HK4 IDENTITY
This eighth grade mathematics lesson focuses on equations that are identities. It is the first session in an eight-lesson unit on identities. The lesson is taught in English and is 32 minutes in duration. There are 42 students in the class.
| Time | Caption |
|---|---|
| 00:00:06 | Okay, okay. Shh. Stand up please. Good morning class. |
| 00:00:19 | Miss Tam, okay, |
| 00:00:23 | returned this for you. |
| 00:00:31 | Okay, we will start a new chapter today. |
| 00:00:59 | On the blackboard, there are two different equations. Okay? Two different equations. It is the equation in X, one unknown only. |
| 00:01:08 | Therefore, I think that you are familiar with this. |
| 00:01:13 | I want two of you, okay, to come out and find the solution for these two equations. Any? None of you? |
| 00:01:25 | [ Laughter ] |
| 00:01:27 | I think that you will like to come out today. |
| 00:01:30 | Kwan Chi Chung, please. This one. Okay, another beautiful girl, right? Chow Suk Fun. |
| 00:01:39 | Yes. |
| 00:01:40 | Okay. You try to use what you have learned in equations to find the value for X. Okay? |
| 00:02:16 | [ Laughter ] |
| 00:02:20 | Some of you laughed, it means that you find some mistakes, which one? Equation one or equation two? |
| 00:02:28 | Two. |
| 00:02:29 | Two? Yeung Cho Yee. You try to correct this. Equation two you found some mistakes. |
| 00:02:46 | Really? [ Laughter ] |
| 00:02:47 | [ Laughter ] |
| 00:02:49 | Okay, it should not be four X. Two X on the right-hand side, to the left-hand side, it should be minus two X. |
| 00:02:59 | Okay? Therefore, left-hand side is zero. |
| 00:03:01 | And 10, positive 10 on the left-hand side, right-hand side? Negative 10, okay? Therefore, 10 minus 10, it is zero. Okay, this, too. |
| 00:03:14 | For the first one, you found that the solution is X equals two. What does it mean? X equals two. |
| 00:03:24 | If I say that X equals two is the solution, what does it mean? What does it mean? |
| 00:03:37 | It means, when X equals two, left-hand side will equal right-hand side. Let's check it. |
| 00:03:49 | Okay, when X equals two, what is the left-hand side? It is two X plus four, okay? Two X plus four. |
| 00:03:59 | Two, X, plus four, what's the result? |
| 00:04:07 | Eight. Eight. |
| 00:04:08 | Eight. All right? And for the right-hand side, it is X plus six. X, we found that X equals two. |
| 00:04:25 | Therefore, it is eight again. Are they the same? |
| 00:04:29 | Yes. Okay? X equals two, then left-hand side right- equals right-hand side. That is the solution for equation one. |
| 00:04:30 | Yes. |
| 00:04:39 | How about the others? Lau Wai Fung, give me one more number for X, other than two. Any one? |
| 00:04:54 | Try to use [ In Chinese ] three [In English]. |
| 00:04:55 | Three. Okay. Let's substitute X equals three. Okay? In equation one. Another value for X. |
| 00:05:09 | The left-hand side, two X plus four. This time, X equals three. What's the value for the left-hand side? |
| 00:05:22 | Ten. |
| 00:05:23 | You will find that it is 10. But for the right-hand side, X plus six, X plus six. |
| 00:05:34 | Nine. |
| 00:05:35 | Nine. All right? They are not equal. Therefore, we will not say that X equals three is a solution. The solution is X equals two. |
| 00:05:49 | All right? Of course, you can test for the others. Okay, how about equation two, I get zero equals zero, what does it mean? |
| 00:06:01 | Do you think that there is no solution? There is no solution. Any one of you say that there is no solution? |
| 00:06:11 | I can't find X, therefore, no solution. No? Then what will be the solution? |
| 00:06:18 | Anything. |
| 00:06:20 | Sorry? Anything. What do you mean by anything? |
| 00:06:24 | Any number. |
| 00:06:26 | Any number. Okay. Let's check it. We have two and three, okay? |
| 00:06:32 | Let's try this two firstly. When X equals two. Left-hand side, right-hand side. |
| 00:06:47 | I try to compare these two when X equals two. Left-hand side is two X plus 10. Two X plus 10. Answer? |
| 00:06:59 | Fourteen. |
| 00:07:03 | Fourteen. Right-hand side? Two X plus five. Two plus five. It is? |
| 00:07:14 | Fourteen. |
| 00:07:15 | Fourteen again. Seven times two. Are the two sides equal? |
| 00:07:22 | Yes. |
| 00:07:23 | Yes. Left-hand side equals right-hand side, therefore, even if I can't find the solution, in fact, two, itself, is one of the solutions. |
| 00:07:32 | How about three? When X equals three. Of course, both the left-hand side and right-hand side, the values will be changed. Okay? |
| 00:07:49 | On the left-hand side, it is two X plus 10. And on the right-hand side, it is two X plus five. For the left-hand side, it is? |
