WEBVTT mathematics/math-analysis/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about word problems.
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This lesson is going to tackle solving those dreaded word problems.
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Hopefully, when this lesson is over, you are going to dread them a little less.
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We are going to start by talking about why we should care about them, and essentially why we should care about learning in general.
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Then, we are going to go over a general structure for approaching and solving word problems, and really any form of problem.
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We will see how it applies to what we are working on now.
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And then, once we understand that method, we will see a variety of different tips and strategies to help us get the most out of it.
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So, we will go over some specific tips that apply to that general strategy that we talked about.
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And then finally, we will work a bunch of examples where we will actually see directly how we are using this step-by-step strategy,
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so we can see how the method gets applied on real word problems.
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All right, let's go: first, what is a word problem?
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The term **word problem** gets used a lot; it is pretty much just a catch-all.
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It is for any problem that is primarily being described with words, as opposed to math symbols.
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That doesn't mean very much; you can use words to describe a lot of things.
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So, it means that there is no one way to approach word problems, because you could be talking about so many different kinds of problems.
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Since there are so many different possible problems, we have to be ready to adapt when we are working on a word problem.
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Trying to answer the question, trying to figure out the one way to solve word problems, would be like saying, "How do you play sports?"
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There are a lot of different sports; there are a lot of different ways to play within each of those sports.
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It depends on the specific situation; that is going to tell us what we have to do.
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The best method depends on our specific situation, so there is no one way to do all word problems.
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But we do know this one fact: pretty much all word problems are going to require us to think.
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We have to pay attention and be creative, because we are being asked to do more than just follow a formula.
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As opposed to a lot of problems that we will get in math, where it is just "here is the method that we do it;
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let's apply that method one hundred times in a row."
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It is going to be "understand this idea, and then do some critical thinking about it."
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So, we have to actually be thinking and paying attention when we are working on word problems.
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We should always be thinking, and we should always be paying attention, when we are doing everything in life.
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But for word problems, it is going to be especially important that we are really thinking about what we are doing, because we are trying to understand it.
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Otherwise, it is just not going to make sense.
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Why are they so hard--why are people constantly thinking that word problems are the hardest kind out there?
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Well, first, there is no simple formula for them: there is no one way that you do it--just plug things in it-- because each word problem can be different.
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So, when we are working non-word problems, we normally have examples that we can follow step-by-step.
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There is some formulaic method that we can just rely on.
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But with a word problem, there is no such formulaic method.
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It is up to us to be paying attention, to be ready to be creative and thoughtful.
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The fact that it actually requires thought--it requires some creativity--that is one of the reasons why word problems are generally harder than non-word problems.
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Second, it is a lot harder to teach word problems.
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It is easy to teach simple, repeatable instructions--perhaps not easy all the time, but it is easier to teach simple, repeatable instructions,
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things like formulas or step-by-step guides--anything like that where the idea is "do what I did; do what I did; do what I did."
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That sort of thing is pretty easy, because you can just follow the "monkey see, monkey do" saying--you do what they led, and it works out.
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But with word problems, you have to teach creativity; you have to teach an ability to understand what is going on.
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You have to really teach analyzing a whole bunch of things at once, and understanding things.
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As opposed to just teaching to follow a method, you have to teach how to understand and how to think.
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That is a much bigger task than just teaching a few quick instructions.
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So often, sadly, they are overlooked, because it is easier to teach that, and a lot of education is based
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on taking standardized tests, so we end up seeing us teaching to standardized tests, as opposed to teaching
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to a larger scale of thinking and understanding, which is the sort of thing
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that is absolutely necessary if we are going to do well on word problems.
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Don't let this make you be worried: don't despair--you can learn how to do them.
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Just because they are a little bit harder, and you don't get much learning about them (usually),
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doesn't mean you can't get great at them--you can get great at doing word problems.
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It is just going to take some though, some imagination and patience.
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Like everything else, it is going to be a skill that you can practice--you just need to practice the skill.
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In this case, if, for example, your teacher doesn't assign you many word problems,
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it might be a good idea to work an extra word problem out of your book.
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Make sure it is one of the problems...most math textbooks have answers to the odd exercises, or sometimes the even exercises, in the back.
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So, choose the one that you have the answer to, but that wasn't assigned to you,
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so that you get the chance to have some extra experience with word problems.
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So, if you know that you have difficulty with word problems, you have to focus on it; you have to work on it a little bit extra.
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I know, I am asking for you to do more; but there is pretty much no other way to get better at something.
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You get better at things because you practiced, not because you just want to be better at them.
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So, word problems are a skill you can practice, and a skill you can definitely improve.
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But you do have to practice it, if you want to improve.
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But if you practice, you will definitely get better; so consider that.
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If you have a lot of difficulty with word problems, just start tackling some easier word problems.
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Work your way up to medium ones; work your way up to hard ones.
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Just sort of do that in the background, as we are working on precalculus.
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It will be a really useful skill that will really help you in a lot of things later--not just math, even--all sorts of things.
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Why should we care about word problems?
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While word problems may be difficult, they are also incredibly important.
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In some way, they are the point of math: word problems turn math into something more than a meaningless series of exercises.
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As opposed to just solving one equation after another, where it is just meaningless symbols, word problems "ground" what we are doing.
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They give it context; they make the math mean something, which can make it, sometimes, beautiful, and usually important to what we are doing.
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Well, when we use math in science, we can better understand and appreciate our universe.
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It is through math that we can have physics be able to turn into equations that we can describe our world with.
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It is through math that we can figure out specific things in chemistry and talk about quantities.
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It is through math that we can do statistics in biology and sociology.
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It is through math that allows us to understand the world around us.
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When we use it in engineering, that allows us to build amazing feats of human ingenuity.
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It allows us to build huge bridges; it allows us to build giant dams, skyscrapers...
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It lets us build devices like phones that fit in the size in your hand...well, yes..."computers" is actually what I meant to say,
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because most smartphones these days are becoming even more powerful and more powerful.
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The incredible miniaturization of technology--what we have right now--the fact that I can be teaching you
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when I am in a totally different place than you, and you are watching it right now...
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Engineering: math has been put in all the engineering, and the computer things when we are building things--we need math
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for us to be able to make all these things: math runs all of the technology that we have.
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At heart, technology is based on what we have been able to figure out in math.
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In many ways, it is applied math through other things.
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And if we use math just on its own, we can find deep truths about the nature of logic and knowledge.
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Geometry: when you were studying geometry, it was all just mathematical ideas.
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They have applicability in the real world, but they are also just kind of beautiful, interesting ideas.
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I have studied a lot of math, and I think there are some really, really cool things that you can see in math, that are just purely ideas in math.
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But to describe them, we need words; just a bunch of symbols isn't going to do it.
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And without language to connect math to these really deep, interesting ideas, we just have symbols.
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So, we need language to be able to give us a deeper context, to ground what we are doing with our math.
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It is through word problems that we find value in math.
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Word problems are our connection between wanting to do things and learn things, and this interesting symbolic language that lets us solve things.
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Word problems are the point of connection between wanting to know things and being able to solve things.
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They are really, really important for that reason; in many ways, word problems are what gives us knowledge.
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Now, not everyone is going to be convinced by my impassioned appeal to the inherent value of beauty and knowledge.
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I think that wanting to learn, and the fact that learning is an amazing, really cool thing--that is a great reason to learn, in and of itself.
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But if you would like a more material reason--grades.
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You can get an A, and you can possibly even get a B, but you are never going to be able to achieve the highest marks
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in a math class if you don't understand word problems.
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To do your best in class, you need to be able to solve word problems.
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Every math is going to include, or at least really should include, some word problems.
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So, it is important that you know how to approach them.
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With time and practice, you can understand them better, and you can improve your grades.
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Remember what I was saying before: if you practice this, you will be able to improve at it.
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And it is not just math class: any standardized test (like if you are going to take the SAT,
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because you are interested in applying to college; or if you are currently in college,
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and one day you want to take the GRE so that you can apply to graduate school)--
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they are going to use word problems in there, as well.
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Pretty much any test that you will take anywhere will have some word problem ideas going in it.
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So, you want the ability to solve problems like that--being able to solve word problems is crucial in a lot of situations.
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Also, if you want to do anything like engineer and build things, or do science,
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or do anything that has really hard scientific connections, or is really analyzing
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and measuring the world around us, you are going to need to understand math.
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So, there are lots of really good reasons to understand math, in just a "material-benefit-to-us" way.
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Finally, there is always that gem of an excuse, "I am never going to use this in real life!"
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"I know that you might say it is useful in science, but when am I ever going to use partial fraction decomposition
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(which you will learn about later)--I am never going to use this in real life, later on, so why should I learn it now?
