This One's For Robert

        Robert Kaplinsky is one of my MTBoS role models.  He is the creative force behind the wonderful collection of problem-based lessons found at Glenrock Consulting, a thoughtful and provocative blogger, and the co-founder, along with Nanette Johnson, of one of the quintessential MTBoS sites: Open Middle.  His lessons have inspired some of our most successful projects, including highway signs and movie theater, among others.
     So it was a thrill for me to finally get to meet him at TMC '15.  I was excited to tell him about another one of his ideas that had sparked a lesson so successful it had even our most math-averse students plugging away for days.
    "Thanks for blogging about it," he said.  I was confused.  This was one that had escaped a blog post.
    "But I didn't..." I started, and stopped when I saw his face and realized he was being sarcastic.  I admit I'm a little slow on the uptake.
     Well, we did the project again last week, and it's not going to get away this time.

     What became known to us as "The McDonald's Project" started with this tweet last September:

Hey #MTBoS, I need your feedback. Skim this vid https://t.co/BGhr3obhtG. I'm thinking about turning it into a lesson on equations. Thoughts?

— Robert Kaplinsky (@robertkaplinsky) September 9, 2014

     I loved the video, and Theresa and I immediately started thinking about how we could use the idea.  After some false starts, here's what we came up with:

• A list of foods would be provided, and the kids would be challenged to put together combinations to equal 1,500 calories (2,000 being the adult requirement.)
• Since we knew most kids liked to eat at McDonald's, we decided to limit the food selection to items on the McDonald's menu.
• We'd use the project in fourth grade, where the kids were working on multi-digit addition.

     Our first hurdle came when we printed out the list of nutrition facts for McDonald's menu items:

The PDF ran 27 pages and contained well over 400 items, with information ranging from the % daily value of vitamin A to total grams of fat.  It was overwhelming.

      The list would need to be whittled down.  Our first idea was to pick out the most familiar items.  But that's not very Danielson, is it?  Why not survey the kids, and see what they like to eat at McDonald's?

Here's the final list.  We added in a few items for balance.  And I know Coca-Cola is misspelled.  That one slipped by the editors.

  What happened next may have been the most important part of the lesson.  Before even explaining the directions, the teacher distributed this sheet...

...and asked the kids to do some noticing and wondering.

    During the discussion that followed, the wondering overwhelmed the noticing because the teacher did something very smart: she left out any units or labels.  So the kids were left with a very pressing question:  What did those numbers mean?
     The first volunteer offered price.  But there was no decimal point or dollar sign!  The class agreed it was unreasonable for a Big Mac to cost $540.00, but certainly $5.40 seemed about right.  Did the teacher leave out the decimal point on purpose?  One student thought the numbers might stand for the amount sold in a year or a month.  Another thought the numbers could stand for the amount left over at the end of a day.  Many students offered their opinion that the numbers stood for calories.  And although they weren't quite sure exactly what a calorie was, they did know that healthier foods, like apple slices and side salads, would have less of them than vanilla shakes and french fries, a wonderful application of inferencing skills that would have made their reading teachers very proud.  Listening to their thoughts as they tried to puzzle this out was fascinating.  It reminded me of this wonderful Graham Fletcher activity, and I made a mental note to try to do more of what Graham calls "undressing tables".
     The class agreed that calories did make the most sense, and after a very brief detour into the world of nutrition, they were asked this question: How many calories does a fourth grader need every day?  We got answers ranging from 1 to 30,000.  We didn't wait long to tell them the recommended daily amount and get them working on the project.

We put them in pairs and gave each a different colored pencil.

There was a lot of trial and error...

   

...along with a lot of addition.

The engagement level was high.

   We wrapped up the lesson by gathering the kids together and asking for strategies.  Many had started with the largest calorie items like the Big Mac and milk shakes, gotten as close to 1,500 calories, and tried to fill in from there.  It was again interesting to listen to their observations about the foods and their calories; for example they were intrigued by the 30 calorie difference between the chocolate and vanilla shakes.
   The kids revisited the project in the following days, which gave me time to prepare an extension:

Some found this difficult, and needed more direct help from the teacher.  But they plowed ahead with gusto:

 

     And this work left me with material to work out a problem set:

Here are some of the problems I was able to generate by removing an item from each equation.  I also added the name of the child who had created the original problem in the margin. 

