Why I am going to let meg and started us off on the lesson for me please. Okay, so our lesson today is about multiplying polynomials. So we're going to talk about different methods and what you do depending on like what you're given. We're going to start off with a little introductory activity that's going to review some topics from like past math classes that we're going to utilize throughout the lesson. So things to just keep in your mind are the Distributive Property, multiplying powers of x, combining like terms. And what the definition of a monomial, binomial, trinomial and polynomial r. So if you all want to get out your phone and go to Kahoo, you can put the game pin in and we can get started. And then I'll turn on the music because the yoga eg, Who's going to get everyone to register? Because anyone nine, we're going to get started. How is awesome? So everyone remembers the distributive property isn't perfectly great. John Tyler. Next question. Okay, so when you multiply X and X squared, you add the exponents. So that's why the correct answer is three, because you would get x to the third. It's okay if you don't remember that, but just like keep that in the back of your mind for when we go into the lesson. That when you're multiplying two X's that have different powers, or just any powers, you're going to add their exponents to get your new degree of the x term. Vu, we have some movement Shan and moved into first play. This is true or false are 43 x and 12-week term. So mostly everyone said no, which is the correct answer because 43 x has the accident and 12 doesn't. If you tried to add them together, you wouldn't be able to. So the simplest form when adding 43 x and 12 would be 40 3x plus 12. And I think Jen and kept her spot at the top. And then another question which is an example of a polynomial. Okay? So polynomial is like the umbrella term for any expression that's in the form of a coefficient times x to a whole number power. So a monomial would just be a polynomial with one term. A binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms. And then once we exceed three terms, they're all just called polynomials, but, so they're all examples of a polynomial. And that was the last question in the cocoon. So a single line, which I'm not again, I like your name. Very strong. Number two. So those were just review topics that we're now going to use throughout the lesson. And we're going to get right into it. So today, our lesson is going to be all about. Working through multiplying monomials, binomials, trial meals, and polynomials. And we're going to multiply monomials by other polynomials using an adaptation of the distributed method. And then multiplying other types of polynomials together. We're going to use the box method, which we will learn throughout the lesson today. So the first step in our progression to multiplying polynomials is multiplying a monomial by a polynomial. So when you multiply a monomial by a binomial, a trinomial, or any other polynomial, we can just use the distributive property, which was the first question and ARCA who it was, we took a and multiplied it by b, and then we multiply it by c and added them together. So our first example is four X to the third times x plus five. So does anyone want to tell me what I'm gonna do with that for x to the third term. You're first going to multiply it by x. So it's going to become four x to the fourth because you're going to add the x and the x to the first power. Awesome. And then what's my next step going to be from someone else? And we're going to multiply the four x to the third times five. Awesome. So firstly most by the four x to the third by x. And then we added that to four X to the third times five to get our final answer. For x to the fourth plus 20 x to the third. So this is really just using that distributed property, but now what we have terms that have x in it. So we're gonna do a quick little monitoring of how well we understand this. So we're gonna put a problem up on the next slide, but we don't want you to submit it. We just want to type in the chat. And then when I say go after everyone has had the opportunity to work through it, then you're all going to submit it once. So just remember, tape it in the chat, don't hit send until I tell you to. And the question you're going to do, he's four x to the third times X squared plus X plus five. So does anyone want to tell me which terms in the parentheses? I'm going to multiply the x for x to the third by, is it just the x squared? Is just x? Does it just by Mary, you raised your hand, you can answer all of them. Awesome. So emerges. Wonderful. And now everyone got through it. I'm gonna give you two minutes for just one minute. I don't think it should take too long. And then I'll tell you when to send your answers. Mm-hm. Is anyone still working? Other question? Yeah, of course. There's no easy way to like show that's an exponent on a keyboard. So like if I just include a carrot where the exponent should be, is that okay? Yeah, absolutely. Perfect demand to say that. Thank you for that question. When you're taping the expression into the chat, anywhere that you have