I use that as a fun math/logic/science exercise with students.
*Q: How much longer do you need to make a rope encircling X if you want to raise it an even foot/meter?
A: 2pi feet/meters
People do find it counter-intuitive, and I don’t know if it helps the less scientific understand the importance of checking your intuition or just reinforces their gut feeling that math/science is “wrong”, but it’s just so fun to see their minds boggle about how increasing the radius of a rope circle the size of Earth’s orbit (or any circle of course) by 1 meter (or any unit) increases the length of the rope by 2*pi meters.
As long as we’re discussing teaching moments about pi (and well, pizza pie ): here’s one my father told me when I was 12.
We were ordering a pizza for the family. They had 6-inch and 12-inch pizzas on the menu.
Naturally, I knew that the 12 inch would provide us with twice as much pizza to eat as the 6 inch… right?
That’s when my father taught me the concept of R squared.
And I learned that math is actually useful in the real world, not just a bunch of abstract squiggles on paper.
Watch for my new venture, the University of Pizza where everything is taught using pizza. We’ll be dividing and ingesting three to six piping hot pizzas every class period, hand-baked by our own Dean Chappachula, Dean of Student Affairs & Mozzarella.
Seriously, I really wish math (and science) were taught with more real-life applications.
Like the time I was standing in front of the egg cooler at the market, trying to figure out the volume of an ovoid in my head. To see if the extra-larges were really worth 20¢ more per dozen.
The answer: “Hey, wait a minute. My time’s more valuable than spending it in front of this egg cooler. At $12/hr, I’ve already spent more than 20¢.”
I remember once, a teacher asked this back when I was in school. The more motivated students started working diligently on figuring it out. Then the teacher said “Here’s a hint: You don’t need to know the radius of the Earth”. At that, my hand instantly shot up, because if you don’t need to know the radius of the Earth, then you can set the Earth’s radius to 0, and that’s an easy problem.
A close variant also showed up once in an Encyclopedia Brown book: If a 6’ man walks around the Earth at the equator, how much further will his head move than his feet?
On a tangential point, I remember in the book “Cheaper by the Dozen” the father, fed up with kids using “million” for any large number, tacked a 1,000 by 1,000 sheet of graph paper on the wall. “That’s a million…”
The math is much easier if you go by weight, not volume.
Ref Chicken egg sizes - Wikipedia, the increments from Medium to Large to Extra Large to Jumbo are all very, very close to 12%. So if the next larger size costs less than 12% more, it’s cheaper by the oz. And vice versa. Adding 12% to something is very easy mental math.
OTOH, most of us don’t make and eat 2 eggs for breakfast if we have Jumbos, and 2.5 eggs if we have Larges, and 3 if we have Mediums. Instead we make and eat 2 of whatever size we have.
If you use them by the each, then you should price them by the each. In that case, the cheapest total price is the one to buy.
Math is nice, but it has to be in service of the correct goal.
Most grocery stores will list price per unit, but the unit isn’t always the same for different products (and occasionally, not even for different brands or packages of the same product, though that’s thankfully rare). For eggs, the unit price is generally going to be “per egg”, not “per ounce”.
What about the weight of the shell? Shell and egg density seem pretty similar but while the weight of an egg has a linear relationship with volume the size and weight of the shell does not. The thickness of the shell can vary a lot but for one type of chicken the shell thickness doesn’t vary much with the size of the egg. One might need to bring a good calculator, scale, measuring vessels, and calipers to the store to work this out correctly.
The reason it strikes people as “wrong” or surprising is that the question is framed with the surface of the earth as a reference point. In our mental model we see the rope stretching off into the distance in both directions and imagine a fairly negligible 3 feet being added to those thousands of miles of rope. We then see the rope raises a very noticeable 6 inches above the surface.
If you remove the earth from the question entirely and ask people to imagine a 24000 mile loop of rope floating in space, adding 3 feet to it and saying that it makes the diameter increase by a foot. That somehow is easier to accept.
I suspect it seems wrong because you intuitively wind up focusing on the increase in area, which does increase by a larger amount depending on the original size. When you picture a circle, you picture the area.
I think a way to see is to add just one unit on each side to a square. It will increase by 2 * the original side + 1, which depends on the size of the original. But it’s still always adding 4 to the perimeter and 1 to the shorter diameter.
Both measures: circumference/perimeter and radius/diameter/side are all linear. They all are measured in, for example, feet, not square feet. So they scale linearly.
Is the Equator the circumference? Or is it a line across the middle? Either way, it doesn’t matter. If the Equator is a line across the middle, then extra rope will just dangle off the edge and tease the turtle. If the Equator is a circumference, then the rope goes out instead of up (and falls onto the turtle). In neither case does the rope go up.