This is about the pumpkin pie question, a truly fabulous question! But I'd like to kibitz about your answer, starting with: "But Circles don't really work like that." Really, the pie being circular is incidental to the problem. Consider the square box the pie would fit into. The square box would ALSO be four times as big. The factor goes from 2 to 4 not because the pie is circular, but because (as you mention later), it's an area problem. The quantity pi also falls out of the equation. This isn't fundamentally about circles at all. It's about dimension. Diameter is one dimensional, area is two dimensional. So the doubled one dimensional diameter is squared to become area, and the area increases by the square of the increase of the diameter, period. As does the area of the top of the square box the pie is packaged in. Thanks for a great problem!
First, thank you for visiting our site and taking the time to write such an insightful note -- one from which, I am quite certain, almost any visitor can learn something new, or least discover a fresh way of considering a particular mathematical situation. (It certainly made me pause and think the whole problem through anew!)
Now, I would like to take this opportunity to share a couple of thoughts that immediately popped into my often-thick head...
Firrst, a disclaimer is in order:
To be perfectly honest, it took me a moment to realize you were referring to a holiday-time word problem which I posted sometime around this season either last year or the year before (believe it or not, this entire website is still a few months away from celebrating its third birthday; yup, we've been enjoying life online for less than three years, albeit with some busy seasons!)
My next challenge was in locating the problem to which you are referring, and although this is usually something I'm used to doing pretty quickly (practice, practice, practice...), I found myself unable to locate it. BUT, FEAR NOT:
I recall the problem, as it really is just my own twist on a very well-known and popular one that has shown up on more than a few standardized tests over the last decade or so. Basically, the question, which I will officially answer before the Thanksgiving break, is this:
A mathematician's pie-making bakery sells two sizes of Pumpkin Pie, one with a diameter of 10-inches and the other with a diameter of 20-inches. (The larger pies and the smaller pies have the exact same height (i.e., thickness)).
If the pie with the 10-inch diameter sells for $1.50, how much should that bakery charge for the 20-inch diameter pie (relying solely on the manager's ability to think mathematically)?
(HINT: The first answer that seems to pop into most students' heads is often not the correct one. So give it a good try!)
NOW, to address (as briefly as I can for now), your very legitimate concern that I had confused my explanation/hint/and/or wording:
While it is true that this is an "area" question, and area questions deal with "square units" (even when the only shapes involved are circles!!), there is a distinguishing factor that makes calculating areas of circles a bit less intuitive than calculating the area of squares and other rectangular figures -- and that (at least to me) is what makes this problem a half-step more challenging than students' intuition might initially make them jump into a way of calculating that is not likely to lead to the correct answer.
Without giving too much away and robbing new readers of the fun of figuring this one out for themselves, let's think it through (You and I) as if we were not adults who have spent at least a few years working with geometry and other related disciplines. In this circle problem, it certainly seems that only one measurement is being changed (here, that number is doubled). BUT . . .
But, with a closed, rectangular book -- such as a closed rectangular telephone book -- when a person opens it up to locate a telephone number, we ALSO have the doubling of one of the figure's measurements: The length of the top edge of one side of the book PLUS the length of the top edge of the other side of the book. Yet, there is more to our story, isn't there????????????
(Shhhhhhhh... Let's not take away any more of the fun that new readers of this classic problem may find beneficial, intriguing, and rewarding to discover on their own.)
I hope this clarifies the way that I (and some, but certainly not all) mathematicians view the pumpkin pie story as one of the "area" problems with a slight twist.
I hope this helps by presenting an alternative view to your intelligent and correct analysis.
Happy holiday season!
P.S. Carol, I really, truly hope you do not mind, but I was so touched and flattered by the email you immediately shot me after the above one, that I am going to do something I do not think I have ever done before: I am going to share it with readers right here:
"I realized after I sent a kibitz on the pie problem just now that I forgot to say, your site is AWESOME! I love your problems, I love how easy your site is to use, I love that it's free and welcoming and kid-friendly, THANK YOU!!!!! I'm the only math teacher in a court school/community school program. I'm always desperate for engaging materials for kids that have not only been turned off by math, but by school, structure, authority, and reality in general. Your site is a treasure trove. Thank you...Carol"
CAROL, THANK YOU.
P.P.S. If the above paragraph is something you'd prefer I remove, by all means feel free to let me know ASAP via email, and I will (regretfully but understandably) press DELETE.