We started a unit on counting principles this week in my 9th grade honors class – permutations, combinations – eventually leading to the binomial theorem. Since my classes had used Desmos Activity Builder a few times and were familiar with the need to enter a 5-character code to start an activity, I planned to ask the following question as a class opener:

**How many different 5-character DesmosActivity Builder codes exist?**

This problem would have likely met my intended goal of having kids think about the fundamental counting principle in a real-world context. It also would have taken about 10 minutes of class time, and have been forgotten about by the next day. It felt like I was missing an opportunity to develop a deeper discussion. A slight tweak to the question added just the right layer:

**Activity codes for Desmos Activity Builder currently have 5 characters, as shown here. When will Activity Codes need to expand to 6 characters?**

And now we have a problem which requires a bit more than a quick calculation. To start, I asked students to work in their teams to make a list of information they would need to help solve this problem. This was not easy or comfortable for them – but a preliminary list of questions emerged from group discussions:

- How many 5-character codes are there?
- Are codes used less on weeekends and summers?
- Can letters repeat in codes?
- How many codes a day are used?

This was a good start to set kids in motion to think about how to solve the problem. I’m hoping they will think about new questions or revise their questions as we go along…the class did not disappoint!

HOW MANY CODES ARE THERE?

As kids worked, clarifying questions came up – some of which I just didn’t know the answer to, and hadn’t really thought about:

Mr. L, are there any zeroes in codes? Kids might confuse them with the letter O.

Mr. L, I don’t see any L’s in the codes?

Excellent observations, and restrictions we need to think about in our calculation. A tweet to the Desmos crew lent some clarity, and added more restrictions!

https://twitter.com/eluberoff/status/806304359331921920

Thank for the intel, Eli!

HOW MANY CODES PER DAY ARE USED?

This was tricky for my class. To help, I reminded students that when we started the semester, codes were 4 characters. When did the Desmos 5-character era begin? A quick scroll through my history (shown here) provides some info. After further interrogation from my class, I shared that Activity Builder started around July of last year with 4-character codes. Add this to our bucket of helpful info.

SHARING IS CARING

Writing a draft solution was the next task for students. But instead of turning it in to me immediately, I formed class teams of 3 where students shared their drafts and ideas. I used this opportunity to build teams of students who I observe don’t often interact or chat. From here, I gave students another day to think about their explanation – keeping in mind that there are no right answers to this question, only answers we can defend. But it still feels like we are missing a key piece in this problem……

DID WE MISS ANYTHING?

The next morning as students were mingling before the bell, I looked across the room at the laptop of Jacob – one of my more insightful, but also introverted, students:

It’s the mother lode!

The google trends graph for student.desmos. Yes! Yes! Yes! Stop everything kids, we need to talk! Jacob – tell us all about this graph. How does this new info factor into our estimates? What should we do with it? Is this going to continue? And with this, I gave the class an extra day to think about their responses, share, and dig deeper. And while many students simply estimated a growth rate by doubling or tripling their computed rate (this is fine with me), I am getting some responses which far exceed my expectations – like Jacob, who developed a growth function and evaluated integrals (did I mention this is a 9th grade class????)

Yep, this was definitely better than my originally intended problem!

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This is what allows a ‘lesson’ to become a learning experience.