Common Core Algebra I.Unit1.Lesson1.Rates, Patterns, and Problem Solving.By eMathInstruction

Hello and welcome to common core algebra one by E map instruction. My name is Kirk weiler. And today we're going to start our common core algebra one journey. Through this course, we have a 101 lessons, a 101 homework sets that will walk you through all the common core standards for algebra one. As denoted by the park end of year standards. If not that really makes sense to you, well, just know this, that if you're in New York State or another state that's using what are known as the park standards for common core algebra one, this course will cover all the targets. All the standards. Today, though, we're going to be doing unit one lesson one, rates, patterns, and problem solving. Before we get into that, let me remind you that you can get a copy of the worksheet and homework set that goes with this lesson. By clicking on the videos description or by going to our website at WWW dot E mats instruction dot com. As well at the top of every one of our worksheets don't forget there's going to be a QR code. That code can be used with a smartphone or a tablet to take you right to this video. But let's get right into it. Today, what we're going to really be doing is looking at some general problem solving and some general understanding of rates and ratios. And the first problem really gets us into it. It's important for you to be able to understand both multiplication and especially division when it comes to the rate at which things are changing or the rate at which one variable changes compared to another. So let's look at exercise number one. This is conceivably a problem that you could have been doing as early as third or fourth grade. So take a look at it. It says answer the following rate slash ratio questions using multiplication and division. Show your calculation and keep track of your units. Well, let's take a look at number one letter a if there are 12 eggs per carton, a dozen eggs per carton. Then how many eggs do we have in 5 cartons? What would you do to figure this out? Pause the video now just for a moment to think about this. All right, let's go ahead. Please note that I'm not going to pause long enough to allow you to actually do the problem. I'm only going to pause long enough to allow you to pause the video. So make sure that when I tell you to pause, pause that video, take as much time as you need. Remember, it's your dime. You know, go ahead, take as much time as you need to solve a problem. So 12 aches per carton, how many do we have in 5 cartons? That's going to be multiplication. So we have 12 eggs in every carton. This is the way that we would write that. Hopefully a little more neatly. Times 5 carbons, notice we can kind of think of something in the denominator in something in the numerator is canceling out, and that will leave us with 60. X we'll look a lot more at units as the course goes on. Take a look at letter B if a car is traveling 65 mph, how far does it travel in two hours? Go ahead and pause the video again. All right, we have another problem involving multiplication, right? Same idea. The car is traveling 65 miles in one hour. Right? How many visit? Traveling two hours, while the hours cancel, and it leaves us with a 130 miles. That almost looks like a three. Take a look at letter C and letter D these are going to be a little bit different. Let's see if you can pause the video now and solve both of them. All right, let's take a look. Letter C if a pizza contains 8 slices and there are four people eating, how many slices are there per person? As probably a problem that's quite easy for you, especially if you eat a lot of pizza like I have, I live in New York. I also lived in Illinois. Chicago and New York, great pizza. Anyway, so this simple enough, right? This is actually division. We have 8 slices of pizza divided by four people, right? Of course 8 divided by four is two, but look at how the units work out. Two slices per person. Really cool. Pause the video again and why don't you work out D if you haven't already? If you have, then we'll go through the solution in just a moment. All right, if a biker travels 20 miles in one hour, how many minutes does it take per mile traveled? Wow. 20 mph, how many minutes does it take per mile traveled? Well, what we know is we know that in 60 minutes, we've traveled 20 miles. Because we really want to know the minutes per mile. So what we see when we do the division is it takes us three minutes. Per mile travel. That one's a little bit trickier. Think about it for a moment. All right, now in these videos, whenever we get to a point where the screen is completely filled up with text, I'm going to always give you a moment to write down something in case you haven't been writing it down the entire time. And then I have to scrub out the text. Okay? So pause the video if you need to. All right, now I'm going to scrub out the text. We just have to get used to the way these videos work in the first one. Let's go on to the next problem. All right, so now let's try to