Ask folks what class they feared the most in high school and college, and I bet you’ll find that “math”, generally, or “calculus”, specifically, is the answer you’ll get more often than any other. Yet math is really nothing more than a) the ability to apply specific equations and get consistent results, and then b) to apply those results to better understand the world around us. So, I think its time to take a look through the Internet and see what we can learn…about math.
Basic Math: If a train left New York and another left Boston…
Most of us know basic math. It’s the math we use when we shop: we pick up half a dozen eggs, we buy 4 steaks, we supply cash for totals and get change back. Its also the math we use at home: we measure out 1 cup of flour, we shape dough into a circle to make a pie, we time how long the pie has baked, and we cut a board in such a way that it fits into a slot on the floor. How about at work, do we use this math at work? You bet: we ask for two packets of sugar for our coffee, the delivery person drops off a gross of pens, we send mail using two day express delivery and know that the mail will be delivered in two days or less.
Basic math is that math that surrounds us and that we use in our everyday world. It is the math that allows us to time events by understanding units of measurement about time, such as hours, minutes, and seconds. It is the same math that then gives us the tools to measure these units and express this measurement as a factor of time elapsed: he ran the marathon in 6 hours, 23 minutes, 3 seconds.
Additionally, basic math is that math we use when quantifying objects, such as 2 apples, 3 people, 4 cats. It is also the math we use with currency and with temperature — though units of measurement can differ here — and with our payroll stubs and income tax.
Basic math consists of addition and subtraction, multiplication, and division, and we can’t forget the most infamous of them all: fractions. It is arithmetic.
Basic math is the math we learn first, and the one that requires us to learn the most and take the largest leap of faith. After all, in algebra we may understand that 2x – y = 3 is a solvable equation, but it is really based on our belief that the number “2” does represent two objects; that two numbers can be multiplied and the result will always be the same; that you can add two numbers and consistently get a third; and that you can then subtract one of the original numbers from the new total, and derive the other original number.
3 + 4 = 7 what will you get if you take 4 away from 7?
Look at that! Your first number quiz.
|So, did you get the correct answer? If you’re not sure, you might want to ask Dr. Math to help you find the answer . How about a different way of learning math? You might want to check out The Clock (Modular) Arithmetic Page for a little learning about math, in the round. Want to have a little fun with math? Then check out the Math Forum Elementary Problem of the week — see if you can keep up with the kids.|
Of course, once we learned basic math, it was time to get into other types of math such as algebra, covered next.
Algebra and the ultimate question: Why?
So what is algebra and why do we need to learn it? Well, something like arithmetic is good when dealing with math of known quantities and objects such as adding two apples together, or measuring a cup of flour. But what if you need to solve an equation involving the addition of 2 quantities of an object, and you only have one of the quantities and the result?
Remember our little math game in the last section:
3 + 4 = 7 what will you get if you take 4 away from 7?
Well, let’s rephrase this question and formalize it into an equation. Instead of saying “if you take 4 away from 7”, say “if you take a number away from 7 you’ll get 4”, and rephrase it again to say “if you add a number to x, you’ll get 7”. Drawing this as an equation, you get:
x + 4 = 7
Algebra is involved with solving the equation for the unknown variable, in this case, x, using a set of rules and procedures to accomplish the task.
For our equation, we first need to isolate the variable, or the unknown value. We can do this by using basic math to eliminate the known value from both sides of the equation:
x + 4 - 4 = 7 - 4 x = 3
Isolating the unknown is the same as solving for the unknown.
See, you just did algebra! That wasn’t so bad, was it?
To summarize, algebra is the ability to solve equations containing one or more unknown variables. The solution is found by applying known procedures such as isolating the unknown variable and combining like terms. Algebra then uses these same rules for more complex equations such as finding ratios, multiplying fractions, graphing results on a coordinate plane, and exponents. Before you click away again, let’s look at each of these and see that there is nothing scary or weird with any of them.
First if all, you use ratios anytime you figure out your odds of winning the lottery (1 in a kagillion), or you read about something such as the “ratio” of women to men of those responding to a survey, for instance, the ratio was 3:5, or 3 women out of 5 respondents were women. If we look at this as an equation, we would have:
x + 3 = 5 x + 3 - 3 = 5 - 3 x = 2 there are 2 men for every 5 respondents
How about graphing? Well, I used to love to graph. I loved the graph sheets, I loved getting my ruler and my pencil and drawing out a nice clean line. Didn’t have a clue why I was doing it, but it sure was fun.
You know graphing: on a number line graph all numbers less than 8. You end up with:
Now, what is there about this that isn’t fun?
