Taking Algebra to the Streets
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
--John von Neumann
In case you haven't heard the news, it turns out that life is not just about solving puzzles, playing games, and having fun. It's too bad, I know. But nevertheless it's true--sometimes we simply need to get stuff done. And it's in those situations that algebra really shines. Why? Well, think of algebra as the highway and road system of the math world. Just as it's tough to drive your car anywhere without roads, it's also tough to solve most math problems without some algebra. But, you might be wondering, I've been solving math problems since I was a little kid, and I don't ever remember using algebra before! What gives? Well, I've got some news for you ... and it may come as a surprise: without even knowing it, you have slowly but surely become something of an algebra expert.
In this chapter you'll discover all the algebra you've already been doing, which should help you realize why algebra is so useful and important (yes, really). I'll also introduce you to some math concepts that are crucial to understanding algebra: things like variables, inequalities, square roots, graphs, and more.
THE SECRET ALGEBRA YOU'VE ALREADY BEEN DOING
To give you a taste of what I'm talking about, let's take a look at a specific example from your everyday life. The first thing I want you to do is put down this book and go get your wallet. Really, go ahead and get it--I don't mind the wait. You back? Great. Now, all you need to do is simply count how much money you have. So go ahead, dive in there and count up that cash.
Okay, do you have your number? Once you do, feel free to safely stow your wallet--we won't be needing it anymore. Just be sure to keep that number in your head.
Why do we need to know that? Well, here's the scenario we're dealing with: You and I are planning to have some friends over tonight for dinner. Our plan is to barbecue a pizza, but we've been so busy lately that we completely forgot to shop for party food. Oops! We've only got an hour to go before our guests arrive, and the refrigerator is a total blank canvas. In other words, it's a cold and cavernous empty space ... which means that we need to buy everything: pizza dough, tomato sauce, cheese, pepperoni, and even charcoal. Which brings us to the point of me having you check what's in your wallet: I'm counting on you to pick up the groceries. So, do you have enough money to buy them?
Presuming you're older than, let's say, ten and that you've done a little shopping in your time, I imagine you could figure this out without any problem. In fact, I'd wager that you stumble across little quandaries like this almost every day, and yourarely--if ever--think twice about them. But what are you actually doing when you solve them? Well, hold on tight for a few minutes, because we're about to go through this in what some might call "excruciating" detail. But, rest assured, there's a payoff at the end--I promise. Okay, here we go. Whether or not you've realized it before, there are basically three steps for answering a question like this:
First, you need to draw upon your past experiences to estimate how much each item on your shopping list will cost. There's no need to be super precise here, we're just looking for a ballpark figure--say, to the nearest dollar. In the case of our party food, I'd guess the charcoal will cost about $5, the cheese and pepperoni will cost about $3 each, and the dough and tomato sauce will cost somewhere around $2 apiece.
Next, you need to add these individual amounts together to estimate the total cost of the shopping trip. In this case it's pretty simple:
$5 + $3 + $3 + $2 + $2 = $15.
Which means that you think your trip will cost around $15. Remember, we're just making a rough estimate of the cost here, so we could easily be off by a few bucks either way.
The final step to figuring out whether or not you need to go to the bank before you head to the grocery store is to compare your total estimated cost--which is $15--to the amount of money you have in your wallet (you do remember how much you have in your wallet, right?). So, if you found $10 in your wallet earlier, then since $10 is less than $15, you know that you'd better go to the bank or else you run the risk of coming up short at the grocery store. On the other hand, if you have $20 in your wallet right now, then since $20 is more than $15, you should have plenty of money for the groceries--even if we were a dollar or two off with our estimate. So, if you and I were actually planning a party, where would you be headed right now? To the grocery store or to the bank?
"Wow, thanks! All of that effort just to solve one little problem that I, uh, already knew how to solve anyway. Great job, Math Dude."
Ouch! The truth can hurt. But, nonetheless, this anonymous reader is correct: taken at face value the (over-)analysis of this problem was over the top and overly complicated. But, as I said earlier, there's a lot more to this problem than meets the eye, and the payoff is coming. Well, actually, it's here.
The truth is that the mathematical part of this problem had nothing to do with pizza or the contents of your wallet. Yes, in practice we were adding up numbers representing quantities of money and seeing if we had sufficient funds to buy the things on our grocery list. But mathematically we were doing muchmore--we were estimating unknown values, assigning them to variables, performing integer arithmetic, and solving greater than or less than inequalities. In other words, we were doing the stuff of real math. In fact, we were doing the stuff of real algebra!
Does that surprise you? Perhaps it does, but hopefully it's a good surprise since you now see that math really does apply to many things you do every single day of your life. And, perhaps more important, hopefully you now see that you already know some algebra. Certainly not all of it, but some of it--and that's a great start.
INEQUALITIES: <, >, =, AND =
I mentioned that one piece of the algebra you were unknowingly using before was the idea of an inequality. I bet you've had a run-in or two with less than and greater than symbols at some point in your life, but a little refresher on the topic won't hurt anybody. Inequality symbols are so named because they, unlike an equals sign, tell us when two things arenotequal. For example, here's how the less than ("<") and greater than (">") symbols work:
And while we're at it, we may as well talk about two additional inequality symbols that we'll run into in the next chapter. The first is called the less than or equal to sign ("="), and it works like this:
The second is called the greater than or equal to sign ("="), and it works like this:
These four symbols give us a quick and convenient way to compare the sizes of two numbers.
WHICH IS BIGGER? SMALLER? EQUAL?
Let's make sure you've got this whole inequality business down pat. Your job is to look at the two things being compared, and then to fill in the symbol that makes the statement true. Each of these five symbols should each be used once: >, <, =, =, and =. If you don't know the answer to a question, the Internet is your friend--look it up!
