10th Grade Modeling with Systems of Equations - By
Transcript
| 00:06 | All right , I'm gonna give you this first . | |
| 00:09 | We're gonna talk about what it says real quickly . | |
| 00:11 | I want to make sure you all our understanding what | |
| 00:15 | it is . Okay ? So just so that we're | |
| 00:21 | all clear . I would really appreciate some help with | |
| 00:23 | reading . So Austin if you would read about the | |
| 00:28 | very first car . Mm Okay . Two very special | |
| 00:32 | cars have been developed steady car with the push of | |
| 00:34 | a button can instantly go from 0 to 1 mile | |
| 00:37 | per Minute and we'll continue to travel at constant speed | |
| 00:41 | of one mile per minute until the button is push | |
| 00:44 | releasing this feature at which point the car is controlled | |
| 00:47 | by the gas pedal eyes in a normal car . | |
| 00:49 | Okay . Would someone like to rephrase that so that | |
| 00:53 | I know that everybody is clear on what that car | |
| 00:55 | can do Goes one mile every minute and does the | |
| 01:01 | speed change at all know what you push a button | |
| 01:04 | ? It stays constant . So it's like a wood | |
| 01:06 | on the car cruz , just like a cruise control | |
| 01:08 | . Okay . Okay , great . Thank you Chelsea | |
| 01:11 | . Would you please read the , about the variable | |
| 01:14 | car ? Variable car , with the push of a | |
| 01:18 | button travels a distance in miles . That equals the | |
| 01:22 | square of the number of minutes travel , divided by | |
| 01:25 | 60 , until the Button is pushed , releasing this | |
| 01:28 | feature , at which point the car is controlled by | |
| 01:31 | the gas pedal as in a normal car . Okay | |
| 01:34 | , Who would like to kind of paraphrase what's going | |
| 01:36 | on with that one ? That's a little harder . | |
| 01:40 | Anyone want to take a large step out jane . | |
| 01:44 | Would you mind ? So how far it goes ? | |
| 01:49 | Gonna Square that and then divided by 16 . Okay | |
| 01:52 | . So when you say how far it goes , | |
| 01:54 | it's not really how far it goes , it's how | |
| 01:55 | long it Travels . How much time has passed , | |
| 01:58 | how many minutes has passed you square that ? So | |
| 02:01 | if it was 36 minutes , you'd square root it | |
| 02:04 | not square root you would square . Remember the difference | |
| 02:07 | between a square root and a square . Okay . | |
| 02:09 | What is the difference ? How are square roots and | |
| 02:11 | squares related guys ? There are what ? To each | |
| 02:14 | other ? Opposite , But what's a better word than | |
| 02:16 | opposites in verses of each other ? They undo each | |
| 02:19 | other . Very good . Thank you Chelsea for clarifying | |
| 02:21 | that anybody else have a question about the cars . | |
| 02:24 | All right . So what I want to do is | |
| 02:27 | put you in groups . Um , pretty much the | |
| 02:29 | groups you've been doing before I think . But anyway | |
| 02:30 | , we're gonna put Kyler Gosh , Kyler Austin and | |
| 02:33 | Chelsea and one group eric if you flip around and | |
| 02:36 | work with Abby and jaden um , I want amy | |
| 02:40 | to flip around . Work with Lauren and Mitchell and | |
| 02:43 | Elliot and if you two will flip around and work | |
| 02:46 | with Elliot . So the question all we should hit | |
| 02:50 | the question real quick first . What is it you | |
| 02:52 | want to do ? So one more person read for | |
| 02:56 | me , please do over . Make sure we're all | |
| 02:57 | clear jaden . So the actual problem ask you to | |
| 03:02 | And raising 30 miles , assuming that both cars start | |
| 03:05 | at the same time at the same place and that | |
| 03:07 | both cars engage their special buttons at the same moment | |
| 03:10 | . Which car would you want to have justify your | |
| 03:13 | solution with multiple representations ? Okay , does everybody understand | |
