Galileo supervises the construction of the wood shed
Note: I am posting this as a lesson to help you understand part of a discussion in the Sept. 2009 issue of The Math Mojo Monthly (“Comes out Quarterly, Mostly”) Newsletter. I’ll also be adding a free Special Report on Basic Plane Geometry soon, in order to help anyone understand this important subject. It will be hand written, and in plain english. Stay tuned for it.
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There are lots of lessons about how to find the area of a plane surface. It’s a pretty straightforward procedure, but like with most lessons, students end up asking the question, “What is this good for?”
Of course, it’s good for a lot of things, but most of those things aren’t interesting to students. There’s the problem. You can tell them the reasons till you’re blue in the face, but if they don’t care about those reasons, you’re sunk. And face it, most of them don’t care, and you didn’t either when you were there age.
Trying to convince someone that something is interesting to them, against their will, is an uphill battle, and I’m no Sisyphus. So I don’t even try. We might as well come clean with the kids and tell them that there is possibly no interesting reason in the world for them at this time. Who cares? Let’s do it anyway. They can figure out the reasons for themselves someday, if they want to.
As far as I’m concerned, if a kid has no curiosity, and no interest in their world, I have no interest in them anyway. But wait … maybe you’re a teacher, or a parent, and don’t have the luxury of telling the kid to bugger off. (Poor you – don’t you wish you’d had a golden retriever instead of that little brat?)
Seriously, though, we only make ourselves look silly trying to cajole kids. And threatening them with, “If you don’t learn it you’ll fail,” seems like a very weak and craven argument.
There is no “reason” that will work for some kids. So let’s just jettison the notion that everything has to have a reason. There really is no reason to play video games, but that doesn’t stop them. Let’s just learn about the area of a square to help them understand more about their world and develop some chops that will help them stretch their minds. That may not interest them either, but it sure beats cajoling and threats. We can’t win over all students, but I think we get a better hit-rate when we just do things to try them, out of curiosity, out of a mild sense of playfulness.
Or maybe not.
On to area:
Surface area (the area of a two-dimensional shape) is generally measured in “square somethings.” You pick the something you like – feet, inches, meters, nanometers, rods – you name it - as long as that something is a unit that measures one-dimensional space.
What is one-dimensional space? (Sheesh, do I have to explain everything?) If you don’t know what one-dimensional space is, stay tuned for an special report on it from Math Mojo very soon. But for now, I’ll just say that lines have one dimension, planes have two dimensions, and solids have three dimensions.
Lines (really line segments, because lines go on forever. If you don’t understand about that, just wait for the special report – it’ll goes into such detail about it that you’ll be sorry you asked) are measured in “somethings,” like the somethings mentioned above. Here’s a picture of a line segment:
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It could be a foot long, a yard, etc. Ii couldn’t be a square foot long, because square “somethings” measure two dimensions, as we’ve said. It couldn’t be a “cubic foot” because cubic somethings measure the volume of three-dimensional solids. It couldn’t be quart, because quarts, liters, etc. measure the volume of liquids. OK, you probably know all this stuff.
You probably also know that in order to measure a rectangle or square, you multiply the measure of the base by the measure of the height.
In the following square, we have a base of 24″ and a height of 24.” (You probably already know that the base and height of a square must be the same. That’s one of the properties that make it a square.)
We’ve been given the measure of the base and height in inches (we say it’s “twenty-four inches by twenty-four inches” or “twenty-four inches square” – be careful, though, because “twenty-four inches square” is not the same thing as “twenty-four square inches“, so it’s easy to calculate the square inches of this square’s area. Just multiply 24 x 24 (or calculate the “square” of 24), to get 576 square inches.
That means that we have a square that measures 24 inches on each side. If we break that square into little boxes, each measuring an 1 inch on each side, we’d end up with 576 little boxes, each measuring an inch on each side, or “576 square inches”.
Here is something to consider:
Since there are 12 inches to a foot, the above square could be measured as “two feet square” or “two feet by two feet”.
That means that the same square measures 2 feet on each side. If we break that square into little boxes, each measuring 1 foot on each side, we’d end up with 4 little boxes, each measuring a foot on each side, or “4 square feet”. So you see that 4′ sq. (which is how you write “4 square feet,” equals 576 square inches.
Another way to think of it is to think that one square foot has 144 square inches in it. (12 inches squared is 122, which is 144.) 4 groups of 144 equals 576, so four square feet is 576 square inches.
I know that this can be confusing unless you give it time to sink in. Don’t worry if you don’t “get” right away. Read it again, think about it. Let it percolate in your head, and pretty soon, “ding!” the light will go on in your head and you’ll feel pretty good about what you are understanding. Just don’t give up.
Now you know the basics about square measurement.
If you want to measure any rectangle, it works the same as a square, just multiply the base times the height. “Square” somethings don’t only measure squares. We can use little square boxes to measure any two-dimensional shape. Think of it this way:
The “squares” in the “square somethings” we are talking about refer to the little squares we are using to measure the area of the shape, they do not refer to the shape itself.
We can even measure shapes that are not simple polygons (closed two-dimensional shapes bordered by line segments) using square inches. For instance, we can measure the area of a circle easily, using pi. But, lucky you, we are not going to do that in this lesson, because it’s getting long and boring enough as it is.
So let’s shake it up a little – let’s actually use what we’ve learned to do something.
Here’s a picture of a woodshed I’ve been building:
See the siding on it? It’s cedar siding. Originally I was going to side it with something cheaper, but I realized that the builders who did the original siding on my home had left a few extra planks that they didn’t use when they did the job. So I figured I’d use that. I hoped it would be enough.
The siding came in 10 foot lengths. Each length was 8″ wide. They looked like this:
Each side of the shed that you see covered with cedar is 5′ 4″ high, and 16″ wide.
As you see, I ran out of cedar siding. I have to order some more.
If I hadn’t had any cedar siding to begin with, how much should I have ordered to be able to finish the area that I wanted to side?
How would you go about figuring that?
To make it a little harder, keep this in mind:
Whey you install siding like this, you first nail in the bottom layer, then overlap the next layer on to it with a 2″ overlap. That means that although the siding is 8″ wide, only 6″ of that is visible, except for the top layer, which you may or may not have to cut down a bit to make it fit.
Happy figuring!
P.S. I hope I get the shed finished by the time I put up the answer in the next post. Wish me luck!
