- So we're going to start off today with a task. This is related to around page 271 in your binder if you want to do that, want to take a look at that. So once you got this, you can go ahead and get started and I'll pass out calculators also. Okay, so put your name on the purple paper and go ahead and get started. I'll come around and help you if you need it. - Okay, so I found this. I did 14 C plus 4 equals 96. Do you need another function so you can do the RREF or elimination or something like that? - Could you do either of those methods with just 1 equation? - No, that's why I put 2 down there. - Okay, could you grab another one if you needed it? - Yes, like 2C plus the next one. - Just another day, yeah. - You could do that? - It seems like if you could do the first day, you could also do the second day. - So you could do three of them and do the RREF thing with the calculator? - Sure, try it out or try it out you could try that RREF on the calculator with one. You could try it out with two. You could try it out with three. - Can you do it with one? - Try it. See if it gives you enough information that you need because think about what you're trying to find. How many things are you trying to find? - The two prices. The child and the adult ticket. - The two prices. So if you try it with just one equation, see if you get all the information that you need, that you're looking for, okay? Yeah. - How would you get why would you have negative 5 adult tickets? - That's actually going to be something that I ask you to write a little story. How could that possibly happen? - They got lost in the sink? I don't know. - The tickets got lost in the sink? - Or they got thrown away so you don't you have five less tickets. - Okay, that's fine. You're going to write that down. So you're doing a matrix equation? Do you guys have the same two equations though? - Yeah, we put it in the calculator. - Okay. Does it matter which two you chose? - We just started with the first one. - Okay. And how did you solve that one? - The inverse. The inverse one, and then we just tested it on these after. - The inverse? Okay, cool. Cool, cool, cool. How are you doing your totals? I'm curious. - Adding all the sales up for the first seven days and then adding all the adult tickets up for the first seven days and so on, and then you can do the same thing up here, both of them. - So you're going to make an equation again with the C and the A? And then how will you solve that if there's only one equation? - Well, you plug in because we already have the answer from the - So you're just going to plug them in and check? - See if it's right. - Cool. I'll probably have you guys do a poster for question two and just sort of be able to talk through what you did on that one. I like it, cool. - Okay. So for this problem, are we adding these up or are we plugging each one into the equation. - I would say you want to add them up because it's saying the totals of the first seven days. - So each one individually and then see if they match up? - Yeah, would that work? - Yeah. - Okay. Ashlyn, you're good? - Yep. - Okay, Roy, why don't you tell us what you're doing. - Just adding them up. - And then how are you getting dollars out of that? - Because I'm multiplying the amount the children ticket was to the amount they bought. - Okay, after you total it or before you total it? What is that 56? Is that the total? Oh, it's 4 times 14, so you're doing each day. Okay, so McKayla is going to total and then multiply it by 4 so if you want to, you can multiply it by 4 for each day and then add those up. That'd be fine. You'll get the same thing, cool. How are we doing up here? Which part of it? - All of it. - Okay, any gut feeling on what it might be asking? - Are you supposed to times the numbers, the children tickets and adult tickets, by three... and two? - Okay, yeah, two and three but really, see that word, respectively? So the first thing that's listed, April 1st, you'll double. The second thing that's listed, April 2nd, you'll triple all of those results, okay? So do you see how that might lead to two more equations? - No. - Okay. Why don't you make a list then of all the information doubled for April 1st, and all the information tripled for April 2nd. - So wait, do you what us just to figure out if the prices are still the same or do you want us to plug in the prices to see if it equals this? - Can one of you do plugging in the prices and one of you solve and see if it turns out to be the same? That'd be good, cool. - So on this problem right here, when it says their returning with double and triple - And then it says this, respectively. - What does that mean? - So that means whatever is listed first here gets doubled and whatever listed second gets tripled, okay? - Oh. Okay, so then do we do a different equation or just plug it in and see how much. - So double everything from April 1st, and see what happens and then triple everything from April 2nd. - Okay, and then what do we do after that? - Well, it looks like Josh is writing some equations. Is that for the first one, Josh? Okay, can you make another one for day two? And McKayla, what do you think you could do with those? - Are we finding out