| 03:17 | the questions ? So in other words , you want | |
| 03:19 | to do what in the race ? When ? So | |
| 03:22 | which car do you want to have ? So that | |
| 03:24 | you will win ? How many , how many answers | |
| 03:27 | do you want to have ? How many answers ? | |
| 03:30 | Oh , you mean how many representations ? At least | |
| 03:32 | two if you can . Okay already remember you feel | |
| 03:36 | free to use anything in the box if you need | |
| 03:38 | it . Some people just like , okay , yeah | |
| 03:45 | , 30 at the same time at the same place | |
| 03:51 | , both have to engage your special buttons at the | |
| 03:54 | same moment . Where do you want to justify your | |
| 03:56 | solution ? Okay , so this car can go from | |
| 04:02 | 0 to 1 ? I'm sorry . Say that again | |
| 04:09 | . Units units are minutes . The time is in | |
| 04:12 | minutes . The distances in miles . So , yeah | |
| 04:15 | , so if you're going to put in , So | |
| 04:17 | if you put in 30 here , that's minutes , | |
| 04:20 | 30 minutes , 30 miles . And so you want | |
| 04:27 | to go 30 miles , the race is going to | |
| 04:29 | take 30 miles . So , and I don't know | |
| 04:32 | how long that will take a 15 minute . How | |
| 04:36 | do you know it's going to take 15 minutes , | |
| 04:39 | but this is 30 miles and you're squaring the number | |
| 04:42 | of minutes , okay ? Yes , that's your variable | |
| 04:50 | . Very good . Very good . So , can | |
| 04:59 | you do that ? Okay then that's what you gotta | |
| 05:03 | figure out . Okay . So think about um So | |
| 05:08 | if the if the time you put in here varies | |
| 05:10 | because that's a variable , then what else is going | |
| 05:13 | to vary depending on what you put it ? And | |
| 05:15 | the answer is gonna be a wet variable ? Yes | |
| 05:19 | , but was that variable representing miles miles miles ? | |
| 05:25 | Right , because you put minutes in here , Right | |
| 05:27 | ? And what you get out , isn't that what | |
| 05:29 | it says travels a distance in miles right here ? | |
| 05:33 | Oh this is like 30 . So then this is | |
| 05:37 | the eggs , that's right here you can do like | |
| 05:42 | this is over one so that you can do that | |
| 05:44 | cross party . Okay . Well I thought it was | |
| 05:49 | really doing fast as one month , so this goes | |
| 05:51 | to some 60 instantly . Yeah . Right . And | |
| 05:59 | then cruz , so it's pretty much like that , | |
| 06:05 | So that's 1.60 mph . pretty much . And this | |
| 06:10 | one's going to tell me why you think that would | |
| 06:15 | be your equation , because this is how many miles | |
| 06:19 | you're going And in your minutes you have to square | |
| 06:21 | your minutes and then divide by 16 . So if | |
| 06:24 | you solve that equation , what would it tell you | |
| 06:26 | ? It will tell you the minutes that it would | |
| 06:28 | take to go To go ? 30 , 30 miles | |
| 06:31 | . Okay . Does that seem reasonable to ? Yes | |
| 06:34 | . Okay , that's what you came up with and | |
| 06:39 | that tells you what , okay , It'll take to | |
| 06:42 | get to speak already . So now until we have | |
| 06:46 | to find a different way to show that . So | |
| 06:50 | you're telling me the second car faster and less time | |
| 06:53 | can car is faster . It only takes 15 minutes | |
| 06:57 | 30 by the second car of us identify which one | |
| 07:00 | you mean ? The variable car ? Okay , So | |
| 07:02 | the variable car would win . So you would want | |
| 07:05 | to be that car . So how do you know | |
| 07:08 | that ? Tell me what these numbers are telling you | |
| 07:10 | ? Because this is this one's going on one mile | |
| 07:15 | a minute and that when you go miles squared , | |
| 07:19 | divide that by 60 but this says equals the square | |
| 07:24 | of the number of minutes traveled . So if you're | |
| 07:28 | traveling , so if you put that in there , | |
| 07:31 | that's saying you're going 30 minutes , not 30 miles | |