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Is that true--"I am never going to use this in real life"?--yes, honestly, you are probably right.
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There is a very good chance that the things you will learn in this course, or any math course,
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will never be the sort of thing that you get paid for doing later in life.
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You will not be having to analyze functions, and that is going to be what gets you your paycheck every day.
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That is not the point: that is not the point of learning these things.
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You are not in school to learn the things that you are going to use later in life.
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You are not in school just to learn those things that are going to get you a paycheck later in life.
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You are there in school to learn how to learn.
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Think about this: musicians do not play scales at concerts; boxers to fight punching bags in the ring; and surgeons do not operate on cadavers.
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But the practice that they get by doing each of those things is absolutely necessary for them being able to perform well later in life.
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So, you might not end up using this immediately in real life, or you might never use it in "real life" (whatever that means).
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But what you are getting now is practice: whatever you do later in life, you are going to need to learn new things over and over.
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So much of any interesting job, so much of whatever you do later in life, is going to be learning new things and getting good at those new skills.
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School is your chance to practice this process, this skill of being able to learn something, get good at it, and use it.
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This skill is what you are learning right now, and it is absolutely necessary if you want a job that pays well, is interesting, or is enjoyable.
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And if you want one that is well-paying, interesting, and enjoyable, you are definitely going to have to be able
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to learn lots of things quickly, in your situation, and do whatever you need to.
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You need the ability to adapt, the ability to learn many things and operate in many conditions.
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The way that you do that is: you practice it now.
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You do not get better by just wishing you were going to be better; you get better by practicing it.
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School, learning, is your chance to practice the learning that you will need for the rest of your life.
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If you do not practice it now, because you are not going to use those things, yes, you are technically right.
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But that is like saying, "I am not going to drive this car later in life; I am going to drive some other car; so I won't practice driving this first car."
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To be able to drive that second car, you are going to have to have learned how to drive a car somewhere.
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So, you can't skip learning how to drive the first car.
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Even though there are going to be some differences between the two cars (they might be completely different cars),
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there are a lot of parallels here--it is going to be very similar driving one car or the other.
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It may be hard to see right now, but please trust me on this.
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I am being as truthful as...this is just from the bottom of my heart: put effort into learning now.
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The process of learning is going to give you skills that you will use for the rest of your life.
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It can be difficult to see how valuable those skills are to you now.
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But trust me, in 5 or 10 years, when you look back later, you are going to be so thankful
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that you took the time and effort to really understand what you were doing--to go through
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all of that practice of learning--because those skills of being able to learn quickly
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and do well and understand things--they are going to make the rest of your life so much better and so much easier; it is really important.
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All right, moving on to actually solving word problems: they are so important, so we really want to be able to solve them.
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Now, as we discussed, there is this problem that we can't just solve all word problems with one method.
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But luckily, there are some general guidelines that will help us work on them.
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We are now going to see a four-step process for approaching word problems.
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If you follow this method, you will have clear, concrete tasks that you can accomplish at every step.
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Now, creativity and thought are still going to be absolutely necessary.
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But these guidelines can be used on virtually any problem that you encounter, so it is a really useful thing here.
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Let's look at it: the very first step is that you need to understand the problem--begin by figuring out what the problem is asking about.
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You don't have to solve anything right now; you don't even have to figure out exactly what you are looking for.
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But you need to figure out what the problem is asking about.
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You just want to have some idea of what is going on--what is the general thing that is happening here?
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Many word problems are going to unfold like a story, in some way; so you just want to understand--"What is this story telling me?"
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Now, this step may seem obvious, like everybody is going to think, "Well, of course I have to understand the problem!"
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But so many people gloss over it, and they just try to hop right into solving the problem.
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But you need to understand it first; how can you solve something if you don't even know what is going on?
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You have to know what is happening in the problem, what the problem is about, what the ideas going on are,
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before you are going to be able to have any chance of actually solving it.
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So first, just get a sense of what is going on.
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Once you have that, second, what are you looking for?
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Set up your variables: what are you looking for?
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Once you know what is happening in the problem, you want to figure out what you are trying to find.
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What are the ideas that are central here, that you need to be able to work out this problem?
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Now, almost always, especially for the next few years, before you get into really advanced math classes,
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if you continue to study math in college--for precalculus, calculus, and even the next couple of years--
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this will take the form of setting up variables: you will set up variables and define any variable that the problem asks for.
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You might also need to define other variables, such as values that are talked about, but never explicitly given.
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Now, I want to make absolutely sure that you write down what each variable means.
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I really want to make sure that you do this; I am not kidding.
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Until you are extraordinarily comfortable with word problems, you should be literally writing out a little reminder of what each variable means.
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For example, if you had a problem where you had to talk about some number of tables, then you might say, before you started, "t = # of tables."
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And then, you also might have to talk about chairs in the problem, so "c = # of chairs."
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And that is enough right there; but very often, students will be working on it, and they will forget what this letter meant or what was really there.
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This is a way of anchoring your work, so you know what you are searching for, what these ideas are about.
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And by writing it down, it will make it that much easier to work on the rest of the problem.
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Really, really, really: write down what the variable means--it should be completely obvious to you,
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either because you will have a picture (which we will talk about later), or because you have literally written out what the variable means.
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That way, you can't forget it while you are in the middle of working on the problem, and get confused.
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So, you set up what you are looking for at this point: what you are looking for, what is unknown, what is going to help you get to figuring out the answer.
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Third, you set up relationships: you want to figure out what the problem is telling you about what you are looking for.
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What is the problem telling you about what you are looking for?
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You are trying to solve something at this point, which you have figured out from #1.
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And you know the sort of ideas that you are looking for; that was #2.
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So now, you need to figure out what the problem is telling you about those things that you are looking for (those variables, if we are doing a math problem).
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Use what you know; use what the problem gave you to set relationships between your unknowns and whatever information the problem gives you.
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Now, in math class, especially for the next few years (once again), this will usually take the form of making equations.
00:17:01.700 --> 00:17:04.800
This is what it is normally going to come down to: you will be making equations.
00:17:04.800 --> 00:17:11.200
You will set up equations involving your unknown variable or variables, and whatever else you know from how the problem is set up.
00:17:11.200 --> 00:17:15.500
The problem will tell you some information, and you will use that information to build equations.
00:17:15.500 --> 00:17:19.900
And if it is something that is not specifically a math problem, or a more general form of math problem,
00:17:19.900 --> 00:17:24.900
you might be just using it to set up things of what you know and what you can rely on as you start to work through it.
00:17:24.900 --> 00:17:29.400
Sometimes, when you are working on these equations, you will realize that you have more unknowns than you originally thought.
00:17:29.400 --> 00:17:32.100
That is perfectly fine; just make up new variables at this point.
00:17:32.100 --> 00:17:38.200
You realize, "Oh, to use this idea, I need to also have this new variable introduced"--introduce a new variable!
00:17:38.200 --> 00:17:43.500
Go back to step #2; make a new variable; write down what it means; and then plug it into the equation where it is going to go.
00:17:43.500 --> 00:17:48.200
Figure out the equations that will involve that new variable that you realize you need to use.
00:17:48.200 --> 00:17:52.000
Finally, once you understand what the problem is about, you know what you are looking for,
00:17:52.000 --> 00:17:56.400
and you have all of your relationships (equations) set up, you are ready to solve it.
00:17:56.400 --> 00:18:02.200
Normally, in many ways, this is going to actually be the easiest part, because it is just like doing any other exercise now.
00:18:02.200 --> 00:18:06.100
You have equations to solve: you are looking to solve for something, and you have equations.
00:18:06.100 --> 00:18:09.500
It is just going to be a matter of doing the math that you have been doing in every other section,
00:18:09.500 --> 00:18:12.200
in every other part of the section that you are working on.
00:18:12.200 --> 00:18:16.200
It will normally have many exercises, and it will only finally culminate in some word problems.
00:18:16.200 --> 00:18:18.300
So, it is just like doing all of the exercises you have been doing before;
00:18:18.300 --> 00:18:23.700
it is just that it had this framing of how we get to those equations--how we get to doing the exercise.
00:18:23.700 --> 00:18:29.400
So, you have some equations to solve or something like that; roll up your sleeves and work on it like a normal non-word problem.
00:18:29.400 --> 00:18:36.300
And as a general rule, to solve a problem, you need as many relationships as you have unknowns.
00:18:36.300 --> 00:18:45.300
For example, if you need to find out three variables, then you will have to have three equations (or more) that relate the three variables together.