the ticket prices for? - You're just yeah, verifying the ticket prices again. - Just double checking if it's the same price, okay. - Yeah, so there'll be bigger numbers for sure. - All right. - That looks good. How are we doing? - I'm just curious if we're actually doing this right? - Tell us what you're doing. - Well, I decided when it says doubling, you're doubling the numbers, which means you're doubling the people, so you're doubling the number of tickets, so that's what I did for these two. I just put them in the equation and I'm doing the same thing that I did with the first one. - Okay. And how is it working out? - So far it's working out pretty well, but We don't know if we're going the right direction. - It sounds okay. So you doubled everything in the charts? Okay, and are the numbers pretty big? - Not really well, the [inaudible] that's okay, I think. - You can still work with it, okay. You do have a calculator. Remember that calculator can do more than just multiply. Cool, what's your story? - Ours is going to say a family with five adults and four children went into the park and five adults got hurt so they sued the company for their money back. I don't know. - Okay, so is the negative five, negative money or what is the negative five? - Yeah, I think it's negative money because yeah, I think it's negative money. - Well, what is the title of that column of data? - Number of adult tickets. They sold negative five tickets, so they - Okay, so I like that story. See if you can tie that into the number of tickets and why that would why negative money might imply that that's going to be negative number of tickets, okay? Any story coming to you? - Working on it. Would it work if I said that they got a refund? - Would that make sense? - They went there and then they wanted their money back because of lack of - Popcorn, okay. So think about this. If they bought the ticket what date does it have the negative? - April 6th. - April 6th. Thank you, so April 6th, if they bought the tickets on that day. - And they returned the money, it'd go back to zero. - It would go back to zero, so could you tweak that story a little bit so that it's not at zero? It actually ends up being - Like they actually lost the tickets or - It could be that you get a refund but could it be think about that part where if they bought it on the same day, that might be a little bit of an issue. - Maybe the tickets, like, fall through the sewer - I mean, that's a different story. It's a lovely story, Carly, but.... what do you have here? - So we said the theater gave five tickets away to students who work hard. - So as like a prize? - Yeah, because they didn't pay for them. - Okay, cool. I like that. That will work. Okay, one more minute. If you got a poster going, let's finish that baby up. - Cause kids don't just go there by them-selves. - So they had to pay. - But the adults didn't. So the movie kind of lost the money for the tickets. - Okay. That sounds good. Another group was talking about a prize like that too. That's good. Okay, see if you can attack number five and then we'll be ready to roll. Okay, we're going to come together here folks. Eye balls this way. Yo. - If you made a poster for question one, we'll have you come up and if they have something on their poster that you don't have on your purple paper, put it on there, okay? So we want to be good listeners and we also want to take notes on the good techniques that they are using, okay? So if you have your poster for question one, if you could slap it up on the board and tell us what's going on, okay. Yeah, that'd be awesome. - Okay. So for the first people, there was 14 children, four adults, and the total was $96. Two, children, 22 adults, 228, same for that and - Hold on just one second. So did it matter which two days or three days that you chose? - No. - So you just picked the first three days? Okay. Did anybody pick different days to make their equations? Okay, sorry to interrupt. Go ahead. - Okay. So then we did the RREF thing and the matrix and we got the inverse thing with the prices for the children and adults, but then we got this extra row on the bottom because we had three instead of two. We only needed to find two things. - Okay, hold on one second there. So the other "I" word, the identity is what you found. Can you guys raise your hand if you used three equations to help you find the ticket prices in question one? Why not? - You only need two. - You only need two, okay. So Lauren is going to kind of talk us through how her group kind of came to that decision. So did you work with the three equations first? - I don't remember. - Somebody, somebody in your group did the three equations and they tried it out on the calculator. They did the reduce row echelon form there and if you guys noticed, she pointed out that there's just an extra row of information in that answer that was given to them. Kind of like what you're saying. You don't need that third equation. You really have some extra information. Okay, keep going, sorry. - And then we did it with just two things. We didn't get an extra row and we got the same 4, 10 with the identity so the child is $4 and the adult is $10. - Cool. And Maddie, do you