| 07:36 | , okay ? So if you go 30 minutes then | |
| 07:41 | you're gonna go how far 15 miles are . So | |
| 07:49 | , Okay . Yeah , Okay , so I heard | |
| 07:57 | what you just told them . So this isn't what | |
| 07:59 | you're doing Lisbon trying to figure out someday . Okay | |
| 08:03 | , you're fine . So this is actually 15 miles | |
| 08:07 | of Ego in 30 minutes , right ? Oh , | |
| 08:10 | it's right . Oh , Okay . We need to | |
| 08:13 | find this one . This is right right to get | |
| 08:16 | you to . Okay , so the study 13 . | |
| 08:23 | Yeah . So this is , So if we figure | |
| 08:27 | it out , you want the exact right or approximate | |
| 08:31 | ? You're asking a question , you just have to | |
| 08:33 | be able to justify it . So this is a | |
| 08:35 | case where approximate might make more sense . Okay , | |
| 08:38 | so this is minutes or it's about 42.43 . So | |
| 08:45 | , because that's like what you figure out for that | |
| 08:46 | . Right . Okay . So what does that tell | |
| 08:49 | you That that this car travels ? So like it | |
| 08:55 | takes 42 minutes for the race . Okay , so | |
| 08:58 | how are we going to answer the question ? You | |
| 09:00 | need to find the other car because you have to | |
| 09:02 | find a push car . Would you want to have | |
| 09:04 | ? Sounds good ? Okay . So what did you | |
| 09:07 | guys decide you'd rather have a study car because it's | |
| 09:14 | double the distance and the same amount of time ? | |
| 09:16 | So tell us , tell me how you found out | |
| 09:19 | that . Yeah , Because it's the one mile per | |
| 09:23 | minute . So you go 30 minutes , you go | |
| 09:27 | 30 miles and then on this one when you uh | |
| 09:32 | square it's 30 , you got like 900 and Evita | |
| 09:36 | provided by 60 . It was going to equal 15 | |
| 09:39 | . So it's only going to go half as far | |
| 09:41 | . Yeah . And the same amount of time . | |
| 09:42 | Okay . So that's why we chose out . All | |
| 09:45 | right . I want to during the other way . | |
| 09:48 | Okay . 45 miles . Nineties study has to wait | |
| 09:57 | 20 minutes . And how would that change ? And | |
| 09:59 | I want you to stop and go back and see | |
| 10:01 | if you can think of another way to prove this | |
| 10:04 | . Is there another way you could have worked this | |
| 10:06 | out and got the same answer . Think about some | |
| 10:11 | different ways to represent the problem . Clara . would | |
| 10:14 | you be comfortable sharing your method here with the class | |
| 10:18 | ? That would be great . I'll have you do | |
| 10:20 | it 2nd . And then um Chelsea , would you | |
| 10:28 | mind sharing this first one with the classes ? At | |
| 10:32 | the class ? With the classes ? Okay , Honey | |
| 10:34 | and everybody kind of stop for a minute . Let's | |
| 10:36 | kind of come together . I'm gonna have Chelsea come | |
| 10:38 | up and show us what she decided for the study | |
| 10:42 | car . It goes one mile per minute . So | |
| 10:46 | we did One times 30 because there's 30 minutes . | |
| 10:53 | And so that would mean you go 30 miles Now | |
| 10:57 | . Why did you choose 30 to start with ? | |
| 11:00 | Because it said in the race you go 30 miles | |
| 11:04 | . Okay ? So if you go , yeah , | |
| 11:07 | one mile . So you figured what the heck ? | |
| 11:10 | Let's just put that in . Okay . All right | |
| 11:12 | . And so for the variable car , the equation | |
| 11:17 | we put was Sorry , you go 30 squared divided | |
| 11:23 | by 60 . And that right there Gives you 900 | |
| 11:30 | And then you divide that by 60 , which equals | |
| 11:34 | 15 miles . And you did , you started with | |
| 11:41 | 30 on this one because because you go 30 miles | |
| 11:44 | and if you square that would have given you 900 | |
| 11:47 | , you said you go 30 miles . But is | |
| 11:49 | that what you put in there is 30 miles . | |
