Common Core Algebra I.Unit 1.Lesson 10.Translating English to Algebra.By eMathInstruction

Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk Weiler. And today, we're going to be doing unit one less than ten, translating English to algebra. Before we begin, let me remind you that the worksheet that's used for this lesson. And a homework set can be found on clicking on the can be found by clicking on the video's description as well. Don't forget, on the corner of each worksheet, we have the QR code that goes along with the video. So scan that with your smartphone or your tablet and I'll take your right to any of our E math instruction videos. Okay, let's begin. The dreaded word problem. So often, in algebra classes, you need to translate English into algebra. Now, this isn't too bad as long as you know a lot of terminology. And you know your order of operations inside now. So what we're going to be doing today is just getting a lot of practice on the terms and on how to interpret things. Listen closely, try to follow along and get practice where you can, okay? So let's jump right into exercise one. It's important to be able to recognize addition and subtraction and phrases. First, let's begin with some numerical work and then transition to expressions that contain variables. Letter a says write a calculation and a result that represents a number that is 5 greater than three. 5 greater than three. Now it's kind of interesting because of course, addition is commutative. So it really doesn't matter which way you write it down. But technically speaking, a number that is 5 greater than three is this. Oh, and I mean green. So that's three plus 5. And of course, the result is 8. I think I'm going to go back to blue. I think you can see it better. Now, you might say, well, I wrote down 5 plus three and got the same thing. But technically speaking, a number that is 5 greater than three is three plus 5. The key word there is greater. All right? Now that's important because when we look at letter B and it says write a calculation and a result that represents a number that is two less than 9, it is very important that we do not write down two -9. In fact, a number that is two less than 9 will be given by 9 minus two. That's the expression. What it's equal to is 7. So two less than 9. A very easy to think, I should just put down two -9, but two -9 is negative 7. Right? There's nothing wrong with negative 7. It just isn't a number that is two less than 9. All right. Now letter C is a little bit easier. Write a calculation and a result that represents the sum of negative three and 8. The sum of that just means we're adding them together. So really there, negative three plus 8, or 8 plus negative three would work either way. We have a result of 5. Letter D, write a calculation that and a result that represents the difference of 20 and 12. Now again, when we're talking about a difference, that subtraction. And we have to go in the order that the numbers were given 20 -12. The difference of 20 and 12 will be 20 -12. Again, the result is 8. All right, now we're going to transition to variables, where it becomes way, way more important. But what I want us to do is to keep looking back at what we did in a through D, because it's going to mimic what we had here. Letter E, if X represents a number, write an expression that represents a number ten greater than X well, if we're at X and we want to number ten greater than it, then that's going to be X plus ten. Make sure you're not writing down ten plus X are they equivalent? Yes. But a number that is ten greater than X is just X plus ten. Letter F, if N represents a number, write an expression that represents a number that is 5. Less than N again, this is absolutely critical. A number that's 5 less than N well, if I had N and I'll subtract 5 to get a number that is 5 less than it. All right, letter G if Y represents a number, write an expression that represents the sum of Y and a number one greater than Y okay, so we're at. What are we adding? We're adding Y and a number one greater than Y so that's going to be Y plus, Y plus one. All right? So the sum of Y and a number one greater than Y I don't want to do anything else with that. It can eventually be written as two Y plus one, but all I'm looking to do here is translate. Not simplify or anything else. All right, letter H if N represents a number, write an expression that represents the difference, then difference. Keyword here. The difference between a number, one larger than N and a number one smaller than Ed. Well, here's a number one larger than N here's a number one smaller than, and here's the difference. Now, it's not quite right though. Any time we're talking about a more complicated number, it sure doesn't hurt to put it in parentheses. And in fact, in this problem, it's absolutely essential. This is very different than this. Well, maybe not very different, but slightly different. This is N plus one minus N minus one. This is N plus one minus N minus one. All right. Parentheses are key parentheses are key. Okay? So I really want you to think about what we did in those exercises, especially in G and H with the parentheses use very, very important. All right? And I am going to clear out the text, okay? So copy down what you need to pause the video. Here we go. All right. Let's go on to exercise two. Now, we need to be able to deal with addition and subtraction, no question. And in addition to subtraction, I have words like some difference, greater less, things like that. But we also need to be able to deal with multiplication. And division. So let's look at a little bit of that. Exercise two, letter a right an expression for a number that is 5 times greater than two. Right next expression for a number that's 5 times greater than two. Well, 5 times greater would just be 5 times two. Now, the actual number is ten, but it didn't actually ask me to do that. That's a number that is 5 times greater than two. Letter B, if N represents a number, then write an expression for a number that is twice it twice N means that we're multiplying by two. So that's just going to be two times N same thing. 