00:18:45.300 --> 00:18:48.700
So, you will need the same number of relationships, the same number of equations,
00:18:48.700 --> 00:18:51.600
as the pieces of information you are trying to find out.
00:18:51.600 --> 00:18:57.900
So, it is like each equation is a key that you can fit into an unknown lock.
00:18:57.900 --> 00:19:05.000
Each equation is a piece of knowledge that you can use to shine the light on some unknown thing.
00:19:05.000 --> 00:19:10.300
In summary, complex problems can be approached by first understanding what you are working with;
00:19:10.300 --> 00:19:17.400
then figuring out what you are looking to find (figure out what you want or need to know);
00:19:17.400 --> 00:19:23.000
third, you work out relationships--how do the things that you know already from the problem,
00:19:23.000 --> 00:19:26.700
and the things that you are looking to find--how are they connected to each other?
00:19:26.700 --> 00:19:30.800
And then finally, you put it all together, and you get the answer.
00:19:30.800 --> 00:19:33.900
So, that is our basic summary of method.
00:19:33.900 --> 00:19:40.700
Now, it may seem surprising, but this method can actually be applied to pretty much any problem--any issue you have, not just math.
00:19:40.700 --> 00:19:44.800
This is such a general tool for approaching complex ideas that you can use it
00:19:44.800 --> 00:19:49.000
for pretty much anything where you are trying to figure out a way to solve an issue.
00:19:49.000 --> 00:19:54.300
You are probably used to doing this in all sorts of situations already; it is just a good way to approach problem-solving in general.
00:19:54.300 --> 00:20:01.600
So, let's actually see two examples that are not connected to math, just so we can get an idea of how we solve problems on a general scale.
00:20:01.600 --> 00:20:07.000
So, consider these two situations: in the first situation, your car is making weird noises when you drive.
00:20:07.000 --> 00:20:10.700
And in the second situation, your friend suddenly stops talking to you one day.
00:20:10.700 --> 00:20:14.600
They are very different things, but both issues that you would want to deal with.
00:20:14.600 --> 00:20:17.300
So, to begin with, our first thing is that we understand what is going on.
00:20:17.300 --> 00:20:22.600
Your car is making weird noises when you drive: well, if my car is making weird noises, there is probably some sort of issue going on.
00:20:22.600 --> 00:20:28.100
And so, now you also have...in a word problem in math, we are going to have a more specific being-told-what-to-do.
00:20:28.100 --> 00:20:34.600
In real life, our issues are normally more amorphous, more uncertain in what we need to do to get to the right thing.
00:20:34.600 --> 00:20:37.700
But with this case, let's say we have figured out that our car has a problem.
00:20:37.700 --> 00:20:40.900
We need to figure out what that problem is, and then we need to fix it.
00:20:40.900 --> 00:20:44.900
Now, let's say, in this world, we don't want to spend the money on taking it to a mechanic.
00:20:44.900 --> 00:20:49.800
So, that means we have to be the one to figure it out, and we have to be the one to fix the problem.
00:20:49.800 --> 00:20:53.500
So, your car is making weird noises when you drive; that means that what we need to do here--
00:20:53.500 --> 00:20:57.900
our understanding of it is that we need to figure out what the problem is, and then we need to fix it.
00:20:57.900 --> 00:21:00.200
That is what our objective is: "Fix the problem."
00:21:00.200 --> 00:21:04.700
In the version where we are talking about our friend who stops talking to us, what we want to do now is:
00:21:04.700 --> 00:21:09.000
we ask ourselves, "Well, do I care about this friend enough to want them to still be my friend?"
00:21:09.000 --> 00:21:13.000
We are going to assume that, yes, they are actually your friend; you want them to be friendly again.
00:21:13.000 --> 00:21:15.500
But this is one of our things where we are trying to understand the problem.
00:21:15.500 --> 00:21:20.000
We actually have to understand and think about what it is--how does it connect to the rest of what is going on?
00:21:20.000 --> 00:21:24.200
All right, if your friend suddenly stops talking to you one day, but you actually just realized that you don't actually like them at all,
00:21:24.200 --> 00:21:27.400
then they are no longer your friend; just make them no longer your friend.
00:21:27.400 --> 00:21:32.500
But in this case, we are saying that, yes, our objective is to get our friend to be friendly again.
00:21:32.500 --> 00:21:36.500
We want them to actually be our friend, not just be this person who won't talk to us.
00:21:36.500 --> 00:21:40.600
All right, now we have our objectives in mind; now we can figure out what I need to know.
00:21:40.600 --> 00:21:45.500
What are the things that I don't know that will help me get to that objective?
00:21:45.500 --> 00:21:51.400
The second step: in the car example again, if our car is making weird noises, and it has some sort of problem (is our assumption),
00:21:51.400 --> 00:21:56.700
we want to figure out what is making that noise; we also want to figure out
00:21:56.700 --> 00:22:00.500
if the thing that is causing the noise is the result of some other issue.
00:22:00.500 --> 00:22:04.800
And if there is some other root issue, has it damaged anything in the car?
00:22:04.800 --> 00:22:07.100
And then finally, how can we fix any and all damage?
00:22:07.100 --> 00:22:15.000
Now really, this is the only thing that matters: we don't really care...in truth, if some genie appeared out of nowhere,
00:22:15.000 --> 00:22:22.200
and said, "I will fix your car, just because I am a nice guy," we would say, "Yes, genie, fix that car--yay!"
00:22:22.200 --> 00:22:26.200
And they would fix our car; so really, if we could somehow get a genie to show up here at the end,
00:22:26.200 --> 00:22:28.500
we wouldn't need to know the answer to these first three things.
00:22:28.500 --> 00:22:37.400
But it is real life; it is likely going to be the case that these first three issues, these first three unknowns, would really help us to fix the damage.
00:22:37.400 --> 00:22:42.100
If we can figure out what is causing the noise, if we can figure out if there is an issue that is making that noise,
00:22:42.100 --> 00:22:46.200
and if we can figure out if that issue has damaged anything, those are all things that are going to help us
00:22:46.200 --> 00:22:48.700
to fix the car and make it stop making that noise.
00:22:48.700 --> 00:22:54.700
So, while our final unknown is the "how do we make it better?" we really want to know other things on the way,
00:22:54.700 --> 00:22:58.400
if we are ever going to be able to figure out how to make it better.
00:22:58.400 --> 00:23:05.700
In the version where we are looking at our friend, we might want to know why our friend is not talking to us, right?
00:23:05.700 --> 00:23:10.600
And then, what has happened recently would probably be a good way to get a sense of why they are not talking to you.
00:23:10.600 --> 00:23:12.600
And finally, what is the best method to make them friendly?
00:23:12.600 --> 00:23:19.000
Once again, we might actually not care about all of the other things--although, with the first two things,
00:23:19.000 --> 00:23:21.200
we would probably be curious about why your friend isn't talking to you.
00:23:21.200 --> 00:23:25.500
But once again, if that genie appeared and said, "I will make your friend be your friend again--no problem!
00:23:25.500 --> 00:23:29.200
I am a nice guy!" you would say, "Thanks, genie--you are the best!"
00:23:29.200 --> 00:23:32.000
This genie would be pretty awesome, and they would make our friendship better.
00:23:32.000 --> 00:23:36.000
But in real life, we have to deal with our problems; we have to make things better.
00:23:36.000 --> 00:23:40.800
We have to figure out how to solve the problem; we don't have some genie who is just going to do it for us.
00:23:40.800 --> 00:23:42.500
So, we have to figure out how to solve it.
00:23:42.500 --> 00:23:47.500
So, while the only thing we really want to know is this final idea, to be able to get to the point
00:23:47.500 --> 00:23:51.500
where we can figure out that final idea, it would probably be very useful to know
00:23:51.500 --> 00:23:56.600
why our friend isn't talking to us, and what has happened recently that frames what is going on right now.
00:23:56.600 --> 00:24:02.600
So, this is an idea of how we set up what are the unknowns that would help me answer that original question.
00:24:02.600 --> 00:24:08.200
For car troubles, if we want to answer what those unknowns are--get a sense of those things--there are a bunch of things we could do.
00:24:08.200 --> 00:24:11.300
We could look under the hood; we could get a service manual for the car.
00:24:11.300 --> 00:24:14.700
A service manual tells us how a car is put together--how that car is put together.
00:24:14.700 --> 00:24:18.900
We would talk to friends and family who know cars well; we might research the problem online,
00:24:18.900 --> 00:24:22.400
or anything else we can think of that would tell us more about what is going on with the car.
00:24:22.400 --> 00:24:25.400
These would help us answer the unknown questions of what is wrong.
00:24:25.400 --> 00:24:31.600
What is the issue? Has there been any damage? And how do we fix whatever damage it is, and fix the underlying issue?