| 11:55 | Yeah , that's what I put Austin . Do you | |
| 11:58 | remember what you guys were doing ? I don't even | |
| 12:00 | know where you asked . Okay . Skyler . So | |
| 12:04 | why did you put 30 in 30 represents the wet | |
| 12:07 | The distance ? The distance ? No . Well , | |
| 12:09 | which 30 represents the distance . This is a little | |
| 12:12 | confusing . So , um , on your very first | |
| 12:16 | one Chelsea , right above the 1st 31 times 30 | |
| 12:19 | . Would you put minutes there ? Okay . So | |
| 12:25 | then the 30 that's down below Austin . What ? | |
| 12:28 | What is that 30 ? Is this 30 minutes miles | |
| 12:32 | or is this 30 miles ? 30 miles Kyler ? | |
| 12:35 | Do you agree ? This is 30 miles . Yeah | |
| 12:39 | . Okay . Now I had another group who did | |
| 12:40 | this the same way and I've got to make sure | |
| 12:42 | that your your accurate and you're thinking here because what | |
| 12:45 | did the formula say ? Read us the formula Kyler | |
| 12:48 | right off of your , off of what it says | |
| 12:51 | on what one question . The variation , the variables | |
| 12:55 | car . Um , with the push of a button | |
| 12:58 | travels a distance in miles . That equals the square | |
| 13:00 | of the number of minutes . Okay , stop right | |
| 13:02 | there . The square of the number of minutes . | |
| 13:06 | So if you're squaring 30 , this 30 is what | |
| 13:09 | minutes minutes . So when you write minutes right above | |
| 13:12 | that one . So we don't get confused that we're | |
| 13:15 | using that number 30 in two different ways . And | |
| 13:18 | the reason you pick 30 Is not because it was | |
| 13:21 | 30 miles . Who can tell us why she picked | |
| 13:24 | 30 ? To put in that second equation to compare | |
| 13:28 | it to the first one . We're gonna see how | |
| 13:31 | how long it takes each one . So we're going | |
| 13:33 | to make sure that they're each using the same amount | |
| 13:35 | of minutes . Okay ? Should we say that again | |
| 13:39 | Jane ? Yeah . So the 30 minutes is comparing | |
| 13:42 | it to the top cars 30 minutes it takes to | |
| 13:44 | finish the race . Okay ? So now if you | |
| 13:47 | put an sc out in front of the one and | |
| 13:49 | A . V . C . On the other one | |
| 13:52 | , just so that we're really clear which one is | |
| 13:54 | which uh into the left to the left of it | |
| 13:57 | . There we go . Super . Okay . So | |
| 13:59 | now what's your conclusion ? Shall see you ? My | |
| 14:03 | conclusion was the city car goes faster because you double | |
| 14:07 | the distance . We're good . Everyone understand . Okay | |
| 14:13 | , thank you Chelsea very much amy will you come | |
| 14:15 | up ? I'm just going to capture all this if | |
| 14:18 | I can get my board to work , wow . | |
| 14:23 | Yes , I want you to show us how you | |
| 14:25 | did . I'm just gonna put this up here . | |
| 14:27 | Same thing to do a good job of standing to | |
| 14:29 | the side and kind of so what are we did | |
| 14:33 | it Like the sc still the same because it's one | |
| 14:38 | minute for miles . So that's the same thing . | |
| 14:41 | But we did a variable for our next one because | |
| 14:45 | we don't know what the minutes are . So we | |
| 14:47 | just put exit and we know it's divided by 60 | |
| 14:51 | And then that's equal to the 30 miles because that's | |
| 14:55 | what it tells us that we get . And so | |
| 14:58 | then you put this Over one and you do cross | |
| 15:02 | products . And then so this equals just X . | |
| 15:09 | Squared . And that's equal to 81 . And then | |
| 15:16 | you square those who get rid of square . So | |
| 15:18 | then x equals about 42.43 . So then we chose | |
| 15:26 | , so then it's uh um steady car would be | |
| 15:30 | better since it takes less time . So let's put | |
| 15:33 | some units on that . 42.343 what minutes minutes now | |
| 15:38 | . Uh And the steady car would take , how | |
