I just finished the latest stage of ADA courseware with an especially hard and fast race to the end under pressure, and my mind needs to work on SOMETHING DIFFERENT for a while!
This article on math teaching caught my attention. I used to teach math as part of teaching electronics. Fortunately we didn’t need to handle fractions. Adding and subtracting are easy enough to understand visually. Multiplying is somewhat easy if you treat it as taking a PART OF.
1/4 * 20 means “a quarter OF 20.”
But dividing is nearly impossible to visualize. I never thought about it. This author proposes a way to handle it, so I decided to think about it before seeing how he does it.
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You have a large farm. You’re getting old and want to stop farming. You decide to split the property into 5 smaller lots and sell them to city folks.
Here’s the whole farm in more schematic form:
After splitting in 5 pieces:
From your viewpoint, starting with the original farm, you’ve divided by 5.
Farm / 5 = Lot.
Each city slicker has bought a 1/5 part of your original property. Part of means multiplication:
1/5 * Farm = Lot.
From the viewpoint of each city slicker, the original farm was 5 times as large.
Lot * 5 = Farm.
Now try the hard way. From the slicker view, the original farm was his lot divided by 1/5.
Nope, doesn’t work.
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Well, let’s go up instead of down. Instead of dividing your property among 5 city folks, you let a city corporation buy up your farm along with 3 other farms to make a big industry. Your farm WAS 1/4 of the industrial property. This is the same as multiplying the property size by 4.
Here’s the industry after the purchase. (Of course this is an ancient fairy-tale, since industry is now in China.)
From your viewpoint, the new industrial property is 4 farms.
Industry = 4 * Farm.
From the industry’s viewpoint, each original farm was a quarter of the industry, so
Farm = 1/4 * Industry.
Trying the divide again, from your viewpoint the industry is
Farm / 1/4 = Industry.
Still doesn’t work. “Dividing by 1/4” simply isn’t a normal part of life. It’s common enough in decimal form, like 500 / 0.25, but we don’t think about it there. We just punch the numbers into the calculator.
The only way to get there is through indirect knowledge. You know that multiplying and dividing are opposite. If you multiply by smaller than 1 the result will be smaller than you started with. Thus, because dividing must go the other way, dividing by smaller than 1 gives you a larger result than you started with.
This is just unsatisfying.
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Footnote on the actual farm. The address is 12120 E Gypsy Lane Rd, Bowling Green, Ohio. It was across the street from a rural house my parents rented in the late ’60s. The farm was abandoned, deteriorating and sometimes squatted. I spent a fair amount of time exploring the house and fortunately took a photograph, figuring it would be torn down soon. Much later I built a digital model, but apparently didn’t show it in this blog. I did use the pumphouse recently.
Amazingly, Google Streets shows that the house and pumphouse are still there, clearly the same floor plan. Somebody must have rescued and renovated it instead of tearing it down. The rest of the property has in fact been divided into lots, and the original swamp has been turned into a nice lake.
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Okay, now look at the cited article. He’s showing it schematically without a real-life image, and he ends up doing pretty much the same trick. You have to understand FIRST that multiplying and dividing go opposite ways, then imagine dividing as the opposite of the understandable multiplication. Still unsatisfying.
Polistra's Mill 