All right, guys. Here we go. Time to dive into math 8, and we're going to do this in a way like what I've done before, but hopefully better. Okay, so we're going to learn together as we use the textbook. I know you're probably like, "Mr. field. I know you've used the textbook before, but usually for my videos, we do online work like or Khan Academy. So, you can see some parts of the textbook we will do specifically in class. Some parts of the textbook we will do now and then review in class and annotate. So, are the are you ready? We'll do that in class so I can watch you do your math. the spark your learning we'll do that in class but let's look at our different examples so we have one a b two 3 all these or these examples are how you learn okay so we need to make sure that you all have it in your notebook so we have a textbook but you can see there's not a lot of extra space. So, we're going to keep a good notebook and let's get started. Okay. So, whatever you see me write down in this notebook, you also have to have it doesn't have to be green, but other than that, it should be the same. Okay? So, we're going to write math 8 at the top. We're going to write the date. So you always when you take notes, you put your class, you put the date, of course, you can put your name, right? I guess you generally put your name up there. We'll just put it over here. All right, there we go. And then right here, the title. Notice the title. I didn't put math or whatever notes. Some people title their notes notes. We titled it module 10 real numbers. And let's see why. Because oh, you know what? This is the reason I titled it module 10 real numbers. But now that I'm looking at these notes, I'm going to put one extra part of my title. I'm going to put lesson one, understand rational and irrational numbers. Now notice how I said it and I am writing it. That is because I do understand that my writing is not very good. Okay. But you can see it right. Lesson one, understand rational and irrational numbers. Okay, now we're going to go to the next page. And the next thing we're going to put in our notes is the vocabulary. So you can see we have one, two, three, four vocabulary terms. So, I am going to take time and you don't have to watch me because I'm going to pause the video. But when you un well after I unpause the video, I'm going to have all four of these terms and their definitions written in my notes. One second. All right. Oh, my hand hurts from all that writing. But now you can see and you can go ahead and just autofocus it here. You can pause this and you can write this as well. If I we're all right. All right, we all do this. Rational number any number that can be written as a ratio in the form a over b where a and b are integers and b is not zero. Okay. So that means you can write it as a fraction. A over B, this is a fraction. Now you can also write rational numbers as decimals. And there's two kinds of decimals. So notice when I did the rational number, I put a closed dot there. And now I'm indenting and I'm writing the two different types of rational numbers. Terminating in in their decimal form. Terminating decimals. That's a decimal that has an ending and repeating decimals. A decimal that has no end but has a repeating pattern. So if a number is not rational, it is irrational. That's a number that cannot be written in the form a over b. So you cannot write this number as a fraction. That means you can only write it as a decimal and not even that because um irrational numbers actually don't have an end. So we can only write their approximation. All right. So where a and b are integers and b is not zero. All right. Sorry about the little grease stains there. Mr. Phil didn't wash his hands after lunch. Okay. Sorry. So rational examples, let's look in the book and we see 3 over8 is equal to 0.375. Let's write that in our rational circle. 3 over8 definitely a rational number cuz it's a fraction and it equals 0.375. And let's just draw an arrow to that. What kind of decimal is that? That's a terminating decimal. So we'll write terminating. Okay. Now let's see if we can find a repeating decimal out of the oh this is another good one. 7 is 7 over 1. That's why that's a rational number. 