How do you turn a bunch of data into a super-handy graph? Keep on reading to learn the final 2 steps in Math Dude's easy 4-step method for making graphs.
In How to Make a Graph (Part 1), the infamous hero of our ancient Egyptian plotline, Knot Dude, taught his pyramid-building father, Papa Knot, a quick and dirty method for figuring out the corner-to-corner diagonal length of a pyramid's base. In particular, he taught his dad the very same 4-step method for making graphs that we're learning.
Last time we talked about the first two steps of Knot Dude's method: drawing Cartesian coordinates and creating a list of ordered pairs. As you might expect, today we're going to talk about the final two steps. What are they? And what can we do with the graph we end up with? Stay tuned to find out!.
Steps 1 & 2 Review
If you missed the first part of this series, you might just want to stop right now and go take a look at it. Because everything we're going to talk about today is based upon the stuff that Knot Dude taught his father in that episode. In particular, he taught his dad that the first step in making a graph is to draw what's called Cartesian coordinates. What's that you ask? Check out Part 1 of this series to find out!
In that episode, we also learned that the length of the corner-to-corner diagonal across a pyramid's base can be written c = √2•a. Knot Dude told Papa Knot that the easiest way to make his graph is to start by using this equation to calculate the lengths of the diagonals, c, for a bunch of pyramids with different sized bases, a, and then to use his results to make a list of values written like (a, c).
Why did he write the numbers he found like this? Because he was making a list of what are called "ordered pairs." What do you do with these ordered pairs?
What Are Ordered Pairs?
Before we find out what Papa Knot did with the ordered pairs he calculated to make his pyramid graph, let's take a minute to talk about what ordered pairs are in the first place. And to do that, let's step back and simplify things by thinking about a graph that contains 3 points. The x and y-axes of this graph both span values from 0 to 5. Let's say point A is located at x=1, y=2; point B is located at x=3, y=5; and point C is located at x=5, y=1.
If you think about it, you'll see that each of these three points is located at a position that's described by an ordered pair of numbers. In exactly the same way that a pair of cross streets directs you to a location on a map, an ordered pair of numbers tells you the location of a point on the coordinate plane. Point A at position x=1, y=2 corresponds to the ordered pair (1, 2); point B at position x=3, y=5 corresponds to (3, 5); and so on.
Which means that all you have to do to describe a location in the coordinate plane is give a location along both the x and y-axes. In other words, all you have to do is give an ordered pair. Now that we know that, we're ready for…
Step 3: Plot Points on the Plane
Knowing that Papa Knot's ordered pairs describe the x and y locations of points on a graph, it's time to go ahead and make that graph. If you use a calculator to find approximate values for numbers from the table like 10√2 and 20√2, you should come up with a graph that looks something like:
Believe it or not, what we have now is the start of what will eventually become the graph that helps Papa Knot solve his pyramid problem. In my version of the graph, instead of an x-axis I've labeled the horizontal axis the “a-axis;” and instead of a y-axis I've labeled the vertical axis the “c-axis.” Why? Because we don’t have (x, y) ordered pairs, we have (a, c) ordered pairs. In other words, since the table we made earlier contains a and c values, our ordered pairs are (a, c) ordered pairs.
Once you've finished plotting all of the ordered pairs from Papa Knot's table, your ready to move from plotting points to plotting a curve. Which brings us to…
Step 4: Connect the Dots to Draw a Curve
After slapping the (a, c) ordered pairs from Papa Knot's table onto your plot, you should have at least six points showing the locations of the ordered pairs corresponding to pyramid sizes of a = 0, 10, 20, 30, 40, and 50 feet. While plotting these points, you should begin to see a pretty clear trend developing.
In this case, you should see that the points appear to be ascending in a straight line from the lower left to the upper right of your plot. So, let’s go ahead and draw a line that connects the dots and shows the overall trend. What does the result look like? In this case, it's a perfectly straight line.
Okay, so what does this graph tell us? Well, Papa Knot's goal was to come up with a way to quickly find the diagonal length of a pyramid's base given only the length of its sides—and to do that without having to use the Pythagorean Theorem every time somebody asks for a different sized pyramid. And that is exactly what this graph does!
For example, if Papa Knot needs to build a 24-by-24 foot pyramid, he can easily find its diagonal length by starting at a = 24 feet on the “a-axis,” then moving vertically up from this point until he hits the diagonal line which connects the dots, and finally moving horizontally to the left until he hits the “c-axis.” The value of c at that location is the diagonal size of the pyramid. With only a quick glance at his graph, Papa Knot can therefore see that the diagonal length of this pyramid must be a bit more than 30 feet.
So, in the future when you need to make a graph, just remember Knot Dude's 4 simple steps:
Okay, that's all we have time for today. 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!
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