What are graphs? When are they useful? And how do you make one? Keep on reading to learn an easy 4-step method for making graphs.
Remember the famous story (that I totally made up) of Knot Dude and Papa Knot? You know, the one where the clever young son (that's Knot Dude) of a big-shot ancient Egyptian pyramid builder (that's Papa Knot) used his mathematical curiosity about knots to discover the Pythagorean Theorem and in so doing to solve his dad's problem of figuring out how to build pyramids with perfectly square bases?
Well, as it turns out, there are more mathematical lessons to be gleaned from this story. To begin with, today we're going to see how Papa Knot used his son's discovery to simplify the process of building perfectly shaped pyramids of any size—an ability which he used to revolutionize the pyramid building industry.
What exactly did he do? He made a graph. How did that help? And how did he do it? Those are exactly the questions we'll be answering today and over the next few weeks..
Why Are Graphs Useful?
Imagine we're once again headed back in time to visit the wonderful world of ancient Egypt. Our trip begins just as Papa Knot receives an order to build a modest yet perfectly shaped 15-by-15-foot pyramid to house the remains of a wealthy family's beloved feline friend. Using either the oh-so-handy rope with specially tied knots that Knot Dude made for him or his son's newly discovered relationship that the lengths of the two sides (dubbed a and b) and hypotenuse (dubbed c) of a right triangle are related by a^2 + b^2 = c^2 (which a few millenia later would be known as the Pythagorean Theorem), Papa Knot was able to figure out the length required for the diagonal across the base of the pyramid.
a2 + b2 = c2.
As Papa Knot's fame grew, the number of wealthy families wishing to intern their dearly departed pets in magnificant pyramids also grew. Some families wanted their pyramid to be 15-by-15-foot in size just as our first family did, but others wanted 21-by-21-foot pyramids, 12-by-12-foot pyramids, and a whole host of other sizes, too.
While using the Pythagorean Theorem over and over again to find the varying lengths of the diagonals stretching from corner-to-corner of each of these pyramids (thereby ensuring they all have square bases) would work, Papa Knot realized that it sure would be useful if he had some way of quickly looking up the length of a pyramid's side and immediately seeing what the diagonal length should be—no matter how big or small it is—without doing any calculations at all.
As usual, when Knot Dude heard about his father's wishes, he used his clever mathematical mind to solve the problem. What did he come up with? He taught his dad how to make a graph. And rumour has it he taught him the very same 4-step method that you're about to learn. Here's how it works:
Step #1: Draw Cartesian Coordinates
To get started, we need to set ourselves up and get ready to make a graph. Which means we need to draw a pair of coordinate axes. How do we do that? We start by drawing a pair of perpendicular lines, like this:
What exactly have we drawn at this point? This pair of lines make up what’s called a Cartesian coordinate system (named after the 17th century mathematician René Descartes), and this Cartesian coordinate system is sitting on top of what’s called the coordinate plane. As we'll soon see, the purpose of a coordinate system is to give us a way to investigate the relationships between numbers. In Papa Knot's case, he needs to look at the relationship between pairs of numbers representing the length of the side of a pyramid and the length of the diagonal across the pyramid—hence the need for two axes in our coordinate system.
The purpose of a coordinate system is to give us a way to investigate the relationships between numbers.
And that’s exactly what we have: the x-axis (sometimes called the abscissa) runs horizontally and will be used to track one of the lengths of interest and the y-axis (sometimes called the ordinate) runs vertically and will be used to track the other. We put little marks along each axis to label the x and y values at various locations along the way. Both axes have a value of 0 at the origin where they meet.
So that’s our playing field—now it's time to start doing something on it.
Step #2: Create List of Ordered Pairs
As we know, Knot Dude's advice to his father for coming up with a way to quickly figure out the diagonal length of any sized pyramid base is centered around the idea of making a graph. But before he can make that graph and ever after rid himself of the need to do any calculations, he first needs to do a little algebra with the Pythagorean Theorem—a^2 + b^2 = c^2. In particular, he needs to figure out the length of the hypotenuse, c, of the triangle formed by half a pyramid base:
To solve for c, Papa Knot first switched around the left and right sides of the Pythagorean theorem to write it as c^2 = a^2 + b^2. He then took the square root of both sides to get
c = √a^2 + b^2
But notice that there isn’t really a side with length b in a square pyramid base since both sides have the same length, a. So Papa Knot used the fact that a = b to rewrite his equation as
c = √a^2 + a^2 = √2 a^2 = √2 • √a^2 = √2 • a
So the length of the hypotenuse of a pyramid is given by c = √2 • a. Which means that Papa Knot could use this equation to figure out the diagonal length of any square pyramid base—he simply plugged in values for a and got back values for c. For example, if we do this for a = 1, we get c = √2. If we do it for a = 2, we get c = √2 • 2—aka, 2√2. And if we do this for a bunch of values of a, we end up creating a list like this:
I’ll leave it up to you to fill in all the missing parts. But wait, what's that stuff on the right? Hmm, that sure is an odd way to write a list of numbers! Actually, those pairs of numbers written in parentheses and separated by a comma are known as ordered pairs…and they’re exactly what we're going to need to make our graph.
Speaking of making our graph, that's exactly what the final steps of Knot Dude's 4-step method entail. But, unfortunately, we're all out of time for today. Which means that finishing up our graph so that Papa Knot can come up with a way to quickly find the diagonal length of a square pyramid base without having to do any calculations whatsoever (except for the ones he had to do to make the graph in the first place, of course) will have to wait until next time.
In the meantime, please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Graph image courtesy of Shutterstock.