﻿Hello there, today, you'll be learning about surds with me, Dr. Saada. For today's lesson, you need a pen and paper. So when you're ready, let's begin. To start today's lesson, I would like you to try this: Decide if the following are rational or irrational numbers and explain why. So in your book, I would like you to write down the numbers that you can see here on the screen next to each number, write down rational or irrational, and explain why. This is something that we have done in lesson one. If you're feeling confident about this task, please pause the video now and make a start. If you're not feeling very confident about this, I'm going to give you some support. Let's recap. What does rational mean? Rational numbers, are numbers that can be written in the format of a fraction a/b, where both a and b are integers, they're whole numbers. Whereas irrational numbers are numbers that cannot be written as a fraction a/b, we cannot write them like this. Rational numbers are whole numbers recurring decimals or terminating decimals, whereas irrational numbers are numbers that carry on forever and do not end, the decimal does not terminate. Now with this little hint, can you please pause the video and have a go at this. Well done. Let's go through some answers. Square root of nine, what did you write down? Excellent, it's equal to three, and therefore it is? Brilliant, it is rational number. Next one: Two multiply by square root of nine. What did you get? Two multiplied by three which is six. Excellent. And therefore, yes, you're correct. It is a rational number, it's an integer, it can be written as six out of one. I wonder what did you write down for square root of eight. Really good, the square root of eight is an irrational number, it does not terminate as a decimal. What about square root of eight, all squared. Well, square root of eight, if square it I'm doing the inverse operation to the square root, I end up with an eight, really good, good job. And therefore, it must be rational, it's an integer. Next one, square root of nine divided by square root of 16 is the same as three divided by four. And remember, three divided by four is the same as 3/4, because that mean column in the fraction is the same as division. 3/4 is written as a fraction, therefore it is rational well done. Next one, let's look at square root of nine, multiplied by square root of 16. The square root of nine is three multiplied by the square root of 16 is four, it gives us 12, it gives us a whole number, and therefore it's a rational number. Let's look at the last one, nine multiplied by the square root of eight. Well, nine multiplied by, we have a rational multiplied by an irrational number. If you remember in lesson one, I told you if you multiply a rational by an irrational number, you always end up with an irrational number, okay? So this would give us an irrational number. Good job, well done. Let's make a start on new learning. Okay, and in today's lesson, we are going to be learning about surds. I need you to understand what a surd is and how to estimate the value of a surd. But before we do that, there are a couple of things that I need you to be really competent with. Let's make a start. Can you list the first 10 square numbers for me? Pause the video and have a little think about it. Well done, this first 10 square numbers are one, four, nine, 16, 25, 36, 49, 64, 81 and 100. Really good job. What is the square root of 20? If we look at our square numbers, we know the square root of 25 is five, we don't have a 20 there, how can we use Our square numbers to help us estimate what the value of square root 20 is? Well, to start with, I know that the square root of 16 is four, 'cause four squared is 16. I also know that the square root of 25 equal five. So the square root of 20 must be somewhere between these two here. So square root of 20 is approximately 4.5. Okay. And now, let's list the first 10 cube numbers. Pause the video and have a little think about this. Okay, so first cube numbers are one, eight, 27, 64, 125, 216, 343, 512, 729 and 1000. Well done if you got this correct. Now, what is the cube root of 20? Again, very similar to what we've done with the square root, we use our cube numbers to help us estimate the value of a cube root of 20. So looking at the numbers that we have here, I know that the cube root of eight is equal to two, because two multiplied by two multiplied by two is equal to eight. And I also know that the cube root of 27 is equal to three. So the cube root of 20, is somewhere between the cube root of eight and the cube root of 27, and therefore it must be between two and three. A good estimate would be around 2.5. So I can now say that the cube root of 20 is approximately 2.5. And this leads us really well into the definition of a surd. So let's read it together. A surd is a number. It's really important that we understand it's a number, it's on the number line, it can be represented on the number line. So a surd is a number that includes a root symbol, so it has a root. It could be square root or a cube root, or root four, any root. Surds are used to write irrational numbers precisely because they cannot be written exactly in decimal notation. So we cannot write the number using decimal because it keeps on carrying... It carries on forever, it doesn't terminate, It doesn't have a specific pattern, so it's not a recurring decimal. So it's not really precise, and we use surd form to write it down precisely. And I'll show you a couple of examples about this. Okay, so a surd is a number that includes a root symbol. Surds are used to write