Students are now shown three numbers: Joe 1 ½, Sarah 1 ¼, and Alex 2. Then students are asked what these numbers could represent. What are the words that go with these rates? Finally students must justify who is fastest.

6th Grade Math - Rates - Lipman School*Joe Condon, Lipman Middle School, Brisbane School District, Brisbane, California*

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00:00 I got these numbers. I got these numbers, about Joe, Sarah and Alex and I have no idea what they mean. No idea what they mean.

00:12 I need you to tell me. I need you to look at these numbers. They are based on the exact same data from the same experiment and they are rates.

00:25 But it's just not clear. Joe has a rate of one and a half. Sarah has a rate of one and a quarter, Alex has a rate of two.

00:33 Could you put a line under the stuff you just finished, write down those three rates, with the kids names next to them and talk with your group for a minute

00:45 about what these rates could mean.

00:53 One and a half…I mean, for Alex it's two seconds, I mean two beans per second. For Joe it's one and a half beans per two…no.

01:08 Because to find the unit rate you divide how ever many beans they counted in however many seconds they counted in.

01:19 And since Joe counted thirty beans in twenty seconds you divide twenty by thirty and then you get one point five beans per second and that's one and a half.

01:28 And then for Sarah you do thirty and twenty –four and you get one and a fourth.

01:35 And then for Alex you divide thirty by fifteen and you get two and that's…

01:40 Two seconds. It's just how many beans they got per second.

01:43 Yeah, it's beans per second.

01:49 Joe gets one and a half beans per second. Sarah gets one an one fourth. Alex gets two beans.

02:04 When she did her unit rate she got one and a quarter, right?

02:12 So that means Sarah's unit rate was one and a quarter. So then Alex's…

02:22 Sarah is the slowest one…because…she could be faster if it was all…

02:30 Then it's Alex and then Joe then Sarah. For the fastest

02:38 For Alex is it two beans per second or seconds per bean?

02:44 Two beans per second.

02:45 O.K.

02:47 Sarah is not fastest because…

02:50 She's not the fastest?

02:52 She's not because Alex has two beans per one second.

03:03 It's like one fourth bean for one second and two beans for one second.

03:13 So these are beans per second so Sarah is not fastest because Alex could get two beans for one second…

03:25 but Sarah could only get one and one fourth bean per second.

03:33 Joe is not the fastest because he is kind of the middle of both of them and Sarah is not the fastest because she can only get one and a fourth bean in a second.

03:46 But Alex is still the fastest because he can get two beans in one second if you still use the…from problems one and two.

03:56 You're throwing me common denominators here? I like it. O.K. I'm clear so…

04:03 What my problem is you've still got these rates written as just random numbers just floating out there.

04:08 Is there any words you can attach to them so that there is meaning?

04:15 Everybody ready to share?

04:20 I think the fractions are just the decimal answers for beans per one second.

04:33 Let's start with Joe. One and a half. What does it mean? What does it mean? Wesley.

04:41 It means his beans per second.

04:47 One and a half beans per second. Yes? Is that what you mean?

05:01 O.K. Sarah. Molly?

05:05 One and a fourth beans per second.

05:08 All right. And Alex. Crissio.

05:17 Two beans per second.

05:18 O.K.

05:23 Which one is fastest? On the count of three. One, two, three.

05:29 Alex.

05:31 So how did you prove it? How do you know? How do you know? Michael.

05:37 Because if they were going…like, if they are having a race I guess…and it was ten seconds…

05:49 O.K.

05:52 Alex would get twenty, if there is ten seconds and two beans per second. He would get the….he would be the fastest.

06:04 So I'm sorry, ten seconds you said, how many beans?

06:08 Twenty.

06:12 O.K.

06:17 Joe's seconds again.

06:18 In ten seconds he could only get fifteen beans.

06:29 O.K. Yeah. You were going to compare…

06:33 Sarah's

06:34 Sarah's yeah.

06:36 Sarah is below Joe so she can't get more than Joe or Alex so she can't win the race.

06:44 How do we know Sarah is below Joe?

06:48 Because…

06:50 Well hold on, let's let somebody else jump in there. I hate to cut you off. Mikey go ahead.

06:59 Because Joe gets one and a half beans per second and Sarah gets one and one fourth beans per second so if you did…

07:13 if you made it so Joe had one and two fourths which is equal to a half Joe would have more than Sarah.

07:24 So you said…I like that. You said one and two fourths and you compared it to one and one fourth. And you said what?

07:40 So Sarah wouldn't be able to win because Joe has two fourths and Sarah has one fourth. If you change Alex's to one and four fourths then…

08:03 Oh I'm sorry. This is beautiful. So Joe's the winner here? Because one and two fourths is greater than one and one fourth.

08:11 So we want to bring Joe down for the next count off. Go ahead.

08:20 So if they had a race Alex would win because if you made that equal, Alex would have two beans per second. Joe would have one and one half beans per second.

08:40 O.K. so you are saying…and I love, you know, the common denominators, you are saying that one and four fourths or two beans per second…

08:50 …is greater than one and two fourths…one and one half beans per second.

08:54 Alex. Well done buddy. Nice bean counting. You guys ready to get started then? Because that was just the warm up.

COACH LINDA FISHER: I like how the lesson study team has a clear goal for the each situation and has made posters to structure the thinking and discussion. This borrows on the Japanese lesson structure of designing the board to tell a story, so that ideas aren't lost as the discussion moves to the next question. While many things can happen with the use of technology, such as overheads or document readers, I appreciate seeing how the ideas unfold across the board as the new questions and ideas are presented. The posters also keep the lesson focused on the important mathematical ideas being discussed, so that the lesson doesn't get sidetracked by random ideas.