Head First Algebra Wouldn’t it be dreamy if Algebra was useful in the real world? It’s probably just a fantasy... Tracey Pilone M.Ed. Dan Pilone Beijing • Cambridge • Kln • Sebastopol • Taipei • Tokyo Head First Algebra by Tracey Pilone M.Ed. and Dan Pilone Copyright © 2009 O’Reilly Media, Inc. All rights reserved. Printed in the United States of America. Published by O’Reilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA 95472. O’Reilly Media books may be purchased for educational, business, or sales promotional use. Online editions are also available for most titles (safari.oreilly.com). For more information, contact our corporate/institutional sales department: (800) 998-9938 or [email protected]. Series Creators: Kathy Sierra, Bert Bates Series Editor: Brett D. McLaughlin Editors: Brett D. McLaughlin, Louise Barr, Sanders Kleinfeld Design Editor: Louise Barr Cover Designers: Louise Barr, Steve Fehler Production Editor: Brittany Smith Indexer: Julie Hawks Page Viewers: Vinny and Nick Nick Pilone Printing History: December 2008: First Edition. Vinny Pilone The O’Reilly logo is a registered trademark of O’Reilly Media, Inc. The Head First series designations, Head First Algebra, and related trade dress are trademarks of O’Reilly Media, Inc. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and O’Reilly Media, Inc., was aware of a trademark claim, the designations have been printed in caps or initial caps. While every precaution has been taken in the preparation of this book, the publisher and the authors assume no responsibility for errors or omissions, or for damages resulting from the use of the information contained herein. No variables were harmed in the making of this book. TM This book uses RepKover™, a durable and flexible lay-flat binding. ISBN: 978-0-596-51486-0 [M] 1 what is algebra? Solving for unknowns... This is so sweet, but how can we possibly afford it? I don’t know... but it can’t be that much, right? The salesman said that twelve times its price was $22,400, and that didn’t seem too much at the time. Do you ever wish you knew more than know? Well, that’s what Algebra’s all about: making unknowns known. By the time you’re through this first chapter, you’ll already have a handle on X being a lot more than a mark where treasure’s buried. You’ll get a handle on equations, keeping both sides of an equation balanced, and why solving for unknowns really isn’t that big of a deal. What are you waiting for? Go on and get started! this is a new chapter 1 algebra is everywhere It all started with a big gaming sale Jo has been watching the game system battles for a while now and has finally decided on the one she wants. Her favorite system’s on sale this week, and she’s ready to buy. But can she afford it? That’s where she needs a little help from you. This is the system Jo’s been waiting for - it’s on sale... but will the system and a few accessories cost too much? S u p e r KillerX 2.0 Gaming System B u y ! The brand new KillerX 2.0 includes full circle entertainment value. special One game controller value included. (KILLX-112) I can handle $199 - but is $199 that really how much the system’s going to cost? special special value value Gaming Headset $39 $49 Big Bag o’ Headset with microphone, Games ideal for online gaming (HS-AL1-867) A variety of games for the KillerX system (HD-ISH-5309) crJoiogm’hset b ageloaennmg ew. asyitstinegm f toor the 2 Chapter 1 what is algebra? What does a system really cost? When you buy things—especially expensive electronic things—there are lots of pieces that add into the price besides just the number on a sales flyer: sales tax, an extended warranty, shipping and handling, etc. So what will a KillerX system really cost? KillerX 2.0 B u y ! There’s tax on the system... Gaming Syst The brand new Killer The base price of the system is $199. After that, 2.0 includes full circle we need to think about taxes, which are 5%. Let’s special entertainment value. figure out how much Jo’ll have to pay in taxes: value game controller includ 5m%u lstailpelys tbayx 0 m.0ea5n..s. .y..ofui’gllu rnee eidt oitu ta.g Raienm ienm ab emri ntuhties,. $199 (KILLX-112) sp $199 # 0.05 = $ If you’re rusty on your decimal math, just turn to wHee rpeu’ sll etdh ef roormig itnahle $a1d9.9 the appendix and brush up! ...and the extended warranty, too. Jo’s about to spend $199 on a game machine, and she wants to purchase an extended warranty plan for an additional $20. Let’s put that in the price, too. What price will Jo need to pay? The original price of the console... $199.00 ...the sales tax you figured out earlier... ...and the extended warranty. +$20.00 Find the Add all these up and you get total cost how much Jo really needs to spend to buy the console. you are here 4 3 solving for unknowns This is stupid. I thought we were supposed to be learning Algebra! Figuring out the total wasn’t just addition! It was solving for an unknown—and that’s Algebra. In this case the unknown was how much everything was going to cost. The original price of the console... $199.00 ...the sales tax we figured out earlier... $9.95 ...and the extended warranty. +$20.00 $228.95 Tosptohamihyse e rwt poair nsiw fctoaehrl ketm hoauatunttiko nJnwo.o