1.The size of the moon is ____ the size of a mountain
2.The numbers 5, 8, 13, and 21 are ____ the number 5.
3.100 ____ 102
4.The size of the planet Saturn is ____ the size of the planet Jupiter.
5.The numbers 5, 8, 13, and 21 are ____ the number 21.
But how exactly was everything we were doing earlier to solve our grocery shopping dilemma algebra? I mean, just saying that it was doesn't actually make it so--we need some concrete details! Fair enough. Let's go over the problem again, but this time let's look at it a bit more algebraically. First, let's think about what to call the amount of money you found in your wallet. Ofcourse we could call it $10, or whatever amount you found. But what if you want to write out a slightly more general "recipe" (no, not the kind for food ... the kind that tells you how to do something) that you can use over and over again each week to figure out if you'll be able to pay for the things on that week's shopping list. Of course, you don't know how much money you're going to have in your wallet next week, or the week after that, but you'd still like to be able to write out your process for figuring out whether or not you'll need to go to the bank that week.
The bottom line is that it's often just plain useful to represent a number without actually knowing what the value of that number is ... at least not yet. So, just to get a taste of how this works, let's represent the amount of money you found in your wallet using the letterm. That's right, we're going to use a symbol to represent a number. In algebra (and all of math), we call symbols likem, variables (we'll have a lot more to say about variables in the next chapter). As we've said, we don't actually care whatmstands for yet--it could be ten cents, ten dollars, or ten thousand dollars. It doesn't matter. We just need to know that there issomeamount of money in your wallet. And once we know that, we call itmso that we can talk about it and, as you'll see, use it to solve problems.
Okay, let's go a step further now and figure out a way to describe the process of totaling up the costs of all the groceries you need to buy. But, again, let's not bother with being specific about which items you need to buy right now and how much they're all going to cost--maybe we don't actually know yet! Instead, let's represent the costs of the various items using the symbolc. And let's also add a little number to eachcto show which item we're talking about on our list. In other words,c1 represents the cost of the first item on the list (maybe it's pizza dough),c2 represents the cost of the second item on the list (perhaps cheese), and so on.
Now, here comes the good part. We can use all these variables that we've defined to write our problem algebraically. For example, you can answer the question "Do you need to go to the bank before the grocery store?" using this
How does that work? Well, in words, this says that if you add up all of the costs on your list (those are all thecvalues), then if that number is greater than the amount of money in your wallet (which we've calledm), the answer is "Yes, you need to go to the bank."
ALGEBRA MAKES THINGS EASIER!
But what's the point of all this complicated notation? The point is that it doesn't matter how many costs you have, or what the actual amount of those costs are, or even how much money you have in your wallet, because the algebra we've come up with can deal with it. Not only can it help you avert a foodless party catastrophe, but it can just as easily be extended to figure out if you can afford something like your normal weekly shopping trip or an upcoming vacation--all you have to do is add morecvariables for the additional costs.
It's okay if this way of thinking seems a little abstract right now ... it is! That's exactly the point and the power of it all. But even though algebra may seem confusing at times, the whole point of it is that it helps us solve problems and makes life easier. I know that might be hard to believe when you're struggling with your homework--it certainly isn't makingyourlife easier--but rest assured that it'll start making more and more sense as we see more and more examples of algebra in action throughout this book.
Finally, I know that some of this stuff might look a little intimidating. But don't let it scare you off. As we just saw, all of this algebra is really just another way of writing down and dealing with something that's completely familiar to you. In other words, with algebra, things are really no more complicated now than they were before; all that's changed is that we have some new fancy ways to write things. So, don't worry if this all looks like a bunch of gobbledygook at this point ... we'll be covering all of these ideas again in a lot more detail.
CAN YOU USE ALGEBRA TO MAKE A BUDGET?
You need to buy certain things every month, but you've only got so much to spend. Can you use algebra to help you figure out whether or not you're going to be able to afford what you need? Start by creating variables to represent your monthly expenses. Then create another variable to represent how much money you make each month--either from a job or from your parents. Finally, write down an inequality like the one we wrote to determine if we needed to go to the bank ... but this time, make the inequality tell you whether or not you can afford what you need to buy! If you need help, check out the solution in Math Dude's Solutions.
Bonus question:Once you answer this quiz question, go back and think about why we used ">" and not "=" in our earlier grocery shopping problem.
WHAT'S THE POINT OF ALGEBRA?
Okay, we've now been introduced to two decidedly distinct ways to think about math. The first was to picture math as a playful and fun-loving puzzle ... a game for your brain, as it were. And the second was to think of math as a useful, practical, and down-to-earth tool. So which is right? Of course, they both are. There are a lot of things in the world that are both usefulandfun: cars and computers, for example. All of which might leave you wondering if these two views ever run into each other? Or do they just quietly live out their separate lives?
Well, to find the answer to this question, let's jump in our algebraic time machine (you'd definitely need algebra to build one of those!) and travel back in time about 4,500 years--back to the time when the Great Pyramid of Giza was being built in Egypt. Imagine, if you will (since no, this isn't a true story), that at that time there was a young dude with a strange hobby: he loved to tie knots in rope. Not surprisingly, everybody else thought this guy (we'll call him "Knot Dude") was borderline bizarre, but he really didn't care because he was absolutely fascinated by his knots.
Now, it turns out that Knot Dude's dad (we'll call him "Papa Knot") was a rather prominent member of Egyptian society. In fact, Papa Knot was the person in charge of figuring out how to build the foundation for the Great Pyramid. Not a job to be taken lightly since this is abigpyramid--about 750 feet long on each side, and 450 feet high! Not having much of a social life, Knot Dude spent a lot of time in the evenings listening to his father complain about the difficulty of his job. In particular, one problem was proving quite vexing: how to line up the four walls of the foundation of the pyramid so they all would meet to form a giant square. After all, without a perfectly square base, the pyramid wouldn't be a pyramid--it'd just be a mess.