| 15:40 | long The city car would take ? 30 minutes . | |
| 15:45 | And so we've shown another way to indicate that we | |
| 15:49 | would choose the steady steady car because it would do | |
| 15:52 | it in less time . Anybody have any questions for | |
| 15:55 | Amy ? Okay . Super . Alright . Spencer would | |
| 15:59 | you do show us what you did for the second | |
| 16:03 | part where the racetrack is 90 miles ? Mm . | |
| 16:14 | Okay . So we used the same thing as Amy | |
| 16:18 | did and so we used a variable , so it's | |
| 16:21 | X squared over 60 equals 19 . And then your | |
| 16:28 | times all this by 60 , Many times this by | |
| 16:32 | 60 And that equals 5400 equals X squared and get | |
| 16:41 | rid of that square . You got a square root | |
| 16:43 | it you have to do to both sides . and | |
| 16:46 | so x equals was at 40 , No 73.5 . | |
| 16:56 | And that would be how many minutes it would take | |
| 17:01 | for the variable car . And for the steady car | |
| 17:06 | it would be 90 minutes because you're going one mile | |
| 17:10 | every minute . That's your steady car . Yeah . | |
| 17:16 | Oh so you want to choose your variable car ? | |
| 17:20 | Very good . Okay , um mm . Jaden would | |
| 17:25 | you go up and show in our last corner the | |
| 17:27 | method that she used to do the 90 mile part | |
| 17:32 | ? Yeah . So if you can't stand another way | |
| 17:43 | , you know , it's kind of rough right A | |
| 17:44 | little bit and then move . So the one side | |
| 17:49 | for the variable car , This is how far it's | |
| 17:52 | gonna go in 90 minutes And I don't know what | |
| 17:58 | that number is . It's 135 . Mhm . So | |
| 18:04 | what's on your paper ? So if they both took | |
| 18:09 | 90 minutes to do the race like over here , | |
| 18:13 | standard card took 1990 minutes then this one go 100 | |
| 18:18 | and 35 miles . So it would be to buy | |
| 18:19 | a long ways . Okay ? But we only had | |
| 18:22 | to go how far to go 90 miles . So | |
| 18:25 | how do we know for sure that at that 90 | |
| 18:29 | miles spot but this one went farther in the same | |
| 18:33 | amount of time . So be it . Yeah . | |
| 18:37 | So it's it's there beforehand no matter what it has | |
| 18:39 | to be because the 90 miles is what the steady | |
| 18:42 | card in in 90 minutes . Okay . That was | |
| 18:45 | a good explanation , Jane . Thank you . Thank | |
| 18:48 | you . Ok , Everyone okay with what we've done | |
| 18:51 | now , just to talk , you have different , | |
| 18:53 | like , different like you can have , like , | |
| 18:57 | decide which one you think goes farther . Like it | |
| 19:01 | doesn't have to be the same , did we ? | |
| 19:03 | Well , I'm not being set , the steady car | |
| 19:06 | goes faster , but we think that the variable car | |
| 19:08 | goes faster in this situation . Did you think the | |
| 19:11 | variable car went faster ? But in ours ? No | |
| 19:15 | , hers was a steady car and Arthur is a | |
| 19:17 | variable . Okay , so , remember we're doing two | |
| 19:18 | different problems . This one this problem up here is | |
| 19:22 | if the race was 30 miles , this one here | |
| 19:26 | was if the race was 90 miles . O Okay | |
| 19:29 | , so , in answer to your question , Erica | |
| 19:31 | , it depends on how long the track is the | |
| 19:34 | races . That's right . That's going to make a | |
| 19:36 | big difference onto uh which one ? Which car we're | |
| 19:39 | gonna choose . Okay . Now , quick thing about | |
| 19:43 | about the algebra . Okay . Amy solved this one | |
| 19:46 | um by doing cross products . Okay . And um | |
| 19:50 | Spencer did this one by multiplying both sides by 60 | |
| 19:54 | . Is that the same or is that different ? | |
| 19:56 | Okay . So how does cross products work ? We're | |
| 20:01 | really just doing what ? Times on both sides ? | |
| 20:05 | By some number . In this case we times over | |