an exponent. You can use the little carrot to represent that, that number is your exponent. Okay, so if everyone's ready, we can all hit send and we can see the answers in the chat. Okay. So everyone looks like you did a great job. Yes. So the correct answer is four X to the fifth, plus four x to the fourth, plus 20 x to the third, because you distribute that for x to the third term to every term in the parentheses, and then you multiply them together. I'm now going to pass it on to Emma to talk about multiplying two polynomials. Okay, so now we're going to learn how to multiply two polynomials by each other. So we're going to start off with this example of 3x plus four times six x minus four. So our first step is we're going to make a two-by-two box on our papers. And it's going to look like this. Now, the reason that we make a two-by-two box, because both of these are binomials, so they each have two terms. Okay? Now for our second step, we're going to add the corresponding terms to each row and column, making sure we don't forget the signs. So what this is going to look like, we're going to take our first binomial, which is 3X plus four. And we're going to write the first term, 3x in front of this box. And then we're going to write the second term four in front of the other box. Following that, we're going to take our next binomial, which is six x minus four. And we're going to put the first term on top of this first box, six X and the second term, negative four on top of the second box. Now when we say Don't forget the signs, if this is a minus, then you carry that over and this becomes negative four. Ok? So for step three, where are you going to do? Is you're going to multiply the first term in row one, which is 3x, by the first term in column one, which is six. So can someone tell me what that would be? You can just unmute yourself and say the answer to it would be 18 x squared. Perfect. Think You, Can someone else tell me what the next box would be? Negative 12 over X? Yes. Thank you. And someone else from an Xbox? 20 for x. And the last box, lingered 16. Yes. So basically we're just multiplying out each of these terms in the rows and columns to fill in every box in that two-by-two box we made in step one. So now that we have all the terms, we're going to go to step four, which is to add all of the products we got into combine the like terms. So for this problem, we're going to get 18 x squared minus 12 x plus 24 X minus 16. You just take all the products from each box and add them together. And don't forget your sign. So if it's a negative, you're going to subtract those values. Now, we do have some like terms in this problem. We have negative 12 x n 24 eggs. Since those are like terms, we're going to combine those to get our final answer of 18 x squared plus 12 x minus 16. Does anybody have any questions? Okay, so now we're going to go on to another example. So we're going to do eight x plus four times 2x minus two. Now, can anyone tell me what the first step was? To back? Yes. Perfect. We're going to make our two-by-two box first. Now remember this is a two-by-two box because we are multiplying two binomials by each other and each have two terms. So our second step was to add the corresponding terms to each row and column. So we are left with eight X plus four down here and 2x minus two on top. We're just taking each of these terms and putting them in front of the boxes. Now we're going to multiply the first term of row one, but the first term of column one. And we're going to repeat this process for every box. So can someone tell me what the products will be for this first row? Kinetic, weird. Yup. And then for the next box. Negative 16 and guess and then Xbox eight execs. Yep. Good job. And last box? Negative a. Yes. So now we're going to go on to our last step, which is to add all of these products and combine like terms. So can anyone tell me what the expression would be when you add all of these products that we just got. 16 x squared minus 16 x. Yeah, and then you'd have to come borrowing term, mmm. So when you combine the like terms, then what are you left with? 1600 square, sorry, minus a x minus a. Perfect, great job. So we just added each of these products, 16 x squared minus 16 x plus eight, x minus eight. And we have two like terms right here are negative 16 x an edX. So those combined to give us negative eight X, leaving us with a final answer, 16 x squared minus eight, x minus eight. Okay, now we're going to do a little activity. So I'm going to send this link in the chat. I just need to share. You guys can see the screen, right? Okay, perfect. So I'm gonna go through the directions and then I'm going to share the link with you guys. So this is just practice for multiplying two binomials by each other. So we're using the box method. There are going to be two practice problems. So for example, if we have x squared plus two times x squared minus five, it's going to ask you to enter the terms into these boxes up here and down there. And you're going to multiply the terms out just like we did. This is basically our two-by-two box that we created. Then you are going to add all of the