I've been on both sides of the issue of Knitting Math: does it exist, or doesn't it? Math is math, right? Or is it like Business Math, and Cooking Math, where some functions are key to everything else working?
I am now firmly a believer in Knitting Math. Designing knitwear is one part sketching, 15 or 20 parts math, one part writing the pattern, and one big part knitting.
If I want to design something simple like a baby blanket, I need to figure out how big I want it. Then I need to figure out what yarn I want to use. Then I knit test swatches to determine the gauge that gives me the look I want. Let's say it's 6 stitches x 7 rows.
Next, I need to multiply the width of the blanket by the stitches per inch. If I want the blanket to be 3 feet wide, that's 36 inches x 6 stitches per inch = 216 stitches. This becomes the cast on number.
If I want it to also be 36 inches long, for a square blanket, I need to multiply 36 inches x 7 rows = 252. Now I can create a graph, and start marking out stitches and color patterns.
Stitch gauge and row gauge are the essential two pieces of information for almost all "knitting math." If you want your blanket larger, you multiply each inch by 6 stitches or 7 rows to know how much to add. Likewise if you choose to make a smaller blanket, you subtract 6 stitches or 7 rows for each inch you want to subtract.
The gauge swatch gives you another critical piece of math to do - figure out how much yarn you're going to need. Make a swatch that is whole inches on both the top and the side (like 4x6). Multiply, and you get 24 square inches.
Now cut the swatch off the ball, and weigh it on an accurate scale. What does it weigh? In this case, 1 gram. The ball comes in 100 yards per 50 grams. For now, disregard the yardage, and just focus on the weight. That means each gram will cover 24 square inches. Multiply that by 50 and you get 1200 square inches of knitting per ball.
Multiply your original 36x36 blanket size out to square inches, and you get 1296 inches. This tells me that one ball of yarn won't quite do it, and you'll need to start a second ball.
DIFFERENCE IN GAUGE
But what if you don't knit to the gauge specified in the pattern? What if you knit to 5 stitches and 6 rows per inch instead? You're going to need a lot more yarn. Those same 216 stitches now represent 43 inches of knitting instead of the original 36. The 252 rows now make 42 inches instead of 36. Now your blanket is 1806 square inches. Now you'll need 1 and 1/2 balls of yarn, because your piece is almost 30% larger than the designer planned for, and that's if your gauge is only off by one stitch!
I know, it seems like a lot to digest. It's not really. Look up any pattern in any magazine, and do the math. You'll see that it works out every time. A little practice (and a calculator) makes it easy to remember what to do and how to do it.