4th Grade Math - Free Educational videos for Students in K-12 | Lumos Learning

4th Grade Math - Free Educational videos for Students in k-12


4th Grade Math - By Lumos Learning



Transcript
00:01 uh , boys and girls . Today's lesson . We're
00:08 going to start off with quick images , so you
00:10 need to have your Mac notebooks out . Let's date
00:12 it for September 14th . We started quick . Image
00:16 is just the beginning of this week , remember ?
00:19 I'm going to show it once , five or 10
00:21 seconds , and I'm gonna hide it again . So
00:25 you need to really visualize what you see and draw
00:28 what you see . I'll give you a few seconds
00:32 . I'll show it again . I'll hide it again
00:35 . Again . You're going to add to your visualization
00:37 of what you see . Then I'll show it a
00:41 third time , and we'll talk about it . We'll
00:43 talk about how we can get it to relate to
00:46 math , math equations . And some people think we're
00:49 gonna probably have a different different ways of viewing it
00:51 . So we'll talk about that too . So are
00:54 your eyes ready ? Hey , here we go .
00:57 First time . Yeah . Let's look at it again
01:11 . Your eyes ready ? I don't want you might
01:14 have missed it . Mhm . Mm hmm . Can
01:24 you think of an equation that might go with this
01:27 ? How did you know how many there were ?
01:29 Can you circle your three groups ? of six ,
01:33 so I can see it even better . Yeah .
01:37 Mhm . Okay , let's see what you got .
01:42 Yeah . Mhm thumbs up if you got it exactly
01:48 right . Items up . If you were really ,
01:51 really , really close and you were on your way
01:56 . So I know sometimes it takes us a little
01:57 bit longer than the time I can give you .
01:59 And folks let's talk about how many this is and
02:03 how you visualize that with your math . Jose ,
02:07 what do you do ? Uh , I'll put three
02:11 times six equals 18 . You got three times six
02:14 equals 18 . Can you come up here and circle
02:18 ? Mm . How you visualize three times six equals
02:20 18 , please . Thumbs up , if that's what
02:22 you saw . Ah , so we've got some thinkers
02:27 that are all thinking the same . Yeah . Now
02:34 he's showing us what he's showing us . What's he
02:38 showing us ? Ethan , say it again , please
02:42 . Groups of three groups of three . And how
02:45 many groups of three did he get ? 66 Good
02:48 job . Thanks . Anybody see it another way ?
02:51 Yeah . Anybody see it another way ? Uh ,
02:56 well , how do you see it ? Uh ,
02:58 Deandra , Nine times 29 times two . And what
03:02 did you get for nine times two , 18 ,
03:05 9 times two equals 18 . Can you come up
03:07 and show it , Please use it A different color
03:10 marker so you can see it . Did anybody think
03:13 like Deandra ? Yeah . Yeah , the teacher .
03:26 Okay , Yeah . Yeah . Okay . I see
03:31 how you connected it . Nine groups of two .
03:33 Did anybody do the community property of that one ?
03:38 Because I really didn't think . Nine groups of two
03:40 . I thought of it as Yeah . What's the
03:44 community that nine times two . Yeah , Yeah ,
03:48 yeah . Say it again . Two times . Two
03:51 times nine . And I'll show you what I did
03:52 . You might think of it like that . Two
03:53 times 92 groups of nine . Come up , Jacob
03:56 . And show us how you saw that way .
03:58 We know that these mean the same thing . We've
04:02 already talked about that , but circling and they might
04:05 show it in a different way . Show me your
04:07 two groups of nine . Yeah . Okay . Mhm
04:14 . Mhm . Good job . Good job . Another
04:19 way . Got another marker . We can do it
04:22 one more way , and then we're gonna go on
04:24 with our math lesson for the day . Another way
04:26 . Uh , what you get ? Mm Care is
04:31 going to show us what she's doing . Oh ,
04:49 good . So you and Jose were doing the community
04:52 of each other ? The flip flops is what ?
04:54 Kindergartners column . Right , But not in fourth grade
04:57 . So a lot of a solid as looking at
04:59 groups up as groups up if I was gonna put
05:03 . And this is just going to extend a little
05:04 bit if I was gonna put all of these together
05:09 . He's still big . Well mm . Grab the
05:17 little eraser out . Uh huh . Does that look
05:23 big , Huh ? That's big . It's quicker .
05:29 If I was going to put all of these together
05:31 and make what am I making ? If I put
05:34 all of my rows and columns and fit them together
05:36 ? What am I making trip ? What am I
05:39 making ? Alright . An array . What would the
05:42 dimensions be if I was going to slide them altogether
05:45 ? Six by three . Good job . Come and
05:47 show us . Yeah . Mhm . And then label
05:58 where you're six would go . Good job . Cool
06:04 . Cool beans . That's the new work . Cool
06:06 beans . Good job . Good job on quick images
06:09 . We're gonna do those . Why are we doing
06:11 them ? We're gonna keep doing them . Why ?
06:12 What's it gonna help us Do especially us that still
06:17 are trying to memorize some things . What's going to
06:20 help us do ? Yeah , Matthew , sometimes it
06:27 can help us . Um , count battle count faster
06:32 . And what's counting has to do with a raise
06:37 . Um , it can help you . Long time
06:40 . Exactly . Very good . Exactly . All right
06:44 . You can keep your notebooks open because you're probably
06:46 gonna need them . But I need your pencils down
06:48 and all eyes up here on our smart board .
06:51 We're gonna do it . Think pair share . We're
06:55 gonna do a lesson today . It's called multiple .
06:58 Okay , tive comparisons . I want you to think
07:04 , What does that mean ? And I mean ,
07:07 when you give you a little bit of hint you
07:09 all know back here on our cafe menu how to
07:13 figure out what words are You've given me these ideas
07:18 how to figure out what they mean . We've done
07:22 nifty thrifty 50 talked about prefixes and root and base
07:27 words and suffixes . Use your schema in reading .
07:32 Help you figure out those math words . Think mhm
07:38 . Mhm . What does multiplication comparisons mean ? Mhm
07:49 . You're thinking pair up with your partners sitting around
07:53 you talk about it . What Do you think it
07:56 means multiplication and , uh , check about multiplication and
08:02 then compare . Compare . We compare . The multiplication
08:07 is to get multiple multiply . What about comparison ?
08:12 Comparison ? What's the word in there like that is
08:15 to say not . What do you think ? What
08:22 did you say that she agreed with said multiplication Paris
08:27 . I know it's hard . It's really like you
08:30 both . Get them , like , steady for each
08:32 for each one you like . Night . Steady .
08:36 So what ? Steady me when you're talking about multiplication
08:39 comparison . Um , seven times three and three times
08:43 seven . Okay , so you think it kind of
08:44 means flip flops ? And you said that the community
08:48 property . I love the way you always try to
08:50 give examples to . Help me understand That helps me
08:52 , like you can still get to compare them .
08:54 Yeah , And you're comparing those two . Okay ,
08:56 What do you think ? It's probably like I mean
09:00 , like , it could be a different ring ,
09:03 a different array . So you're going to compare one
09:06 array to a totally different array . Okay . Okay
09:15 . Does anybody want to share their thoughts ? Their
09:17 ideas ? Chiana , me and my partner shares that
09:22 watch application . Compare multiplication of comparison multiple . Kate
09:29 of comparison . We think it means that you compare
09:35 . Want one multiple case ? Ethan , You can
09:39 help her out if she wants your help . We
09:42 believe that , um mm . That means that we
09:46 that you compare one multiplication equation with another . Okay
09:51 , Okay , that's a good idea . Good thinking
09:56 . What do you think ? I think that you
09:59 have to put two numbers together and see what are
10:01 equal to any two numbers . No . And you
10:07 said see what it equals . Well , let me
10:08 rephrase my question . What are you going to do
10:10 with those two numbers when you put them together ?
10:13 You gotta tell me what put together means ? Yeah
10:17 . So we can do a lot to numbers .
10:19 Think of your word . Yeah . Think of the
10:22 work . Do you see any words ? Small words
10:28 that jump out at you in that big , long
10:30 word multiple multiplication of comparison . Compare . Okay ,
10:38 but you were talking about doing something to two numbers
10:41 . Do you see another word out there that might
10:43 help you decide what you're gonna do to those two
10:45 numbers besides compare them share ? Share them . What
10:50 are you thinking , Trent ? You have your hand
10:52 in the back . I was thinking that it might
10:56 , You might have to have two numbers and you
10:59 have to find the multiples . Okay . And why
11:01 do you say to numbers with multiples ? What do
11:04 you do ? Multiples ? Because it's as multiple .
11:08 So you're you're chunking . You're seeing a little word
11:11 in that big , long word . That's one of
11:12 the things to be said about accuracy . That's one
11:14 of the things we're learning about . 50 50 50
11:16 . Um , de janeiro . Uh , you take
11:23 two different months of for months of multiplication equations and
11:29 see if they could are saying okay and I'll have
11:33 time for one more , and then we're gonna go
11:35 on , and we're gonna see if what we were
11:37 thinking is really what we're going to be doing with
11:40 multiplication comparisons . One more . Um , how about
11:47 Olivia ? Oh , sure . Multiply means multiplying by
11:52 numbers . So you think multiplication means multiplying numbers ?
11:56 Okay , we're going to compare them . Sure .
12:00 Okay , we'll track . All right , let's focus
12:03 back . We've done I think , pair share .
12:06 And I want to show you a problem that it
12:12 is an example of multiplication of comparison . And it
12:18 sounds like this . Deandra picked seven apples . Reese
12:25 picked four times as many apples . How many apples
12:30 did he pick before I even start to think about
12:35 it ? Like math . I think about it .
12:37 Like reading Were the characters in our story Who are
12:42 the characters in our story ? Kayleigh , Deandra and
12:50 Tianjin . Reese . What are t Andhra and Reese
12:54 doing ? Shayla at the count of Apple's ? Uh
13:04 , they're counting apples . Do you think they might
13:06 eat them ? What can you infer they might do
13:08 with them besides , eat them like this is done
13:13 and Yeah . Okay , Mike , plant the seeds
13:19 out of them regular apple school . All right ,
13:21 what is , uh , your problem ? Because in
13:24 math , the story problem becomes your problem . So
13:30 what do we need to figure out here , Alex
13:40 ? Do we need to figure out how many apples
13:42 they have ? Do you need me to reread it
13:46 for you ? Okay . It says Tianjin picked seven
13:51 apples . Reese picked four times as many as as
13:55 many as apples . How many apples did he pick
13:59 ? So who are we looking to see ? We're
14:01 looking for weeks . We're looking for racist . Apple's
14:04 good job . Good , good , good . Good
14:06 job . All right , Hands down . I'm starting
14:10 to learn how to use the smart board . You're
14:11 gonna think this is pretty cool . Your third grade
14:13 teacher may have already showed it to you . I'm
14:15 going to show you the apples that Deandra picks .
14:20 Okay , you can count with me . 12 Oh
14:34 , What's to say exactly ? I'm gonna make this
14:38 page just a little bit longer . Alright ? Seven
14:41 apples , T . Andres , you have fun picking
14:43 your apples . I like apple . You like Apple
14:45 School ? All right ? Reese comes along and he
14:49 the story says he picked how many more Or how
14:53 many times ? More . Four times . Who wants
14:57 to show me the one time ? The one time
15:02 one time ? As many as Reese . Jacob .
15:06 All you gotta do is your apples are gonna be
15:09 red Reese . 200 years of yellow . So you're
15:12 gonna show me one times as many , and this
15:15 is a Well , we'll see what happens . Floor
15:20 mhm three . Why ? Good job . Are you
15:32 done ? Good job . That four step back because
15:36 we're in the thing . This shows that one times
15:41 as many as who T Andhra . That's one .
15:46 Okay , alright . Trends ready . Jacob said he's
15:50 ready , so j trend . You're going to show
15:53 us two times as many , but You've got to
15:56 use what Jacob already has on there . Okay ,
15:58 Keep the Richard do your stuff . Do your thing
16:01 . Do your thing . Uh , great . Oh
16:13 , Oh , just click off . There you go
16:17 . She says hi . Seven fish . He said
16:29 , Yeah . Tell me why I should show me
16:31 two times as many we can put . Show me
16:35 two times as many seven apples . You did seven
16:40 apples . Uh , is that a group of seven
16:43 apples ? It's not a group . You tell me
16:49 . You think it is a group ? Why is
16:51 it considered to be a group ? Because I put
16:54 seven apples and you put on we could put a
16:56 circle . Did Jacob make a group of seven apples
17:01 ? And what did I ask you to show ?
17:04 Did I say show two groups of seven ? You
17:10 are interpreting that , and that's exactly what I want
17:12 you to do . That's exactly what I want to
17:14 do . But what did I really say ? What
17:19 did I say ? I said show me two times
17:22 , but listen to their language . They kept saying
17:26 to groups up , Hmm ? All those synonyms ,
17:30 maybe maybe think about that , Jose , You want
17:36 to show me three cars as many I'm going to
17:39 extend this page a little more . What ? Okay
17:49 . Oh , seven . You caught yourself . Why
18:00 did you go back ? Because all I heard eight
18:04 hurt . But he wasn't paying attention . You probably
18:08 must have been . He didn't because he stopped .
18:10 He didn't pay attention . He would have kept going
18:11 , so you didn't pay attention . So tell me
18:13 what you got here , Jose ? I get 21
18:17 . You have 21 . How do you know ?
18:20 Because I know that seven times 47 7 times to
18:24 14 . And if you add seven more than 14
18:29 , you'll get Oh , 21 . Cool you got
18:37 can I'm gonna say , Do you have seven or
18:40 you have How many groups up here ? That's what
18:43 I was . How many groups you have ? 33
18:45 So could I say you have three groups of seven
18:48 . Can Does anybody know the synonym of what I
18:51 asked him in the beginning to do ? What did
18:55 I tell him ? I didn't come up here and
18:57 show me three groups of seven . What did I
18:59 say ? Giovanni ? Three times seven . I did
19:07 show me three times a day , three times a
19:11 day job . Okay , Someone , Dijon A I
19:15 need you to show me four times as many using
19:18 already what they have . Why three ? Are you
19:32 finished ? So can you explain what you have up
19:35 on the board using all your peers stuff too ?
19:39 You have four times seven . Do you have ,
19:46 uh , four times as many as to Sandra ?
19:49 Yes , you do . So tell me , are
19:52 we finished ? How do you know ? I think
19:57 are we finished with the story ? How do you
20:01 know ? I'm going to read it out loud ?
20:04 Because I think some people might need that . Deandra
20:07 picked seven apples . What color ? Her apples ,
20:10 everybody . Reese picked four times as many apples .
20:16 How many apples did he pick ? Are we finished
20:20 with our story ? Did we solve our problem ?
20:25 No . Talk to me about that . Talk to
20:27 us because we have to figure out what it equals
20:31 . Have we done that ? How many of you
20:34 think we've shown what it equals ? What should go
20:40 with our picture to make it very clear that we
20:43 have an answer ? We've got some equations up here
20:48 . We've got a visualization . A picture up here
20:51 . What else would even make it better ? Total
20:55 ? What is the total Kayleigh 21 . Yeah .
21:00 Mhm . Jax . Kelly , You What were you
21:05 gonna say ? Good morning , Orange . 21 .
21:08 Jose , Did you say 21 when you had three
21:12 groups of seven ? Another four groups of seven .
21:15 So what we need to do ? Oh , 10
21:22 . It's four groups of seven . What do we
21:24 need to add to the three groups ? Seven more
21:26 . Seven more . Get it going . Waiting .
21:30 28 . Very good . 28 . What ? 28
21:34 apples . Should we label ? Well , why should
21:37 we label every answer in a math story ? So
21:42 the person knows what you're talking about ? Absolutely .
21:45 What's the story about ? We don't wanna put oranges
21:48 . We don't want to put raindrops . We want
21:52 to put apples . So if I was going to
21:55 answer the question , how many apples did he pick
21:58 ? Using our visualization , the manipulative on the smart
22:03 board , using the math equations that you gave me
22:07 , what would we put for our answer ? How
22:11 every and should be up ? Because good listeners should
22:13 have heard Caylee . She worked hard on that .
22:20 Hi , Ana . 28 28 1 28 . Apples
22:24 . Very good . 28 . Mhm . That is
22:32 an example of multiplication comparison . Are you changing your
22:37 predictions a little bit . Are you still going ,
22:39 huh ? How many of you like that , huh
22:43 ? It's okay . Like , uh okay . Don't
22:48 try . How many ? You're still wondering anymore ?
22:50 Help . I want to see it again . Guess
22:54 what ? You're gonna see it again ? Because I
22:55 have a feeling most of your uh huh . Brand
22:58 spanking new . I want to show you another problem
23:01 . Okay . It's an example of multiplication comparisons .
23:05 Think about what you were predicting . It was all
23:07 about . Let's see . Yeah . Yeah . You
23:13 change your mind . All right , here's the thing
23:17 . Go back to what you were thinking . Multiplication
23:20 comparisons is all about . And I want you to
23:25 focus on the repeated words . I love it because
23:30 repeated words what we were doing in writing focus on
23:34 the repeated words . And Aaliyah's little sister is 2
23:38 ft tall . Her dad is three times as tall
23:41 as her little sister . Katz follows her dad .
23:45 Deandra picked seven apples . Respect four times as many
23:50 apples . How many did he pick ? Are there
23:53 some repeated words in both of those problems ? Are
23:56 there some repeated words in both of those problems ?
24:03 Are there some repeated words ? Mhm , Erin ,
24:12 are there some repeated words ? Yes . What do
24:16 you see ? Okay . Picked . You see Pic
24:20 . Oh , okay . I see what you're doing
24:22 . I wouldn't know . Pick is in your twice
24:25 , but I don't know if there are some repeated
24:26 words in both stories . You're absolutely right . So
24:29 let me rephrase it . Comparing both stories . Are
24:32 there some repeated words ? Mhm . Would you like
24:36 some help ? Are you thinking more ? Time to
24:39 think or help ? Hell , calling somebody that can
24:44 help . Yeah . So , Deandra mhm in the
24:53 white structure . Grief ? Yeah , The great shirt
24:57 . Oh , I on uh , yes . Okay
25:02 . Where are they ? Little sister and apples .
25:06 Little sister and apples are all the same words .
25:08 Oh , I want the words in this one to
25:11 be in this way . You are making repeated words
25:14 in one story . Times time . Is that all
25:19 ? And there's spoilers , video work . We should
25:24 forever town . Uh huh , pal . No ,
25:29 I'm not thinking that when the car goes with times
25:33 , her dad is three times as tall as her
25:36 little sister respect four times as many . It will
25:41 be work as many again as many as many times
25:48 as times as this one's times as tall . This
25:53 one's times as many . Okay , think about your
25:57 fair share . Think about multiplication comparison . Go ahead
26:02 . Okay . Like taking a drink . I think
26:06 . What multiplication comparison , You know . What does
26:11 that mean ? Uh , still , I think it's
26:15 , um it's where you can like the community and
26:20 try to make them like bounce bows . And ,
26:25 uh , I think it's the same thing for you
26:29 can give them balance . It's like a story problem
26:34 and that it can help you with story problems about
26:40 county and multiplication . A . I think it's actually
26:48 like where you can . I'm like China . Chunky
26:51 , chunky . It's like it's like in the word
26:54 like , um uh , like , compare it I
26:59 compare for , you know , again , Do you
27:06 have anything else ? Wait a minute . When I
27:37 look at the picturesque two on this side and six
27:39 on this time when you were talking about your dad
27:41 is that balance ? So what do you mean ?
27:43 You have to explain balanced way . Mhm . Ah
27:51 . So you're saying balance for another word for communicative
27:56 catch ? Okay , give me five . Anybody want
28:00 to share their thoughts now ? Drew thoughts about multiple
28:03 comparisons . Change because I know in the beginning a
28:06 lot of people say we're going to compare multiplication problems
28:12 . Okay . Is that really what we did ?
28:15 Did we compare multiplication problems ? Yes . Or now
28:20 ? Thumbs up . If you think we did compare
28:21 multiplication problems . Okay . Explain why we did that
28:26 . How we did that . We were comparing multiplication
28:29 problems . Can you explain it ? You have evidence
28:33 to explain it ? Yeah , Matthew , we can
28:39 power towards think that will both end them . You
28:44 compare two words in them . Those two words have
28:47 to do with multiplication . Yes . Okay . Did
28:50 we compare multiplication problems ? Because if you said we
28:53 did it , you've got to prove it or I'm
28:55 not going to believe it . So you people that
28:57 just raise your hands , you better be thinking about
28:59 how you're gonna prove it . If you can't prove
29:02 it , you might say I don't know so much
29:05 if I really believe that's what we did . I
29:08 agree with you . We compared the problems , the
29:10 word , the story problems and found a repeated work
29:15 from what do you think we really did ? Because
29:18 when I think and maybe I'm thinking different than you
29:20 did when you all say we compare multiplication problems ,
29:22 I'm thinking I compared seven times 4 to 2 times
29:27 three . Did I do that ? No . Well
29:31 , that's what I'm thinking . Because that's what that
29:33 means to me . Who can tell me what they
29:37 think multiplication comparison means and prove it with evidence .
29:42 Just like when we infer in reading , we must
29:45 prove it with evidence . Mhm . Sure . Okay
29:52 , Ethan . Nice and loud . I believe that
29:58 multiple creative comparisons means when you compel the two factors
30:07 in the multiplication occasion say it again . Nice and
30:10 loud . Um , our thinking is when you compel
30:16 two factors of a multiplication equation . So I'm going
30:21 to bring up this one , and you talk to
30:27 us about what you just said . Like , what
30:31 would be the fact ? Come on . Okay .
30:35 All eyes , This press student , Ethan , the
30:39 teacher battles would be full . And how do you
30:47 know that ? Um , I know that because ,
30:52 um , seven times four equals 28 . And did
30:57 our pictures show that , um , there are pictures
31:01 show seven times four or the community four times seven
31:05 . Did it ? Can you circle that so people
31:08 can see where it says four groups of seven or
31:12 seven groups of four ever . You want to look
31:14 at it good . Why don't you circle that one
31:19 ? Because that , um mm is , um ,
31:25 the There's a number more fun . Yeah , it
31:28 is . And who's bad about Tian Tian Draw .
31:31 And what is the rest of the story about ?
31:34 We've got to use this information , but what is
31:36 the question focusing on , um , it's focusing on
31:39 how many apples picked and which one Which one of
31:43 these colors shows Reese , Um , would , Yeah
31:50 , so he said and let me restate it .
31:54 And if I'm not saying it right , you help
31:56 me . Because sometimes that kids understand your language better
32:00 than they understand my language because they're used to hearing
32:02 their peers speak . So you're you're saying that multiple
32:06 active multiplication of comparisons are when you're really comparing the
32:11 two factors in a story . To get a product
32:20 , go back to your seat . Good job .
32:23 To sum it up , I want to look at
32:24 the different ways you can say £4.17 28 and we
32:34 know the community property seven times four is 28 .
32:38 Let's look at the next week and ordered people come
32:40 up with We can say it this way . Let's
32:42 say it way we could say it this way .
32:55 It this way Going to show you this . And
33:07 I want you to read it carefully . There's another
33:10 one after it . I'm gonna I'm gonna show it
33:11 to see if you can guess what it is we
33:13 could say . Or 27 they get , they get
33:26 What could the other way be shouting ? Trump ,
33:33 What are you thinking ? Wow . Seven , seven
33:38 times as many . Seven is 28 to us ,
33:45 as many as well , Does that make sense the
33:48 7 . 28 times as big as four . Do
33:53 you think about the pictures ? I mean , he
33:54 had 28 fours . 28 4 is gonna give us
33:57 seven . Look at how 2028 is four times as
34:03 many as seven natural factors . Mhm . Yeah ,
34:11 we switched out factors . So what , are we
34:13 gonna switch out here ? Yeah , 70 . No
34:18 , 28 Factors I gotta think of factors and four
34:27 . Yeah , because because four times seven is 28
34:30 so we switched them out . We gotta switch them
34:32 out here in the community . We could say Yeah
34:36 , yes , that's what that says . Factors as
34:47 many . Say it really loud . Everybody these ,
34:54 these these are groups . What we wanted in third
35:01 grade . This is an array . This is what
35:04 we've learned in 3rd and 4th grade . And this
35:10 it's called . What ? What we talked about today
35:14 . Who read it like this ? We are doing
35:19 What ? How would you say Why do it ?
35:26 Oh , you're saying multi multiplication . Multiple kind of
35:31 comparison sounds here . What is it ? Everybody with
35:38 a with you're gonna learn that word like you learn
35:43 medical ignition . You're gonna work on this either tomorrow
35:47 or after lunch . We'll see what happens .
Summarizer

DESCRIPTION:

Grade 4 lesson demonstrating CCSS Math Standard 4.OA..A.1,2, using literacy strategies to understand word problems. Lesson progresses through prior understandings of multiplication as groups of objects to multiplicative comparisons.

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