| 20:07 | here by 60 times 60 to both sides , and | |
| 20:10 | actually times one to both sides . But times and | |
| 20:12 | by one doesn't change anything . So we're good . | |
| 20:14 | Alright . Just want to do a real quick thing | |
| 20:16 | on the algebra . Now , is there another way | |
| 20:19 | I actually do several representations ? You guys all This | |
| 20:22 | is actually kind of two different ideas . Well , | |
| 20:24 | these two , both of both of those groups worked | |
| 20:27 | with starting with a number and figuring out how far | |
| 20:31 | it would go , figuring out . Starting with time | |
| 20:34 | , figuring out the miles . And this group figured | |
| 20:37 | out using a variable and started with the the distance | |
| 20:42 | through the yeah , the distance . So could we | |
| 20:45 | create a table ? And I want to create a | |
| 20:49 | table for , for the minutes traveled and then I | |
| 20:53 | want the distance for steady car and I want the | |
| 20:57 | distance for variable car . Okay , that's kind of | |
| 21:01 | just do this together because I don't think it's gonna | |
| 21:03 | be too hard . What kind of units do you | |
| 21:04 | think we should use ? How , how often should | |
| 21:06 | we figure out what's going on based on what you've | |
| 21:09 | been thinking ? Is everybody understand the questions so I | |
| 21:11 | don't embarrass anybody . Okay , so let me ask | |
| 21:14 | the question again , I'm going to , should I | |
| 21:16 | put up here 1234567 minutes ? Or should I go | |
| 21:20 | every 10 every 20 every 30 every 50 every 100 | |
| 21:24 | . What's , what's the intervals we're gonna do . | |
| 21:26 | So let me pick on somebody here , Elliot . | |
| 21:31 | I was thinking in every like 10 because You would | |
| 21:36 | go like 30 miles . You want to see How | |
| 21:40 | many it goes each mile and then you can and | |
| 21:44 | then you can go all the way to 90 easily | |
| 21:46 | . Okay , and why are we interested in an | |
| 21:48 | idea , Elliot ? Because we're also doing 90 miles | |
| 21:53 | just to compare how far it goes over a long | |
| 21:56 | period of time is too short period of time . | |
| 21:58 | Okay , um , Jesse is on our back . | |
| 22:01 | Um , you guys can just follow along well actually | |
| 22:05 | I would like you to do it on the back | |
| 22:06 | , so yes . Thank you . Um , Spencer | |
| 22:08 | , I'm gonna give you the really tough job where | |
| 22:09 | you come up and fill in all of the distances | |
| 22:12 | for the steady card . I know that's really a | |
| 22:13 | big challenge for you . Okay . Um and then | |
| 22:17 | for the distance for the variable car , awesome . | |
| 22:20 | Let's start with you . Will you do the 10 | |
| 22:23 | Chelsea ? You do 20 ? I want you to | |
| 22:24 | figure out how far the variable car is going to | |
| 22:26 | go in 10 and then you do it in 20 | |
| 22:29 | . We've already got 30 Kyler , you do 40 | |
| 22:32 | mitch do 90 and Elliot and Amy and Claire . | |
| 22:37 | I want you to pick anyone you want and check | |
| 22:38 | and make sure everybody does things right ? You should | |
| 22:41 | check . Okay , give you every minute to do | |
| 22:45 | those . Yeah , he's great . Mhm . So | |
| 22:56 | what does that tell you about about the race ? | |
| 22:59 | About the cars ? Okay . Say that again , | |
| 23:05 | Jaden longer the races , the car gets passed . | |
| 23:09 | Okay . Does that make sense for everybody ? So | |
| 23:12 | why is that happening ? Why is that happening ? | |
| 23:17 | Make sure you got that number 90 ? You know | |
| 23:21 | what to do ? Okay . So what's the formula | |
| 23:24 | we're using for finding the the minutes or the yeah | |
| 23:28 | , the miles . If you know the minutes this | |
| 23:30 | is this is 90 . Would you come up and | |
| 23:35 | do it on my calculator up here , Mitchell ? | |
| 23:37 | Would you come to it up here so we can | |
