Instead I taught a concept that only relied on counting and identifying numbers as even or odd. I thought that my students' brains could handle these second grade concepts after a two-hour comprehensive test. I combined my AP Stat class with a Calculus class and a PreCalculus class with an Algebra 1 class and we set out to discover what the conditions needed to make Euler paths and Euler circuits. I encouraged other members of my department to do this lesson and several of them did. One of my colleagues noted that she was surprised how uncomfortable many of her students were with an open-ended question with no immediate solution method.

Here is the outline I sent to my department.

1) Pose bridges of Konigsburg problem. Have kids
determine if a walking tour of the bridges is possible or not possible and
explain why.

2) Ask if we could remove or add a bridge to make this
possible and describe the location of the bridge.

3) Show how the bridge problem could be modeled with line
segments (edges) and dots (vertices).

4) Have them work in groups of 3-4 at the boards to create
a set of edges and vertices that can be traced where you start and end at
different vertices. (Could give restriction like, you must have at least 5
vertices and 5 edges.)

5) Ask students to count the valence or degree of each
vertex. (This is the number of edges going into a vertex.) Have each
group report out on the number of even and odd vertices. Amazingly
enough, each group should have exactly two odd vertices! Try to have a
student or two articulate why this makes sense.

6) Now have students create a new graph where they would
start and end at the same vertex. Go through a similar process to have
each group report on the number of even and odd vertices. Amazingly
enough, each group should have all even vertices! Again, call on a few students
to try to explain why this makes sense.

7) Have students come up with a class generalization about
Euler Circuits and Euler Paths. For example, to start and end at the same
vertex, all vertices must be even. This allows us to have an Euler circuit.
To start and end at different vertices, there must be exactly two odd
vertices. This allows us to have an Euler path.

8) Finally, give each group one of the problems from the
attached materials to do. They do their work on the board together and then
each group shares out the solution to their problem with those at their seats
critiquing the solution.

Below you can see some of the graphs made by my students.

We only had 10 minutes of the 60 minutes of class time left, which was not enough time to complete item number 8. Instead, I beat a few volunteers in several games of Nim. There was a game show atmosphere reminiscent of the Price is Right, as students called out suggestions for the next move in the game.

Were the goals of the lesson achieved? If the goal was to expose students to new ideas in mathematics that they have never seen and would probably not see in a traditional curriculum, the answer is yes. If the goal was to have students work together to solve a non-routine problem, the answer is yes. If the goal was to have students learn about one of the most prolific mathematicians of all time, the answer is yes. Teaching after a Two Hour Exam? It can be done and the sizzling of brains was avoided.

We only had 10 minutes of the 60 minutes of class time left, which was not enough time to complete item number 8. Instead, I beat a few volunteers in several games of Nim. There was a game show atmosphere reminiscent of the Price is Right, as students called out suggestions for the next move in the game.

Were the goals of the lesson achieved? If the goal was to expose students to new ideas in mathematics that they have never seen and would probably not see in a traditional curriculum, the answer is yes. If the goal was to have students work together to solve a non-routine problem, the answer is yes. If the goal was to have students learn about one of the most prolific mathematicians of all time, the answer is yes. Teaching after a Two Hour Exam? It can be done and the sizzling of brains was avoided.

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