products that we just got from these boxes and combine your like terms to get your final answer. Yes. So Meghan, just send the link to the activity so you guys can go to that link and make sure when you go to the second problem, you click Next. It's up in the top right corner. And that will bring you to problem number one and problem number two. And I'm going to give you guys about five minutes to complete this. And if you have any questions, feel free to unmute yourself or send a message in the chat. All right. I'm going to give you guys a couple more minutes just to finish up the second problem. So someone had asked if there was a way to submit a you don't need to do anything. The answers get automatically sent to us and we can see them. So there is no final submit. Okay. Does anyone else need extra time to finish the second problem? Now? Okay. So after looking at everyone's answers, everyone did a really good job on filling in the correct terms. So now we are going to move on to the second activity. Okay? Okay, so now we're going to do this activity. I'm going to send you the link. And the second, this is basically going to be very similar to the last activity where you're still multiplying two binomials by each other. We are only going to do questions 12. If you do not have Google Chrome. To open this link, I will have the a screenshot of what the questions are and what the site looks like so you can still answer the questions. So here's the link. Isn't doing an agenda maroon. Thank you. So just make sure you guys are only answering questions 12. And make sure you keep track of which answers you chose because we are going to send our answers in the chat once we are done. And I'm going to give you guys two to three minutes to complete this popular screen. Right now we see the paths. So if you don't have Google Chrome, this is what the website looks like. And it's just asking you these two questions which are at the bottom of the screen. Does anyone need anymore time to do either Question one or two? Okay, perfect. So now we're gonna go over those. So we have the first question. Which is, what is the area of one of the single rectangles that make up the corkboard. So what I'm gonna have you do is send which letter answer you got in the chat. They might be different from the order of them might be different from your when you pulled up the website. So just make sure we're clicking the right one. Perfect. Everyone got C, which is the correct answer. Awesome. Okay. Now we're gonna go over question to which asked, which of the following expressions represents the area of the corkboard? So again, you can send your answer in the chat. Awesome. Everyone got the correct answer for this one as well. Which was d eight X squared plus 56 X plus 48. Does anyone have any last questions about using the box method to multiply two binomials by each other. Okay, I'm going to transfer the lesson to Andy now. Alright, sweet, let me get my screening hearing. Alright. Let's see I can everyone see the PowerPoint? All right, sweet. So as am I just went through, we're going to be doing a very similar process. But now instead of strictly doing a binomial by a binomial, we're going to be doing any polynomial multiplication. So we're kind of widening our capability, but we're still going to be using the box method. So our steps are going to be similar. So step one, we're going to identify how many terms are in our first polynomial. So I have an example polynomial up here. It's 3x squared minus 2x plus four. So could someone tell me how many terms there are in this polynomial? And you can just unmute and say, yeah, thank you. It is three. Let me that. Oh, yes. So we have the 3x squared is our first term. The negative 2x is term two in the positive four is our third term. All right, so we're going to do the same thing. We're going to identify the terms in our second polynomial. So this time we have negative X cubed plus 2X squared minus six, X plus ten. Can someone tell me how many terms are o? Well, yes, I'll just show you guys. It's four. So can someone just tell me what the four terms are? There's negative three or negative X cubed, then there's positive 2x squared, there's negative 6X, and there's ten. Great, thank you so much. All right, step three, you are going to create a box again. So similarly to what we did before with Emma, we were making a two-by-two because we're doing binomials, but now we're not strictly doing binomials. So our box, we set it up by saying the number of rows is going to equal the number of terms in our first polynomial. And the number of columns will be the number of turns and our second polynomial. So just to remember, we said that there are three terms and polynomial one in four and polynomial two. So then can someone tell me what the dimensions of our box would be, meaning how many rows, how many calls? 