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Question

 

Dear Mitch,

I am a middle school teacher on Long Island. This June I celebrated my twentieth year in teaching in my current school.

A few years ago, you were hired as the guest lecturer for our district's annual Teacher Inspiration Week ("TIW"), which marks the important beginning of our 4-day faculty week. In those days you were best known for the power of your guest lectures, and the fact that you somehow made each lecture unique. In light of that, you might not recall the speech you gave to our faculty that day. I sure do! Our entire math department changed the way it taught several fundamental topics. You influenced even some of the old-timers who can be very 'set' in their ways and protective of the way they do things. Here's my question to you, sir: After you presented a comprehensive collection of "hands on" methods using materials that most of us had never even considered using, and some of which I have used each year since, what I found just as intriguing was the last quarter or so of your lecture which you set off by announcing that we were now going to switch gears and demonstrate how powerful the opposing style can be; no hands-on exploring, just storytelling.

Our district was exploring many of the acclaimed math story books, particularly for younger students, but you took an approach that is almost never taken: rather than read from a book written for that purpose, you told a few stories which were clearly true about ordinary experiences and how math or problem-solving skills made a huge difference to everyone in such a situation.

And there was one you described that had something to do with a small closet or cabinet you had built in your home that had some kind of problem that became clear the moment it was finished. And that's where I'm stuck. I recall enjoying the story, and, of course the problem, but I can't recall how you came up with a set of numbers that work. And as lovely and as funny as the tale was, without a set of numbers that work even half as well as the ones you had it doesn't have the same impact.

Sincerely,

Mrs. McFree

Answer

 

Dear Mrs. McFree,

I know the one you're thinking of!

You were correct, it was a closet in the kitchen.

And, I promise you, the event occurred exactly as I set it down here, and every word is true. You may notice in fact that I am careful to leave out all names other than my own.

We had a cupboard built underneath a staircase to store canned foods and other small items.

Unless we were willing to have major reconstruction done to the house (we were not), it was to have a height limit of five feet. (divided vertically into two matching sides, like a narrow refrigerator/freezer combo that's split vertically). It was only six inches deep because the whole point was to build it flush with the wall. It was designed with adjustable shelves with standard clips, and for each side I measured a few favorite products and after trying a few different arrangements, settled on using 8 shelves on each side, some allowing for tall items, and some not. (A Heinz ketchup bottle would be the tallest, and mustard, for example, would be one of the shorter items. The cupboard was sealed by two plane matching doors, each a couple of feet wide. Like this:

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I was pleased with it, and thought it would be one of the most useful changes we had made to the house that year. But there was a problem.

I was informed that although the closet was a delightful idea, unfortunately, as luck would have it, it turned out to be just a bit too small. And since the items that didn't fit into it would have to be stored in the basement, the closet failed in relieving the person cooking from having to venture down there. The closet had to be enlarged. Two carpenters were called in to give opinions on enlarging the closet. They both agreed that the obvious solution would be to add  six additional inches, vertically. The closet would then reach inside one of the steps but would cause no problem because the strength of the step would be restored by a technique involving adding steel supports to reinforce the six inches of wood which would have to be cut away. No problem.

Estimates:

Carpenter number one: $760.00 (remake doors to fit enlarged front, recut wall and additional support to top, painting, finishing, and if we were to give him an immediate Yes, we would get a commitment that he'd do his best to start the project by August 1st. The magnetic calendar on the refrigerator indicated that we were in December.

Carpenter number 2: Not $760.00. No, nothing like that.

$860.00.

He could probably get it done in a month, but don't quote him on that.

But by all means, he said, we should feel free to quote him on the price. $860.00.

$860.00.

QUOTE:

"$860.00."

I thought of all the benefits of keeping certain foods in a basement environment.

"Like wine in a wine cellar," I said.

To which someone passing behind me replied, "But you don't drink wine."

"Still, it's nice to have around at holiday time in case guests show up."

"But you don't like guests."

"Not when they ask for wine, that's true."

She nodded. "Eight-hundred sixty dollars" she mumbled softly.

"I'm thinking there's got to be a way to do this without so much energy."

"Energy?" she said, "You mean the eight-hundred, sixty dollars."

"Energy," I repeated. Then I mumbled, "And yes, eight-hundred-sixty is some number."

"Why do you say that -- is it a special math number?"

"No, it's just some number."

"Oh."

Later, when I was the only person awake, I sat down in front of the closet. I leaned forward and opened the doors and thought.

Often, when trying to solve a problem, the process could involve hundreds of different attempts over a ten year span. More often, though, hundreds of different ideas are attempted and discarded within a hundred minutes. And that, of course, is less than two hours (which is 120 minutes). But to solve a real-world problem and implement the plan, which in a case like this might require an experienced carpenter, such periods of time would be considered miraculous. And, in addition to the cost in time, implementing a change would almost certainly require accepting a period of noise and mess -- from cutting the wall open to accommodate the taller doors, to getting the smaller doors into a dumpster and hauled away. And then there's sawdust...

In fact, this puzzle was so perplexing that I had to stare straight into it for just a few seconds shy of 175 minutes before my entire skull arched into a smile. 175 minutes! And that's almost three hours (which would be 180 minutes). But, by the end of that period, the problem was not only solved, but the solution was implemented and the last bit of sawdust swept up. (There wasn't much to sweep).  And the best part is that the following morning when the sun came up and shined into the kitchen the problem appeared to be right where they'd left it, though now without the merciful cover of night.  You just couldn't take a break from its newly painted trim, making it even harder to ignore. Fortunately, though, the instant its doors were

                                       ------------ opened -----------------

 

 

        the solution presented itself.

 

*

 

......STAY ......... TUNED...........

 

NEXT WEEK, SAME MATH TIME.... SAME MATH CHANNEL....

 

 

-- Mitch