| 00:08:05 | Sixteen. |
| 00:08:06 | Sixteen. Six plus 10. But for the right-hand side? |
| 00:08:12 | Sixteen. |
| 00:08:13 | It is also 16. This time it is two times eight, is it equal? |
| 00:08:19 | Yes. |
| 00:08:20 | Yes, the left-hand side is still equal to the right-hand side. Not no solution, in fact, at least we have found two. Okay? |
| 00:08:31 | More than one. How many? From the book, you still have three trials, try to test whether these three are the solutions or not. |
| 00:08:45 | Page one-four-four. Page one-four-four. In fact, the equation listed is the equation two. Okay? |
| 00:08:54 | Two X plus 10 equals two brackets, X plus five. Test for the other three solutions of X. Part one, part two and part three. |
| 00:09:04 | X equals zero, X equals negative one and also X equals negative one over two. Zero, negative integer and negative fraction. |
| 00:09:15 | Test for the left-hand side and right-hand side, okay? Are they equal? Do it now. Just mark it on your book. |
| 00:09:26 | And answer the question, whether they are equal or not, for the left-hand side and also the right-hand side. |
| 00:10:09 | It's better not to use a calculator, okay? But if you use it, just use it to check the answer. It is simple calculation only. |
| 00:10:44 | Errors? |
| 00:11:06 | Don't use this, this kind of ball pen, you can't see it clearly. |
| 00:11:15 | Have all of you finished? Okay, let's check the result. Page one-four-four. The three values for X. Uh... okay, Mak Pui Ling. You are nine. |
| 00:11:29 | Tell me the result, when X equals zero, what will be the left-hand side and right-hand side? Left-hand side? |
| 00:11:36 | Equals 10. |
| 00:11:37 | Equals 10. How about the right-hand side? |
| 00:11:39 | Equals 10. |
| 00:11:40 | Then is the left-hand side equal to the right-hand side? |
| 00:11:43 | Yes. |
| 00:11:44 | Yes. Okay? We have test the third value for X. When X equals zero, it is still left-hand side equals right-hand side. |
| 00:11:54 | Okay, how about the fourth trial, when X equals negative one. Sung Wai Ling, okay. |
| 00:12:02 | Left-hand side equals eight, right-hand side equals eight. |
| 00:12:06 | Therefore, do you think that they are equal? |
| 00:12:08 | Yes. |
| 00:12:09 | Yes. When X equals negative one, both the left-hand side and right-hand side equal eight. Okay? |
| 00:12:17 | Therefore, it is still left-hand side equals right-hand side. How about the fifth trial? This time, Lee Shan. |
| 00:12:28 | Left-hand side equals nine, right-hand side equals nine. |
| 00:12:32 | Okay. Therefore, equal. This time, when X equals negative one over two. Both the left-hand side and right-hand side, the result is nine. Okay? |
| 00:12:44 | Therefore, we have the same result. Left-hand side equals right-hand side. How many solutions now? |
| 00:12:53 | Five. |
| 00:12:54 | Five. Okay? Two on the blackboard with the three in the book, you have five results. Do you think it is only five? |
| 00:13:03 | No. |
| 00:13:04 | No. It has many many. Infinitely, many results. Why? Okay, let's use another trial. |
| 00:13:17 | This time, this time, we just simplified these two parts. Okay. Left-hand side and right-hand side. |
| 00:13:31 | In the expressions, you have learned two forms. The one, all the terms add or minus together. |
| 00:13:40 | It is called? It is called? How do we call them? Add or minus together, it is called? |
| 00:13:56 | Expanded form. |
| 00:13:57 | Expanded form, okay? Expanded form. You have other ways to express the terms, for example, like that. |
| 00:14:12 | This time, the terms are times together. Of course we will, we will not call them terms, we should call him- call them? |
| 00:14:23 | Factors. Therefore, this is called? Factorized form, okay? |
| 00:14:38 | You may express different expressions in expanded form or factorized form. Now we try to change them, with the same kind of form. |
| 00:14:51 | Which form, is more easy for you? Expanded form or factorized form? |
| 00:14:59 | Factorized form. |
| 00:15:01 | Some say expanded, some say factorized. In fact, if you want to find expanded form, what are you doing? Just multiplication. Okay? |
| 00:15:12 | But if you want to find the factorized form, you need to find common factors, or maybe groupings, etcetera. Okay? |
| 00:15:21 | Therefore, usually, expanded form will be more common, more usual. Just use multiplication, expand it one by one. Okay? |
| 00:15:33 | We'll try to change both sides, to be expanded form and compare. Left-hand side, is it expanded form? |
| 00:15:42 | Yes. |
