Permutations Of Minions

We’re getting ready for my daughter’s Minion themed birthday party.  She’s turning 8.  Where does the time go?  Since we had a snow day today, we were all home during the day and making preparations.  My wife made the poster below.

Kayla couldn’t figure out which minion was which.  She kept saying their names in different orders.

“Kevin, Stuart, Bob”

“Bob, Kevin, Stuart”

Not expecting much conversation I asked how many ways the three could be arranged.  Without shying away from the challenge, she sat and thought for a minute.  Deciding she needed to write down her thinking, she got some paper and made the lists below before confidently announcing her answer of 6.

I praised her for both showing her thinking and the organization of her method.  She had been stating aloud, “well there are 2 ways that they could stand with Kevin in front, then there are two ways that they could stand with Bob in front, and then there are 2 ways that they can stand with Kevin in front”.

Curious how far I could I push, I asked her about including another minion, Dave.  She struggled more with this one, initially listing 17 possibilities.  When I told her that she missed some possibilities, instead of going back to her list, she surprised me by trying to go into some arithmetic.  She started with a “4 times 3 times 4….48”.  I asked her how she got that and she started explaining that there are 4 minions and then you multiply by 3 because there are three left, but she couldn’t explain where the other 4 was from.

She revised her thinking to just the 4 times 3, but then realized that she had already created more than 12 possibilities.  I asked her what she knew about arranging 3 minions.  She said she knew there were 6 ways to do it.  I asked what happens when we add another minion.  She connected the dots and was able to realize that there would be 4 times 6 possibilities for arranging 4 minions, where the 4 was “the number of minions that could be first and 6 was the number of ways that the other 3 could be lined up”. 

To build on the momentum, I asked about adding a fifth minion, Jerry.  Here she struggled again, so my wife asked her what patterns she noticed.  We started listing the ways to line up 2 minions, then 3, then 4.  Kayla recognized the recursive nature of the pattern that a 5^th minion would mean 5 times as many possibilities as 4 minions, so she set to work on multiplying 5 by 24, which was immediately followed by 6 times 120, at which point her desire to do multiplication stopped.

My wife then gave Kayla a brief explanation about what a factorial is, how to calculate it and how to represent it.

 

Besides being ridiculously excited that I could bring Kayla into this conversation, I reflected that this is exactly the type of You do, We do, I do model of teaching that I espoused in my Ignite talk. 

You Do – Kayla had a prompt about lining up minions and wrestled with the task with no input from me.

We Do – In the extension problems of 4 and 5 minions, my wife and I asked Kayla questions to understand her thinking and refine her thought process.  By asking about the origins of the values in her arithmetic, she was able to independently identify flaws in her reasoning.  Some direct guidance was also provided.  I told her 17 was not correct, but didn’t offer any additional strategies.

I Do – Once a pattern was identified, Kayla was in a place where she could attach some new vocabulary (factorial) to an experience.