Of course, once we mastered graphing on a linear line, the next step is to try graphing within a coordinate system. This is a graph where the X values are plotted along a horizontal line and the Y values are plotted along a vertical line. The Y-axis intersects the X-axis at the point where X is zero, and the X-axis intersects the Y-axis at Y’s 0 point. Then, individual points on the graph are plotted at the point where the X value and Y values intersect. So, if you have an X value of 3 and a Y value of 3, your point will exist in the upper right of the system. If you have many points, such as those drawn for an equation and using different values of X or Y in the equation, you can connect the points and you actually have a line. From this you can determine not only what values are from an equation for given values of X or Y, you can determine what all values of X or Y will be. Why? It’s in the graph!
So, we know that basic arithmetic isn’t scary, and algebra can be fun, are you ready to try something a little stronger? Say, Geometry?
|If you want to know about algebra, have I got some sites for you. First up is Math for Morons Like Us. Don’t let the name chase you away, this really is an impressive site providing an overview of pre-algebra, algebra, geometry, and calculus. Math for Morons was created for the ThinkQuest program. ThinkQuest is a competition held every year where students or adults who are teachers or studying to be teachers can create Web sites, all based on knowledge and education. There some pretty impressive Web sites from this project. For instance, another Web Site is Volcanoes Online, created by students from all over the World.
Now, doesn’t all this sound like fun? Well, to make it even more fun, James Brennan from Boise State University has created an interactive Java applet called the Graph Applet. Try it out.
Well, you’re probably pretty comfortable with addition and subtraction and even equations, about now. Time to up the ante and take a look at geometry.
First of all, to ease your anxiety, and to keep you from clicking out of the page, geometry is not only fun, it is really based on the same mathematical foundation you worked with in the basic math and algebra sections. Now, those sections weren’t so bad, and this one doesn’t need to be either.
So, what is geometry? Well, it has to do with shapes. All kinds of shapes, from lines to circles to triangles to spheres to what have you. Geometry gives you the tools to do such things as find the volume of a sphere or to find the circumference of a circle.
You don’t think you need this kind of stuff? Well, sure you do.
For instance my husband and I walk around a water reservoir behind our place that has a diameter of about .4 miles. We were curious about the actual distance we traveled so we dusted off our geometry and found the formula for finding a circumference of a circle given the circle’s radius:
C = 2(PI)r
Well, a diameter of a circle is twice the size of the radius, so the radius of the lake would be .2 miles. Plugging this in for r, and remembering that the value of PI is 3.14159 — five decimal places is more than enough, we ain’t building a rocket here – we would have:
C = 2(PI)r C = 2(PI).2 C = 2 x 3.14159 x .2 C = 1.25663
Hey, 1.25 miles! A nice little jaunt.
Geometry is very big in the computer animation business. Did you like A Bug’s Life or Antz? Well, geometry is a basic tool used in creating these types of animations. Geometry also forms the basis for work accomplished with VRML — Virtual Reality Modeling Language.
If you like Geometry, then you might want to look more closely at trigonometry, covered next.
|Where to begin when it comes to learning about Geometry. You can go back to Math for Morons Like Us, which has excellent coverage of Geometry in addition to Algebra. You can also go to the Geometry Home Page, which has some very nice tutorials. There’s also the Geometry Center, with documents, multimedia, and software about geometry. This site led me to another site, called Science U, which has its own Geometry Center. Science U has several interactive demos and games, related to geometry and astronomy. Site also has an online store with some unusual items for sale. There aren’t many places where you can create your own fractal design and then have it made into a T-shirt.
Wait, there’s even more sites. I mentioned the use of geometry with computer animation and VRML. Only fair to mention some sites for these topics. First of all, the grandmother of VRML sites is The VRML Repository. Two other essential links are VRML Consortium, and The VRML Specification. And you can’t mention VRML without reference to the SGI VRML page.
For computer animation, try out The Shape Modeling and Computer Graphics page, from the University of Aizu in Japan. Webreference, a favorite of mine, has a nice site called the 3D Animation Workshop. And the king of computer animation is, of course, Pixar.
Oh, and don’t forget the Antz and A Bug’s Life official Web pages.
Okay, you had some fun looking at all the pretty computer generated animations and graphics. Let’s get back to the real reason you’re here: to learn more about math. Right?
First, trigonometry — or “trig” as it is affectionately known — is based on angles. It is this, which distinguishes trig from the rest of geometry.
Why learn more about trig? Well, if you are interested in astronomy, you should be aware that it is trig, and the trigonometric tables, that provided the basis for early star charting. Engineering is dependent on trigonometry. When you see surveyors along the road at construction sites, what do you think they are using to plan the work? Why, trigonometry, of course.
Consider a building. Can you measure how tall it is? You could climb to the top of the building and drop a line of rope down from the roof until it touches the ground and then you could measure the rope. However, this doesn’t sound like a very efficient method, and what if you are trying to measure a mountain peek, or a balloon in the air?
A better approach would be to use our friend, the right triangle, and the trigonometric functions.