| 23:38 | get that taken care of ? Yeah , Mitchell come | |
| 23:42 | do it up here . Excuse me . Okay . | |
| 23:51 | Yeah . Yeah . Yes , Divide This by 60 | |
| 24:00 | . Mhm . John 66 Just back here aren't just | |
| 24:06 | like you're doing your calculator right there . One more | |
| 24:10 | . Mhm . Okay . So Reddit and what happened | |
| 24:17 | to my table ? Okay . All right . So | |
| 24:33 | everyone clear now are miles good ? So again , | |
| 24:38 | these are miles . This is miles . This is | |
| 24:42 | minutes . Okay . No clear . Okay . Now | |
| 24:47 | , so we can see in fact , at what | |
| 24:49 | point does it not matter which car you have ? | |
| 24:53 | At which point does it not matter , wow . | |
| 24:57 | Um Abby 60 . And why is it at 60 | |
| 25:01 | that we know that ? Because they're all the same | |
| 25:03 | . They're all the same . Okay . And it | |
| 25:05 | really didn't matter that the minutes where is the same | |
| 25:07 | ? But it mattered that the two distances were exactly | |
| 25:10 | the same . Okay . So we got those two | |
| 25:12 | together . All right . So why is this one | |
| 25:15 | doing what it's doing and this one is doing what | |
| 25:17 | it's doing ? What do you think ? Spencer ? | |
| 25:19 | Spencer ? It's because of the squared because of the | |
| 25:25 | squared . What does the square due to a graph | |
| 25:28 | ? It makes it forever . It makes a parabola | |
| 25:31 | . Okay . If we were going to grab these | |
| 25:34 | , in fact , can we write the functions ? | |
| 25:36 | I would like you all to write on your paper | |
| 25:38 | , the two functions for each one . So if | |
| 25:40 | I was to take my chart here and kind of | |
| 25:46 | extended over here a little bit so that I had | |
| 25:49 | here one more column and I said for m minutes | |
| 25:54 | , what would I put in ? What would be | |
| 25:57 | the function or the formula ? That would allow me | |
| 26:00 | to determine how what the distance is for either one | |
| 26:03 | . Okay . Everyone understand the question . Okay . | |
| 26:06 | The question is for any type of minutes , how | |
| 26:11 | do I find out the distance of the steady car | |
| 26:13 | ? How do I find out the distance for the | |
| 26:15 | variable car ? Okay . Tyler , I talked about | |
| 26:20 | like minutes squared divided by 60 I am . And | |
| 26:23 | which then which one does that go to ? What | |
| 26:26 | do you ? Uh the variable . Okay , so | |
| 26:30 | a minute square divided by 60 . Okay , Kyle | |
| 26:32 | has done that one , Clara , what's the other | |
| 26:34 | one ? It's just the minutes . Okay , those | |
| 26:38 | are our two variables . And so what we can | |
| 26:40 | write is we can write one function is distance equals | |
| 26:44 | minutes . This is the steady car . Distance equals | |
| 26:48 | minutes squared over 60 . There's a variable car . | |
| 26:51 | Those are two functions that we have going . Okay | |
| 26:54 | . What do the graphs look like the graphs look | |
| 26:59 | like ? Yeah , copy this over so we can | |
| 27:08 | do this all on one page . Mhm . Mhm | |
| 27:21 | . The ground , Yeah . Why don't you everyone | |
| 27:23 | grab a piece of grid paper out of your bucket | |
| 27:25 | . And why don't you do a graph of this | |
| 27:29 | and see what you can come up with a a | |
| 27:34 | . Oh yeah , we have to do exactly . | |
| 27:39 | Um well you want to like , yeah , because | |
| 27:43 | we're rounding some stuff anyway . So yeah , you | |
| 27:45 | don't have to have a one for one . You | |
| 27:47 | can make every grid be fat when I did this | |
| 27:50 | one , I made every grid . How much dan | |
| 27:53 | . So whatever you want to do , always make | |
| 27:58 | your graphs be workable for you Mitchell . Come come | |
| 28:10 | where put where you think the fixed point in the | |
| 28:13 | fixed line are approximately . Just just as a good | |