00:24:31.600 --> 00:24:34.700
With the friend, we want to think about what happened to your friend recently.
00:24:34.700 --> 00:24:37.200
We want to think about your recent behavior to your friend.
00:24:37.200 --> 00:24:42.100
We want to ask our other friends if they know what is wrong--perhaps they have some information that we are unaware of right now.
00:24:42.100 --> 00:24:47.800
We want to pay attention to how our friend is acting when we are not around, or when we are not interacting with them directly.
00:24:47.800 --> 00:24:52.200
That will give us some sense of what is going on, and be able to help us understand the situation.
00:24:52.200 --> 00:24:58.300
And we can use all of those pieces of information to help us figure out how to make them friendly again, and make things better.
00:24:58.300 --> 00:25:04.400
With all of this information, we now just figure out what is wrong with the car and what we need to do to fix the issue; and then, we do it.
00:25:04.400 --> 00:25:06.700
In the friend version, we use all of this information to figure out
00:25:06.700 --> 00:25:10.800
what would put our friend in a good mood to be friendly towards us again; and we do it.
00:25:10.800 --> 00:25:15.600
We use the information to give us many things that we can now to reach our objective.
00:25:15.600 --> 00:25:19.300
That is what is going on in complex problem-solving, just generally all the time.
00:25:19.300 --> 00:25:23.200
We figure out what the problem is; we figure out what we need to know to deal with that problem.
00:25:23.200 --> 00:25:26.300
We figure out how we can answer those things that we need to know.
00:25:26.300 --> 00:25:30.600
And then, we use all of that information to get what we want.
00:25:30.600 --> 00:25:37.100
Math-specific method...that is really great for any complex problem, and it can be used on anything at all that you need to solve.
00:25:37.100 --> 00:25:39.700
But what about what we are going to see exactly in math?
00:25:39.700 --> 00:25:44.900
This general method is great, but let's see what we are going to use in the next couple of years.
00:25:44.900 --> 00:25:46.700
Let's look at the specific approach that we will be using.
00:25:46.700 --> 00:25:52.000
Once again, we understand what the problem is about: we want to just get a sense of what is going on.
00:25:52.000 --> 00:25:58.400
Then, we set up and name variables: variables represent our unknowns, the things that we are going to need
00:25:58.400 --> 00:26:02.100
and have to have a handle on to be able to answer the problem.
00:26:02.100 --> 00:26:10.000
Then, we set up equations; we use the information that is given to us in the problem statement
00:26:10.000 --> 00:26:14.400
to get us the equations that we need to be able to answer the question.
00:26:14.400 --> 00:26:18.400
And then finally, we have equations; we solve for that answer.
00:26:18.400 --> 00:26:22.500
We have equations; we know what we are looking for because we understood the problem in the beginning.
00:26:22.500 --> 00:26:25.600
We just work it all out, and we get an answer.
00:26:25.600 --> 00:26:29.700
There are going to be a few word problems, probably around 5 to 10 percent (it depends).
00:26:29.700 --> 00:26:32.500
But these are going to be mostly concept questions and proofs.
00:26:32.500 --> 00:26:39.600
So, there are a few word problems that won't use this exact formula of "Understand, Variables, Equations, Solve."
00:26:39.600 --> 00:26:42.200
All right, that is our formula in general when we are approaching word problems.
00:26:42.200 --> 00:26:46.100
But sometimes we are going to have to prove things; or sometimes it is asking a general concept question
00:26:46.100 --> 00:26:51.600
of "Is this true? Is this false? Why does this happen?" and in those situations, we won't be able to use this exact method.
00:26:51.600 --> 00:26:57.900
But we can fall back on that more general method, where we just think in large-scale, complex terms.
00:26:57.900 --> 00:27:02.700
Tips: let's start talking about tips that help us improve the efficacy of this method.
00:27:02.700 --> 00:27:06.100
We understand how the method works now; it is "Go through and understand;
00:27:06.100 --> 00:27:10.200
set up the things that you are looking for (your variables); set up what you know (your equations);
00:27:10.200 --> 00:27:12.900
and then finally put it all together and solve it."
00:27:12.900 --> 00:27:15.800
Here are some tips that will help us to use this method.
00:27:15.800 --> 00:27:21.700
First, draw pictures: this is probably the single most useful tip in all of this.
00:27:21.700 --> 00:27:26.800
Draw pictures: it may not be possible for every problem--some problems are going to be completely abstract, and you won't be able to do this.
00:27:26.800 --> 00:27:29.400
But when you can, it is going to help massively.
00:27:29.400 --> 00:27:35.400
Basically, if a problem is asking about geometry, shapes, or something that is physically happening--
00:27:35.400 --> 00:27:39.900
if something is a real-world thing that we can imagine--it is really useful to draw a picture.
00:27:39.900 --> 00:27:44.400
If the problem doesn't give you a picture right away, draw it yourself--just draw a picture.
00:27:44.400 --> 00:27:50.500
And don't worry about making the picture look nice--you just want something that you can sketch in a few seconds, and that makes sense to you.
00:27:50.500 --> 00:27:55.500
Cars can become boxes; people can become little stick figures; you can change things to dots and just straight lines.
00:27:55.500 --> 00:28:00.100
You don't have to draw an entire road, if you are talking about a road; it is enough to just have a street line.
00:28:00.100 --> 00:28:04.300
You just want something that you can use as a reference point, so that you can see it in your mind's eye.
00:28:04.300 --> 00:28:09.300
It is not absolutely necessary to do this; but a visual representation of a problem can help so much.
00:28:09.300 --> 00:28:12.700
When you are not sure how something works or what to do, sometimes a quick sketch
00:28:12.700 --> 00:28:17.500
(so we can see what is going on) can help clear things up so much--it can be really, really useful.
00:28:17.500 --> 00:28:20.500
Drawing pictures is a great way to approach word problems.
00:28:20.500 --> 00:28:26.200
It really helps us understand it, helps us figure out what we are looking for, helps us set up our equations, and it can help us solve it.
00:28:26.200 --> 00:28:29.200
It is really, really useful stuff.
00:28:29.200 --> 00:28:33.700
The next tip is breaking things into pieces: Imagine you have a plate in your hands.
00:28:33.700 --> 00:28:35.800
You drop it, and it breaks into many pieces.
00:28:35.800 --> 00:28:39.700
If you collect all of the pieces and you put them back together, you will have a plate again, right?
00:28:39.700 --> 00:28:42.700
And if you pull apart the rebuilt plate, you will have all of the pieces again.
00:28:42.700 --> 00:28:46.800
You can take something into the pieces that make it up, and you can put it back together to make the whole.
00:28:46.800 --> 00:28:50.500
The whole is made out of its parts, and the parts make up the whole.
00:28:50.500 --> 00:28:54.100
The sum of the parts equals the whole; the whole equals the sum of the parts.
00:28:54.100 --> 00:29:00.300
That is what this idea is expressed by; it is a simple idea, but it is important, and it comes up in math a lot.
00:29:00.300 --> 00:29:04.700
Consider this problem right here: if we want to figure out what the area of the shaded area is--
00:29:04.700 --> 00:29:11.700
we want to figure out how much shaded area there is--by breaking this figure into its pieces, we can figure out how to do this.
00:29:11.700 --> 00:29:16.800
We see that that shaded part and that semicircle--they come together to form a square.
00:29:16.800 --> 00:29:20.600
Well, I know what the square is; I can figure out what the semicircle is.
00:29:20.600 --> 00:29:23.100
And so, I can use that information to figure out the shaded area.
00:29:23.100 --> 00:29:29.200
You can break things into their pieces; and this will normally be applied to geometrical things, where we are talking about shapes and objects.
00:29:29.200 --> 00:29:33.600
But once in a while, it will actually be applied to mathematical things, where it is just in terms of letters and things.
00:29:33.600 --> 00:29:37.700
And we will actually break things into their pieces then, and use each of the pieces on its own.
00:29:37.700 --> 00:29:42.600
We will see those things later on, potentially; and we will talk about it then, if it happens.
00:29:42.600 --> 00:29:48.800
But normally, you are going to end up seeing it in pictures, where we are dealing with shapes, geometry, and things like that.
00:29:48.800 --> 00:29:52.400
But it is a really useful idea and comes up in a lot of different places.
00:29:52.400 --> 00:29:57.500
Next, try out hypothetical numbers: sometimes, it can be hard to find out what is going on in a problem,
00:29:57.500 --> 00:30:00.100
because we are not working with numbers--we are really good at working with numbers,
00:30:00.100 --> 00:30:03.000
because we have been doing it for a long time at this point.
