Filed under: Education, SAT Logic, SAT Math, SAT Prep | Tags: Andrew Turner, Archimedes, fractions, SAT Prep
Ask 1,000 students if they know what a number is and you will get 1,000 affirmatives. Ask the same students to tell you what a number is and you might get 500 correct answers. Now ask those students if they can properly dissect a number and explain it piecewise…Sadly, in the modern school curriculum’s race to Calculus many students cannot answer the most fundamental questions about mathematics. This will serve as the second step in understanding the nature of fractions and why they cannot be done away with.
Every number has five components: a base (b), a coefficient (c), a divisor (d), a power (p), and a root (r).
As a simple example, 24 is a number: It exists as an independent, specific, and unique value and has eight “factors” or “numbers that are evenly divisible” into 24: 1, 2, 3, 4, 6, 8, 12, and 24. The amount of 24 is the same as 1 x 24, and this is such an important idea in mathematics that we have a name for it: The Multiplicative Identity. So whereas we started with 24, now we have:
We call a number that sits just out front of the a set of parenthesis a “coefficient”. If you break down that word it gives you more of a clue to what it means: “with” + “efficiency”. It is a simplified way of dissecting and thinking of a number. As an example, rather than thinking of the number 24 as a solid, concrete value of “24″ it might be better to think of it as 8 x 3. In this case, I could think of it two different ways: 8(3) where 8 is my coefficient and 3 is the base number or 3(8) where 3 is my coefficient and 8 is the base number. This is all, of course, very simple – and that is exactly the point. This is the first step in understanding what numbers are. We have covered 2/5 of the components of a number which brings us to our third idea: the divisor.
Any number you can imagine is a fraction, because at the very least every imaginable value is always “over 1″. The number 5? It’s really 5/1. The number 14? It’s really 14/1. -8? You guessed it: -8/1. Even fractions (i.e. 3/5 = (3/5)/1), constants (i.e. π = π/1), imaginary numbers (i.e. 2i-3 = (2i-3)/1), or even algebraic variables (i.e. x = x/1) may ALWAYS be written as [insert every number you can imagine]/1. In the case of our number 24, it is equivalent to write it as:
1(24)
—–
1
Symbolically I might code this as:
c(b)
—–
d
where d = divisor
Even the obelus (“old school” division sign: ÷) looks like a fraction! Mathematicians are so literal, but that is the advantage of mathematics! There really is no guessing involved. The study is filled with very simple ideas which, when put to use on a test like the SAT can turn out to be very powerful…which reminds me.
The last two components are the power and root. These deserve their own discussion so I won’t get into them right now, but the general philosophy is that powers are a quick way to multiply in the same way that multiplication is a quick way to add. Roots are not the “opposite” of powers because the statement, “roots are a quick way to divide in the same way that division is a quick way to subtract” is not correct. Roots are better defined as “inverse roots” than “rooted in subtraction.
What is important is to walk away understanding another reason why not teaching fractions in school is just a bad idea. Contained within the very definition and understanding of a number is the fraction.
– Andrew Turner