What's the key to getting the foundation to come out right? Well, Papa Knot knew that if he could come up with a way to make sure that each of the four walls came together and formed perfect corners known as "right angles," then his foundation would also be perfect. But for the life of him, he couldn't figure out how to do it!
* RIGHT ANGLE
Extra details about the origin and meaning of some math terms--aka "jargon"--will be given in "Algebra Decoder!" sections like this.
Why is it that we call this type of "square" intersection between two lines aright angle? Well, the quick and dirty explanation isfound by looking at the similarity between the phrase "right angle" and the word "rectangle" (think rect-angle). They almost look and sound the same, right? And this makes perfect sense because all four corners of a rectangle form right angles! On a related note, the symbol used to indicate that an angle is a right angle looks kind of like an "L" (which you can see on the pyramid foundation drawing). Again, this symbol makes sense since it looks like one corner of a rectangle.
HOW MANY RIGHT ANGLES CAN YOU FIND?
Write the number of right angles in each shape on the line. Only count right angles on theinsideof each shape. In other words, we're not talking about the total number of corners in each shape! This distinction really only matters for one of these shapes--which is it?
When Knot Dude heard his father describe his problem with building the pyramid foundation, he immediately got excited because he realized that he had made a discovery several months earlier during one of his knot-tying frenzies that solved the problem. All Papa Knot needed to do was get a single piece of rope and tie thirteen evenly spaced knots in it--one at each end, andthen eleven more spaced evenly between the two ends. And that's it ... that's all he needed!
Well, actually, Papa Knot would need to do a little more than just tie the knots--he'd need to use them in a rather clever way too. The trick was to lay out the rope in a straight line, and then fold one end up after three even segments, and the other end up after five even segments. The result was a triangle. But it wasn't just any old triangle, it was averyspecial triangle. In fact, Knot Dude found that as long as the rope was pulled nice and taut, two sides of this triangle will always come together to form a perfect right angle.
Ding-ding-ding! That's precisely what Papa Knot needed--a right angle that he could use to set the four corners of the pyramid's foundation and ensure that it formed a perfect square. Of course, after Knot Dude told his father, great celebration ensued and a merry time was had by all since the great dilemma had been resolved ... with math, I might add.
Remember when we found all those perfect squares--numbers formed by squaring whole numbers like 1, 4, 9, 16, and so on? Those were good times, right? Yeah, well, the good times are back because I've got a question that's related to perfect squares: If I give you a perfect square, can you figure out what number was squared to get that perfect square? For example, let's think about the perfect square 49. The fact that 49 is a perfect square means that there must be some unknown number that when multiplied by itself gives 49. So, how can you figure out what that number is?
As you may know, we can find that number by taking what's called the square root of the perfect square 49 ... which we usually write. And, since we already found that 72 = 49 when we looked at perfect squares, we can immediately figure out that= 7 . Okay, that makes sense, but isn't it also true that (-7)2= 49 So shouldn't I really have said thatis equal to either 7or-7. Aren't both of these correct solutions? Well, not exactly. Here's what I mean: In algebra, you'll sometimes see problems that look like
And, as we'll look at in detail in the next chapter, the solution to this problem is eitherx== 7 orx= -= - 7 (you should try plugging both these values ofxinto the equation tocheck that they both work). But saying that this equation has two possible numbers that make it true isn't the same thing as saying thatitself is equal to either 7 or -7. The bottom line is thatis just a number ... and it's a positive number!
So, that means that= 7. That was pretty easy to figure out, right? Perhaps you're thinking that square roots aren't so tough to calculate after all. Well, yes ... it's not too difficult to find the square root of a perfect square. But don't get too excited because life isn't always so simple. In particular, it's not nearly as easy to find the square root of non-perfect squares. For example, what's the square root of 60? That's a lot harder to figure out because there is no whole number that you can multiply by itself to get 60. So what can we do?
HOW TO CALCULATE SQUARE ROOTS
Exactly how you go about calculating the square root of a non-perfect square number depends upon how accurately you need to know the answer. If you just need to know roughly what something likeis equal to, then you can come up with a very rough estimate simply by figuring out which two perfect squares 60 falls between. In other words, since 72 = 49 and 82 = 6 4, we know thatmust be between 7 and 8. And we could also guess that the real answer is a little closer to 8 than 7, since 60 is closer to 64 than 49. Make sense?
Of course, this works for much larger numbers too. For example, what's the approximate square root of 10,400? Well, we know (or at least we can quickly figure out) that 1002 = 10,000--which is already pretty close to 10,400. But it's not quite enough. So what's 1012? If you do the multiplication, you'll find that it's 10,201--closer yet, but still not close enough. Okay, how about 1022? Well, that equals 10,404--a little bit too much! But nonetheless, 1022 is very close to 10,400 ... so we can conclude thatis equal to a tiny bit less than 102.
But what if an estimate just isn't good enough? What if weneed to know the answer with a high degree of accuracy? In that case, your best option is to use a calculator or computer. Sure, there are methods that you can use to help you calculate better and better approximations by hand. But, in all honesty, those methods were developed hundreds of years ago ... when there weren't machines that could do the job. But now there are ... and those machines are much faster and more accurate than you are. So doesn't it make sense to use them? After all, we've got better things to do with our brains!
The Peril of Square Roots
There are some things in life ... and math ... that you really want to avoid. And those are exactly the kinds of things you'll find in Watch Out! sections.
Before you run off and start calculating square roots, I should warn you that not every number has one! Which numbers are out of the club? Well, let's think about it. When you square a positive number, the answer is always positive. And when you square a negative number, the answer is also always positive. Which means that the square ofanynumber is always positive. And, if you think about it, you'll see that this means that a negative number does not have a square root--since there's no number that you can square to equal it (at least none that you know of at this point). So, consider yourself warned--you can't take the square root of a negative number!