extend that idea of rate and try to work a little bit where we know that somebody is traveling at a constant rate. Let's take a look at exercise two. Take a moment to read it. All right, now I'm going to read it. A runner is traveling at a constant rate of 8 m/s. Make sure you really understand that, right? That means after one second, the runner has traveled 8 meters, right? And then after another second, another 8 meters after another second, another 8 meters, et cetera. We're trying to answer the question how long does it take for the runner to travel 100 meters? Okay? Let it raise this experiment solving this problem by setting up a table to track how far the runner has moved after each second. Pause the video right now and see if you can do this on your own and then we'll fill it out, okay? All right, let's go through the problem. Well, if the runner is traveling at 8 m/s, then after one second, the runner has clearly traveled 8 meters. After two seconds though, the runner has traveled 8 times two or 16 meters. That's what it means to do 8 m/s. It's sort of like the eggs in the carton, right? 12 eggs per carton, 8 m/s. So after 5 seconds, the runner has traveled 40 meters. Actually, even though the units are denoted here, I'm going to put them down again just so that we really have them. And after ten seconds, the runner has traveled. 80 meters. Now, by the way, we're trying to figure out how long it takes for the runner to travel a hundred meters and clearly what we know is it takes more than ten seconds. Now here's where the tools of algebra start getting used. Letter B asks us to create an equation that gives the distance big D that the person has run if you know the amount of time T may have been running. What we always want to do when we create an equation like this, when we're modeling a real world situation, is we want to take the data that we've already come up with, the pattern that we've already established. And what we want to do is just write it down symbolically, right? And what did we do? We kept calculating the distance that the runner traveled by taking this 8, which was a constant, and then multiplying it by how much time we had been traveling. So to write that down symbolically, we would say that the distance that we've traveled will always be that 8 times the amount of time we've been traveling. Remember, if we want multiplication, we simply put the two numbers or the two variables beside each other. So there it is. Distance equals 8 times time. Letter C says now, set up and solve a simple algebraic equation based on B that gives the exact amount of time it takes for the runner to travel 100 meters. Pause the video right now if you think you know how to do this and if not, we'll be showing you how in just a moment. All right, let's go through it. Well, algebra says, look, if I know that the distance I want to run is a hundred meters, then I'm going to substitute it right into this equation 100 equals 8 times T now in order to solve this and we'll be doing a lot of equations solving in the future. We're going to divide both sides of the equation by 8 because 8 divided by 8 is one. We get our T all by itself. A hundred divided by 8 might not be the nicest number, but you can use your calculator to figure that out. That'll end up being 12.5. So it takes 12.5 seconds, never forget your units. To complete the race. That seems pretty reasonable. Pretty fast runner. Pretty fast. All right. Well, I'm going to scrub out the text so pause the video now if you need to write anything down or think about anything. All right, here we go. It is scrubbed. Let's keep moving on. All right. So mathematics and algebra are used to model situations. A lot of times in those situations you have multiple variables that are changing. And you want to try to model using equations, variables, constants, rates, and all of your basic operations, addition multiplication division subtraction. You want to model what's going on. So let's take a look at a slightly more complicated scenario. We've got a man walking across a 300 foot long field at the same time that his daughter is walking towards him from the opposite end. Okay. So we got the two people walking towards each other. The man is walking at 9 feet per second, right? So for every second that passes, he's going to travel 9 feet. And the daughter is moving at 6 feet per second. What we want to determine is how many seconds it will take before they meet somewhere in the middle. So they're booking along, and at some point in time, boom, they meet in the middle. All right. So let's draw a little diagram. Diagrams and pictures are really helpful. Students tend to dislike them because it causes them more work. But I love them. So we're going to see how wonderful of a drawer item. Here's the guy, the dad, here I'll give him a little beard. See? That's a beard. Okay, here's his daughter, and I'll make her a little bit shorter. And they're walking towards each other. He's walking at 9 feet for every second. And she's walking at 6 feet for every second. And somewhere, they're going to meet in the middle. All right? We want to figure out how long that's going to take. So let's do a little bit of what we did before. What I'd like to do is I'd like to fill in this tape. I'd like to fill in how far the father has walked after one second, how far the daughter has walked after one second, and then what the total distance was that they walked. Okay? And that's pretty easy. After one second, the father has walked 9 feet, and the daughter has walked 6 feet, so they've walked a total of 15 feet, right? 