#MakeOverMonday ... Will the sauce spill over?

It's the fifth week of Dan Meyer's MakeOverMonday challenge!  Each week Meyer chooses a textbook problem from middle school math curriculum and invites anyone who would like to participate to suggest how to transform the problem.  He doesn't define the goal of the transformation other than to make it better!

This week is about volume.  The problem asks a great question ... "Will the sauce spill over when the chef adds the meatballs to the pot?"

The problem requires multiple calculating steps.

At first glance the problem reminds me of the book, Mr. Archimedes' Bath by Pamela Allen.  The book is quite primary and yet I find that sharing literature in math class helps to lighten the classroom and helps some students connect with the math.  I might share the story in preparation to work on this problem.

I would start to work on this problem by sharing the context and showing the picture of the sauce and meatballs.  Then I might set out a pot of water with two dozen small balls (or blocks).  "What do you think?" I'd ask.  Will this pot of water spill over if I add these balls to the pot?  I'd ask students to choose, "yes" or "no," will the water spill?  Then I would ask the students who say, "no, the balls don't fit" to estimate how many balls will fit.  And for the students who say, "yes, the balls will fit" I would ask them to estimate how many  more balls might fit.  I'll record their estimates on our chart paper.

Next I'll ask students to think for a minute about what information they have - both in my display of water and balls and in the sauce textbook problem, what information they wish they had, what step they think they would work to solve first.  I'll ask students to pair up and discuss their plans.

Then I'll challenge teams of students (2 - 3 students working together) to determine how to prove their estimations.  As they work I'll circulate to ask questions (in no particular order below ... )

• What is the volume of the pan?  How do you know?
• Why is knowing the sauce is 2 inches below the top of the pan significant?
• What is the volume of a meatball?
• How does the volume of one meatball affect the volume of the pot?
• What is the formula for the volume of a cylinder?
• What is the formula of a sphere?
• How are radius and diameter related?

Last ... we'll demonstrate ... we will toss in the balls and watch to see if the water spills.

What are you thinking?  What would you do with this problem?