00:30:03.000 --> 00:30:08.100
So, you can try plugging in hypothetical numbers to help you understand what is going on.
00:30:08.100 --> 00:30:13.400
Now, if you try a hypothetical number, make sure you try out numbers that follow the rules set in your problem.
00:30:13.400 --> 00:30:18.100
If your problem says that the number is even, you want to make sure whatever hypothetical number you use
00:30:18.100 --> 00:30:24.700
is something like 2, 4, 6, 8, so that you are following the rules that they tell you, so that you can go along with that path.
00:30:24.700 --> 00:30:27.500
This is a great way to test the equations you set up.
00:30:27.500 --> 00:30:32.500
We can set up an equation and say, "Well, this should tell me how much money the company made."
00:30:32.500 --> 00:30:38.100
And what we do is plug in some numbers and say, "Well, what if they made 10 widgets?"
00:30:38.100 --> 00:30:42.600
Well, we could probably figure out what it should be for 10 widgets, and so we could make sure that our equation
00:30:42.600 --> 00:30:46.600
is giving out the same thing that we know 10 widgets should make them, in terms of money.
00:30:46.600 --> 00:30:49.400
And so, we can make sure that, yes, our equation seems to be checking out.
00:30:49.400 --> 00:30:52.100
It is working when I use it on simple numbers, that I can easily understand.
00:30:52.100 --> 00:30:56.900
So, now it is up and running, and we can use it on variables and trust it.
00:30:56.900 --> 00:31:02.100
So, if you have difficulty setting up equations, try plugging in hypothetical numbers to understand what is going on in there.
00:31:02.100 --> 00:31:08.500
It is easy to make mistakes when you are setting up equations, so check them afterwards by plugging in hypothetical numbers, if you are not sure.
00:31:08.500 --> 00:31:11.500
You can plug in a number where you understand what should happen there.
00:31:11.500 --> 00:31:17.100
And then you will see that that makes sense; and we will see some examples when we work on the examples part of this lesson.
00:31:17.100 --> 00:31:22.500
If things don't work out with your hypothetical number, you know you made an error in your equation, or possibly an error in your hypothetical number.
00:31:22.500 --> 00:31:28.300
And there is something you need to go back and work on; make sure you fix it before you move on and try to solve it.
00:31:28.300 --> 00:31:31.100
Student logic: this is an interesting idea.
00:31:31.100 --> 00:31:36.300
The hardest part often can be figuring out what relationships are going on.
00:31:36.300 --> 00:31:38.900
But you have a secret weapon--you are a student.
00:31:38.900 --> 00:31:46.200
What I mean by this is that you can use student logic: you can be pretty much certain that whatever your problem is going to be about,
00:31:46.200 --> 00:31:49.500
it is going to use what you are currently learning.
00:31:49.500 --> 00:31:53.800
If you are a student, they are not going to have you do something that you haven't learned.
00:31:53.800 --> 00:31:57.000
And they are probably mainly going to be focusing on the things that you have been working on.
00:31:57.000 --> 00:32:02.300
So, whatever you are currently learning is exactly what you will probably use to set up the relationships.
00:32:02.300 --> 00:32:05.100
For example, if you are currently studying parabolas, you know that the problem
00:32:05.100 --> 00:32:09.100
can almost certainly be solved by using something that you learned about parabolas.
00:32:09.100 --> 00:32:11.900
So, you will set up your word problem, and then you will say, "Oh, right, what do I know about parabolas?"
00:32:11.900 --> 00:32:18.200
"Oh, the top of a parabola happens at this equation," and so you will see if that would be useful here; and it probably would be.
00:32:18.200 --> 00:32:21.600
Try to figure out how the problem is related to what you have recently been studying.
00:32:21.600 --> 00:32:26.400
How is this connected to what you have been working on in this section, in this lesson, in this chapter?
00:32:26.400 --> 00:32:28.800
And then, use those ideas to help you set up the equation.
00:32:28.800 --> 00:32:32.900
You don't have to normally worry about things that are completely unrelated to what you have just been learning,
00:32:32.900 --> 00:32:36.900
because your teacher is normally trying to teach you the things that you have been working on.
00:32:36.900 --> 00:32:41.400
So, they are going to be using those ideas in the word problems, as well.
00:32:41.400 --> 00:32:46.900
Jump in: this is a good suggestion--you don't want to just get paralyzed.
00:32:46.900 --> 00:32:49.200
Working on word problems can sometimes freeze you, and you are thinking,
00:32:49.200 --> 00:32:53.900
"I don't know what to do! I don't know what to do!" so just calm down and try something.
00:32:53.900 --> 00:32:58.700
You are not sure where to start; you don't know exactly what you are looking for; you don't know how to solve it, so you get scared.
00:32:58.700 --> 00:33:05.000
But instead--that is OK; that happens to everyone; instead, throw in something; try to do something.
00:33:05.000 --> 00:33:07.600
If you can't set up the equation right the first time, that is OK.
00:33:07.600 --> 00:33:12.400
If you pick the wrong variable, that is OK; if you plug in the wrong hypothetical number, that is OK.
00:33:12.400 --> 00:33:15.200
You are probably going to learn from your mistake.
00:33:15.200 --> 00:33:18.800
Normally, by making a mistake, we think, "Oh, well, that didn't work..."
00:33:18.800 --> 00:33:23.200
Oh, but because that didn't work, this other thing would work...and then you are right on the way to doing it.
00:33:23.200 --> 00:33:27.600
As long as you pay attention to what you are doing, and you think about what makes sense and what doesn't make sense,
00:33:27.600 --> 00:33:32.500
and you are double-checking your work and thinking "Is this sane? Is this a reasonable thing to be doing?"--
00:33:32.500 --> 00:33:34.800
you are normally going to see where you went wrong.
00:33:34.800 --> 00:33:38.400
And by seeing where you went wrong, you can realize what you need to do in the problem.
00:33:38.400 --> 00:33:41.700
The fastest way you can learn is often by just making a mistake.
00:33:41.700 --> 00:33:46.900
So, reach in; get your hands dirty; it is OK if things go wrong at first, because eventually they will work out.
00:33:46.900 --> 00:33:49.900
If you just stand back and keep thinking, "I don't know how to do this! I don't know how to do this!"
00:33:49.900 --> 00:33:54.900
yes, you are right--you are not going to know how to do it, because you are just saying one thing; you are pulled back.
00:33:54.900 --> 00:33:59.900
You need to try something; trying something is almost always going to work better than doing nothing.
00:33:59.900 --> 00:34:03.700
So, just jump in and get started, if you get stuck.
00:34:03.700 --> 00:34:13.600
All right, let's look at some examples: a four-part method: first, understand; 2) Set up variables; 3) Set up equations; 4) Solve the thing.
00:34:13.600 --> 00:34:17.200
So, first, we need to understand this; #1--we read through it.
00:34:17.200 --> 00:34:24.900
"Sally has a job selling cars. Her monthly base pay is $2,000, along with a 1.7% commission on all the cars she sells."
00:34:24.900 --> 00:34:30.200
"If she earned $5,315 in March, what was the total cost of the cars she sold in March?"
00:34:30.200 --> 00:34:33.000
So, the first thing we have to do is understand what is happening here.
00:34:33.000 --> 00:34:35.800
Sally has a job selling cars--that makes sense.
00:34:35.800 --> 00:34:45.300
She gets paid some base pay; she gets paid some amount, and then..."along with"...she gets paid $2000, plus some other thing, this commission.
00:34:45.300 --> 00:34:48.100
Now, we might say, "I don't know what a commission means."
00:34:48.100 --> 00:34:55.500
So, what do we do? We look it up! You put the word "commission" into an Internet search, or you look up "commission" in a dictionary.
00:34:55.500 --> 00:34:56.900
It would probably be a good first step.
00:34:56.900 --> 00:34:59.200
If you don't know what the word "commission" is, you look it up.
00:34:59.200 --> 00:35:05.000
We look up "commission," and we find out that it is normally a percentage fee that you get for selling something.
00:35:05.000 --> 00:35:11.800
So, if somebody has a 10% commission, if they sell $100 of things, they personally get $10 back.
00:35:11.800 --> 00:35:15.900
So, a commission is a way of making a profit off of what you sell to other people.
00:35:15.900 --> 00:35:20.400
A salesman gets a commission on their sales, generally, to encourage them to sell more.
00:35:20.400 --> 00:35:28.000
So, she gets a 1.7% commission: 1.7% of whatever she sells, she gets back; OK.
00:35:28.000 --> 00:35:33.900
So, she gets $2000, plus 1.7% of the amount of the cars that she sold in the month.
