## Wednesday, June 25, 2014

### Minecraft Babies and Concrete Canoes

One of the hardest concepts I currently tackle with my ninth graders is Buoyancy.  It's the culmination of the mass vs. volume vs. weight vs. density vs. pressure unit.   We learn to distinguish mass from weight (by thinking about what they are, how you measure them, and what they mean on the International Space Station).  We find a relationship between mass and volume (by measuring them with a balance and a graduated cylinder and plotting them against each other to find the density of some aluminum rods).  Then we think about a thumb tack and Newton's Third Law to define pressure and think about it in solid-solid interactions and fluid situations (swimming pools, airplanes).  We measure the weight reduction of prisms of the same volume when submerged in water (it's the same!  even if the prisms are different materials!).  Then we roll all that together into two derivations that we work out together, both of which blow my students' minds (some in a good way, some in a bad):  first the depth dependence of fluid pressure, then the fact that the difference in pressure on the top and bottom of a container reduces to be equal to the weight of the fluid displaced by the container.

This unit does not immediately "work".  I know, because I do Standards Based Grading, and the buoyancy concept is the one I do the most additional help and retesting on (although my Measured Uncertainty, Calculated Uncertainty, and Sig Fig concepts are close on its heels).   Eventually, with further small group discussion and additional hard thinking on the part of my students, most of them internalize that for floating objects (e.g. concrete canoes) the buoyancy force must equal the gravitational force on the object.  And some of them grok the connections between the volume of the displaced liquid, how the density of that liquid functions as a conversion factor between volume and mass, and how the gravitational field strength functions as a conversation factor between mass and weight.  But it is painful.

Every year I think about omitting buoyancy, and every year (so far) I've decided to mix things up instructionally and give it another shot, because it's such a beautiful combination of all these challenging, interrelated, but distinct concepts (volume, mass, density, pressure, force).   This comes to mind both because one of yesterday's Hestenes readings bemoans the lack of a solid understanding of buoyancy in physics students, and because so many of those tricky fundamental concepts feature in tonights' Arons reading.

All of which is to say, the weakness that Aron mentions are certainly ones I've seen in my students.  I see how we've incorporated counting squares for area and displacement volume measurements in the Unit 1 labs.  But do density and buoyancy come in anywhere?  A casual flip through my binder of Modeling Mechanics units doesn't seem to include buoyancy.  Is there a Modeling unit that deals with these basic ideas about different ways to describe how big something is or how much of it there is (length vs. area vs. volume vs. density)?  And that addresses fluid forces?  Is it off in some Physical Sciences set of materials?

My favorite exercise for addressing the scaling issue is a warm-up I developed after the first time I read some of these sections.  I call it my Minecraft Baby warm-up, as to figure out the right answer, I guide them through sketching a Minecraft baby and then increasing each of it's dimensions one at a time.  It goes as follows:
"Consider an newborn who weighs 40 N.  During a year she grows so that each dimension of her body (length, width, height) increases by 40%.  How much will she then weigh?  (Assume constant density.)"
It is accompanied by cute pictures of my daughter as a newborn and at almost-a-year.  (This may be why it is one of my favorite warm-ups.)

Would anyone like to guess the most popular answer (by far)?  And the second most popular answer?  (Hint: the answer to at least the first is in the Arons section on Scaling!)

#### 1 comment:

1. I don't know.. The baby seems the same size to me in both pictures. I used my thumb for scale.