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For example, here's how I understand fraction multiplication through visualization. If we have 2/3 times 1/3, we're saying we want 1/3 to 'exist' 2/3 times. Because we're not multiplying 1/3 by a whole number (1,2,3, etc), we're going to end up with a fraction which represents less than 1/3. I then visualize a pizza (or any object/conceptual group) broken up into 3 equal groups. I realize that when we consider the symbol '1/3', we mean we're selecting 1 of the 3 visually separate divisions on our pizza/whatever. We're then saying we want that '1/3' of the pizza to 'exist' 2/3 times. Therefore, 2/3 of that section of the pizza will be selected. So then, I break up the third of the pizza into 3 equal portions, selecting 2 parts. To express the new 'selection' of parts, I break up the rest of the pizza into equal parts as well, resulting in 9 equal parts. Therefore, my new selection is 2/9. 2/9 is 1/3, 2/3 times.

However, while this method works and makes visual sense to me, the common method of fraction multiplication doesn't. I'm watching TTC Basic Math training videos, and the teacher says the numerator is the digit, while the denominator is the place value -- and that this is why the traditional multiplication method of fractions works the way it does -- where we multiply numerator by numerator and denominator by denominator, getting our same answer -- 2/9.

First of all, I don't understand why the numerator is the digit and why the denominator is the place value. I understand that in the case of a number like 1293, each of the numerical symbols are in specific place values, which represent 'amounts of 1's' (a thousand 1s, 200 1s, 90 1s and 3 1s). I understand the digit is just the numerical symbol in a specific place value, indicating said amounts of 1s (again, 1000's of 1s, etc). However, I've yet to understand how the numbers which comprise a fraction connect to the concepts of 'digit' and 'place value'. I understand fractions in this way -- that we have all the necessary symbols (1,2,3, etc) to complete various functions with what we consider '1' (whether '1' represents a group of 10 trees, a forest or a piece of pizza). These functions are those like multiplication, division, subtraction and addition. However, we want to be able to deal with parts of what we consider 'whole' or '1'. Therefore, we introduce the concept of the fraction, breaking up what we consider '1' into a set of equal parts. This set of parts is called the denominator of the fraction. The numerator tells us how many parts of the entire set of equal parts are being selected (or how many are left). The issue is that this is how I understand fractions -- that we simply needed a way to convey parts of what we consider '1', and therefore pulled the '/' symbol out of nowhere, placing one number on the top and another on the bottom. I understand the '/' symbol represents division, and understand that every proper fraction will result in a decimal, which curiously represents the same thing as the fraction. However, this is about as much as I understand, therefore, the traditional fraction multiplication method doesn't make one bit of sense (other than the fact that, somehow, quite magically, it replicates exactly what I did when I multiplied fractions in my long and tortuous visual method).

This is the only reason I've been able to come up with for the traditional mechanical method of fraction multiplication is that, for some reason, it just works that way -- multiplying numerators by numerators and denominators by denominators somehow reproduces the exact set of steps I slowly took to determine the answer -- 2/9.

Thanks for the help.