Ok, I’m still really overly excited about Graph Theory (and teaching in a way more aligned to my philosophy!).

While Kogut and I were at the AERA conference, my awesome student teachers led students through the Bridges of Konigsberg and the beginning of a snowplow route project, where students had to find a more efficient route through a neighborhood (in the future … smaller neighborhoods!).

*(Fenway Park neighborhood)*

This week, we moved on to attempting to discover the rule for how to tell if a graph has a circuit or path and then a choose-your-adventure project.

**Awesome Moment #1**

On Friday, for the Do Now, I asked students whether the graph below (drawn by a student on a HW) had an Eulerian circuit, path, or neither. The student had submitted it as a graph without a circuit (S/E) or path (GPS).

Every single student thought the graph had neither a circuit or path. Except! One student, who barely talks to the large group, figured out it was a circuit. I whispered to him, asking him to go up to show the class. We started with a few students explaining why they thought it was neither. Then I called up this student, who showed the class his method of going in the circuit. All the students cheered and many gave him fist pumps as he went back to his seat. Sweetness.

**Awesome Moment #2**

Today, we started the aforementioned project, inspired by Dan’s class set-up and this post about students teaching each other. My student teachers and I generated a list of “open questions” in our class about graph theory — some have come up, some are just about theory we haven’t studied yet.

The students had to choose 3 questions they would be willing to work on, and then they formed groups n < 5. They spent the period answering the questions and all (almost all?) the students were really working on the math, discussing with each other, etc. I barely did any work or talking. Students asked each other questions, checked their work, and arrived at accurate responses. One group went to the board with our collection of graphs to determine whether it matters where you start on a graph with a path.

Other groups got whiteboards to write up their answers or draw graphs to analyze. And they’re going to write lessons and teach each other about these questions. I’m so pleased that incorporating ideas such as choice in topic, process, and group; autonomy; and open-ended products at least seems to be leading to the type of classroom I aspire towards. Let’s hope the rest of the project keeps going well! (I definitely just jinxed it.)

**Awesome Moment #3**

Students got to the following rules for how to tell if a graph is an Eulerian circuit or path (basically by themselves) …

– Circuit’s vertices have even degrees.

– Path’s vertices have some odd degrees.

– Graphs with neither a path/circuit have MORE vertices with odd degrees.

Despite the fact that they were so.close. to the final theorem (and the adults in the room were making awesome faces over students’ heads as they danced around the answer), we did NOT give away the answer. I’m hoping it might come up during the student-led lessons.

*SL*

Wow. This is amazing. I would have been a totally different student if I had been in your math class!!