want to talk to us about how you started off with one equation? - Yeah, I tried it with just the first day and I got just two answers and they were all decimal numbers so - And what did you try with the first day? - I tried 14, 4 and 96 in the matrix. - Okay, so what size matrix did you work with? - I did a 1 by 3. - Okay, and you said 14, 4, and 96. So that first row of information, she made a tiny little augmented and do you remember what it said? - It said one point something and then six point something. - So weird stuff, okay. Could we figure out the ticket prices with just one equation? Fisher says yes. - You can do it with just one but it's easier if you have two or even three, like Lauren did. - Okay. And Ashlyn, you first circled something on your paper, the one. Which April day had the one on it? - April 5th. - Okay, and what did you tell me about if there was just the one there? How did you think you could proceed? Don't remember? - No. - Okay. Could anybody think why that day might be an important day to work with if that was just the one thing that you had? If you just had one day's worth of information, could you figure it out? Okay, that's cool. Now, we'll move on. Did anybody get different ticket prices than four and ten? Okay, poster for number two, who's got a poster for number two? Thank you, poster number one. Right up next to it would be great. - Okay, so for problem number two, you had to add up all of the totals for everything. So if you added up all the number of the children's tickets, it was 43. If you added all of them for the adult tickets, it was 60. And then we just plugged in to see if it'd work, the $10 for the adult tickets, like right there, and then the $4 for the children's tickets, and it worked because 172 plus 600. So 772. - Cool, thank you. I saw some other groups do something different for that second problem. Did anybody try to solve a new system of equations on problem two? You have one equation? Everybody just plugged it in, substituted in to test it out? Yeah, go ahead Logan. What did you do? - I just took the I don't know, I guess I did a long equation, where I did 14 and then so in parentheses, 14 times 4, parentheses plus parentheses 2, and then continued on so it was like added all of them up and then - Added each one individually instead of doing the total first and then multiplying by 4? Cool, got it, thanks. Thank you, Fisher. Any questions on two? Okay, poster for three? Do we have a poster for three? Come on up. You can put it over the matrix in the middle. - So what we did was we doubled the 1st day so the first day was 14 C plus 4 A equals 96. So that would give us 28 C plus 8 A equals 192 and then we tripled the second day and it gives us 6 C and then 66 A and then the total is 684 and then we put that into an augmented matrix and then did the RREF and then got 1, 0, 4, 0, 1, and 10. - So what did that tell you? - That the ticket prices would still be the same. - Cool, cool, thanks. And Maddie, you doubled both didn't you, doubled both days, and how did it work out? - It didn't work. - It didn't work, okay. So this is what I'd like you guys to do and I want to double check Maddie's work here. So if we take these first two days, April 1st and April 2nd, and double the first day and double the second day, could you take two minutes right now and double both and then check the system and see how that turns out. - I doubled both of mine and it worked out perfectly. - And it worked out okay? What method did you use? - Elimination. - Elimination, so she didn't do it on the calculator Maddie. So maybe that could be our discrepancy. Okay, how about we take 2 minutes? Everybody check doubling both, okay. Take two minutes and if get something kind of wacky that we don't expect, raise your hand and let me know. Are you done, did you double them? - Yeah, I doubled them. - Where? - Right there. - You doubled one and tripled this one. So right now, you're supposed to be doubling both and checking. - Doesn't it not work just because this one says triple for this day? - Right, and so that is what the wording is saying, but Maddie started doubling and I thought that that was interesting to take a look at so it is a little bit different than the instructions so we're just checking that out. Yes, sir? - We got it so it works. - You got it so it works. Okay, what method did you use, elimination or the matrix? - The RREF. - Okay, and it still came out four and ten, okay. Okay, Maddie, are you trying it again? - I'm trying it again, the A inverse B. - Okay. - And the A worked. - And it worked, weird. Maybe it was just something that was entered in a little bit funny way. Anybody getting something strange, other than four and ten? It's working okay? Okay, so let's have our creative story team come on up and ooh, actually before that sorry I forgot. So when I was talking to McKayla before about this question number three that we're on, you wanted to share what you said? Why would it not matter what we did to the equations. - It doesn't matter as long as you do it on both sides of the equal sign because if you only do it to one, then your problem will be off. - Okay, what do we call that when we do something to both