equation fit the data at first, so it’s a fun way to introduce quadratics.

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I would like a little more information. Do they ever write the equation? Is it a teaser and then you come back to it later? At what level do you use this activity?

Great questions! I do this with 9th graders in Honors Algebra 1. Usually by the end of the class we have come up with the equation. Once we determine that it’s not linear, I guide the students toward looking at the 2nd differences in the table. At least one student in every class remembers that when the 2nd difference is the same that means it’s quadratic. At this point in the year they’ll only be able to do some guess and check to figure out a quadratic equation that works. Later in the unit, I try and go back to the problem when we’ve learned how to write quadratic equations from two points.