| 00:15:43 | It is already expanded form. Two X plus 10. |
| 00:15:48 | The left-hand side, it is factorized form. |
| 00:15:57 | What will be the expanded form for the right-hand side? |
| 00:15:59 | Two X... |
| 00:16:00 | It will be? |
| 00:16:01 | Two X. |
| 00:16:02 | Two X. |
| 00:16:03 | Plus 10. |
| 00:16:04 | Plus 10. Constant terms, both are the same, ten. X term, the same, two X. Therefore, will they be always the same? |
| 00:16:21 | Yes. |
| 00:16:22 | Yes. In fact, on both sides, the expressions are exactly the same. Or we say that they are identically the same. |
| 00:16:34 | Therefore, no matter what's the value of X, it is- you substitute for X, the changes will be the same. |
| 00:16:41 | Therefore, you will get the same value. Okay? You cannot see it very easily because at first, they appear in different forms. |
| 00:16:54 | But if you change them to be the same, same form, then you can see that in fact, they are identically the same. All right? |
| 00:17:03 | Therefore, not just one solution, you have many many solutions. |
| 00:17:11 | For this kind of solution, uh, that's- this kind of equation, we will give them a name. |
| 00:17:24 | Identity. Identity means that they are exactly the same. Okay? Follow me. Identity. |
| 00:17:34 | Identity. |
| 00:17:35 | Identity. |
| 00:17:37 | Identity. |
| 00:17:38 | Okay? And therefore, for this kind of identity, we will give it a symbol, this time, not just two lines. |
| 00:17:53 | We use three lines as a symbol. It means both sides are identically the same. |
| 00:18:03 | We say that, two X plus 10, is identically equal two bracket, two, uh- X plus five. Okay? It's identically equal. |
| 00:18:18 | They are in fact, exactly the same. Okay? All right, then how to prove identity? Do you think that we try all the values for X? |
| 00:18:34 | First try, second try, third try, and then, oh, five trials. Then I can conclude they are identity. |
| 00:18:42 | No, because, that maybe the sixth trial- it fails. All right? Therefore, to prove identity, we will use this method. |
| 00:18:57 | We will try to change the left-hand side or right-hand side to be expanded form and then compare each term. |
| 00:19:01 | When all the terms are the same, then we say that it is an identity. |
| 00:19:10 | But if there are some different terms, then we will not say that it is an identity. Then it will be a normal equation only. Okay? |
| 00:19:21 | All right, I will give you some examples, who's on duty please clean it. |
| 00:19:27 | Clean the blackboard please. |
| 00:19:36 | Can you see the blackboard clearly? |
| 00:19:38 | Yes. |
| 00:19:40 | Yes? |
| 00:19:58 | Just leave the word identity, okay? |
| 00:20:15 | Therefore, the difference between identity and equation, equation it may be only one solution, two solutions. |
| 00:20:23 | But for identity, you have infinite many solutions. |
| 00:20:28 | It will be always true, okay? For any value of X. |
| 00:21:13 | Okay here, I have two other equations. Of course, now, they are equations only. Okay? We don't know how many solutions for each one. |
| 00:21:25 | Therefore, they are still equations only. I want to prove that, whether these equations are identities or not. |
| 00:21:36 | Are they identities? Or are they just equations? |
| 00:21:43 | The main steps will be, we try to expand the left-hand side and right-hand side, and then compare the terms. Okay? |
| 00:21:52 | If they are, okay, they are expanded form already. No need to simplify it. But if they are not, simplify it one by one. |
| 00:22:01 | And then compare the sides. Okay? You can start with the left-hand side or right-hand side. No matter. |
| 00:22:10 | Okay? It doesn't matter. Left-hand side... expand it. It should be... |
| 00:22:22 | Five X. |
| 00:22:24 | Five X. |
| 00:22:25 | Minus 15. |
| 00:22:27 | Minus 15. |
| 00:22:28 | Minus three X. |
| 00:22:31 | Minus three X and... |
| 00:22:32 | Plus three. |
| 00:22:33 | Plus three. Therefore, how many X? |
| 00:22:39 | Two X. |
| 00:22:40 | Two X only. And the constant term? |
| 00:22:43 | Minus 12. |
| 00:22:45 | Minus 12. Okay? Expanded form. Simplify that expanded form. And for the right-hand side, after expansion, it is two X. |
| 00:23:02 | Two X. Minus 12. |
| 00:23:07 | Are they equal? |
| 00:23:08 | Yes. |
| 00:23:14 | All right? They are equal. Therefore, do you think that it is an identity? Or an equation? |
| 00:23:21 | Identity. |
| 00:23:23 | Identity. Therefore, you can write down the result like that. Five X minus three minus three X minus one. |
| 00:23:34 | Two X minus six bracket. With this symbol. It's identical to the left- uh, right-hand side. |