CALCULATING SQUARE ROOTS
Now, it's time for a test of your root-taking talent. For the following problems, I want you to solve one-quarter of them exactly, another one-quarter of them by estimating the square root, another one-quarter of them using a calculator, and, finally, don't solve one-quarter of them at all! You can choose which method you want to use for each problem. But choose wisely ...
Hold on a minute! What do those squiggly equals signs mean? Those little guys indicate that the thing on the left isapproximately... but notprecisely... equal to the thing on the right. In other words, p˜3.14 says that the number p (pi!) is approximately equal to 3.14, but not exactly. Does that help you figure out which one-quarter of the problems you should answer exactly?
THE PYTHAGOREAN THEOREM
Almost everybody has heard of the Pythagorean theorem. Heck, even the brainless Scarecrow fromThe Wizard of Ozknew about the Pythagorean theorem! Remember? When he receives his diploma from the Wizard, he declares: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy ... rapture! I've got a brain!" Well, the only problem is ... the Scarecrow got it wrong! So, what does the Pythagorean theorem actually say?
In words, the Pythagorean theorem says that the lengths of the two legs ofanyright triangle have a very special relationship to the length of the long side of the triangle (which is called its hypotenuse). Specifically, if you square the lengthsof the two legs and then add the resulting numbers together, that number will always equal the square of the length of the hypotenuse. Okay, words are nice, but this is an algebra book, so let's express the Pythagorean theorem algebraically. Here it is:
What do those symbolsa, b,andcmean? Well,aandbrepresent the length of each leg of the triangle, andcrepresents the length of the hypotenuse. Think about it for a minute, and you'll see that this formula "says" the exact same thing as that other more wordy description--but it's a lot more concise.
The beauty of the algebraic form of the Pythagorean theorem is that I can give you the lengths of the legs of a right triangle (aandb), and you can tell me how long the hypotenuse must be (in other words,c). Just to make sure I'm not crazy, let's check and see if this works for Knot Dude's rope. Remember, Knot Dude's rope triangle hada= 3 andb= 4. That gives usa2 = 9 andb2 = 16, which means that
Now, how aboutc2? Well, in Knot Dude's trianglec= 5, which means thatc2 = 25. And, of course, that's exactly the same answer that we got fora2 +b2. So, who would've guessed it: the Pythagorean theorem actually works! (For the record, I might have guessed it.)
* WHAT IS A "PYTHAGOREAN"?
Just in case you're wondering where the word "Pythagorean" in "Pythagorean theorem" comes from ... well, here's your answer: Pythagoras was a Greek dude who lived areallylong time ago ... about 2,500 years ago, in fact! Given this length of time, it shouldn't be too surprising that many of the details of Pythagoras' life are a little fuzzy. But we do know that he was a philosopher and a mathematician, and that he started a kind of religious movement known as "Pythagoreanism," where, among other things, everybody was a vegetarian and lived in awe of the awesomeness of math. So, now you know!
THE WONDERFUL WORLD OF "PYTHAGOREAN TRIPLES"
If you're an overachiever, you're going to love these sections. While you don't have to know all this stuff, you'll definitely be glad that you do!
Clearly, Knot Dude's 3 - 4 - 5 triangle is special. But how special is it? I mean, are there actually any other combinations of three whole numbers that can satisfya2 +b2 =c2? Maybe there just aren't any others. Well, actually, there are others--plenty of them. In fact, there are an infinite number of them! And I can say this without even having to plug in a single number to check. How can I be so sure? Take a look at this:
Do you see why now? The only difference between the left and right triangles is that I've scaled the one on the right up to be twice as big as the one on the left. In other words, the legs that were 3 and 4 units long on the left are doubled to be 6 and 8 units long on the right, and so on. And that means that all the sides of the bigger triangle are still whole numbers. So, therefore, Knot Dude's 3 - 4 - 5 triangle isnotthe only one of these special right triangles with whole number sides.
But does the bigger triangle on the right still work with the Pythagorean theorem? It had better, but let's try it out just to be sure. For this triangle,a= 6,b= 8, andc= 10. So,a2 +b2 = 62 + 82 = 36 + 64 = 100. Andc2 = 102 = 100. Thank goodness,a2 +b2 =c2. It works! If you think about it, you'll see that instead of doubling the sides of the triangle, we could have multiplied them all by 3, 4, or any other whole number too. Each of these scaled-up triangles would have satisfied the Pythagorean theorem, and they all would have had whole number sides.
By the way, all of these special right triangles with whole number sides we've been talking about are known as "Pythagorean triples." So the three numbers (3, 4, 5) describe a Pythagorean triple, as do (6, 8, 10), (9, 12, 15), and so on. But, in truth, these scaled-up Pythagorean triples we've talked about so far are all kind of trivial cases. And by that I meanthat they're not really very interesting since once you know about one of them, you know about all of them--in other words, "Big whoop!"
So, are there other Pythagorean triples besides (3, 4, 5) that aren't "trivial"? Yep, it turns out that there are. And, again, it turns out that there are an infinite number of them. One way to find them is just to start plugging in numbers. The first triple of numbers you'll find after (3, 4, 5) is (5, 12, 13) since 52 + 122 = 132. We can check this out to make sure it really does satisfy the Pythagorean theorem: 52 + 122 = 25 + 144 = 169, and 132 = 169. So it works! But looking for Pythagorean triples by brute force like this is pretty tedious work. And, as it turns out, there is an easier way. But that's a topic for another day.
SECRET AGENT (in Training) MATH-LIB
WHAT IS PYTHAGORAS' MESSAGE?