| 28:16 | review where we've been before . Uh huh . Six | |
| 28:24 | point , that's the vertex . Where's the fixed point | |
| 28:28 | ? What do we call the fixed point ? Everyone | |
| 28:30 | focus . Focus . Where's the focus gotta be ? | |
| 28:33 | I don't know to be honest . Just if you | |
| 28:37 | had to guess , where would you put the focus | |
| 28:42 | somewhere ? Put it there . Yeah . How many | |
| 28:46 | agree Mitchell ? Just did a really good job there | |
| 28:48 | . Okay . Now where's the fix Help him out | |
| 28:51 | . Abby has to be where it has to be | |
| 28:53 | directly above the vertex . Yes , Yeah , somewhere | |
| 28:58 | . And then if that's where your fixed point is | |
| 29:01 | , where does your direct tricks have to be your | |
| 29:03 | fixed line ? And what do we know about the | |
| 29:05 | position of it ? Same distance you need a line | |
| 29:16 | . Okay . All right . Yeah okay so my | |
| 29:25 | next question is I'm watching clock because we're getting close | |
| 29:28 | to bail so I want to kind of kind of | |
| 29:30 | finish up with just this and then we're gonna start | |
| 29:32 | with this again tomorrow . But where is ? So | |
| 29:37 | can you can you can someone just kind of show | |
| 29:39 | us how close mitch got in other words can we | |
| 29:43 | kind of look at the distances to see just and | |
| 29:47 | then following what I'm asking amy you wanna come up | |
| 29:50 | and draw on and kind of show us what I'm | |
| 29:52 | trying to say . Here was that ? Yeah yeah | |
| 29:57 | like an X five point . So then you pick | |
| 30:00 | a point and you dry down to your square and | |
| 30:06 | then you dry two they're not too there you have | |
| 30:10 | to go to where you want . It gives the | |
| 30:12 | definition girl . There we go . That was the | |
| 30:15 | point because those are equal distance and you have to | |
| 30:18 | find the distance . So how do you do do | |
| 30:19 | you think those are equal mm All right . They're | |
| 30:23 | all right . Not too bad . And we really | |
| 30:26 | should be on the problem . Right . You should | |
| 30:28 | be right there . Yeah . Sorry . Okay . | |
| 30:31 | All right . Well okay with that . Okay . | |
| 30:34 | So what we have here is we actually have a | |
| 30:38 | system of equations . We have two equations . We | |
| 30:41 | have a problem . We have a line , we | |
| 30:43 | have that system and you guys solve the system when | |
| 30:47 | we solve a system . Where do you think the | |
| 30:48 | solution to that system is your solution ? Isn't . | |
| 30:52 | What we talked about is the answer to that first | |
| 30:54 | question . Which car do you want to own ? | |
| 30:56 | The solution is where or what ? Erica what do | |
| 31:01 | you think ? Where is the solution to this system | |
| 31:08 | ? Um what's more , I don't know where where | |
| 31:17 | would you guess if you were gonna say where is | |
| 31:19 | the solution to the system ? What do you think | |
| 31:24 | ? Yeah . And we want to help her out | |
| 31:26 | ? I don't get the question . Do they want | |
| 31:28 | to have an idea what I'm going for ? Abby | |
| 31:31 | ? No . Clear the place where they meet up | |
| 31:37 | the places where their solutions The same . In other | |
| 31:39 | words , when we solve a system and for this | |
| 31:41 | particular problem , we're actually asking the question at what | |
| 31:44 | point is the time For each car and the distance | |
| 31:49 | for each car the same ? And where does that | |
| 31:52 | happen , 60 , 60 ? Where else does it | |
| 31:55 | happen ? 00 when they start out ? |
Summarizer
DESCRIPTION:
10th grade math lesson on modeling with systems of equations. The lesson involved high school modeling standard A.CED.A.2 from the MA Curriculum Frameworks for Mathematics, which specifies creating equations in two or more variables.
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