First, a right triangle is one which has one 90o angle. The angle opposite the right triangle, along the horizontal axis is written as q, and is called theta. The side of the triangle opposite and adjacent to q are known as, respectively, the opposite and adjacent sides. The side opposite the right angle is known as the hypotenuse, as shown in the figure below.
- sin q = opposite / hypotenuse
- cos q = adjacent / hypotenuse
- tan q = opposite / adjacent
- csc q = hypotenuse / opposite
- sec q = hypotenuse / adjacent
- cot q = adjacent / opposite
Now, considering the right angle and the trigonometric functions, how can we measure that building? Well, you start with a protractor, a small plastic semi-circular or circular disk that allows you to measure angles. You walk 100 feet from the building and then measure the angle from yourself to the top of the building using the protractor. Let’s say this angle is 60o.
At this time you have some known values. You know that q is 60o, and you know that the adjacent side is 100 feet. Now, to get the value for the opposite side, we’ll use the trigonometric formula to compute the tan or tangent of the angle:
tan q = opposite / adjacent tan 60o = opposite / 100 feet tan 60o = 1.73 1.73 * 100 feet = opposite / 100 feet * 100 feet opposite = 173 feet (approximately)
There you go, you found the height of the building all by yourself, with a cheap plastic tool and no long rope. Pretty darn good — and all thanks to trig.
Well, now that you have found that trig is fun, time for pulling in the big guns. Time for calculus.
|I just can’t believe how many Web sites there are on math, including trig. First of all, check out the Free-ed Net, specifically the section on Trigonometry. Free-ed Net is a very hot Web site focusing on free educational resources on the Net, and in the Trig section, they list some nice trig resources. First of all is S.O.S. Mathematics, which provides an overview of Trig, and provides a table of trigonometric identities. Then there is the Math Abundance Trigonometry Introduction, which is very extensive. Very.
Sorry, I’m back. I was sidetracked by Net-Ed’s Astronomy section. Where was I? Oh, yes, trig resources. A great trig resource page is at Study Web’s Math page. I can guarantee that if you go through all the resources they list, you will be a math wiz. Angles, are your friends.
Do you want to order a protractor of your very own? Then check out k-12source.com which has most school supplies for sale. Check out the engineering and drafting supplies.
Oh, and if you want to know how to measure the height of a rocket, check out the University of Nebraska page on measuring a rocket’s height, from the N.E.R.D.S (Nebraska Educators Really Doing Science) project.
Bring on the tanks: Calculus
Well, you’ve made it this far so you deserve a real treat: Calculus!
What is calculus about? Well, first of all it takes what you know with the other math types, and goes a bit farther, or nearer as the case may be. The Excite online encyclopedia, InfoPlease has the following definition for calculus:
"branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit the notion of tending toward, or approaching, an ultimate value. The English physicist Isaac Newton and the German mathematician G. W. Leibniz, working independently, developed the calculus during the 17th cent. The calculus and its basic tools of differentiation and integration serve as the foundation for the larger branch of mathematics known as analysis. The methods of calculus are essential to modern physics and to most other branches of modern science and engineering."
Calculus isn’t just one subject, it’s many. There is differential calculus, integral calculus, there is statistics, and probability, and so on. However, it is also about the world around us. It is not an exercise in seeing how many equations one can stuff into a sophmore’s brain before it explodes.
For instance, could you see needing to know the volume of a sphere? Sure you could. How does one measure the volume of a sphere?
Well, going with empirical method, you could fill the sphere with water and measure how many cups of water fit into the sphere. But, this technique is kind of wet, possibly messy, perhaps not very scientific, or even accurate. Wouldn’t you really rather use a formula?
Borrowing from integral calculus, the formula for calculating the volume of a sphere is:
V = 4(PI)r3/3
So, given the sphere’s radius, you can now find its volume. You can find its surface, too, with the following formula:
S = 4(PI)r2
I won’t lie to you and say that all calculus is this easy. I still think parts of calculus are a joke perpetuated by math majors on the rest of us (“let’s string them along…see when they break”), but calculus can be met face to face at the least, and even mastered (gasp) at the most.
Now, I think that’s enough for me to say on calculus. I’ve forgotten way too much on this subject and if I say anthing more, I’ll embarrass myself. Time to follow this article’s links … and learn a little math.
|Well, I have to go back to Math for Morons like us for their coverage of pre-calc and calculus. Boy, I wish they would change the name. But they aren’t alone with the names, as another good site on Calculus is Help with Calculus for Idiots (like me).
The best reference page with listings on calculus is StudyWeb’s Calculus page. Another great resource on Calculus is Calculus Net.
An example of calculus applied to mechanics is nicely illustrated at Calculus and its Applications to Mechanics. A fun site is a page full of Calculus Java applets where you can change values and observe results for calculus equations.
You can calculate the volume of most shapes with the ABE Volume Calculator page. You can find the calculations used at Calculations for Volume.