2011 May 15: Major updates on account of useful critical feedback in the comments.
Let’s see how well the storytelling framework holds up.
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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.]
I recently took your nesting dolls activity and here’s what I did:
At the beginning of the unit on exponential functions, I followed your process fairly closely, except I used pictures of the dolls. I asked kids to predict the patterns, etc. Most kids went with exponential, though a few were strongly in favor of linear. At the end of the unit was where I believe the magic appeared and is what I will use in the future. By this point, kids had done work with linear and exponential functions and some kids had studied quadratics. I had 7 different sets of nesting dolls in the room. Kids were told they could pick any of the sets, but had to identify them. Their job was to determine an equation to model the growth/decay pattern of the dolls and use math “tools” to convince me that their equation did an adequate job at modeling the dolls. They had to do all of the measuring…some kids chose height, some volume, some girth.
I got a huge array of problem solving. Some kids used graphs to visually show more of a regression to see whether linear or exponential had a better fit. Some kids developed both linear and exponential equations and then used tables and graphs to see where each went off track. Some recognized a constant second difference in growth and used systems of equations to develop an amazing quadratic equation that appeared to fit their data perfectly.
This project really allowed students to take the problem as far as they wanted with an entry point for everyone. And the kids loved the nesting dolls so they were really engaged. I strongly recommend actually using the dolls rather than video-taping them as well. It adds a tactile dimension which is really valuable to many students.