Good evening, agent in training __________(name of fruit or vegetable). Your mission today, if you choose to accept it, is to figure out the __________(silly adjective)secret message that Pythagoras left embedded in his name for new math secret agents to __________(unnecessary adverb)discover. If you succeed, you will graduate from secret agent math academy and become a full-blown secret agent. __________(overly enthusiastic exclamation)!
So, what's in store for your final training session? Well, you might not be surprised to learn that it has to do with one of dear Pythagoras' most __________(judgmental adjective)discoveries: the Pythagorean theorem.
Here's how it's going to work. First, I need you to write down the whole numbers that correspond to the positions in the alphabet of each letter of Pythagoras' name. For example, "A" corresponds to the number 1, "B" to 2, and so on. So, since the first letter of Pythagoras' name is "P," the first number in your list is 16. Write down each number above the corresponding letter here:
Now, here comes your real secret agent math test. Remember the Pythagorean theorem? (How could you forget!) It's a little algebraic relationship that says:a2 +b2 =c2. In the problems below, I give the lengths of the two legs (aandb) and the hypotenuse (c) of six right triangles that satisfy the Pythagorean theorem. Well, almost. I don't actually give you the lengths of all three sides of each triangle--I only tell you the lengths of two of the three sides. Your job is to figure out the length of the third side, and then to match that to the corresponding letter from the name "Pythagoras" above.
For example, in the first problem that follows, the valuesa= 12 andc= 20 are given. Your task is, therefore, to find the value ofb. If you do the calculation, or just notice that this must be a scaled-up version of the 3 - 4 - 5 triangle (all the sides are multiplied by 4), you'll find thatb= 16. What letter does that correspond to in the name "Pythagoras" above? The very first letter: "P". So you then write that in the space to the right of the problem below. Got it? Good. Remember to look for problems that are just scaled-up versions of the 3 - 4 - 5 triangle, and to look for problems that are repeats of ones you've already solved but that ask you to find a different one of the three sides. (Hint: number 4 below looks awfully similar to number 1, doesn't it?) Now have at it!
If you've got those six problems solved and six letters from the name "Pythagoras" written on the right, then you have my heartiest congratulations! Now, I've got three more problems for you--but these ones are a lot easier. Just take the numbers that are the solutions to the six problems above, and use them to do the following arithmetic problems. For example, "(5)" and "(6)" indicate that you should substitute in the numerical solutions for problems 5 and 6 above.
Excellent--you've done it! All that's left to do is decipher the secret message with advice for aspiring math secret agents that Pythagoras left hidden in his name thousands of years ago. What's the secret to finding that? Just fill in the letters in the following message that correspond to the solutions of the indicated problem above.
Pythagoras' secret message for math secret agents is ...
Okay, not quite what you were expecting, right? Well, nobody ever said it was going to begoodadvice--perhaps Pythagoras was getting a little eccentric in his later years. Also, just in case you'rewondering what I mean that Pythagoras left this message for you "in his name," the letters in the words "SPY," "OATH," and "ARG(H)" (that last "H" was a bit of a cheat) can be rearranged to spell "Pythagoras." That's right--it's what's known as an anagram!
Congratulations! As of this moment, you have officially graduated your training program and are now a full math secret agent. Good luck in the field, agent __________(name of fruit or vegetable).
WHY ALGEBRA MATTERS IN THE REAL WORLD
So what's the point of our tale of Knot Dude and Papa Knot? Well, it's certainly not to suggest that this is how things actually went down 4,500 years ago. No, the real point is that something happened in this story that also happens all the time in the real world--and it's a very important something. What is it? It's that some creative person spent time (perhaps too much) thinking about seemingly abstract and crazy things like: How many ways are there to tie knots in a piece of rope? And what interesting arrangements can I make from them? In other words, some creative person spent time thinking about pure abstract math. And, as we saw, every now and then these pure abstract thoughts make their way into the real world.
Typically what happens is that somebody realizes they have a problem that they don't know how to solve. Perhaps, like in our story, they need to figure out how to build a great pyramid that has perfectly square corners. But our story was not typical (in more ways than one) since Papa Knot was lucky enough to have a budding mathematician with an answer to his problem living right under his own roof. Instead, what usually happens is that a physicist, biologist, economist, or some other persontrying to understand something seeks the help of a mathematician. They ask the mathematician if they've ever seen anything like their problem before, and the answer is usually one of two things:
•"No, let me think about it."
•"Yeah, actually--I played around with something a while ago that behaves exactly like that. Have a seat, let me show you how it works ..."
But this process doesn't just help out mathematicians, engineers, and scientists. Believe it or not, it has even helped you out! Think that sounds crazy? Okay, well it turns out that people didn't invent the mathematical concept of a greater than or less than inequality just to help you figure out if you have enough money to buy the ingredients for a pizza. Nope! What actually happened was that a long time ago people started pondering inequalities and the relationships between numbers simply because they thought it made for a pretty good time. And, lucky for you, all these millennia later, those ideas turned out to be useful!
So there you have it--that's the story of how the abstract world of math that mathematicians spend lots of time in, and which we dabbled with in the prologue, runs smack into the real world. And this, math fans, is also precisely the point at which math moves from the clouds down to the gritty streets and into the realm of "reality." In other words, this is the point at which abstract math is transformed into the math that you actually apply and use in your daily life. Speaking of which, here's a real-world situation for you to think about ...
HELP, MY DECK IS CROOKED!
Imagine you've just paid someone to build a new deck in your backyard. The design was simple--it was just supposed to be a 15-by-15-foot square. But, unfortunately, when all was said and done, your simple square deck turned out looking like this:
And that's most definitelynota square! So what went wrong? Well, clearly the folks building your deck hadn't heard the story of Knot Dude, Papa Knot, and the Great Pyramid of Giza. Otherwise, they would've used a piece of rope with knots in it to help build a deck with square corners. After seeing the deck, you're so angry that you decide to tear it out and rebuild it yourself--properly this time! But while you're fumbling around trying to make a triangle like Knot Dude's, you come to the realization that rope is pretty floppy and isn't a very practical tool. So you stop and think for a while ... and you come up with something ingenious: a way to use your rope to build a square deck that doesn't require tying a single knot!
