A couple weeks ago I had a little fun with BC on Halloween. I liked the mechanics of tying a math problem to some bigger puzzle. This is no different than those escape box kits you can buy for way too much money.

For the first time we were looking at a week off for Thanksgiving, so I wanted something to do on Friday that would be a nice closure. Most people go test here, but I was not in the mood to take a big to do list home. In AB we had made some good progress of curve sketching. Students knew how to start at an original and sketch the two "lower" functions as well as start at a second derivative and sketch the "upper" functions. We'd talked about critical points of the first and second derivative as well as what they represent.

### Setup

In this activity students would work with a partner and be given a random 5th degree polynomial as a starting point. On the back was a word, seemingly random.

I started with 5th degree polynomials because it offered a lot of variety. I mixed up the types of behaviors, repeated x-intercepts, and included some with their reflections to ensure students took the time to analyze the functions as we had done. The students objective was to start with their randomly assigned original, find its first derivative from a bank of choices, and then find the second derivative from another bank of choices. All their cards would have a word on the back.

The hallway set up like this:

Minimums and maximums were marked on the f cards. Minimums, maximums, and x-intercepts were marked on the f' cards. X-intercepts were marked on the f'' cards. As I made a lot of similar functions on purpose, I wanted some anchor to help students determine if they were on the right track. They'd have to find the functions with the appropriately matching critical points as well as behavior in order to complete their set.

My largest AB class has 28 students, so there were 14 sets. SOURCE GRAPHS

### In Action

How would they know if they had completed their set?

The words on the back of the cards are seemingly random, but they aren't at all. Some time ago I watched a video on the Remote Associates Test (RAT), a mechanism for studying intuition. When presented with three words, they trigger a related fourth word as a response. Some fourth words are more difficult to determine than others. Sets that comply with the RAT methodology are available right here. Students wishing to validate their set would read me the three words they collected. If they formed a valid RAT item, I proceeded to let them guess the fourth word. If one of the words was incorrect, I would tell them which one.

For example, if a student told me ACTOR, FALLING, DUST they had a valid set. The hurdle to their prize was to correctly guess STAR from the prize wall.

Now in an official RAT test, it's not a multiple choice thing. I wasn't really out to do any kind of behavioral study here, so all the 14 possible final words were available for them to see. To prevent random guessing, students had two chances to guess their fourth word. Failure to do so and they forfeit the prize. If they successfully guessed the fourth word, they could open the locker and retrieve the candy inside.

### Results

Students would take pictures of their original and walk between the f' and f'' sets while having discussions. Some would retreat to the classroom and sketch the set and go hunting for something that matched. Some theorized they could try to reproduce the function in Desmos, though most abandoned that idea when they realized there were faster methods. The idea was for them have a discussion about function behavior and how f, f', and f'' are linked by critical points.

All students were able to claim their prize, some in as little as 10 minutes. The longest anyone took was about 20 minutes. Most groups produced a valid set on their first try, those failing to do so usually had the second derivative incorrect. In a handful of cases students made a mistake at the first derivative but found the corresponding (though overall wrong) second derivative, meaning their 2nd and 3rd words, while matching each other, were not valid when grouped with their 1st word. One poor kid had the set right the entire time, but had misremembered the first word, leading me to intervene to figure out what the heck happened.

As a point of comparison, I let BC loose on the task too (mostly because I over bought candy and needed a way to get rid of it). They did curve sketching a solid month ago so I was curious to see what they remembered. Unsurprisingly, all of them cracked it in 5 minutes or less.

### Conclusion

This passed with flying colors. Wrapping my head around the decoding scheme took some time, but the use of RAT was really clutch here, a ready made puzzle that provided just enough of a pause point to add a nice challenge at the end. Some students spent many many minutes deliberating what their fourth word was, dancing around it the whole time. The two guess rule really put the pressure on. Designing functions that were similar but not too similar took the most time, as well as all the screenshotting, printing, and cutting. Oh, and a lot of nervous labeling. One misplaced word could derail the whole thing, so would losing the sets I used. I clutched that sucker TIGHT.

Loved it though. I'm not sure I've had a first time task go so smoothly.