0.2 is equal to 1/5. So see you can start out with a fraction and find out its decimal or you'll learn to start out with a decimal and find out its fraction. So they're both are two equal ways of writing the same quantity, right? And this one right here, 1 nth is a fraction. So put it in the rational circle, but it has a decimal that goes on forever. One, one, one, one, one. All these ones, and they never stop. So we put a dot dot dot. or you could put 0.1 with a bar over it. Both of those mean that it's not going to stop. Now, irrational numbers. Pi is your first one. You should know about pi. It's 3.14159 26535897932 and it goes on forever. Talk more about pi when we talk about circles. and it's not exactly 3.14. So, we put the wavy lines, but it's first three numbers are 3.14. So, that's an approximation. Um, and I'm going to put one other example even though I don't see it in the book. I know it as a math teacher. The square root of two is an irrational number and so is the square root of three. And you might say, what about the square root of four? But actually the square root of four is rational. Why? Because it's two. So these ones are decimals that are irrational. If you put them in the calculator, you'll see that. All right, let's move on. I can just talk all day about these things. So now, let's go. Like I said, we're going to do the four examples. And I have a time stop. So that's good for you and me. So if we don't get through all four, we'll get through what we can. Is every number rational? Well, we already know the answer is no. Consider the decimal 1.3453 blah blah blah. Does this decimal appear to have a repeating pattern? Okay, instead of saying blah blah blah, let's look a little closer. 3 4 5 3 4 5 6 3 4 5 6 7 Well, it did have 345 and then 345 again, but then I was saying no. Oh my goodness, what am I doing writing in the text? Okay, so zero. So, we're going to write when we do the textbook examples, we're going to draw a little line indicating I'm done with my vocabulary and we're going to write example one and then we're going to copy down the number. Now 1.345 345 634 567. Okay. And the question is is this rational? And the answer is no. There is not a repeating pattern. There's not a repeating pattern. It only repeats twice. When we say repeating pattern, we mean repeats over and over again. So that's a no. But now, do you think oh it's rational? So yeah. So we just answer both questions. That was a and now we see b. Here you learned that pi is the ratio of the circumference of any circle to its diameter. The decimal value of pi is shown below. Pi is an irrational number. But it can be written as a ratio. How could this be? Well, that's a good question, isn't it? So whenever we try to write so what they're saying is a good question. So, let's talk about Whoa, no, somebody. Okay, let's talk about pi. And so, here we are. Pi. Um, we know if we have a circle all the way around the circle is called the circumference. And the and the diameter of the circle is all the way across the circle. Okay. And so pi is equal to the circumference C divided by D. But we cannot actually get the exact data. We can't actually measure the exact circumference and diameter because if we did then we would have a ratio. And so what we find is every time we try it ends up being an irrational number. So the closer you get to the actual circumference and diameter, the closer you get to pi. It's a very interesting situation. So, we could just say either C or D must be approximate, meaning you didn't actually measure the thing exactly. Okay, now let's keep going. There are two ways to write repeating decimals. You can use an ellipsus or you can use the over bar. So, we talked about that. So, let's just turn the page in our notes and we'll just make a section just for that here. We're going to get used to this here. All right. So, right here, there's so much to put. So, I'm going to take and I'm going to write ellipses. How do you spell that again? Ellipsis overbar. Okay, so we got 0.111 dot dot dot. We could do that or 0.1 without we could write 0.235 235 dot dot dot or 0.235 like that. And we could write, let's just do one more, 0.2 4 44 dot dot dot or 0.24 and just the overbar is over the four. Okay? So we don't have to have the whole thing repeating for it to be rational, but it has to end up repeating forever. If you remember the first example, we only had the pattern repeat twice and then it stopped repeating. In this one, it doesn't start repeating, but it ends up repeating forever. And that's okay. All right, guys. We're doing good. We got about five more minutes. And so, let's look at step it out. Okay. A basketball player's free throw percentage is 82.5% or 0.825. Write this as a fraction. And so you see they gave us um they gave us some boxes to fill in and they also gave us some vocabulary but we should know that vocabulary. Okay. So let's draw our line and we'll write ex2. And then I'm going to write 0.825 = 825 over Whoops, there we go. And do you know what is over? Well, this is a terminating decimal. So I'm going to indicate that. And so anytime it's a terminating decimal, you look at the place where it terminates. And what place value is that five in? That is in the thousands. So we're going to put it over 1,000. Oh, I thought you were going to be opposite yourself. All right. Did they give us any other examples? So now now here uh what are they doing here? Well, we have equal division. It says write the con the fraction and lowest terms. Identify the greatest common factor GCF of the numerator and denominator. Divide the numerator and denominator by the GCF. So you know what we got to do? We got to reduce this fraction. Right? So 825 and a,000 what is the GCF of both those numbers? You may or you may not know. If you ever need to figure out what to divide a number by, you can always start with two, right? 