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.

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