Filed under: SAT Logic, SAT Math | Tags: Andrew Turner, Archimedes, fractions, mathematics, SAT Logic
The four basic operations: Addition, Subtraction, Multiplication, and Division. Every student knows what they are and how they are used, but few students can describe how the relationship between two numbers changes as we exchange a “+ or -” for a “x or ÷”. As an example, the difference between 3+5 and 3×5 is much more than simply amount.
In the case of 3+5 we are saying that 3 and 5 are two complete separate numbers that exist independently from one another. Each has its own unique identity and we combine the two to get one value.
In the case of 3×5, however, this is not the case. 3×5 is actually one number to start with, not two numbers. It is just an illusion that, when variables are thrown into the mix, creates confusion.
The Distributive Property gives us hints of this. Consider the case where we have
n(xy+z) = nxy + nz
Notice, of course, that the n is applied to the xy only once. This is because xy is one number, just as 3×5 is one number.
Even the way the obelus (÷) looks is very telling. In the case of 3÷5, it reminds us that this is really just a fraction: 3/5.
So this is the first step we come to in response to this talk of taking fractions out of school. To take away the fraction is to take away 1/4 of arithmetic.
– Andrew Turner