HOW TO MAKE A SQUARE - METHOD ONE
So, can you figure out what the "trick" is? If the answer isn't jumping out at you (and it very well might not be), try backing up and rethinking the situation from the beginning. You've got a bunch of rope and four long boards that have been nailed together into a squished square, and your task is to unsquish the square until you have anactual square. Take a few minutes to think about how you could do it (it may help to try sketching out your solution on paper).
Don't worry if you're not getting it. Believe it or not, you're not really supposed to figure these things out right away. That's right--they're not supposed to be easy! Taking a few minutes to think about the puzzle, and then walking through the solution with me is the key to getting your brain thinking mathematically--and that's our real goal! Eventually the answers will just start to jump out at you.
So, got any ideas? If not, here's a hint: start by thinking about what makes a square a square. So, what is it? Well, first and foremost:
1. The lengths of all four sides have to be equal.
It's absolutely impossible to make a square otherwise! And this means that you'd first better be certain that the lengths of each of the four boards that make up the frame of your deck are exactly the same length. That means you should start by using your rope to measure the length of one side of the deck, and then checking that each of the other three sides match this length.
But will that alone ensure that you've got a nice square deck? Well, take another look at the squished square shape of your crooked deck. It's not a square, but there's no reason that all four sides can't be the same length, right? So the answer must be "no"--four equal sides doesn't guarantee anything. Which brings us to the second critical thing that makes a square a square:
2. All four corners must form right angles--just like the corners of a pyramid.
Is that it? Do those two things guarantee a square? Yes, that really is it. Go ahead and think about it for a minute and you'll see that if both of these things are true, then you're guaranteed to get a square. Are you now thinking, "Okay, that's great. But how do we actually use that second requirement to make our deck square? I mean, it's nice to know what makes a square a square, but how exactly do I build one?"
Oh right ... that's a pretty important detail. Well, given all that we've discovered about squares, one option would be to create one of Knot Dude's triangles with a right angle and use it to check that each of the four corners of the deck are square. But you say, "That's exactly what we didn't want to have to do in the first place because Knot Dude's triangle rope is a major pain to work with!" So what can we do?
HOW TO MAKE A SQUARE - METHOD TWO
Okay, it's time to reveal the key idea that will help us move forward and stop talking in circles. While having equal length sides and right angles are all that is required to guarantee that a square is really a square, these properties are not unique. "Whoa!" you might be saying, "What does that mean?" Well, it means that there are other sets of properties that will also guarantee that our square is a square. Yep, sorry to break it to you, but the world is a complex place ... even the seemingly simple world of squares. Which means that rethinking how you define what makes a square a square might open up a new way for you to fix your deck. In particular, think for a minute about this pair of properties and try to figure out whether or not they guarantee a square:
1.The lengths of all four sides must be equal.
2.The lengths of the two diagonals stretching from opposite corners of the square must also be equal.
What do you think? Do these two things do the trick? The answer is ... yes!
Actually, the fact that the two diagonals have equal lengths can be used to show that each of the four corners also must form right angles, and vice versa ... just one of the many things you can learn from geometry. Which means that the two sets of properties we came up with are actually different ways of saying the exact same thing. If you're having a little trouble seeing why the two diagonals have to be the same length in a square, take a look at the differences in the diagonals of these two increasingly "squished" squares.
So that's the answer to our puzzle. All you have to do to fix your deck is use your rope to measure the length of one of the diagonals from corner to corner, and then check to see if the length of the otherone is the same. If it is, you're done! If not, you need to make an adjustment to further unsquish the square, and then remeasure the diagonals to check your progress. In the end, when all is said and done, the two diagonals of your perfect deck must be exactly the same length. And there you have it--yet another example of the very real way that abstract mathematical ideas can help solve problems in the real world of your own backyard.
ALGEBRA IN YOUR BACKYARD
But wait, there's algebra in the deck puzzle we just solved too. By now you're probably getting the impression that algebra pops up everywhere in life ... and that's pretty much true! For example, let's take another look at your newly rebuilt and now perfectly square deck.
If you just look at half of your deck, what do you see? That's right ... it's a right triangle! The lengths of the legs of this triangle,aandb,are both 15 feet, and we can find the length of the diagonal stretching from one corner of your deck to the other using the good old Pythagorean theorem. How? Well, first we know that
a2 +b2 = 152 + 152 = 225 + 225 = 450.
And sincea2 +b2 =c2, this means that
Finally, we need to take the square root of 450 to get the value ofc. Which, after running the numbers through a calculator, gives us a number that's approximately equal to 21.21.
So what's the point of this whole discussion? Well, the point is that we can use a little bit of algebra to learn that a perfectly square deck with two equally sized 15-foot-long sidesmusthave a corner-to-corner length of about 21.21 feet. And that means that instead of using a rope to check that the two diagonals of your deck are the same length, we could instead
• Measure the width and length of the deck using a tape measure.
• Then use these lengths in the Pythagorean theorem to calculate the length of the hypotenuse stretching from corner-to-corner of the deck.
• Finally, use the tape measure to check and make sure that the length of the deck's hypotenuse is correct.
Either method will work and ensure that your deck ends up square. That's some pretty handy math, right?
HOW TO MAKE A GRAPH
Throughout the book, you'll find sections like this containing "algebra tutorials." Each of these sections gives an in-depth look at a particular type of algebra problem and includes a step-by-step guide showing you how to solve them. Some of the steps are left for you todo, but if you get stuck you can find solutions at the end of the book in Math Dude's Solutions.
