 All right, well let's try to add two fractions. So here we'll try to add the fractions two-thirds plus one-quarter. So I might start off by drawing a model for two-thirds. So I'll take my whole, divide it into three pieces, and take two of those equal pieces. So there's my model for two-thirds. And then I want to draw a model for one-fourth. Now, for convenience, it's helpful if I alter the direction of the division. So here I started by finding my two-thirds by making my divisions vertically. For the one-quarter, it'll be convenient if I make my division of the whole horizontally. So I'm going to take my whole, divide it into four horizontal pieces and take one of those. And so now I'm going to add together this two-thirds to this one-quarter. All right, so let's see. Well, they're different sizes. So I am going to learn nothing if I just consider the fact that there are three of them, because it's of different sizes and I can't combine them. So what I'm going to have to do is I'm going to have to do something to make them of equal size. And here's where the vertical and horizontal splits come in useful because if I horizontally split this into four pieces and vertically split this into three pieces, I get the same division of the two holes. And so each of these little rectangles, regardless of which box I'm in, is of the same size, which is not what I could have said for these. These rectangles and these rectangles are of different sizes, whereas these little rectangles here are all the same size. And now it's actually useful to count them. So I'll count them up, or if I want to do this purely visually, I'll just go ahead and move these three over here to one side here. And so all together there's my two-thirds, there's my one-quarter, this time run vertically, and I have a total of 11 pieces. There's 12 all together in this box, so my sum is the fraction 11 over 12. What about a different subtraction? So for example, four-fifths plus one-third. So again, I'll set down my model for four-fifths. Again, I'll divide this unit bar vertically this time. And my model for one-third, again, I'll divide it horizontally this time, and so I now have the model for four-fifths plus one-third. And again, I'll try to divide the two in the same way. Again, you notice that this piece is much bigger than any of these pieces, so it doesn't do any good counting the total number of pieces yet. But if I divide both bars in the same way, then the little rectangles in both are exactly the same size. All right, so here we run into a slightly different situation here. If I put the green together with the blue, then what I have is a little bit more than a whole. So I get four-fifths, and then these five pieces here, well, three of them fill out the one, and then two more are left over. So what do I have? Well, I have a total of 12 and 5. I have 17 parts each of size 115th, and that gives me a whole and two-fifteenths as my total. Of course, at some point we want to graduate to working with the concrete representation, to working with the abstract representation. So here's a convenient formula for adding two fractions, two fractions a over p plus b over q. I can add them by multiplying aq, bp. Those will be our new numerators. Our common denominator will be p times q, and where, if possible, when I get the sum, I can reduce that to the lowest terms. For example, if I were to add one-sixth plus three-tenths, that's 10 plus 18 is 28 over 6 times 10, and there I can reduce that this 28 has two factors of two that could be eliminated, and I get my quotient 7 over 15. Now, you might be surprised that I haven't mentioned anything about lowest common denominators and all that, and that's because it's generally a waste of time to find them. So, for example, three-twenty-eighths plus five-twelfths without finding the lowest common denominator, three by 12, five by 28, I can add those two together, one-seventy-sixth, look for common factors, four divides this, in fact, four times four divides both of them, and so that reduces to 11 over 21. And there's my quotient without bothering to find the lowest common denominator. On the other hand, if I find the lowest common denominator, I factor 28-12, find the LCD, 28 times 3, and then convert both fractions to the equivalent forms, get 4D4 over 28 times 3, and then let's see, still have to do that factorization at the end, and so that gets me my quotient 11 over 21. However, it's worth pointing out that if the LCD is immediately obvious, go ahead and use it. So, here's a problem three-fifths plus seven-tenths, and here the denominators are just multiples of each other, so the lowest common denominator is just the larger one, so three-fifths is the same as six-tenths, and now I can add six-tenths plus seven-tenths at getting my final answer as one and three-tenths.