3y form. Yeah. Great. Thank you so much. So our box is going to look like this, just three by 43 rows, four columns. And next step we need to put some labels on our box. So again, similar process, we're going to take the terms in polynomial one and those will go by each row. And then we're gonna take the terms in a polynomial to four columns. So starting with the rows, can someone tell me what term is going to go next to Rwanda using this polynomial up here? 3x squared. Yeah, thank you. And then how about the next row? Minus 2x? Yes. And then the last row, o minded. Well, okay, yes, positive for. Now. Those just go here. And now we're going to do the same thing for the columns. So here's our polynomial and can someone tell me our first column label? Negative 3x cubed? It's just negative x cubed, but yeah, like ice cubes. Now, no worries. Okay. And the next column. Kingwell? Yeah. And what about our third column? Negative 6X? Yep. And does someone want to just tell me the last one? And yes. Thank you. Okay. So we're going to add those up here for our column labels. And our next step is where the math is going to happen. So you're going to do the same process. You're going to be multiplying the first term of row one by the first term of column one and do that for every row. So some of you might have done my multiplication tables and school before. Punnett squares. It's the same concept. So going back to our square, I'm going to go slowly just to our first row to make sure we're all on the same page. So first, what we would do, 3x squared times negative x cubed, which is negative three x to the fifth. Remember we're adding those exponents and then 3x squared times 2x squared is going to be six x to the fourth. Then 3x squared times negative six x is negative 18 X cubed. And lastly, for this row, we have 3x squared times positive ten, which of course is 3x squared. So if you guys can just take like 60 seconds and fill out the rest of your table. I'm going to ask for two volunteers. I'm going to call on two people. So I'm going to have one person read me this middle row and then someone reading the last row. So b1, just take 60 seconds, fill out your box and then I'll call on people. Just take a bow. All right. Tyler, a would you mind sharing with us what you got for this middle row here? Yeah, of course. So for the first box, I've got 2x to the fourth and then second box I've got negative four x to the third, third box I got 12 x and then the final box I got negative 20 x. A suite. I think the only thing is this third box because it's negative two x, I'm saying six x i would be squared, but that was really great. Thank you so much. And then let's see. Can caitlin can you share with me what you got for your last row? Or the last row I got for the first one, negative four X cubed. And then I got eight x squared. Negative 24 acts. And 40. Great ink. You both so much for that. So again, we gotta combine these like terms to find our final answer. So just a reminder, the like terms just mean that the degree of the exponent is the same. So if we went too fast, but yeah, I just kind of color-coded it in the box so we can see how we're getting those like terms. But it's, I find it helpful to write them all in order, starting with the highest degree all the way to the constant and then group them right next to their liked terms. Just so that way, the simplification is a little bit easier. So just for time sake, I'm going to go ahead and give you guys the simplified answer is, what you see here, 3x to the fifth plus x to the fourth minus 26 x cubed plus 50 x squared minus 44 X plus 40. So we're going to do one more quick example. Right here you can see our problem. X squared minus four x plus eight times x to the fourth, minus 3x squared plus x minus six. So quickly I'm going to get the ball rolling for you guys, but you guys are actually going to be doing this a little bit more independently. It will actually, I might just scratch this because we are a little bit short on time and I want us to be able to do a little bit of group work. And some individuals have at the end, but basically, we'll just quickly go over this. It's going to be a similar, a similar process. You're gonna have to identify your terms in the polynomial is create your box, do the multiplication. So, yeah, we could see that there's three terms in the first one or the second, meaning our dimensions would be a three by four box. And then our box would look like this. And then we do the same thing we just did. So we get all of these terms, combine them into like terms, right? I'm out, simplify it. So we're going to move on to this problem and I'm gonna put you guys in breakout rooms for about four minutes. And you're going to do this. Now notice we're multiplying three things here. So your hint is going to be to multiply the monomial by the trinomial first and then multiply what you get there by the last polynomial. So I think Professor Wong Wilson, you guys into groups. Oh, also very important, don't want to say is make sure you copy this problem down because once you're in your breakout room, oh my gosh. The answers are wondering a breakout room, you are not going to see it. And just use the box method. Okay, maybe. I have to choose rooms. Who? Mr. Megan, Do you guys know how to do that? Think they're joining the rooms now. So I just chance I saw them still writing. I waited until they got the problem before I opened up the breakout rooms. On the chat