5 times greater than two, twice the number. Ah, here it gets a little bit tricky. Quotient or ratio. That's division. Asian. Very important. Right in expression for the quotient or ratio of 12 and three. Lots of different ways we could write this. We could write it with a little colon notation. We could write it like this, but the most appropriate way is to write it with a fraction bar. So the way that you're going to see it 9 times out of ten algebra. So anytime you see quotient or ratio of blank and blank, the first one goes in the numerator, the second one goes in the denominator. That's just conventions, just the way it is. So letter D it says if X represents a number right in expression for the ratio of X to 5, well, simple enough. X divided by 5. Okay? Well, it's a space in that problem that I probably didn't need. Anyway, at least it gives me somewhere to put my head as opposed to the last problem. Always an issue on these screencasts. So I'm going to clear out the text pause the video if you need to. All right, here we go. Okay. Next exercise. Translate each of the following statements into algebraic expressions. So now these are going to become a little more complicated. We have to think about order of operations and keep in mind. Keep in mind. That multiplication and division will always occur before addition or subtraction and less you somehow circumvent order of operations so you get around order of operations by using parentheses. Okay? So let's take a look at that array. If X represents a number, then write an expression for a number that is three more than twice the value of X so we want a number that is three more than twice the value of X well, three more, that's going to be addition. Twice the value of X, that's going to be multiplication. So a mass otherwise told we should do that multiplication first, and there's twice the value of X and there's three more. All right? Not three plus two X, but two X plus three. Three more than twice the value of X let her be. If N represents a number, then write an expression for two less, then one fourth event. All right? Two less than one fourth event. Well, again, one fourth event can be thought of as either division or multiplication. I think I'm going to keep with multiplication. One fourth of N is one fourth times N and then two less means we're subtracting two. That's really important because in terms of the wording, the two less comes first. But when we find one fourth of a number and then something that's two less than it, we will find the one fourth first. Okay? Letter C, if S represents Sally's age, and her father is four years less than 5 times her age, then write an expression for her father's age in terms of the variable S okay, so wait, wait. Her father is four years less than 5 times her age. While her age is S 5 times her age would be 5s and four years less would be minus four. So there it is. 5s minus four. Letter D, if X represents a number, then write an expression for three times the sum of X and ten. Now this is interesting. Because I want to do three times. That's easy enough. The sum of X in ten. So I want to add X in ten. That's the sum, and then I want to multiply by three. Please note that's not the same as three times X plus ten. These aren't the same here. This would be ten more than three times X this is three times the sum of X and ten. All right, real important those parentheses. Okay, pause the video if you need to to write those down. All right, I'm going to clear it out. Gone? And continuing on. Let's keep doing it. So the more this translation we work with, the better it will get. It can be tough, though, okay? Keep at it. What I'd like you to do in these last four, they're not the last four, but in the next four is to try them on your own. So pause the video now and see if you can do a little bit of translation. All right, let's do it. Let it read. If N represents a number, then write an expression for 7 less, then four times the difference of N and 5. So we're going to find the difference of N and 5, we're going to multiply it by four, so that's four times the difference of N and 5. And then we're going to have 7 less, wow. Look at that. That is definitely tricky. 7 less than four times the difference of N and 5. Let's look at letter F if X represents a number, then write an expression for the ratio of three less than X to two more than X well, all right. Here's three less than X ratio means division. So the ratio of three less than X to two more than X. Here we go. X minus three divided by X plus two. Letter G if X represents a number, then write an expression for the sum of twice X with twice a number one larger than X okay, so we're adding some things. What are we adding? We're adding twice X that's easy. With twice a number, one larger than X well, here's a number one larger than X and here's twice that. Really important on the parentheses, and then I'm going to sum them together. So there's that sum, there's twice X and twice a number one larger than X again, we could simplify that, but nothing in the problem tells us that we have to. Let's take a look at letter H if N represents a number, then write an expression for the quotient, again, division. Of twice N with three less than N that's not so bad. Here's twice N with three less than N. All right. Wow. Some of these are quite complicated. Like letter E and letter G half an H, not so bad. But E, G, special E, look at that. All right, let's clear this out and attack the last two in this particular problem. Here we go. I'm going to scrub it. All right, two more. Translate each of the following statements into algebraic expressions. Again, I'd like you to pause the video right now if you haven't already tried letter I and letter K. You know, one wonders why I left the letter J out. Weird. Apparently I just don't like the butter J anyway, translate each of the following statements into algebraic expressions. All right, so pause the video and take a shot at it. All right, let's go through it. Number three letter I if Y represents a number, then write an expression for three quarters of the difference of Y and 8. Well, the difference of Y and 8, that's pretty easy. That's going to be 8 minus Y now if I want three quarters of of is a good word to look for when it comes to multiply. Right? So when we say one fourth of or one half of or three quarters of we're multiplying. So we're taking Y -8, we're multiplying by three fourths. Three quarters. All right, let's look at letter K if X represents a number, then write an expression for one half, the sum of X and four. All right, well, the sum of X and four is easy enough, X plus four, and then if I want one half of it, I'll just have that. Likewise, we could just do this X plus four divided by two. That's also a good way to do it. But kind of sticking with the last problem. I think that makes some sense. All right, so watch for that word of sometimes it means multiply, sometimes it doesn't, as in this situation, right? The sum of X and four. All right, I'm going to be clearing out the text. So pause the video if you need to. All right, let's get rid of it. Moving on. All right, the last problem we're just going to play around with a pattern again. See if we can see some structure in a problem, little pattern coming out that we might be able to prove algebraically. Exercise four. Need patterns can occur repeatedly when you play around with numbers. A fairly easy one occurs when you add a number to one less and one more than the number. All right, so let's make sure we understand what we're doing. When you add a number to one less than the number and one more than the number. Do this for a few numbers X and record the results. Then prove a general pattern by running an expression for the sum of a number with one less than a number and one more than it. So watch, let's grab a random value of X, let's go with 5. Okay? And what this pattern, what we're looking for is four plus 5 plus 6, right? A number less than it. A number one more than it. Four plus 5 is 9, plus 6 is 15. Let's go with a different number. Let's go with, I don't know, 8. So then we'll have 7, a number one less than it, plus 8, plus 9, number one more than it. 7 plus 8 is 15 plus 9 is 24. Okay. Um, I'm not sure I see the pattern yet. Remember, what we're trying to do is think about how the 5 relates to the 15, how the 8 relates to the 24. Let's go with another one. Let's go with ten. A number one less 9. Plus the number ten plus number one more 11, 9 plus ten is 19. Plus 11 is 30. And why not? One more. The calculations I was supposed to put down here, but I just totally ignored it. I don't know. Let's go with 20. Why not? Number one less. Plus 20. Plus a number one more. 19 and 20 is 39. Plus 21. Is 60. So what's the pattern? It looks like we always get three times. Our original variable. Or three times our original value, right? Three times 5 is 15. Three times 8 is 24, three times ten is 30, three times 20 is 60. So here's how we'll do it. We'll let N be our number. That's simple enough. So then what we're doing is we're taking a number one less than it. Plus the number lesson number one more than it, right? Now, we don't really need these parentheses here. We can remove them due to the associate property associative sorry property. Not the associate. The associated property. Then we can use the commutative property to start rearranging things. N plus N plus N plus negative one plus positive one. Those three put together and will be three N and these put together will be zero. So there it is. Right? We will always get three times our original number. And that's kind of neat. We can see that in the pattern. So let's clear out the text and wrap this one up. All right, so today we worked at trying to turn English into algebra, specifically turn English descriptions of algebraic expressions into algebraic expressions. This is going to be critical as we move forward and we look at a variety of word problems over the span of this course. So work on it hard, make sure you do that homework. And you'll get better at it. I promise. All that takes is hard work, but every single person can get better at math if they just keep practicing at it. All right, no matter how weird it may seem. Okay. Well, thank you for joining me for another common core algebra one lesson. My name is Kirk weiler. And until next time, keep thinking. And keep solving problems.