Observers who were not part of the lesson planning team reflect on their observations. The teacher shares the development of the unit by a group over time. The students were able to self-correct because of the context and use of sentence. Other observers share stories about how students thinking about units changed over time during the lesson.

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

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00:00 The next part is comments from the observers.

00:02 And if you remember, the one question they wanted us to focus on was that one:

00:06 Where we're looking at the labels for the ratios or in the sentence.

00:10 If you observed a group and would like to share something that you observed, and give feedback to the team, that would be great.

00:20 One thing that I thought, first I keyed right into it because it was so interesting to me.

00:25 She put "12 beads," and then they went and they counted every single bead, and they put "over 81."

00:32 12 over 81 was their rate, or ratio. And I tried to keep an eye on her and see, because they didn't label, okay, maybe if she had labeled it she would have made more sense out of that.

00:45 That it wasn't per the time, or that she didn't include anything about the time.

00:49 And then all of a sudden, I looked back over, and I talked to a couple people who were also observing with me, and she had changed it.

00:57 Like it had just made more sense to her, and all of a sudden she put, I think she labeled it as well then, right?

01:02 Well, at that point, her partner, Sammy, said, oh yeah, you've got to label that.

01:09 She had, she goes, "I think it's over 30," and Sammy says, "Yeah, you've gotta put b over s. To label it beads and seconds."

01:16 At that point, they were kind of correcting each other, that they needed to be labeled.

01:20 At the way they got the 81 was they counted all the beans?

01:22 Right. Right.

01:23 So it was a fraction of the beads. What fraction did they put on the rope, or on the string?

01:28 But it was interesting that that happened pretty quickly///

01:32 Yeah, it was just, like instantaneously, somehow it didn't look right to her. Something. Or maybe she overheard some of the conversations from the other groups.

01:39 I don't know.

01:40 That was exactly what happened. They were finishing up, and they were at that point, where she'd only put 4 beads on her string,

01:47 And they were in the arithmetic of, "Now, how do we figure out what this ratio is. If there's 81, but you had 12, and I have 4,

01:55 Because they didn't really have 12 to 81, they had 12 beads on a string, and 81 other beads.

02:01 Okay.

02:02 They didn't really count the total beads. And then they realized that they hadn't done that.

02:05 And she started looking around and went "Oh! We should have time…" It was just like that. "We should have time." And they switched what they were doing.

02:13 Right there.

02:15 And then Joe talked about, like, are these three, are they all ratios? And someone said, "Yeah, they're all ratios."

02:24 Then some students said that it's not, only the first one is a ratio. And then you did say that, okay, the unit is important.

02:31 You have to write something after the number to make it make more sense out of it.

02:36 So maybe they were using that into the activity? Because the 600 over 15 was done before the activity.

02:44 Could be, they might have. It's my assumption they might have taken that into their activity.

02:49 Sammy, who were comparing each of their, rice totals. So it was 16 to 11. I got 16, you got 11. 16 to 11 then is our ratio.

03:02 So … yeah. So there were a couple that were … not clear on the …

03:13 And the definitions didn't go, you know, typically I would spend more time trying to ferret out the clearer definition of rate.

03:26 I think in one of the classes I taught something similar to, had come up with something like the number of items over a measurement, or something like that.

03:39 A comparison of two different measurements, or something like that. And so you're working with a little bit more meat when you, but there was nothing for them to really hold on to.

03:49 So I could see, yeah, there would be a few people that would go off track.

03:53 Alex, I believe, and Alex was the born leader. He was in charge and in control, and he accepted that title willingly.

04:01 She said, "You're going to be in charge because I'm not the smartest."

04:03 And actually, then, she started working with her partner across from him. And when they were doing their recording, whether it was beads or rice,

04:11 They were writing their ratio as 4 over ? 7 over ? They weren't sure what they were recording. They just knew that they had to record that 4 or that 7.

04:21 And then after, I think, the second activity, um, I think it was Tyler, he said, "No, you have to change it, it can't be a question mark. It's the number of seconds."

04:33 Once they put the labels on, then it made sense what the ratio was gonna be.

04:36 And they changed their question marks to "over 10 seconds" because that was their denominator.

04:42 It's easier with groups because you have people to help you. That was nice. But they were having a disagreement when they were talking about the rate.

04:49 About how fast you'd been going. He said, immediately, it was 3.5 over 1.

04:55 And Allie was insisting that it was 7 meters per 2 seconds because she "didn't want a fraction on the top." She didn't want a decimal on top of the fraction in the numerator.

05:06 She was real clear that she didn't want that. She was still uncomfortable with that, the rate being 3.5.

05:17 This concept of fractions versus rates and ratios, I know the percent, when we're asking for the definition, they keep coming to the "tip percentage."

05:25 Like that is what this rate is going to be. And just, giving students a chance to struggle with that, and figure out what each one really is, to apply it to problems,

05:37 I think that's really meaningful, and that's one thing you would get out of this. I wish I'd seen the last 15 minutes, but you'd get out of that conversation.

05:44 When you're saying "Who's the fastest?" Talking about the unit rate even more in depth. Why is it okay to have a decimal in this case?

05:52 So I loved how there were many different opportunities to bring in all the different things that the kids have thought,

05:59 And finding those misconceptions at the beginning, because I know you said this is the beginning of this unit, this is the beginning of the discovery.

06:05 So I really appreciated seeing this, and seeing what kinds of ideas kids come in with.

06:10 And I shouldn't be, because I've seen it over and over again, as Becca and I put this lesson together, we really wanted to emphasize terms for the units.

06:23 In as many ways as we could, and that's where the whole sentence thing came in. Let's put it in words, let's put it in a sentence before we put it in a ratio.

06:30 And the number of kids that were writing down the rates without terms, it didn't—so, and it was equal, or easily equal to the first iteration of the lesson,

06:47 When we didn't stress putting the terms down, it was just naturally writing down fractions with no words next to them,

06:54 Even though we tried to really stress that. So they still really struggled with writing terms down.

07:00 The idea that this was sort of a beginning place to think about it, so that they would have something to refer to.

07:09 So as they learn future, more formalized ways of thinking about rate, they could go, "Oh yeah, it's like when we counted rice. It's like when we were stringing the beads."

COACH LINDA FISHER: I value the lesson study process and how it allows us to step back and watch learning happen. Observers can concentrate on a pair of students for the whole lesson, rather than in teaching where we hear only bits and pieces as we move around the round.

Listening to the groups, I am amazed at how students tried to think about fractional ideas, number to total or number used to number not used. It gives me insight into why these mathematical ideas are confusing to students. We write fractions and rates with the same representation, but are expressing very different mathematical ideas. This can help us think about the types of questions to ask in the classroom to help push student thinking or the types of activities needed to bring out these differences in a way that students can compare and contrast fractions and rates.

I was intrigued by the group that focused on a single number. "I counted four or seven." As an adult, it seems so obvious that two numbers are involved objects and time. But for students that is a huge leap from counting to a compound unit of objects and time. What activities help students move from single number thinking, such as I have 3 pieces of pizza versus 3/8 of a pizza, or I counted 4 cubes versus 4 cubes per second?

I am also fascinated by the idea of students being uncomfortable with 3.5/1 and wanting to change to 7/2. Do we provide students with "comfortable numbers," whole numbers, too much? When numbers always come out even in the book, what message does that give students? The observational data helps me think about what activities or experiences students need as the unit is developed and what questions I should be asking.