Why egg math
? Leave alone the why
for now. Let us get to the how
first. In order to know how
, one must know what
Known: Jumbo's waist: Ø46.8mm; medium's waist: Ø41.9mm. Question: How much bigger
is Jumbo compared to Medium?
To younger children, the question can be ambiguous or even confusing. But ambiguity is inherent in the real world. Do not get me wrong: Formulation of mathematic question from the "real world" is a higher level skill, a skill that goes beyond modeling. (The latter of which grade schools do teach.) I am not proposing to add this to your SAT. (Don't even add to GRE. Seriously. I see lawsuits on the wall if GRE adds questions like this. After acceptance, however, graduate students routinely get such from their supervisors.) But I am proposing that every mom and dad do this with your children.
At first look - metaphorically and
literally speaking, the difference is marginal. 46.8mm - 41.9mm = 4.9mm. Obviously this is not the answer to the
question. (Some ingenious children may even get to circumference. But that's still not the question.) Lesson #1
: It is not a big stretch for children to understand that real world questions about "how much bigger" is often about fractional comparison. In fractions, 4.9mm is merely 12% of 41.9mm. Is Jumbo 12% larger than Medium? (Alternatively, is Medium 10% smaller than Jumbo?)
Enter practicality as a guidance to scientific problem solving. In the 18th century, European traders discovered that goods they brought from tropical Africa weighed less in their home port. Did someone cheat them on scales? Did someone stole the goods at sea? After careful calibrations, they finally realised that they were using a questionable assumption that content of goods was proportional to their weight
. Thus was born the scientific concept of mass. Why? Because what the traders truly care was how much content was in the goods, not how much force the Earth pulls on them.Lesson #2
: Similarly, when someone asks "how much bigger is a jumbo egg than a medium one," the person is perhaps not interested in Humpty-Dumpty's waistline, but interested in its nutritional value, which is proportional to its mass, which is proportional to its volume.
Now, an egg is not a perfect sphere or a perfect cube whose volume can be computed by a single dimension. Can we determine the egg's volume from mere waist measurement?
This brings us back to Lesson #1: fractional (or relative) comparison. Although we do not have sufficient measurements in order to determine their absolute volumes, they have similar geometries, therefore it is sufficient to compare the volumes in fraction.
On to math. Lesson #3
: Volume of similar shapes (not restricted to oval) is proportional to the cube of any single dimension. (At their age, this has to be taken with a leap of mathematical faith. But then, a lot of what they learn at the age are presented as mathematical/geometrical facts, without proof.) Fractional difference between volumes can thus be established by (D2^3 - D1^3)/D1^3 = (D2/D1)^3 - 1. In our case, (46.8/41.9)^3 - 1 = 39%. (Without algebra, there is no need to present the general formula.)
So, a Jumbo egg is 39% larger than a Medium egg in nutritional value, for which we make payment.
That is a very long and winding way to compare two lousy eggs! Why the trouble? Review of the three lessons:
1. The person who asks "how much bigger is this egg than the other egg" is probably interested in relative comparison, a fraction.
2. The person who asks "how much bigger is this egg than the other egg" is probably interested in volume, not waistline.
3. Volume of similar shapes is proportional to cube of any one of its linear dimensions. (Extension: Area of similar shapes is proportional to square of any one of its linear dimensions.)
In the real world, you must combine practicality with mathematical modeling in order to answer a question. No grade school is ever going to teach this.
The big deal? My local grocer had a sale: a dozen medium eggs for less than 1/2 the price of jumbo eggs. I told my daughter that we should buy medium instead of our usual faire of jumbo. Lesson #4
: I told her that we pay for nutrition, not for size. "With this promotion, if we eat two medium eggs, we get more nutrition than from one jumbo but pay less," said I. But is my gut feeling about sizes justified? After all, I did not bring USDA with me on a shopping trip. This little exercise confirms my gut feeling.
Wait! The statistician in you must be crying out. How can you derive relative sizing from a random sample from two cartons and claim this is scientific? The two cartons are not even from the same brand. How'd you know they are even comparable? How do you account for dissimilarities in shape? OK. This is not rocket science, just an estimation. But this is hands-on. Something your child can easily do at home in one minute. And comparability between brands is established by trade regulations.
How far is it from real science? USDA defines minimum net weight per dozen in http://www.fsis.usda.gov/wps/portal/fsis/topics/food-safety-education/get-answers/food-safety-fact-sheets/egg-products-preparation/shell-eggs-from-farm-to-table/#17
. Jumbo: 30 oz; Medium: 21 oz. If both just register at the minimum, the increase would be 43%; if the Jumbo registers at the minimum and the Medium falls just 1oz below Large (minimum 24oz), that would make an increase of 30%. As Jumbo does not have an upper limit, let's just say that chickens are not ostriches and cap the difference at 48%. Our home math isn't too shabby. (Dissimilarities are not
accounted for, although a true nerd can prove that shape variation is a very minor contributor, commercially speaking.)
The next time your grocer sells medium eggs at <1/2 the price of jumbo, jump on it!