00:35:33.900 --> 00:35:37.400
If she earns $5,315 in March, what was the total cost of the cars she sold in March?
00:35:37.400 --> 00:35:41.700
Oh, so we have a piece of information about March; we are looking to know about the cost of that.
00:35:41.700 --> 00:35:47.600
So, we are looking for a connection between how we know her money breaks down and what the cost of those cars must have been.
00:35:47.600 --> 00:35:50.800
All right, so number 1 is done: we understand what it is about.
00:35:50.800 --> 00:35:53.500
#2: What are the things that we need to know?
00:35:53.500 --> 00:35:58.000
Well, we want to be able to talk about her pay--how much does she get paid?
00:35:58.000 --> 00:36:02.700
So, we want to have some way of being able to talk about how that varied based on her commission.
00:36:02.700 --> 00:36:09.900
How about we make c equal, not commission, but the cost of cars sold.
00:36:09.900 --> 00:36:13.700
This is the amount of money she makes off of the cars she sells.
00:36:13.700 --> 00:36:20.600
So now, let's figure out a way of being able to make this $2000, along with 1.7% commission, into something.
00:36:20.600 --> 00:36:28.500
She gets $2000; so #2 is done--we figured out the things that we need to talk about, her pay and the amount of the cars that are sold.
00:36:28.500 --> 00:36:30.400
Those are the two variables that we are really looking at.
00:36:30.400 --> 00:36:35.100
#3: We set up equations--we want to have some way of being able to talk about her pay.
00:36:35.100 --> 00:36:43.600
$2000 + 1.7% of the cost of those cars that are sold...how does that work out?
00:36:43.600 --> 00:36:47.300
You might say, "Well, I don't really remember how percent works," so let's test some things.
00:36:47.300 --> 00:36:56.300
We know that 10% of 100 should end up being 10; we know that 5% of 100 should end up being 5.
00:36:56.300 --> 00:37:04.000
We know that 1% of 100 should end up being 1; so, we might say, "Oh, right, you move the decimal 2 over, and then you multiply."
00:37:04.000 --> 00:37:08.600
0.017; let's check and make sure that that makes sense.
00:37:08.600 --> 00:37:19.800
If she sold $100 car (a cheap car, but...) if she were to sell $100, 1.7% of 100 would be one dollar and 70 cents.
00:37:19.800 --> 00:37:29.200
So, she should make $1.70: .017...let's check that out: 0.017 times 100 would end up being...
00:37:29.200 --> 00:37:38.300
we move the decimal 2 over...we get 1.7, or $1.70, which would be a dollar and 70 cents.
00:37:38.300 --> 00:37:46.800
That makes sense, so .017 times the cost of the cars sold is equal to the amount that she gets paid in any given month.
00:37:46.800 --> 00:37:49.500
Now, what month are we looking at specifically? March.
00:37:49.500 --> 00:37:55.000
We know that p, in this case, is equal to $5,315.
00:37:55.000 --> 00:38:06.000
So, we take this information; we plug it in here; we have 2000 + 0.017 times the cost of the cars = 5315.
00:38:06.000 --> 00:38:14.400
From here, it is just step #4: we set up our equations; we know how the things interact.
00:38:14.400 --> 00:38:17.400
We know all this; now we just have an equation to solve.
00:38:17.400 --> 00:38:22.600
We want to know c; c is what we are looking for; it asks what was the total cost.
00:38:22.600 --> 00:38:28.500
We are looking for c = ?, so we just solve for what c has to be.
00:38:28.500 --> 00:38:38.200
Subtract 2000 from both sides; we get 0.017c = 3315.
00:38:38.200 --> 00:38:53.700
And then, we divide both sides by .017; so we get c = 3315/0.017.
00:38:53.700 --> 00:38:59.800
We plug that into a calculator to make it easy for us, and we get 195,000 dollars.
00:38:59.800 --> 00:39:10.100
So, she makes $195,000 in terms of what she sells, which brings her a commission of $3315, so a total pay of $5315.
00:39:10.100 --> 00:39:15.200
The total cost of all the cars she sold was $195,000; great.
00:39:15.200 --> 00:39:19.700
Example 2: the first thing we want to do is understand what is going on.
00:39:19.700 --> 00:39:24.400
The very first thing...we have a semicircle of radius 8 inscribed inside of a rectangle.
00:39:24.400 --> 00:39:30.300
So, we might say, "What is a semicircle?" Well, we look at the picture--a semicircle is half of a circle.
00:39:30.300 --> 00:39:32.700
Semi- means half; half of a circle makes sense.
00:39:32.700 --> 00:39:39.900
So, it is radius 8; from center to edge is 8; great--it is inside of a rectangle, so it is inscribed.
00:39:39.900 --> 00:39:44.100
We see exactly what it looks like; what is the area of the shaded portion?
00:39:44.100 --> 00:39:47.200
We want to figure out how much is the shade in here.
00:39:47.200 --> 00:39:53.000
We understand what we are looking for: we have half a circle held inside of a rectangle, so it barely touches the edges;
00:39:53.000 --> 00:39:58.900
and we are looking to figure out what is the part that isn't the circle, but is still contained in the rectangle--what is that shaded portion.
00:39:58.900 --> 00:40:03.200
So, #1 is done; #2--what would allow us to know this?
00:40:03.200 --> 00:40:21.500
Well, if we knew what the area of the rectangle was (area of rectangle); if we know what the s for the shaded area;
00:40:21.500 --> 00:40:29.100
and if we knew the circle's area; we would be in pretty good shape to be able to figure this out.
00:40:29.100 --> 00:40:32.300
So, in #2, those are the three things we are looking for.
00:40:32.300 --> 00:40:39.400
R equals the area of the rectangle; S equals the shaded area; C equals the circle's area.
00:40:39.400 --> 00:40:44.700
#3: we start looking for some equations--can we figure out what the area of a rectangle is?
00:40:44.700 --> 00:40:50.800
We say, "Oh, yes, it is length times width"; so R = length times width.
00:40:50.800 --> 00:40:53.200
So now, let's go back, and let's look at our picture.
00:40:53.200 --> 00:40:59.800
Well, if this is 8 from center to edge, look, over here is the edge of the circle; so this is 8;
00:40:59.800 --> 00:41:09.400
over here is the edge of the circle, so this is 8; so the entire length is 16.
00:41:09.400 --> 00:41:18.200
Vertically, you have...we go from here, and we go directly up; then this is 8 here, so it must be 8 on this side, as well.
00:41:18.200 --> 00:41:27.000
So, we have length times width; we know that it is 16 times 8, which comes out to be 128.
00:41:27.000 --> 00:41:30.500
So now, we know what the area of the rectangle is; great.
00:41:30.500 --> 00:41:33.800
What about the area of the shaded area--do we know what that is?
00:41:33.800 --> 00:41:38.200
No, that is what we are looking for--it is what the problem asked, so that is our question mark--it is our unknown.
00:41:38.200 --> 00:41:44.300
C: can we figure out what is the semicircle's (let me write this out...it is the semicircle, not the circle,
00:41:44.300 --> 00:41:48.500
because it is half of a circle)...C equals the semicircle's area.
00:41:48.500 --> 00:41:52.200
Now, we say, "Well, what is the area for a circle?"
00:41:52.200 --> 00:41:55.000
If I knew the area for a circle or a semicircle, I would be good.
00:41:55.000 --> 00:41:58.600
Well, we probably remember that the area for a circle is πr².
00:41:58.600 --> 00:42:06.000
And even if we don't remember what the area for a circle is, we say, "I have learned this before; let's type it into an Internet search!"
00:42:06.000 --> 00:42:10.100
You type in "area of circle"; you have it right there.
00:42:10.100 --> 00:42:14.700
Or if you have a math book, you can possibly look in the cover, and it will already have that formula right there.
00:42:14.700 --> 00:42:27.300
The area of a circle is πr²; so the circle equals πr²; the area for a circle is equal to πr².
00:42:27.300 --> 00:42:33.400
Now, notice: this is r, not this capital R; the area of our rectangle is very different than r.
00:42:33.400 --> 00:42:36.400
What is little r here? We look once again, and try to remember--what is it?
00:42:36.400 --> 00:42:40.200
Oh, right, it is the radius of the circle, from the center of the circle out.
00:42:40.200 --> 00:42:44.700
The radius of our circle is 8, from the center of the circle out.
00:42:44.700 --> 00:42:48.900
Now, there is one difference between the area of a circle and the area of our semicircle.
00:42:48.900 --> 00:43:01.700
What is a semicircle? It is half of a circle, so it is 1/2 times πr², so 1/2 times π times 8².