sides of the equal sign? Anybody remember? - Property of equality. - Property of equality, we got it. So that's exactly right. We could be doing anything, like adding up the first seven days of the week, and we're still going to get those same ticket prices because we're adding the same thing to both sides of an equation, or if we're doubling both sides of an equation or tripling both sides of an equation. We're not really changing the solution to that system, so very cool, awesome, all right. Story, come on up here, please, and then if somebody will have some exciting story you'd like to share, also that'd be great. So we had some weird information from April what day was that? April 6th had a negative number, okay, and some of you noticed that right away and said, "How could that be possible?" So let's hear a couple of possible explanations. Okay, what do you got, Carly? - So we thought maybe that somebody reserved five tickets but never picked them up, so and then the movie theater was never able to sell them so they were out five tickets. - Very nice, thank you. Any other explan-ations? There were a couple of good stories. What's the story? Didn't you have one about the prize? Yeah, what was it? - The school would give away five tickets to students as a prize. - As a prize, okay, and so that would be why would it be negative then? - Because they're not gaining anything. They're free. - They're free. They're not gaining anything. Nobody's paid for it, cool. And that back group, you had a similar story, okay. What was your situation there? Sequoia, you want to tell us? - On the 6th of April, five adults won five movie tickets in a raffle and then on the same day, four children went with the parents to the movie. - And they had to - Pay for the children but not the parents. - Okay, so we had a negative five because they were also free. Cool. Okay, so to kind of tie this up, on that last question, could you just make up any data that you wanted? Stick to the table and what was that? - The ticket prices never changed. - Ticket prices never changed. So this group chose 1 and 1 and then 2 and 2. Another group chose 1 and 1 and 2 and 2, which was strange that you chose the same thing. But you could have chosen anything in the world as long as you kept those same things. Okay so a couple things that I want you to write at the bottom of the purple paper to kind of tie this up and what really goes back to all of these different things that you did. As long as we are doing these same operations to both sides of the equation, using our properties of equality, we're going to have a system that has the exact same solution, okay. - So if we use the properties of equality I don't know if you can read this the solution to the system will not change. All right, so take a minute, copy that down, and then I want to show you one more thing on the computer screen. Okay, so first of all, it was totally impressive that you guys right away started writing equations. In no other classes that I've done this did everybody immediately start writing equations. That was impressive, so great job on that, and I have a question for you about those equations. What kind of functions are those that you wrote? You wrote about 10 of them in the last half an hour. What kind of functions are they? - Linear. - Are they linear? - Yeah. - How do you know that? - Because they're staying the same. - What's staying ... - If you buy one, it's going to be $10. If you buy two, it's going to be $20 just adding that. - Okay, so that rate will stay the same. Okay, so linear functions here. I think that's also worth noting. Linear functions are everything that you wrote down today. So before you came in, what I did was I took all of these possible linear functions from April 1st all the way to April 8th and I think I even put some other information on from possible April 9th and 10th and I graphed them. I graphed all of those linear functions and I just want to show you really quickly what that looks like. So I need to have uh oh. My computer, take a minute. Okay, no that' all right. So while this is loading, at the bottom of your purple paper, I want you to pretend like you're opening your next movie theatre on your own and you're going to choose the prices. So pick a child's price and pick an adult price because we're going to use that in just one second, okay? Pick a child price and an adult price. If you want it to be the same as those in your group, that's fine. - Okay, so assuming you have something written down, take a look up here just for one second. Take a look up here eyeballs, eyeballs this way. So this is all of the this is a collection of linear functions from April 1st to April 8th and maybe a little bit beyond. All of these linear functions, what do you notice about them? - They're straight. - They are straight which makes sense because they're linear. What else? - It looks like they all meet in the center. - They do all meet. Where do you think this is? What do we call that? Equilibrium point and what is it for this story? Four dollars and $10, okay, so no matter what we do to these equations, like McKayla said, as long as we do something to both sides of the equation, it might