2011 May 15: Major updates on account of useful critical feedback in the comments.
Let's see how well the storytelling framework holds up.
Download the full archive [5.5 MB].
Play the question video.
Ask your students what question interests them about it. Take some time here. This is the moment where we develop a shared understanding of the context. If a student has some miscellaneous question to ask or information to share about the dolls, encourage it. That isn't off-task behavior. This task requires that behavior.
Then ask them to write down a guess at how many Russian dolls they think there are. Ask them to write down a number they think is too high and too low.
Offer your students these resources:
After you show them the first set of two dolls, ask them how big they predict the third will be. As one of the commenters mentioned, they need to discover the fact that these guys aren't decreasing by a fixed amount every time, that a new model is necessary.
Once they have this new model in mind, they'll keep applying it until they reach a doll height they think is impossibly small.
That task isn't going to win anybody a Fields medal. As students finish, ratchet up the demand of the task with this sequel. Say:
I need you to design me a doll that's as tall as the Empire State Building and is made up of 100 dolls total. Tell me everything you know about that doll.
Ask them to generalize. Ask them to graph.
Host a summary discussion of the activity. At this point you've identified different solution strategies around the room. Have those students explain and justify their work to their peers. Everyone is accountable for understanding everyone else's strategy.
Then show them the answer video:
Find out whose guess was closest.
[h/t @baevmilena who gave me the idea when I met her in Doha.]