While using the Pythagorean theorem to find the length of the diagonal line stretching from corner-to-corner of your 15-by-15 foot square deck is nice, we can do a lot more with what we've learned so far than that. For example, imagine that you build decks for a living. A lot of the decks you build are square, but most of them are not exactly 15-by-15 feet in size like the one we just looked at. Some of them are 21-by-21 feet, some of them are 12-by-12 feet, and so on. As a deck builder, it'd be useful if you had some way of quickly looking up the size of a deck and then immediately seeing what its diagonal length should be ... no matter how big or small it is. How can we do that? Well, the first thing we need to do is learn how to make a graph.
STEP 1: DRAW CARTESIAN COORDINATES
Let's start by talking about how to get set up so that we're ready to make a graph. Which means that we need to draw ourselves a pair of coordinate axes, like these:
What exactly are we looking at? Well, this is what's called a Cartesian coordinate system (named after the seventeenth-century mathematician René Descartes), and it's sitting on topof what's called the coordinate plane. Its main purpose is to give us a way to investigate the relationship between numbers (more on exactly what this means later). In this case, we want to look at the relationship between pairs of numbers, which means that we need two axes in our coordinate system. And that's exactly what we have: thex-axis (sometimes called the abscissa) runs horizontally and they-axis (sometimes called the ordinate) runs vertically. The little marks along each axis label thexandyvalues at those locations. Both axes have a value of 0 at the origin--the place where the two axes meet. So that's our playing field ... now let's start doing something on it.
STEP 2: CREATE LIST OF ORDERED PAIRS
In case you haven't guessed it by now, our plan for coming up with a way to quickly figure out the diagonal length of any sized deck is centered around the idea of making a graph. But before we make that graph, we need to do a little algebra with the Pythagorean theorem. Our goal in this problem is to figure out the length of the hypotenuse,c,of the triangle formed by half a deck:
Since we want to solve forc,let's start by switching around the left and right sides of the Pythagorean theorem and write it likec2 =a2 +b2. Then, let's take the square root of both sides to get
But notice that there isn't a side called"b"in our drawing of the deck. Instead, both sides are labeled"a"since this deck is shaped like a square--which means that both sides have the same length. So let's use the fact thata=bfor this deck to rewrite our equation like this
Don't worry if you don't understand every single step here yet--we'll go over solving this type of problem in much more detail in the next chapter. But for now the really important thing to notice is that the length of the hypotenuse of our deck is given byc= v2 *a
Which means that we can use this equation to figure out what the diagonal length ofanysquare deck will be ... we simply plug in values fora, and in return we get values forc. For example, if we do this fora= 1, we getc= v2 If we do it fora= 2, we getc= v2 * 2--which is more often written 2v2 (we can omit the "*" since it's clear we're talking about multiplying 2 by v2). And if we do this for a bunch of values ofa, we end up creating a list like this one:
I'll leave it up to you to fill in all the missing parts. But what is all that stuff on the right? That sure is an odd way to write a list of numbers! Actually, those are called ordered pairs ... and they're exactly what we need to make our plot.
STEP 3: PLOT POINTS ON THE PLANE
Now we're ready to plot some points. But why would we want to do that? Well, as I mentioned earlier, plots help us see the relationships between numbers. Each point on what's called thex-yplane (which is just another name for the Cartesian coordinate system we set up) is defined by a single pair of numbers. So by plotting several points, we can begin to investigate how these pairs of numbers are related to each other. How does this all work? Well, before we tackle our deck problem and the ordered pairs we just came up with, let's take a look at this graph:
The positions of each of the three points plotted on this graph are 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. All you have to do to describe a location on the plane is to give a location alongboth thexandyaxes. Ordered pairs are typically written inside parentheses with thexcoordinate given first. For example, the location of the origin is specified by the ordered pair (0, 0), the location of the point labeled "A" atx= 1 andy= 2 is specified by the ordered pair (1, 2), and the location of the point labeled "B" atx= 3 andy= 5 is ... well ... I'll leave it to you to finish up writing this ordered pair. And while you're at it, go ahead and fill in the ordered pair for point C too.
Okay, after doing that you should now understand exactly what those ordered pairs that we came up with earlier were all about! They simply describe the locations of points on a graph. But which graph? Well, this one:
What you see here is the start of what will eventually become the graph that helps us solve our deck problem. Notice that instead of anx-axis we have an "a-axis," and instead of ay-axis, we have a "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 containsaandcvalues, the ordered pairs we get from it are (a,c) ordered pairs. So far I've plotted the ordered pairs (0, 0) and (30, 30v2 ). I'll leave it up to you to plot the rest of the points from the table we made earlier. And while you're at it, go ahead and add the two points fora= 40 anda= 50 that aren't in the table. Feel free to use a calculator to help you figure out the approximate values of all those numbers multiplied by v2. Once you're done with all that, you'll be ready to move from plotting points to plotting curves.
STEP 4: CONNECT THE DOTS TO DRAW A CURVE
All finished with slapping those (a, c) ordered pairs on the plot? Okay, there should be at least six points on your graph showing the locations of the ordered pairs corresponding to deck sizes ofa= 0, 10, 20, 30, 40, and 50 feet (you might have some additional points too). While plotting these points, you should start to see a pretty clear trend developing that tells you about how these points are related to one another. In this case, you should be able to see that the points appear to be ascending in a straight line from the lower left to the upper right of the plot. So, let's go ahead and draw a line that connects the dots and shows the overall trend. We'll learn more about this in chapter 3, which will help you understand why the points in this problem end up in a straight line.