everyone and you're through the chat still there so you can get when you get to your breakout room. So I only have six breakout rooms. So five with 34. Okay. Yeah, that's perfect thing. Yeah. Yeah. Some people didn't shall act out which you guys can go check out. Scanning. Yes, we can check on them. It would be I don't know if eliza was assigned a breakout rooms and she's still in law. And the girls. I thought I put herring, I break out of them, but I could be wrong. I don't think we can join them. Oh, she's in she's been assigned a breakout Well, I think she just one were during breakout room. Does a pop up for, you know. Yeah. Okay. S5. I didn't now, I've never done this before. It works. Sometimes it doesn't work. Depends how you, So Professor Wang for like the next group, if you want them to be able to pop thrill. When they join the meeting, make them a co-host. Once they join, you guys are all calls. ****, but if you'd like, you have to like on co-host them and recover them because I like I'm a TA like through your day and I like, jumped through the breakout rooms. But in order to give me that ability, it doesn't work when I just joined an automatically the co-host, she has to uncover me and remake because if I think about him, that like it doesn't always jive well with like the co-host assignments, but if you do that, then they won't be able to jump through. Ok. That must be a bug. And then guys, AND I feel you should do the other final integration TO now because you had to skip that one example. Group. Can stitch you. Did your group get an answer? We didn't finish. We didn't have enough time to finish. Okay. Yes. Sorry about that. Did any group finish? I can tell you what we did first if you want me to like the first oh, yeah, that'd be helpful thinks. All right, so we distributed the x first and then we got 3x cubed plus 2x squared plus 7X suite. And then we set up the box after that. Good, yea, that's a great star and that will definitely lead you to the right answer, assuming you carry out that multiplication, right? So we don't have a slide with it all worked out, but as Can she has helped us out here. That is exactly the process. You would distribute that acts like she said. And then you would have your two polynomials do your rows, column labels, not forgetting the signs. Then multiply everything out and you should come to this answer here. I'll give you guys a second if you just want to jot it down or take a picture or anything, just so you have it cut about ten seconds. Alright? We are going to do a little activity to finish. Also, this y is going to be independent. I believe Megan has or is going to send this link in the chat. It's to a Google form so I can show you. It's gonna take you here. And there's two problems, but since we are a little bit short on time today, we're just going to be doing the second one with not the a, B plus C, not that one, just number two. So take about three or four minutes and do this problem here. And you're going to take a picture of it and add it where it says Add File and then submit it to us. And that will be the last thing you have to do for us. And there's all there's a problem. Yeah, what's up? It says dynamic link not found. What happens if I don't think you can copy the your link. I think you have to go to a button that says like Sharon link, it's asymptotic especial on I can't be it the way you go back to the original form, but not the actual way. It looks like an assessment that in actual this one. Okay. There we go. Awesome. Sorry about that. Everyone in Washington I resolving. You're just doing the second problem. You don't need to worry about the first one. Mm-hm. Okay. Is it two separate problems or a multiplying the top line and bottom line. It's just one problem that you're doing and you can just ignore that top part, this bullet here, and just this part. Okay, so Andy, once you guys to finish that up, I know some of you might still be working. But once you're done, we're just going to have you send the chat. One thing you learned in today's lesson, and you can send it to any of the three of us. And we just wanted to thank you all for being so attentive and participating so much. You guys really made it much better. Okay, and starting to get a little bit of the next group command. So I want to take all three instructors for their lesson, for teaching us about how to multiply polynomials. So can we get around of applause for them? And for the til 5258 students, I'm going to ask you to provide feedback on the lesson. I've put the link here and you can copy and paste the link. And once you have a you've copied and pasted the league, feel free to leave so that we can get the next before the next group, the people start coming in. Okay. Guys. Thanks, Mary. Thank you guys. So I'm 413 students. You're also going to provide feedback for each of the lesson. So using that same week, we have a few minutes here. If you guys want to go ahead and start while we wait for the rest of the people come in. So we have Tyler ethnics. Oh.
Microteaching Lesson: Emma Megan Andie
From Carol Wong November 30, 2020
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