   I'm sure there's a lot more gold we get out of this task, and Robert has great ideas for using the calorie lesson with middle school students over at his site.

     So thanks, Robert, for your inspiration, encouragement and polite but firm way of pushing me to think and work outside my comfort zone.  Looking forward to more collaboration, and to connecting with you again at TMC '16!

At Play in a Mathematical Sandbox

     One of the highlights of this past summer's Twitter Math Camp was the chance to attend a session led by Federico Chialvo.

Friday afternoon from 4:00-5:00.  It was a tough call because Fawn Nguyen and Matt Vaudrey were presenting Barbie Bungee at the same time.  An example of the difficult decisions one had to make at TMC 15.

     Federico is the Director of Mathematics at the Synapse School in California.  He also plays professional ultimate frisbee for the San Francisco Flame Throwers.  (Who knew there was such a thing?  He even had to leave TMC early because he had a play-off game!)  I had read some of his blog posts, and we tweeted back and forth a few times last year.  I knew we shared an interest in the potential that games and game-like activities have to engage kids in learning.  So I was excited to meet him in person and learn something new to bring back to my school.
     One of the activities that caught my attention was called Subtraction Reversal Mysteries:

 
     Federico gave us time to experiment with the game, and led us through some guided discovery.  He showed us student work samples he had collected from a post he wrote describing his experience using the game with his class.  I was attracted to this activity for several reasons:

• The directions were simple and easy to understand.  Everyone in the class could participate.
• There were some important content standards embedded within.  I saw subtraction with regrouping and place value at work.  
• The data collected from playing the game would lead to some very interesting mathematical discoveries.

      I felt it would be perfect to use with a third grade class at the beginning of the year, both to review multi-digit subtraction and introduce the idea of looking at data and making conjectures.
     Back in school I approached Shannon, whose third grade class I'm working in this year.  She agreed to let me try it out, and armed with some 10-sided dice I gave it a go.

The kids picked it up quickly.

They got about 15 minutes to play.  I collected their sheets, looked them over, and brought them back the following week.  I wanted them to play for at least another 15 minutes in order to collect more data.

One pattern I saw in their mistakes involved regrouping when there was a 1 in the ones place:

Kids who, under other circumstances, could regroup correctly would make this mistake.

Here it is again.  

    
     So we were able to gather some good formative assessment information on their abilities to subtract, and jump in with some direct instruction.  Several days later I brought their papers back and let them continue to collect subtraction problems.
     During the third session, I asked some kids if they noticed anything interesting happening with their subtraction problems.  One girl explained that when she rolled two numbers that were right next to each other, the resulting subtraction problem had a difference of 9. I asked her for an example:

I put this up on the board, and asked the kids to pair up.  I wanted them to collaborate, looking through their data to see if they could come up with any more interesting observations.

I asked them to write a statement and back it up with evidence.

I explained that they should not limit themselves to the combinations they had collected during the investigation.

For their first time, I thought they did well.

This student worked alone.  He didn't want to collaborate and I didn't force the issue.

The same student came up with this: "The difference of the difference added together equals a round number."  Can you see what he means?

At the end of class, one girl proudly showed me this.

   
     During the session, Federico spoke about the importance of providing students with a "Mathematical Sandbox" in which to explore, play, and create:

     I like everything about this, and an activity like Subtraction Reversal Mysteries is a wonderful example.  Everyone in class is playing in the same sandbox.  Some are playing together, some alone.  Some are building very intricate castles, others are simply filling pails and digging holes.  But they're all in there together.  It exemplifies the MTBoS ideal of an activity with a low barrier to entry that scales up high.
     Federico had lots more examples of sandbox activities that he had us try out during his presentation.  I look forward to experimenting with more of them as the year unfolds.