Okay, so what does this graph do for us? Remember way back in the beginning of this tutorial when we said that we wanted to come up with a way to quickly find the diagonal length of a square deck given only the length of its sides? Well, that's precisely what this graph does for us! For example, let's say you're building a 24-by-24-foot deck. You can easily find the diagonal length of this deck by starting ata= 24 feet on the "a-axis," then moving vertically up from this point until you hit the diagonal line that we drew, and finally moving horizontally to the left until you hit the "c-axis." The value ofcat that point is the diagonal size of thedeck. With just a quick glance at the graph we can see that this value must be a bit more than 30 feet (the actual number is 24 v2 ... which is close to 34 feet). So, in the future, when you need to make a graph, just remember to follow the four steps we used in this tutorial:
How to Make a Graph
1.Draw Cartesian coordinates.
2.Create a list of ordered pairs.
3.Plot ordered pairs on the x-y plane.
4.Connect the dots to draw a curve.
Before finishing, check to see if you can figure out the approximate diagonal lengths of square decks with 15- and 45-foot-long sides (bonus points if you can also calculate a more precise estimate using a calculator). Once you've got that all figured out, you're done with this tutorial! But don't worry, while this graphing tutorial may be over, rest assured that we're nowhere near done with making plots ... not by a long shot.
CONNECT THE DOTS
Speaking of doing more graphing, I think it's high time for you to make a graph of your very own ... from scratch. Here's what I want you to do: Make a graph showing the relationship between the perfect squares and the whole numbers you multiply together to make them. In other words, make a graph showing the ordered pairs of points (n,n2). Don't worry,nhere isn't anything crazy--it's just representing allthe various whole numbers. So the first ordered pair forn= 1 is (1, 1), the second ordered pair forn= 2 is (2, 4), and so on. After you come up with a list of ordered pairs (going up ton= 10 should do the trick), plot the points on the axes, and then draw a curve that goes through the points and shows the overall trend.
Bonus question:What does this curve you've drawn mean? In other words, all of the perfect squares are sitting nicely on the points you plotted, so what do the parts of the curve that are located between the points represent?
Okay, now that we have a better understanding of the big picture of what math really is and how it gets applied to the real world, and now that we've put together an impressive collection of mathematical tools for our journey, it's time for us to head out and start our trek into the real bread and butter of this book. And trust me, that's definitely a journey you want to take. Because when we're finished you'll be able to do things that you simply can't do right now. For example, here are three problems that, barring some painful brute-force expenditure of effort, you simply can't solve right now:
1.Pick a whole number greater than 2. Now multiply it by 2, and then subtract it from the square of your originalnumber. Then add 1 to this result. After that, take the square root of this number, and then add 1 to the result. What's the answer? After all that work, did you get back exactly the same number you started with? How can that be? Magic, right? No, it's algebra.
2.You are told that the product of two consecutive whole numbers is 1056. Can you figure out what those two numbers are? If you think about it, this seems like a nearly impossible problem to solve without simply trying to multiply a bunch of numbers together until you get the right pair. But how can we do it smarter and faster?
3.You drop a rock into a well and hear a splash two seconds later. Can you figure out how far it is from the top of the well to the surface of the water? There's a lot going on here. We have to worry about the physics of gravity, the motion of the rock, the speed of sound, and how this all comes together mathematically. Do you think we'll be able to do it?
By the end of this book, I assure you that we will be able to solve all of these problems ... and many more too. So, I'd like to formally challenge you to keep going until we solve them. Of course, before we can do that, there are a few things we need to learn about: variables, expressions, equations, exponents, polynomials, factoring, quadratics, and more. In other words, we need to learn algebra.
So, have a seat, let me show you how it works.
* ALGEBRA IN THE EVERYDAY WORLD
1. We've seen how algebra can show up in our shopping carts, our monthly budgets, and even in our backyards. What other ways does algebra enter into your daily life? Take a few minutes andthink about it. It doesn't matter how big or small your list is--what really matters is that you start to recognize the math that's lurking in your life ... and then embrace it!
2. Fill in the blanks with >, <, =, =, or = to make each statement true.
Some of these questions can have more than one correct answer. Can you figure out which those are?
* SQUARE ROOTS
3. What's the square root of a square? Um ... huh? Okay, I know this one sounds a little strange, but here's what I'm talking about:
In other words, the big square here is 4 little boxes wide and 4 little boxes tall, which means that it contains a total of 4 * 4 = 16 little boxes. So, what's the square root of a square?
* THE PYTHAGOREAN THEOREM
Building square decks isn't the only backyard problem that algebra--and in particular the Pythagorean theorem--can help you with.Imagine that you need to buy a ladder to climb onto the roof of your nine-foot-high house. How tall of a ladder should you buy? Should it be nine feet tall?
After seeing this picture, it's clear that a nine-foot-tall ladder won't do since the ladder is going to form the hypotenuse of a triangle. So, high tall should it be? Well, the answer will depend on the height of the roof,b, and how far you need to position the bottom of the ladder from the house to keep it from falling over,a. Given all that, can you figure out the answers to the following two questions?
4. If the height from the ground to the roof is nine feet and you decide that the bottom of the ladder should be positioned three feet from the house, how tall of a ladder do you need to buy to be able to get on the roof?
5. You have a twelve-foot-tall ladder in your garage, and you've determined that the bottom of the ladder should be positioned a distance one-third the height of the ladder away from the house. How tall of a roof can you climb on using this ladder?
* GRAPHING POINTS AND LINES
6. Plot the two relationshipsy1 =xandy2 = 2 *xon the same set of axes. In other words, first choose a bunch ofxvalues (thewhole numbers from 1 through 10 will do), plug thesexvalues into these two relationships to find ordered pairs for (x,y1) and (x,y2), plot these points (use different symbols for each relationship), and then draw lines connecting the dots. What effect does the 2 iny2 have on the result?
THE MATH DUDE'S QUICK AND DIRTY GUIDE TO ALGEBRA. Copyright © 2011 by Jason Marshall. All rights reserved. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.