## The Magic Octagon – Dan’s, Andrew’s, and mine

15 Nov

I had saved Andrew’s post in my folder for a recent lesson, which was about Dan’s video.

We paused halfway in, and students decided where it would be. They answered a Quick Poll to let me know, and by the time they had all answered, some had changed their minds.

We quickly looked at the responses, and they decided using time would be easier to decipher than some of the other descriptions.

I sent a second poll. I waited for everyone to answer, even the ones who wanted to take their time thinking about it.

And then we continued to watch.

We paused for the last question, they discussed with their team, and then we finished watching.

Good conversation. But we didn’t get to the sequel proposed by one of Andrew’s students: If the front side arrow is pointed at 5:00, would the other arrow point at 5:00, too? Why or why not?

So I emailed that question to my students.

• Yes, the two points move like opposite hands on a clock moving closer to each other and overlapping at 5:00. At about 11:00 they would overlap again. Otherwise, there is no overlap.
• They would be at 5:00. This is because when he flips the magic octagon, the back arrow also flips, causing the new time to be 3:00 instead of 9:00. This means that if you were to find a line of reflection, you could flip the octagon on that line and the arrow would always land right where the previous one did. If this was on transparent paper, you can see that if one arrow points to 5:00, then the other one would be pointing at 7:00. But if you were to flip the octagon on the reflection line which intersects 12:00 and 6:00, then you would continuously get 5:00 because of the reflection.

As I got the responses from students, I realized that I wished I had asked a different question. While I did include why or why not, and it was obvious from the responses that students didn’t just answer yes or no, I wish I had asked “At what time(s), if any, are the front side and back side arrows at the same time?”

I am reminded of something I can no longer find that I read in a book. A group of teachers observed a “master” teacher for a lesson and then went back to their own classrooms to teach the lesson. The teachers asked the same questions that the master teacher asked; however, the lessons didn’t go as hoped. The teachers were not asking questions based on what was happening in their own classrooms; they were asking questions based on what had happened in the other classroom.

I love reading blog posts and learning from so many mathematics educators. They give me ideas that I wouldn’t have on my own. In fact, as my classroom moved toward more asking and less telling, I used to say that my most important work happened before the lesson, collaborating with other teachers and deciding what questions to ask. I’ve decided otherwise, though. My most important work happens in the moment, not just asking, but also listening. And then, if needed, adjusting what I planned to ask next based on the responses of the students in my care. And so the journey will always continue …

Posted by on November 15, 2016 in Geometry, Rigid Motions

### 7 responses to “The Magic Octagon – Dan’s, Andrew’s, and mine”

1. November 16, 2016 at 4:01 am

Thank you so much for sharing your insight! My 13 yr old students have just started a consolidation unit on time- just to make sure they know the basics. It’s a pretty bland topic, and difficult to make exciting in the second last week of school for the year (Australia). So I used the ‘magic octagon’ as the opener! What a treat! There was intrigue, genuine curiosity, disbelief, confusion! And then on there own, a determination to find out ‘how’.
We even used two students back to back, arms pointed to the clock, so that all students could appreciate what was happening. Surprisingly, after that physical demonstration, the challenge question was simple- my students found the ‘same time’ answers quickly, confirmed by ‘back to back’ students!
Your post made the bland topic of ‘time’ much more inviting- because the students had been rewarded with satisfying their own curiosity.

• November 16, 2016 at 5:11 am

I’m very glad it was helpful. Finding what makes students curious and determined to find out how is always good. I like that the students eventually demonstrated what was happening.

2. November 23, 2016 at 7:37 pm

Thanks for posting Jennifer. You put a fabulous, relatable spin on the magic that happens in the moment. I am reminded that I need to make more notes about what happened, vs what I plan to happen.

• December 15, 2016 at 3:31 pm

A reminder to us both! Luckily, blogging does help me remember what actually happened and even pay closer attention to it while it’s happening.

3. December 2, 2016 at 12:08 am

Great post, Jennifer. Your last two paragraphs are spot on:
“The teachers were not asking questions based on what was happening in their own classrooms; they were asking questions based on what had happened in the other classroom. My most important work happens in the moment, not just asking, but also listening. And then, if needed, adjusting what I planned to ask next based on the responses of the students in my care. And so the journey will always continue …”

I totally dig this. I know what you mean.

• December 15, 2016 at 3:33 pm

Thank you, Andrew. It is always good when someone understands what you mean.

I wish I could remember the book that first made me notice this happening. Let me know if you ever run across it!