irrational numbers precisely because they can not be written exactly in decimal form. For example, square root of two, square root of 53. What is the square root of two? If you put that into your calculator, it gives you something like 1.41421356 and it keeps going on and on. So if you want to say that value to someone, and if you want to write it down, it's not precise. How many digits are you going to write after that decimal point? When will you stop? So it's not very precise, it's approximately that much, because you're not going to write every single number there. So the more precise way of writing it down is square root of two, to write it down with the surd notation with the root and then the two underneath it. Now remember, that here, the root symbol is actually telling us or representing a number, it's not representing an operation. I'm not telling you got and square root it for me, it's telling us the number, the value of that number. Non-examples, so these are not surds. The square root of nine is not a surd, because it has a value of three and three is a rational number, it's not an irrational number. The square root of 100. Again, it's 10, it's a rational number. So square root 100 is not a surd. The cube root of eight, what multiplies by itself three times to give me eight? The answer is two. And again, two is not an irrational number, it's a rational number, therefore, the cube root of eight is not a surd. In order for it to be a surd, it has to be a decimal that carries on it does not terminate. Okay, and now it's time for you to have a go at the independent task. Question number one, which of the following is not a surd. I've written some numbers there for you, I want you to have a look at each of them, write them down in your book and write down next to each one, surd or not a surd. Question two, estimate the value of the following surds. How many ways can you represent this? I've done couple of examples about estimating the value of a surd with you in the previous slide. So if you need to, you can always go back to it and double check. Now, pause the video and have a go at this, when you're done press play again so you can mark and correct your work. Okay, let's mark and correct this together. Square root of nine. What did you write down? Three, excellent, and therefore, it is not a surd. Well done, good job. Square root of 100. Good job, it's 10 and therefore it's not a surd. Square root of 18. Good job, this one is a surd. Square root of 81. Nine, and therefore, it is not a surd. The square root of 36. Really good job. That's six and therefore it is not a surd. Square root 57, 57 is not a square number, therefore, it's going to be a surd. So the square root of any number which is not a square number is going to be a surd. Square root of five, excellent this is also a surd BEcause that five is not a square number. What about the cube root of 125? What number multiplies by itself three times to give you 125? Really good, that's five and therefore this is not a surd. And what's the cube root of a 1000? What number multiplies by itself three times to give you 1000? Good job, so 10, and therefore, it's not a surd. And now let's have a look at this one. Okay, it was not part of the questions, but it's a really good question that students really struggle with a bit or make silly mistakes with. So if we look at square root of one, is this a surd or not? The square root of one is equal to one, therefore, it is not a surd, okay? It's really, really important that we understand this 'cause the square root of one is one. And now let's have a look at question number two. Estimate the value of the following surds. How many ways can you represent this? First one is square root of 30. In order to do this, you need to think about the square numbers, that are around the number 30. So I know that the square root of 25 is five, I can write that down. I know that square root of 36 is equal to six. Therefore, I know that the square root of 30 is approximately 5.5. It's somewhere between the five and the six. Now, how many ways can we present this? Well, I can write it down as approximately 5.5, I can write it using inequalities. So I can say that the square root of 30 is greater than five and is less than six. I can use inequalities for that. I can also draw this on a number line or present it on a number line. So if I have a number nine here, and I have five, and I have six there. I know that it's somewhere between the five and the six and with inequalities, if you remember how we would represent them, it does not equal to five, it does not equal to six. So I draw two circles and they are empty, I do not colour them in, and that's how I present it. Now let's have a go at the second one. The square root of 110. Again, think about square numbers that are around 110. What are they? Really good job. The square root of 100 is equal to 10, and that's very close to square root 110. Square root of 121 is equal to 11. Therefore, I know that the square root of 110 is somewhere between 10 and 11. And I can say that it is approximately 10.5. Similarly, if I want to represent it, I can represent that using the inequalities, so I can write down that the square root of 110 is greater than 10 and less than 11. I can do exactly the same thing on a number line, I can say if this is 10 and this is 11 on my number line, the square root of 110 is 10 is greater than 110... Sorry, greater than than 10, less than 11. So I can represent them using inequality on a number line. Let's look at the last one. What is the cube root of 100? Now, because it says cube root, we need