9 plus 6. Is 15. What I'd like you to do is finish filling out that table. Think about what we did in second problem, and it's very, very similar. All right, let's go through it. Well, after two seconds, the father has walked 9 times two or 18 feet. The daughter has walked 6 times two or 12 feet, and together they have walked a total of 30 feet, right? 18 plus 12. After 5 seconds, and again, these times are just picked at random. What we're trying to do is we're trying to establish a pattern after 5 seconds, the father has walked 45 feet, the daughter has walked. 30 feet, and so they've walked a total of 75 feet. And finally, after ten seconds, the father has walked. 90 feet, the daughter has walked 60 feet. And together, they've walked 150 feet. All right, so we're just getting a feeling for what's going on physically here. Okay? Part B, read that over. Then we'll read it together. Part B asks, what must be true about the distances, the two have traveled when they meet somewhere in the middle. So when they meet at that magic moment that we're looking for up here, right? At this point in time, what must be true? I'd like you to pause the video. And take a while to really think about this. And let me give you a hint. They haven't met yet after ten seconds, and I can tell that. By simply looking at this value. Pause the video now and think about letter B. All right. Well, hopefully what you figured out is that when they meet in the middle, the total distance. Must equal 300 feet. There's a total of 300 feet that they're traveling, right? There's 300 feet that separates them. So when they meet in the middle, the daughter and the man must have walked a total of 300 feet. All right, I'm going to clear out the text, scrub it out. So write down anything you need. All right, here we go. All right, let's keep going on this problem. So I've recopied the problem at the top of the page. Often I'll do this because although you might have the worksheet sitting in front of you, I can't get it all on the screen. All right, so just kind of travel with me along this journey. Letter C says create equations similar to exercise three to predict the distance the father has traveled and the distance the daughter has traveled. So this is going to be very, very simple. All right? We want one equation for the father. And we want one equation for the daughter. Now keep in mind there are equations here. We want an equal sign. We want the quality of two things. So for the father, this is going to be simple, right? The distance he's traveled will be the rate 9 times the amount of time T so why don't you write down the distance the daughter has traveled. All right, that should be easy. The distance the daughter has traveled, the 6 T all right, finally, letter D says create and solve an equation to predict the exact amount of time it takes for the daughter and father to meet in the middle. All right, so why don't you pause the video now and see if you can figure that out. Go all the way, create an equation. Solve the equation and tell me how much Italian it takes for those two to meet in the middle. All right, let's go through it. Well, what we knew was that their total distance had to be equal to the total distance must be equal to 300 feet. But the total distance is given by the father's distance, 9 T plus the daughter's distance, 6 T so when I add 9 T to 6 T, I act to get 300. If you remember a little bit from 8th grade math, 9 T plus 6 T, we can combine what are known as light terms to get 15 T, that will equal 300. And then we can pretty easily solve that by dividing both sides by 15. And what we find is that it will take 20 seconds for them to meet in the middle. All right. That's it. A little bit of modeling with some simple algebra. Some simple rates. I'm going to scrub out this text, so right now anything you need to and then we'll conclude the lesson. All right. So a basic appreciation and understanding of multiplication and division are very important in algebra one. This, though, was just an introductory lesson to get our feet wet, if you will. To sort of think about variables and modeling and problem solving and some patterns. We're going to be exploring a lot more about the basics and fundamentals of algebra in the coming lessons. So if a little bit of this was confusing, if you didn't remember it all from last year, that's okay. I want to thank you for joining me for the first common core algebra one lesson by E math instruction. Until next time, my name is Kirk weiler, and I want you to keep thinking and keep solving problems.