| 00:23:47 | Okay? Or you can write it as a written form. It is an identity. Okay? But of course it is not so clear, to just write down it is. |
| 00:23:58 | Therefore, I think this is better. With the symbol, one, two, three, three lines. Okay, how about the second one? |
| 00:24:18 | How about some helper? So Wing Chung. You need to practice some more about your handwriting. |
| 00:24:32 | Try to simplify the left-hand side and right-hand side. |
| 00:24:46 | How about the bracket? Are you too nervous? |
| 00:24:51 | Yes. |
| 00:25:09 | He is very careful. |
| 00:25:11 | [ Laughter ] |
| 00:25:13 | Very very careful. [ Laughter ] How about the right-hand side? Is he correct for the left-hand side? |
| 00:25:26 | Yes. |
| 00:25:27 | Yes. |
| 00:25:36 | Okay, thank you. Firstly, is he correct for the simplification in the left-hand side and right-hand side? |
| 00:25:47 | Yes. |
| 00:25:48 | Okay. After simplification, you have two expanded forms, are they equal? No. |
| 00:25:58 | Although it is the same five X, but one is positive, another is negative. Okay? |
| 00:26:05 | Even the 10 is the same, it is not identically equal. But of course now, it is not the same, negative 10, positive 10, okay? |
| 00:26:14 | Therefore, we can say that, the left-hand side is not equal to the right-hand side. |
| 00:26:23 | Then can we say that it is an identity? |
| 00:26:25 | No. |
| 00:26:26 | No. Then your answer, you may just write down. Or you complete this, okay? Uh, write down the whole equation. |
| 00:26:44 | We have the conclusion, this equation is not an identity. Okay? If it is not identity, we can still call it an equation. Okay? |
| 00:26:57 | It is only an equation, not an identity. If it is an equation, it may be one solution only. Because it is one unknown, one equation. |
| 00:27:08 | Therefore, you may find the solution for it. Just one. But for identity, you have, in fact, infinite many solutions. Okay? |
| 00:27:19 | It will be satisfied for all the values of X for identities. Understand? Know the difference between identity and equation. |
| 00:27:31 | And for identity, in between the two sides, you can use a new symbol with three lines. And we read it as, is identical to. |
| 00:27:43 | Or you can say that they are identically equal. Okay? You have some class practices here. Page one-four-seven, page one-four-seven. |
| 00:28:01 | Seven equations are given to you. Okay? Seven equations are given to you. Some of them are already in the expanded form. |
| 00:28:10 | But some are still in factorized form. Use the method, okay? Use the method listed on the blackboard. |
| 00:28:19 | Tell whether they are identities or not. Understand? Try to prove whether they are identities or not. |
| 00:28:29 | Number one to number seven. Those simple ones, just write down the answers in the book. |
| 00:28:35 | But if you need to simplify it, for that kind of equation, please do your work on your class workbook. |
| 00:28:45 | All right? Class practice number one to number seven. Any more questions? Number one to number seven. Please complete that. |
| 00:29:24 | If you need to expand it, simplify it, please do it in your classwork book. Don't just write yes or no. |
| 00:29:36 | Just in the case, both sides are in expanded form, you can completely- |
| 00:29:41 | Uh, you can directly compare it, then you can write down the answer. |
| 00:29:46 | But if they are not exactly the same, in different forms, in your classwork, show some steps. |
| 00:29:54 | Okay? How to simplify the left-hand side, how to simplify the right-hand side. And then compare the terms. Okay? |
| 00:30:02 | Remember, before the conclusion, write down the result. Whether the left-hand side or right-hand side are equal. Okay? |
| 00:30:11 | Before your conclusion, you should have the result. Left-hand side equals, or does not equal right-hand side. |
| 00:30:24 | (inaudible) |
| 00:30:25 | Classwork book. |
| 00:30:40 | [ Bell ] |
| 00:30:52 | Is there any question? Write down whether it is left-hand side or right-hand side. Okay? |
| 00:31:06 | Stop for a while. After the expansion, remember, you must tell whether they are equal- or not. And then your conclusion. |
| 00:31:19 | Okay? If the left-hand side equals the right-hand side, then it is an identity. |
| 00:31:25 | But if they are not equal. Then this one, or you can simply say that, it is not an identity. |
| 00:31:39 | All right? Okay, finish the work at home and we will check it tomorrow. Stand up please. |
| 00:31:50 | Bye class. |
| 00:31:55 | Yes. And please say thanks to Miss Tam. |
| 00:31:58 | [ Laughter ] |