to think about the cube numbers instead. So what cube numbers do we know that are close to 100? I've got 64, so, the cube root of 64 is equal to four. Next one up is the cube root of 125 and that is five. So now I know that the cube root of 100 is somewhere between four and five. So I can say that it's approximately 4.5. Okay, now I can do exactly the same thing with the representation. So I can say that the cube root of 100, so remember to write the three here for the cube root, is greater than four, and less than five. And I can do exactly the same thing about that number line, I can write four here, and I can have my five there and I can say that it's somewhere in-between. Obviously, I'm going to leave the two circles unshaded because it does not equal four, it does not equal five, okay? And that's my presentation. Good job if you've done these correctly. And this brings us to our explore task. How many surd lengths can we draw? I'm going to hint here, what I would like you to do is to draw lines on it that have a surd length. So it's not a rational number, it's not two or three units. I want it to be surd length. I'm going to give you a little hint here, think about by Pythagoras theorem, pause the video and have a little go if you're confident about this. If you need a bit of support, don't worry, I'll give you help. You can draw a line like this one, and think about Pythagoras theorem. It says that in a right-angled triangle, the sum of the squares of the shorter sides is equal to the square of the longer side or the hypotenuse of the triangle. So if I have this green line here, I can draw a triangle around it, a right-angled triangle. I can start with this short length here and then this short length there. The first short side is one unit, the second one is three units in this case. Now I want you to think about the square of those shorter sides. The square of this one is one squared, and what would be the square of the second... Of our shorter side? Well done, it's going to be a three by three, and that's nine. I can call this green side here, the diagonal that I drew, I can call it x, because I don't know the length of it, and I'm trying to find out. Now let's write this down. So I have one squared plus three squared is equal to x squared, well done. Now, I can simplify this and say I have one plus nine equal to x squared, but I don't want the x squared, I want the length x. So to find that out, I know that 10 is equal to x squared, therefore the length of x is square root of 10. And there we go, I've got what my first length here, that side here that I drew, and it has a surd length. Now, can you please pause the video and draw some other lines similar to this and work out the length. Try and choose ones where the length is going to be or give you a surd length. There are so many side lengths that you can draw on this grid that will give you surd lengths. I wonder how many you managed to get and draw. Now I'm going to go through one really interesting one, okay? So if I have this line here, and this time I'm going to call it m. I know again that the sum of the two shorter... The squares of the two shorter sides is going to be equal to m squared. So I'm going to start by constructing a triangle, right angled triangle around it. So if I go across here, that gives me three units, and if I go this side, it gives me four units. So I know that the square of three plus the square of four will give me squared of m, so I'm going to write that down. Four squared plus three squared equals m squared. Now four squared is 16 and three squared is nine. So I can write this down, 16 plus nine is equal to m squared, and 16 plus nine is 25. So I can write 25 equal m squared. Now, I need to find m, so the square root of 25 equal m. And I know what the square root of 25 is, it's five. So five is equal to m. And in this case, the m here, this length is not a surd. Whereas in the first example, x was a surd the square root of 10. So some of the lengths that you're going to draw, some of the lines that you will draw will end being surd lengths and some of them are not. Be really interesting to see which ones give you a surd onset and which ones do not. So, you will have an opportunity to draw many lines and practise with this question. I would love to see which lines you've drawn, and what side lengths did you get? Okay, and this brings us to the end of our lesson. Well done, fantastic learning today, you should be really, really proud of yourself and the work that you have been doing. What I want you to do now is make sure that you complete the exit quiz for me, please. And if you would like to share your work with Oak National and you would like me to see it, please ask your parent or carer to share your work on Twitter tagging @OakNational and #LearnwithOak. That's it from me today, have a lovely day and see you in the next lesson. Bye

In this lesson, you will learn what a surd is and how to estimate the value of a surd.

Maths

Y9 Maths

Key Stage 3

Year 9

Introduction to Surds

Surds and trigonometry

Number

THEME

UNIT

Rational and irrational numbers

Surds and triangles

Manipulating surds

Varying the ratio of side lengths in right angle triangles

The case of 30 and 60 degrees

The sine ratio

The sine and cosine ratios

The sine and cosine ratios for 30 and 60 degrees

Finding missing sides using sine and cosine for any angle

Using inverse sine and cosine to find missing angles

Sine and cosine graphs

Lesson overview: Introduction to Surds

Intro Quiz

Video

Presentation

Worksheet

Exit Quiz

Transcript

12 lessons in Surds and trigonometry: