10th Grade Modeling with Systems of Equations - Free Educational videos for Students in K-12 | Lumos Learning

## 10th Grade Modeling with Systems of Equations - Free Educational videos for Students in k-12

#### 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
00:28 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
01:14 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
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
10:11 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
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
12:45 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
15:52 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
17:16 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
19:31 , 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 .
22:23 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
22:59 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 .
30:54 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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