# Lessons from teaching gifted elementary students (Part 7b)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprising depth of mathematical knowledge.

Here’s a question I once received after a really big hailstorm:

How big would a 1000-pound hailstone be?

The guesses from the students ranged from the size of a small car to the size of a large pick-up truck. Here’s how I gave a reasonably accurate answer using only mental arithmetic. This kind of describes the way that I try to size up things when only an approximation (and not an exact answer) is necessary.

First, I had some real-world experience that quickly told me the answer was going to be deceptively small. When I was young — maybe 10 or 12 years old — I was getting ready for a picnic, and I was assigned cut a block of ice — maybe a cubic foot of ice, if memory serves — into smaller chunks. (The party organizer bought a block of ice instead of a bag of ice to economize.) I remembered how incredibly heavy that block of ice was even though it wasn’t much larger than a basketball… several of us kids had a lot of trouble lifting the block of ice as we prepared to chop it into pieces. So, for the sake of argument, if that cubic foot of ice weighed about 100 pounds, then 8 cubic feet would weigh 800 pounds. So, based on that chance encounter with a block of ice when I was a kid, my guess would have been that the hailstone would measure 2 feet across.

Back to the problem at hand.

First, I converted to metric. I knew that there are about 2.2 pounds in a kilogram, and so I knew that the block would weigh something like 400 or 450 kilograms. I knew that I would be making plenty of crude approximations, so I just went with 400 kilograms and didn’t worry too much about immediately calculating $1000/2.2$.

Next, I knew that metric units were originally defined so that a cubic centimeter of water weighs a gram, so that a 10 cm-by-10 cm-by 10 cm cube of water weighs one kilogram. Ice (hail) is slightly less dense than water (after all, ice floats in water), but for crude approximation purposes, I ignored this.

So, if a cube of ice with a side length of 1 decimeter (10 cm) weighs 1 kilogram, then a cube of ice with a side length of $\sqrt[3]{400}$ decimeters would weigh about 400 kilograms.

How big is $\sqrt[3]{400}$? Well, I have memorized that $7^3 = 343$ and $8^3 = 512$, so it’s between 7 and 8 someplace… say 7.5. So the answer would be a cube of side length 7.5 decimeters. Also, I have memorized that 1 decimeter (10 cm) is approximately 4 inches, so the cube would have side length $7.5 \times 4 = 30$ inches.

Finally, hailstones are more spherical in shape than cubic, and a sphere of diameter $d$ has less volume than a cube of side length $d$. So the answer should be a bit larger than 30 inches, so I just rounded up to a nice even number: 36 inches (one yard).

This calculation took me about a minute to do in my head and another half-minute to re-do to make sure I didn’t botch the arithmetic. So I held my hands about a yard apart (perhaps the crudest part of this calculation), pretending to hold a ball of diameter a yard across, and announced, “The hailstone would be about this big.”

Of course, a more thoughtful analysis produces the actual answer. The density of ice at the freezing point is 0.9167 grams per cubic centimeter, and 1 pound converts to 0.453592 kilograms. So:

$\displaystyle \frac{4\pi}{3} r^3 \times 0.9167 \frac{\hbox{grams}}{\hbox{cm}^3} = 1000 \hbox{pounds} \times \displaystyle 0.453592 \frac{\hbox{kilograms}}{\hbox{pounds}} \times 1000 \frac{\hbox{grams}}{\hbox{kilograms}}$

$r^3 \approx 118127 \hbox{cm}^3$

$r \approx 49.1 \hbox{cm}$

$r \approx 49.1 \hbox{cm} \times \frac{1 \hbox{inch}}{2.54 \hbox{cm}}$

$r \approx 19.3 \hbox{inches}$

Therefore, the sphere would have a diameter of twice that, or 38.6 inches.

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