Scientific equations and the metric system

A formula with fractions. Can we just decimalize everything?

I’ve been told (as in second-hand information) that many countries that have switched to the metric system don’t really need to teach fractions anymore because pretty much every fraction can be decimalized. Additionally, I’ve had first-hand conversations with middle school teachers and students who find teaching and learning fractions is a nightmare. Do we really need fractions or, once we switch to the metric system, can we just lay them aside? Over time, I’ve come to question that and here’s my current thinking…

Ultimately, a fraction is part of a thing

We will always need the concept of a less than a full measurement unit. Just as a pound cake was known for its ease because it was based on the ratio of a pound each of butter, sugar, eggs, and flour, it was pretty crude in terms its result. Whether we only need part of a measure of fabric or a piece of wood that is less than a meter long, it is important that children learn (and adults understand) the notion of something that is less than “one” of something.

A fraction has a built-in “math problem”

Maybe some of the resistance to fractions is the inherent idea that it is, as its heart, a “math problem.” (Would we think about things differently if they were called “math challenges” rather than “math problems”? Let’s see…) A fraction is a division “quest.” It can ask a practical question, as in “If I have a half a cup of flour left in this bag and I need a whole cup for my recipe, how much more do I need to borrow from my neighbor to get the full amount needed?” It is a division “question” that needs to be solved if we pair it with anything else (as in add, subtract, divide…).

Consider the following two math “dares”:

A traditional math “problem” on the left that includes fractions with uncommon denominators. On the right is the same math problem decimalized. The top number in the decimalized addition has only been carried two places to the decimal point, otherwise, it would go on forever.

When I hold up “flash cards” side by side with both types of problems shown above during my metric system demonstrations, almost invariably, people choose the one that has been decimalized because it eliminates the issue of uncommon denominators that are such a stumbling block for both children and adults. It also eliminates the steps to get to common denominators because all decimals already have common denominators in the form of 10s, 100s, 1,000s etc.

For the decimalized addition, just add up the columns as you would any math “action,” just making sure you keep track of where the decimal point goes in the final result. Pretty easy if the original equation is properly aligned as above.

In answer to the question “Can we get rid of fractions altogether?”

No. While most things will work just fine if you even go two or three numbers to the right of the decimal point, for some things it just won’t work since many decimals are frequently “rounded” and don’t fully express a numerical concept. While I am not a scientist, I do work with quite a few and when I posed the question of just decimals, “No” was the answer that came back to me. That’s because many fractions just don’t work as decimals for scientific formulas. Consider the fraction in the second math image. If you try to convert it to a decimal you have a problem because, technically, it trails on forever as in .3333333333333333333….

Scientists and mathematicians can’t work like that and need the compactness of fractions to visualize and express their work. (See the graphic at the top of this page for equation fractions.)

That said, let’s keep them where we really need them and stop needlessly torturing students, teachers and our population in general.

Even U.S. stock markets no longer report losses and gains with fractions down to the 16^th. It changed a few years ago when the Securities and Exchange Commission ordered that all stock reports convert to the decimal system prior to April 9, 2001.

Why did it use fractions of quarters, eighths, halves, and sixteenths? According to the article from Investopedia, it dates back to the “pieces of eight” that Spain used some 400 years ago when it decided to exclude the thumbs for the purposes of counting…

Thanks for reading,

Linda

Dividing We Fall: Base 12 and Resistance to Metric System Adoption

I’m not a math person, I’m a words person. That’s one of the reasons I find the metric system so appealing—it’s easy to learn, use and divide with. However, my readers should be aware that one of the metric counter arguments that will arise is the ease with which the foot and its 12 units can be divvied up as a reason to hold onto our current (if illogical) units.

Here’s the point that will be made: The foot—consisting of 12 inches—is superior to metric system units (of 10) because 12 is divisible in so many ways. The numbers 1, 2, 3, 4, 6 and 12 all go into it quite easily, while base 10 units are really only easily divisible by 1,2, 5 and 10.

That’s a crap argument and here’s why:

THE FOOT IS OUR ONLY POPULAR MEASUREMENT UNIT OF WHICH THAT IS TRUE.

Frankly, if all our measurement units were based on 12, I could almost see an argument that we should stay where we are. Not the best argument, mind you, but a stronger one. However, that is far from the case. Let’s take a look at a few of our other weights and measures and see where else the number 12 comes into play.

Teaspoons in a tablespoon = 3
Cups in a quart = 4
Pints in quart = 2
Quarts in a gallon = 4
Ounces in a pound = 16
Feet in a mile = 5,280
Pounds in a ton = 2,000

Hmmm, no luck so far. While I’m not saying there isn’t possibly another U.S. customary unit that uses 12 of something, it’s probably pretty obscure.

To interject some of my extensive research, in A History of Mathematics, Carl B. Boyer and Uta C. Merzbach point out (I used brackets below to include the numbers referenced in words to make it easier to follow):

A study of several hundred tribes among the of American Indians, for example, showed that almost one-third used a decimal system [10], and about another third had adopted a quinary [5] or quinary-decimal system [think abacus]; fewer than a third had a binary [2] scheme, and those using a ternary [3] system constituted less than 1 percent of the group. The vigesimal system, with the number 20 as the base, occurred in about 10 percent of the tribes.

As the book also points out:

As Aristotle had noted long ago, the widespread use today of the decimal system is but the result of the anatomical accident that most of us are born with ten fingers and ten toes.

And, according to the book, the number 10 even takes the upper hand when it comes to language:

The modern languages of today are built almost without exception around the base 10, so that the number 13, for example, is not described as 3 and 5 and 5, but as 3 and 10.

Given that we pretty much come with our own built-in decimal system it only makes sense (to me at least) that the metric system is the system that prevails around the world.

From http://www.roman-numerals.org/chart100.html

If those points aren’t enough, allow us to consider the Romans and their numerals. They were all about units of ten as you can see from the large number of “Xs.” (See chart from http://www.roman-numerals.org/chart100.html)

As a friend pointed out, there are other things that come in twelves, such as our months and recovery programs (as in 12 steps) but these are not numbers that we need to divide so go ahead and use a dozen of something where it makes sense, just not in our measurement system please.

Thanks,

Linda