1 2 3 4 5. Can we divide this by two? The answer is well a th00and you can divide by two. That's 500. But this number, no. So then you could try three and I don't think you'll have a lot of luck because that five there. And then you could try four, but maybe you already knew five is a good one. And if you know the GCF, you know, maybe we can divide them both by 25, then you could do that. But if you can't figure out the GCF, you can always just start low and then divide again. So let's do 825 / 5 is 165 and 1,000 / 5 is 200. And you should notice, hm, I think that I can divide both of these by five again because of the they that one ends in five and that one ends in zero. So 165 / 5 is 33 and 200 / 5 is 40. And now we have our reduced fraction 33 over 40. So 0.825 825 is 33 over 40. And guess what? Wait, you can check. If you do 33 / 40 on the calculator, you should get 0.825. If our math is correct, look at that. Look at that. Boom. Boom. Boom. I did it. Okay. So happy. All right. Now we're going to do example three and we're going to call it a day. We'll have to do example four in class. So example three is converting a repeating decimal to a fraction. And this is an important one. So let's read it and then we'll write it in our notes. Write 0.5 repeating as a fraction. Let x be the given decimal. Write the first few digits few first few repeating digits. Multiply both sides of the equation by 10 so that the repeating digit appears to to the left of the decimal point and it's just a repeating part. Subtract an expression equal to x from both sides. Solve for x. All right, let's do it. So this is an important one. This is example three. So, we're going to write example three. And our first one is 0.5. We want to write this as a fraction. So now we're going from a repeating decimal to a fraction. So what we do is we say let's let we don't know what the fraction is, so we call it x. Okay. And now we need to multiply this both sides of this equation by 10. And then that will give us x * 10 is x * 10 is 10 x and 0.5 repeating time 10 is going to be 5.5 repeating because you take that decimal and you move it over to the right one. This is this 0.5 repeating is really 0.55 five55. So if we multiply it by 10, we're moving the decimal point over one and then we get that 5.55. So or 5.5 repeat. So now I have two equations this one and this one. And what the book is saying is I can take 10 x = to 5.5 repeating and I can subtract from it that first equation x = to 0.5 repeating. And what what happens when I do that? First of all 10 x - x is not 10 because this is actually 1 x. 10 x - 1 x is 9x and 5.5 repeating minus 0.5 repeating is 5. And now I have a one-step equation. So I'm going to divide both sides by 9. x = 5 9th. And again I'm just going to prove to you that it's true. 5 / 9 is 0.5 repeat. Pretty cool, right? And of course, we don't have to use a calculator to check our work. I could do the long division uh 9 into five, but I'm not going to do that right now. I'm just saying you could. All right. Two pages now is pretty good. CMIT. We're going to do another page. We're going to do uh example 3B. Not the whole page, just one more. And this one is writing 0.18. So 0.18 repeating we want to write as a fraction. Okay. So again we set it up equal to x and this time we need um we need to move that decimal place. Now, if we only moved it one time, seeing as our pattern is two digits, we would move it into the middle of the pattern and that's not enough. We need to move it the full length of the pattern. We need to move it two times. That means we need to multiply by 100. So you need to see the length of your pattern and move it that many step in four minutes and just not okay so now that means we have 100x = 18.18 repeating and now I'm going to subtract these two so 100x = 18.18 repeating - x = 08 18 repeating. And when I do that, I get 99x = 18. Divide both sides by 99 and you get x = 18 over 99. All right. So in class we will add success criteria to these examples so that you can ask questions and we can kind of write out some steps and and criteria you can use to be successful on every problem. But you should come into class with these notes. So you coming into class with the vocabulary some examples of the v vocabulary. Example one 2 3 A and B and you're ready to go. All right.