00:43:01.700 --> 00:43:09.600
That equals 1/2 times π times 8²...8² is 64, so that gets us 32π.
00:43:09.600 --> 00:43:17.800
Now, that is 2 pieces of information: we know that r equals 128, and we know c equals 32π.
00:43:17.800 --> 00:43:22.000
But we are still looking for this other thing: we have three unknowns to start with, R, S, and C.
00:43:22.000 --> 00:43:26.400
And we have two pieces of information; so we need some way to connect S to R and C.
00:43:26.400 --> 00:43:32.000
Well, we look at this, and we think, "Oh, the area of the rectangle, the shaded area, and the semicircle--they are all connected!"
00:43:32.000 --> 00:43:38.700
The area of the rectangle contains both of the other ones, put together, so it is S + C.
00:43:38.700 --> 00:43:43.600
So, at this point, we are ready to go on to step 4: we have all of our equations set up.
00:43:43.600 --> 00:43:49.100
Step 4: R = S + C--we have numbers, so let's get S on its own, because that is what we are looking to solve.
00:43:49.100 --> 00:43:58.700
So, R - C = S; the area of the shaded area is equal to the rectangle's area, minus the area of the semicircle.
00:43:58.700 --> 00:44:08.800
So, we plug in the area for the rectangle; that is 128, minus the area for the semicircle--that is 32π; and that is equal our shaded area.
00:44:08.800 --> 00:44:13.700
And that is our answer, because there is no way to combine 32π and 128--they speak different languages.
00:44:13.700 --> 00:44:18.700
If we want, we could plug it into a calculator and get 32 times 3.14, and then get something.
00:44:18.700 --> 00:44:22.500
But 128 - 32π...that is a great answer; there we are.
00:44:22.500 --> 00:44:28.700
The next example: Tobias has precisely 17 coins in his pocket; so the first thing we are doing is understanding what is going on.
00:44:28.700 --> 00:44:33.700
#1: Tobias has some coins in his pocket--that makes sense; 17--cool.
00:44:33.700 --> 00:44:38.000
Coins come in three types, so he has three different coin types: quarters, nickels, and pennies.
00:44:38.000 --> 00:44:43.100
Now, if we didn't remember, we would say, "Oh, what is a quarter?" and we would look it up; a quarter is 25 cents.
00:44:43.100 --> 00:44:47.300
What is a nickel? 5 cents. What is a penny? 1 cent.
00:44:47.300 --> 00:44:54.500
He has a total of 2 dollars and 17 cents in coins; if he has 2 pennies, how many quarters does he have?
00:44:54.500 --> 00:44:59.200
Let's see...everything here makes sense; the kid has some change in his pocket.
00:44:59.200 --> 00:45:03.200
They come in three different types of coins that make up that change.
00:45:03.200 --> 00:45:08.000
And he has a total amount of money, and now we want to find out what the specific number of quarters is.
00:45:08.000 --> 00:45:14.000
So, what might be useful to know? First, we are certainly going to need to know what is the number of quarters.
00:45:14.000 --> 00:45:20.400
That is our very first thing; so here in step 2, the first unknown that we definitely have to figure out is number of quarters.
00:45:20.400 --> 00:45:23.400
Well, we will probably also want to know how many nickels and how many pennies he has,
00:45:23.400 --> 00:45:26.200
because they are connected to the other pieces of information right here.
00:45:26.200 --> 00:45:37.400
n = number of nickels; and finally, p = number of pennies.
00:45:37.400 --> 00:45:40.000
Those little reminders serve us to understand what is going on as we work through.
00:45:40.000 --> 00:45:43.700
Now, #3: How can we connect these ideas together?
00:45:43.700 --> 00:45:48.700
Now, we notice...well, how can I talk about the number of coins in his pocket?
00:45:48.700 --> 00:45:52.200
Well, I know what the number of quarters is; I know what the number of nickels is; I know what the number of pennies is;
00:45:52.200 --> 00:45:56.100
do I literally know what they are?--no, but I have names for them.
00:45:56.100 --> 00:46:00.500
So, we can talk about...well, how can we say how many coins he has, in terms of these variables?
00:46:00.500 --> 00:46:06.400
Well, he is going to have his number of quarters, plus his number of nickels, plus his numbers of pennies.
00:46:06.400 --> 00:46:10.000
That should describe the amount of coins--the number of coins in his pocket.
00:46:10.000 --> 00:46:12.400
Let's check and make sure: let's do a really quick check.
00:46:12.400 --> 00:46:15.800
What if he had 2 quarters, one nickel, and one penny?
00:46:15.800 --> 00:46:24.200
Then that is 2 quarters for q, 1 nickel, and 1 penny (p); so 2 + 1 + 1 is 4 coins, and he would have 4 coins.
00:46:24.200 --> 00:46:31.900
Great; that makes sense; in this case, we are told that he had 17 coins, so q + n + p = 17.
00:46:31.900 --> 00:46:34.300
What else were we told? We were also told that he had $2.17.
00:46:34.300 --> 00:46:37.300
Oh, so how can we figure out how much money he has?
00:46:37.300 --> 00:46:40.600
Well, if he had two quarters, that would be 50 cents.
00:46:40.600 --> 00:46:47.700
Let's make things easy, and let's, in fact, instead of talking in terms of dollars...we will talk about 217 cents.
00:46:47.700 --> 00:46:51.800
We will think just in terms of cents: if he has just one quarter, how many cents does he have? 25.
00:46:51.800 --> 00:47:02.600
If he has two quarters, he has 50 cents; it is 25 times the number of quarters, so 25 times q is how much money, how many cents, he has from quarters.
00:47:02.600 --> 00:47:05.800
What about nickels? Well, 5 times the number of nickels--let's check and make sure.
00:47:05.800 --> 00:47:10.400
3 nickels would be 15 cents; 5 times 3 would be 15; it works out.
00:47:10.400 --> 00:47:17.500
So, 5 times nickels, plus...what if we had pennies? Well, what number would we multiply by pennies?
00:47:17.500 --> 00:47:20.700
Oh, pennies would just be by 1; so it is just the number of pennies.
00:47:20.700 --> 00:47:29.800
That is going to be the amount of money in his pocket; in this case, we know how many cents he has--he has 217 cents, $2.17.
00:47:29.800 --> 00:47:34.800
2.17 dollars is the same thing as 217 cents; we are thinking about this in terms of cents.
00:47:34.800 --> 00:47:36.700
That way, we just don't have decimal numbers.
00:47:36.700 --> 00:47:41.200
Finally, do we have any other information?--because currently, we have three unknowns and three relationships.
00:47:41.200 --> 00:47:48.500
Oh, yes, he has two pennies--so we know immediately that p = 2.
00:47:48.500 --> 00:47:53.200
Three pieces of information; three unknowns; we are good to go.
00:47:53.200 --> 00:47:59.500
So, #4: we start working things out--well, we are going to have to somehow use p = 2 in one of these two.
00:47:59.500 --> 00:48:09.200
So, let's plug it into this one first: so q + n + 2 = 17; that means q + n = 17, or...
00:48:09.200 --> 00:48:12.600
let's go with n; we will solve for n, because we want to work out q over here.
00:48:12.600 --> 00:48:16.100
So, we will plug it in here, and we can keep working up through substitution.
00:48:16.100 --> 00:48:23.400
n = 17 - q; great; we will take that information, and we plug it in over here.
00:48:23.400 --> 00:48:36.900
So, 25q + 5...remember, we have to substitute with quantities...(17 - q) + p = 217.
00:48:36.900 --> 00:48:41.900
Do we know what p is? Yes, we certainly know what p is; p is just 2.
00:48:41.900 --> 00:48:54.700
So, 25q...we will distribute that 5...+ 5(17)...5 times 17 is...oops, I made a mistake here...
00:48:54.700 --> 00:48:59.700
sorry about that; hopefully you caught that and were thinking, "What are you doing?"...
00:48:59.700 --> 00:49:06.900
q + n...we subtracted 2 here; this is a good example of why we have to make sure we do the exact same thing on both sides.
00:49:06.900 --> 00:49:14.500
15; 15; so this shouldn't be 17 - q over here; it should be 15 - q over here.
00:49:14.500 --> 00:49:24.200
Great; so 25q + 5(15 - q)...we get 75 - 5q; and then, we will also subtract 2 right now, just to get rid of it.
00:49:24.200 --> 00:49:30.900
So, we will get...equals 215; great, 25q + 75 - 5q.
00:49:30.900 --> 00:49:38.700
Let's consolidate that: we will get 20q, and then let's also move the 75 over to the other side; -75, -75...
