“Invert the divisor and mul-ti-ply!  Invert the divisor and mul-ti-ply!”  Imagine an entire class chanting that phrase in a sing-song voice, and you’ll understand why that is one of my two strongest memories of Mrs. Schumacher’s 8^th grade math class. 

From those days forward, if any of us were ever confronted with a divide-one-fraction-by-another exercise in a dark alley, we instantly knew what to do.  (In my later career, I realized we didn’t know WHY that worked, but I’m not here to quibble.)I still can’t think of good ole Mrs. Schumacher’s class without silently breaking into that refrain. But it’s a more personal memory from that class that we’ll focus on here.

It must have been a Friday, because I remember that the weekend was ahead of us.  As our math class was finishing that day, Mrs. Schumacher referred us to a Bonus Problem in our book.  She promised the class that we could get the ever-desired ‘extra credit’ if we solved it. 

I only remember the gist of the problem, so don’t fact-check me here, but it went something like this: “The local TV store wants to run a ‘sale’ on a certain TV, but they want to receive $100 for the transaction.  What starting price do they need to put on the sign in order that their ad can say ‘Take 20% off!’ and the result is $100.”  (Ignoring tax, as word problems so often do.

For some weird reason, I decided I really wanted to get that problem.  Over the weekend, I tried everything I knew how to do on that problem and couldn’t make any progress. 

The key phrase there is ‘everything I knew how to do’.  Ultimately, the problem was one of those that succumbs easily to some early straightforward algebra, but of course we didn’t know ‘how’ to do algebra yet. I don’t think any of us solved it.

Looking back, we still could have solved that problem, you know, and without algebra.  Indeed, I’d like to think the incident helped me learn an important lesson.  (It probably didn’t, but I’d like to think so.)

By that time of the school year, most of us were semi-decent at doing percentages.  So, we COULD have started at $105 (or $150) and worked our way up (or down) to the right answer. (Which is?) We could have taken 20% of our ‘estimate’, subtracted it and see if we got $100.  If not, move up (or down) to the next estimate.  Yes, you’re right: trial and error.  (Or guess/check, if you prefer.)  We could have done that.  But no.  We (or at least I?) thought we were supposed to know how/where to start, and therefore didn’t move even one step out of the rote-procedure box and think about it.

To this day, there’s a lingering suspicion among teachers about working problems with trial and error, in class or in life.  Even up to the day I retired (and in fact, beyond), I’d still get debates from (mostly secondary) teachers that it shouldn’t be allowed.  It was somehow cheating. 

Granted, it’s rarely the shortest or most elegant approach.  And it rarely uses shortcuts/tools (like algebra) that are or soon will be available.  But it still works.  (And then, when later showed a ‘shortcut’ tool, we remember it better!)  Sometimes there’s no cleaner – or enlightening – way to get started on a problem.  

I’d like to think Mrs. Schumacher would have been pleased, regardless.