change the way that this is tilted a little bit, but that equilibrium point is not going to change, okay? So that's kind of cool. This is kind of a nice visual. So if you would take 30 seconds and make a really rough sketch of this on your purple paper and maybe make a note what this point is that it's 4, 10. That's our idea of equilibrium that we've been talking about. That would be great. And I have one more question for you about this picture up here. So in what I have, everything seems to be kind of tilting this way except for this one line right here. What day is that? - April 6th. - The fateful April 6th, the one with the negative. Okay, so in some way, that would influence the slope of that line, okay? Cool. All right, so do you all have your new chosen movie ticket prices? Okay, so what I'm going to do is give you a blank table now for April 1st through 8th and you're going to fill in numbers of people that came to the movie theatre, preferably not all zero every day. That's not a really great movie theatre. And then fill in the totals, okay? So I'm going to give you about two minutes to do that. So hopefully you have your ticket prices and a calculator ready. I'm going to give you a new table, okay? Okay, two minutes here to fill in your chart. Create some data. You use your new prices that you selected, okay? On your chart guys, don't put the prices on there. Don't put the prices on there. Sorry, sorry, I should have told you that you're going to crumple this up. If you put them on there, that's all right. Maybe cross it out. - Cross these out? - No, your set ticket prices that you wrote on the purple paper. Okay, even if you don't have your table totally filled in, I want you to think if you have enough information on that table for somebody to figure out the ticket prices. Crumple it up. So think about how many days worth of information you need to make this a successful activity, okay, and we'll do a three count, chuck it up here to the front board, okay, and then instead of everybody running up here, let's just send one person from your group and get as many snowballs as you need for your group, okay? So 1, 2, 3. Almost hit Ren, okay. Go send one representative up. Send one representative up and get enough snowballs for everybody at your group. You choose your method and I want you to solve their system, okay? Figure out how many days you need. Even if the answers are on there, go ahead and take two minutes and solve a system, okay? Use whatever method you want to use. Go to it. - Bailey, what do you think? How are you going to solve that? - Functions. - Okay. So what are you going to do first? - The functions. - Okay, didn't you already do that? And now, what are you going to do? - Solve them by elimination. - Elimination, okay, cool. You're going to use a matrix, cool. All right, let's come back together with our discussion. We had a couple of interesting cases here. Maddie's testing something out where it looked like did it look like there was a negative ticket price. - Negative 75. - Negative $75 ticket price. She's going to test it out and see if that actually works. And then Mason had an interesting set of data where every day the data doubled, right, and so what did you get when you were trying to solve it? - Zero. - Zero... which method did you use? - I tried the matrix and I tried elimination and neither of them worked. - Okay, what did you get with your matrix? - 1.4 no, no, no 1, 0.4, then 9.4 and then 0, 0, 0. - Zero, zero, zero. Bless you, sneezy. So if we have something like this, okay, this is certainly not the identity that you were looking for and you said you got zero, which is like saying zero equals zero. Will there be a solution to this system? Will there be an equilibrium? Uh-uh, so what could the ticket prices even be? - Zero? - Zero could work. What if you tried $1, 1 and 1 or 2 and 2? I'd like you to try some numbers out in there and we'll report back to that on another day. Anybody else get something sort of interesting to take a look at? - I got $1 and $2. - One dollar and $2, and you were not surprised were you, Carl? No, okay, cool. Well, we're at about the five-minute mark and what we're going to do for your homework is two things. First of all, I'm going to give you one more task different theater. And this time you're going to consider children's prices, adult prices, and then a senior price. So think about what you would need to solve a system in that situation and what method you would prefer to use that. And then also, if you remember, we're going to have a test on this stuff next time. So I have another review paper for you as well, okay? So I'm going to hand out task three, which is really related to this, and if you think right now about what you're going to need to solve something for those three different pieces of information, you might need something that's in this classroom to help you do it hint, hint, hint, a calculator. So I'll go ahead and pass those out and if I were you, I'd probably get started on that paper first. It's green. And then I'll pass out the review paper, okay? Great job. Please do the green and follow-up with your studying for Friday and bring your triangles back, all right?