00:49:38.700 --> 00:49:47.400
So, we get 20q =...215 - 75 is 140, so q is equal to 7.
00:49:47.400 --> 00:49:53.900
The number of quarters he has in his pocket must be 7 quarters, because we were able to work things out from those original equations.
00:49:53.900 --> 00:49:58.000
And if we wanted to figure out what the number of nickels he had was, we could just plug it right here, and we would get,
00:49:58.000 --> 00:50:07.200
"Oh, he must also have 8 nickels"; so if n = 8 and p = 2, q = 7, we could do a really easy check.
00:50:07.200 --> 00:50:14.300
Is 7(25) + 5(8) + 2 equal to 217; yes, it ends up being that that does check out.
00:50:14.300 --> 00:50:18.800
If we multiply the number that the quarters should be worth and that and 7 + 8 + 2,
00:50:18.800 --> 00:50:24.200
that equals 17, so everything checks out; our answer makes sense; q = 7; great.
00:50:24.200 --> 00:50:26.800
Final example: we have a tank for holding water.
00:50:26.800 --> 00:50:30.300
The first step we need to do: we need to understand what is going on.
00:50:30.300 --> 00:50:34.900
The tank for holding water is shaped like a circular cone with its point towards the ground.
00:50:34.900 --> 00:50:44.700
What does that look like? Oh, yes, OK...you draw something circular, and then it comes to a point, like a cone.
00:50:44.700 --> 00:50:49.300
That makes sense; the tank is 10 feet tall, and has a diameter of 6 feet at the top.
00:50:49.300 --> 00:50:57.600
Let's put that into this: it is 10 feet tall, and has a diameter that is 6 feet across.
00:50:57.600 --> 00:51:00.000
What is the volume when the tank is full?
00:51:00.000 --> 00:51:03.300
And then, we have a second part, which is "What is the volume when the water is only 5 feet deep?"
00:51:03.300 --> 00:51:08.300
So, let's just start by breaking it here, and we will answer, "What is the volume in the tank when it is full?"
00:51:08.300 --> 00:51:15.900
Our first thing makes sense: we have this cone full of water; we fill it up to the top; how much water is going to be in it?
00:51:15.900 --> 00:51:20.700
The second part, second idea: we want to get what we need to know here.
00:51:20.700 --> 00:51:25.100
It seems like it would be useful to talk about the height; well, we actually know what the height is.
00:51:25.100 --> 00:51:34.500
Height h will equal the height of the cone; and let's say d equals...we were told the diameter, so let's say the diameter,
00:51:34.500 --> 00:51:39.500
even though these are really just going to be values; we can talk about them as if they are like that.
00:51:39.500 --> 00:51:43.100
And that seems like everything we need to know right now, so let's go and let's see...
00:51:43.100 --> 00:51:46.300
do we have a good way to relate these two things together?
00:51:46.300 --> 00:51:50.100
How can we relate the height and diameter of a cone to its volume?
00:51:50.100 --> 00:51:55.500
When I think, "Well, how do I get the volume of a cone? I have learned this before--they told me in geometry..."
00:51:55.500 --> 00:52:00.200
Right, it is this: we remember the formula--or maybe we don't remember the formula, like,
00:52:00.200 --> 00:52:05.400
"Well, I know I have been told it..." if you have been told it, it is out there on the Internet, right?
00:52:05.400 --> 00:52:11.900
Or it is in a math book; either crack open a math book, or do a quick web search, and you will be able to find it really quickly.
00:52:11.900 --> 00:52:20.000
And you find out that the volume of a cone is equal to 1/3 times what the volume of its cylinder would have been.
00:52:20.000 --> 00:52:29.000
The volume of its cylinder would have been the area of the top, πr², the circle, times the height of the cylinder.
00:52:29.000 --> 00:52:34.800
OK, now at this point, we think, "Wait a second; we are talking about volume; we want to know what volume is."
00:52:34.800 --> 00:52:37.600
So, volume = ?; that is what we are really searching for.
00:52:37.600 --> 00:52:45.200
But did we have r show up before? No, we didn't have r show up before, so let's make a new one: r = radius.
00:52:45.200 --> 00:52:51.000
We were told some things here: we were told the diameter is 6 feet, so how does a radius connect to that?
00:52:51.000 --> 00:52:59.500
Well, we say, "Oh, right, the diameter is just double the radius; the radius is half the diameter; so r equals 3."
00:52:59.500 --> 00:53:08.100
Halfway would just be...if the whole thing is 6, then halfway is going to be 3; so r = 3.
00:53:08.100 --> 00:53:18.500
Our h equals...our height was 10 feet tall; so h = 10; now we have only one unknown left in this equation here.
00:53:18.500 --> 00:53:22.800
V = 1/3 times πr² times height: we know what the radius is; we know what the height is;
00:53:22.800 --> 00:53:27.400
π is just some number; 1/3 is just some number; the only thing we are looking for is the volume--really easy.
00:53:27.400 --> 00:53:31.900
We just plug our numbers in at this point; so, the fourth step is just solving it.
00:53:31.900 --> 00:53:40.300
Our volume is equal to...plugging in the values...1/3 times π times 3² times 10.
00:53:40.300 --> 00:53:45.700
That is going to end up being...the 1/3 here will cancel the squaring that we have here.
00:53:45.700 --> 00:53:52.000
So, we have π times 3 times 10, or 30π; and we have to talk about it in terms of units.
00:53:52.000 --> 00:53:57.400
So, if it is volume, and we have been doing feet before, it must be cubic feet.
00:53:57.400 --> 00:53:59.700
And there is our answer; all right.
00:53:59.700 --> 00:54:04.000
Now, what if we wanted to this other portion of it?
00:54:04.000 --> 00:54:11.400
We will change over to a new version; now, it is nice, because it is going to follow a lot of parallel ideas.
00:54:11.400 --> 00:54:13.800
So, we can just use what we have already figured out.
00:54:13.800 --> 00:54:21.600
Instead of being 10 feet tall, it is only 5 feet tall; so, the water really only comes up to here.
00:54:21.600 --> 00:54:29.400
Now, if that is the case, if the diameter is 6 up here, is the diameter going to be the same down here?
00:54:29.400 --> 00:54:34.100
No, that makes no sense: the diameter can't be the same, because it is a cone.
00:54:34.100 --> 00:54:37.100
It shrinks down, the farther down we get.
00:54:37.100 --> 00:54:40.600
So, if we were all the way at the bottom, the diameter would be 0; if we go all the way up to the top, it would be 10.
00:54:40.600 --> 00:54:45.900
It makes sense that the diameter is going to be half of what it would have been before, because we are now at half the height.
00:54:45.900 --> 00:54:55.900
So, the diameter in the middle is 3; so that means our radius is equal to 1.5; our height is 5; and that same formula from before,
00:54:55.900 --> 00:55:06.600
our volume formula for a cylindrical cone, still works; so volume equals 1/3 times πr² times height.
00:55:06.600 --> 00:55:15.800
We plug in 1/3 times π times (1.5)² times 5.
00:55:15.800 --> 00:55:35.300
We work that out: 1/3 times π times 11.25 simplifies to 3.75π; and it is in cubic feet.
00:55:35.300 --> 00:55:41.100
And there is our answer: we are basically following the same outline we did before.
00:55:41.100 --> 00:55:45.700
So, we don't have to worry about doing it step-by-step, because we can just work from our previous idea.
00:55:45.700 --> 00:55:50.400
We figured out how it was done in the complex way; now it is just a matter of using new values
00:55:50.400 --> 00:55:53.600
and making sure that the values we are using are right.
00:55:53.600 --> 00:55:57.100
r changes, because we are at a different place; height changes because we are at a different place.
00:55:57.100 --> 00:56:00.600
But all of the relationships are still the same, which makes sense--we want to be thinking about it,
00:56:00.600 --> 00:56:03.900
but it makes sense that all of the relationships are the same, because we are still just looking
00:56:03.900 --> 00:56:07.600
to figure out what is the volume inside of this cone.
00:56:07.600 --> 00:56:11.100
It is just now a cone inside of a cone.
00:56:11.100 --> 00:56:15.800
All right, I hope that made sense; I hope that gives you a slightly better understanding of how to approach word problems.
00:56:15.800 --> 00:56:20.200
Don't be that scared by them; it is just a matter of breaking it down and understanding what is going on.
00:56:20.200 --> 00:56:23.900
setting up what you are looking for, figuring out the relationships that connect
00:56:23.900 --> 00:56:29.000
what you are looking for to what you know, and then finally just solving it like a normal math problem.
00:56:29.000 --> 00:56:31.000
All right, we will see you at Educator.com later--goodbye!