Dice Game and other things

I was still feeling bad about being snippy to one of my geometry classes last Friday, and I gave them a test on Monday so I didn’t get to address it. But today at the top of class, I apologized to them. It felt good to do.


At our department meeting today, a colleague shared a problem: At 3:00, the minute and hour hands form a 90 degree angle. What is the next time that happens?

What I loved is how many different approaches we teachers had for coming up with the answers. I went to brute force. But my answer had a beautiful simplification, and it gave me all the times when a clock’s hands form a 90 degree angle. And guess what? It had this beautiful step where I recognized a trig sum of angles formula (expanded) and that helped me collapse something, which them made the solution fall out like PUTTY.


At our department meeting today, another colleague shared a game he plays with his class. Kids draw six blanks (“____”) on a sheet of paper. Then the teacher rolls two dice, and reads the faces of them. Students then have to write the product of the faces in one of the blanks. Once it is written, it cannot be erased. Then the teacher does it again. And the student has to fill in another blank. Sounds easy, right?

Here’s the catch: the students have to decide which blank to place the product, so that the numbers in the blanks read from smallest to largest.

So, for example, say the blanks look like:

___  ___  ___  ___  ___  ___

Now say the teacher rolls a 3 and a 2. That’s a low product. Maybe you put it in the second blank.

___  _6_  ___  ___  ___  ___

Now say the teacher rolls at 2 and a 1. Phew. We still have a blank on the left.

_2_  _6_  ___  ___  ___  ___

Now say the teacher rolls a 2 and a 2. ARGH! You’re out of the game, because you can’t get the six numbers in the blanks from lowest to highest!

What would your strategy be?!

(NOTE: If the teacher rolls the same product a second time, we ignore it and roll again.)


After the department meeting, I was working on prepping my lesson for precalculus tomorrow. I don’t know why, but I had a stupid simple insight that I never had before (a pedagogical thing) that would help kids evaluate finite geometric series more easily. It isn’t an insight into the topic or anything, but I really love this slightly different way to reframe the formula (that the kids derive). I’m too tired to type it out here, now… but I’m guessing you probably already do this and I’m just super late to the party.

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