## Thinking with Units

I sign up at the golf course as a single and am paired with a couple of strangers.  On the second tee the inevitable question arises, “What do you do?”  I return, “I teach math at the university,” expecting to hear, “I was never good at math but my niece (daughter, grandson) got straight A’s” and such. Instead Norm says, “I know forty divided by one-half plus fifteen is ninety-five because the computer says it is, but how can eighty halves plus fifteen wholes add up to ninety-five.  The units don’t work. How can you explain that to a kid with candy bars?”  My jaw (mentally) dropped.  I told them I love arguments about units and that I might be quiet for a while but I would explain it before we got to the eighteenth hole. [I am sure this is some internet controversy somewhere.]

We tee off and while we are waiting to hit the next shot, I explain, “You get into a taxi.  Just for sitting there they charge you fifteen dollars.  They charge one dollar for each half mine. You go forty miles.  What do you pay?”  Both said, “Ninety-five dollars,” immediately.  “Yes,” Norm says, “But you can’t explain taxi rides to a kid.  What about the candy bars?”

I walk to the third hole, some two hundred yards away.  There is time to think since those two drive ahead in a cart.  As we are waiting to tee off, I say, “Little Joey is having a birthday party.  Lots of kids are coming.  He wants to give each kid a half of a candy bar.  So he sits at the kitchen table cutting candy bars in half.  He cuts eight bars in half and eats of the halves.  He has forty bars left to cut in half.  How many guests did Joey invite to his party?”  Both golfers calculate ninety-five half bars and therefore ninety-five guests.  Norm is still skeptical but we have a pleasant game for the rest of the round, most of the conversation of the “good shot”, “nice putt”, “you’re away” variety.

Those two old guys knew how units worked, I am sure, based on life experiences.  The key to a suitable explanation is noting that the fifteen counts half bars the same as forty divided by one-half counts how many half bars are in forty whole bars.  The units of the one half is whole bars per half bars: one whole bar per two half bars.  Now let’s play some more golf.