Scientific equations and the metric system

Fancy images with cosines and fractions.

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”:

combined

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 16th. 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

2 thoughts on “Scientific equations and the metric system

  1. “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…”

    This is not true. Fractional math is a component of Algebra and still needs to be taught, but there doesn’t have to be any extra, specialised teaching of fractions that take up too much classroom time in order for students to manipulate inches. Even with this extra time taken, fractional math is often forgotten. The American student then enters adulthood unable to do fractional math and unable to function well using inches, but has no problem adding, subtracting, multiplying and dividing decimal numbers which are the core of the metric system he/she doesn’t know how to use and despises.

    ” it trails on forever as in .3333333333333333333….”

    Bad numeration practice here. It should be properly written as 0.3333333333….

    “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.”

    We need to get away from using the term “pound” cake and instead calling it as a “four part” cake. Because any units can be used as along as the four parts have equal mass. The French call it “Quarte-quarts” and the German Rührkuchen. Others languages too use names that refer to its four-partness. We need to do the same.

  2. I call my own favourite cake recipe the “fifty, fifty, fifty, egg” cake: 50g. flour, 50g. sugar and 50g. fat for every egg (plus bicarb if not using self-raising flour, water to mix and any additional flavourings as required — vanilla essence, blueberry preserve, cocoa powder, chocolate chips or whatever). And since a large egg in its shell fortuitously weighs about 50 grams, the main ingredients can be weighed out using only pan-type scales, with no standard weights. (Though I usually use electronic scales; weighing the ingredients straight into the mixing bowl on the scale platform and zeroing the display after adding each ingredient.)

    I’m 100% with you on getting rid of mixed integers and ratiometric fractions. Call it 3.75 or call it 15/4, but please don’t write 3 3/4. It makes sense to keep ratios as ratios when they don’t simplify neatly as decimals, or if you are going to attempt cancelling or taking out a common factor, but a final answer need not be given with excessive precision anyway (certainly not to more digits than the givens at the beginning of the problem) and writing a fraction in decimal notation is simpler than trying to convert a ratio greater than one into an integer and a fraction in its lowest terms. And especially easier than trying to round to a different denominator. Dividing up a length given in feet and inches and giving your answer to the nearest sixteenth of an inch is not a particularly contrived problem; it occurs often in construction when attempting to position things evenly in a space and mark the wall using a tape measure, yet it is a lot harder than dividing a length given in metres and giving your answer to the nearest millimetre …..

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