wit Ynwho au-ts hkyganooteuiw n cg hot anthdsoe o ralheelal, vwtaeahn sedt o that was the unknown! Algebra is about solving for unknowns Algebra is about finding the missing information that you’re looking for by using the information you already have. The unknown could be the cost of a car loan, the quantity of soda you need, or how high you can throw a water balloon. If you don’t know it, it’s an unknown. All the other things that you’ll learn in Algebra are just ways to jiggle things around to help you find a piece of missing information. There are rules about when you can multiply things or when you can bump something from one side of an equals sign to another, but at the end of the day, they’re all just tricks to help you find that missing piece of information you’re looking for. 4 Chapter 1 what is algebra? Jo’s got more unknowns So Jo knows how much it will take to buy an awesome gaming system, including an extended warranty. But she still doesn’t have any games... or another controller... or a headset. Jo started with $315.27 in her bank account. Now that she’s paid for the console, how much can Jo spend on accessories? Let’s start by writing this out in words: Abaclcaonucnet – Ccoonsst oolfe = Macocneesys foorrie s WiYsno ruaumit bgdinreogern as’a tt a ptnwr eaoteybhd li testm oos t gowaeugtoter .srvtyea rrabtbaeollduyt. We know how much the console costs ($228.95), and we know how much Jo has in her account ($315.27). Now just fill in the blanks, and we can figure out Jo’s accessory budget: Wonit ahc mceosnseoyr ietso, Jspoe nisd a happy gamer... $315.27 - $228.95 = Here’s what’s in Fill in the Jo’s account. Here’s the cost of the unknown! console you figured out earlier. you are here 4 5 x means unknown Console Pricing Up Close Let’s take a closer look at what we just did. First, we’re going to swap out the unknown box for the standard Algebra version of an unknown—an x. $315.27x- = $m2on2e8y. 9fo5r a=cc$e8ss6o.3rie2s= x Ttjlhuheseettr t elete’strh t enu eosmr e modxsa t ig-n ic c Aot tmlhgoame tbo’rsn a . Here’s what’s in her account. Here’s the cost ...and here’s our of the console you unknown. figured out earlier. We can also turn th$ixs a8=ro6u.n$3d8 2to6 b=.e3:x2 Wtdtrheohee’ ligslnl aaegtt r aadefrlrlokeo r suim nsnn odootr wfhea e nwi asdcbh htoeoauthtptahhtt eewe rrrt h .thyt heT ryseiheco exkua s ir cimsyea o pnotuo n hrs cwteta aahnsnpea t m e, right or left. Understanding a problem and finding the unknown, x, is working with Algebra. Using tricks like writing the problem out with words and flipping things around are just ways to make finding that unknown possible. Cool! $86.32 is still plenty to spend on games and a headset. Solving for any unknown is Algebra. 6 Chapter 1 what is algebra? X marks the spot unknown Stands in for acsdrsolCeomosiotmgg us bhel tadt,y och oo kbiursn ee go nwh nheo w.oewyewdo i gumtmrhuo suc b,c hpyrhh uooo iluttlw r ha wem urinla’lsk n nyo wn. X Hwcahasan t’rt u yldeosoou c rDtowc hualoiealtendtths tn a bs’eintteard .ys ah. nayve o ttoh ebre lext; titer Cnoaufnm bnstuemra nboder r fsao. rw hjuoslte bonune ch Cina ltlehde teqheua vtaiornia.ble x is just a user-friendly stand-in for the unknown box we used earlier. x is easier to write and it’s what you’re looking for when you solve an equation. The unknown in any given situation is called a variable. In the real world, problems present themselves every day; translating them into mathematical equations allows you to solve them. Q: Will the unknown always be x? A: Nope. As you progress with math, you’ll see x, y, and z pretty often. You can use any letter that you want, though. Q: Back on page 6, how come you could just flip that equation? A: All we really did was switch the same equation around, called manipulating the equation. There are rules about exactly how you can work with equations without changing any values, and we’ll learn lots more about them in the rest of this book. you are here 4 7 equations are math sentences Equations are math sentences Equations, like the one you used earlier to figure out how much Jo could spend on accessories, are just math sentences. They’re a mathematical way of saying something. So when we talked about Jo’s account balance, we were actually using an equation: Abcaclaonucnet – Ccoonsts oolfe = Macocneesys foorri es The account minus how much ...equals how much balance... we spend on the we have left for console... accessories. Ttdhhifeesf esea rmmenaett th fh soinergnm.tTse hnoecfey s’tr hesa ety w o Our equation means “The account balance minus how much we spend on the same problem. console equals how much we have left for accessories.” So, that means that the account balance must equal the cost of the console plus the money for accessories. If we write that sentence as an equation it looks like this: Account Cost of Money for = + balance console accessories Tbahlea naccec..o.unt emquucahls wheo wsp end .w..ep luhsa vheo wle fmtu ch Equations can on the console... for accessories. be rearranged Both sentences mean the same thing; they’re just phrased differently. Over the next few pages, you’ll learn how to rearrange math like sentences. sentences and make sure that you don’t change any values. 8 Chapter 1