CF: My name is Christine Fairless. I teach 8th grade math in Union City, California. I worked in curriculum development for three years and then I became antsy to get back into the classroom. I really missed the interaction with kids. Using Pathways materials is just so night and day compared to teaching in a traditional way. I mean, it just really opens up not only things you can do as a teacher, but also allow students to show you their knowledge in such a variety of ways.

CF: Anyone know much about Antarctica? Angela?

Angela: It's cold.

CF: Okay, well we're going to be learning about Antarctica and we're also going to be learning about architects. Does anybody know what an architect does for a living?

Student: I think it's like the designer, to design buildings?

CF: Okay, good. You guys are going to be architects while we're in this project.

JK: In every school or district where we've been able to collect data on student achievement, we've seen student achievement improve as the program is used and implemented more fully. The centerpiece of our program are a series of design units, where kids take on the role of some kind of math-using professional.

Student: Greetings and welcome to Antarctic.

JK: What we do is find a problem in a world for which kids need to design a solution.

Student: (?) specializing in designs for cold climates. We hope you'll feel right at home in your new job.

Student: We also hope that you're ready to get right to work, because we've got a hot, or rather cold new client.

JK: Mathematically we centered the materials around two big math topics. One of those is proportional reasoning, and the other one is algebra and function.

Student: The more you know about your clients in Antarctica, the better your design will be.

JK: What kind of a building are we going to be designing?

Student: A research lab for Antarctica.

CF: Well, luckily for us we happen to have a little bit of video that we're going to show you that we can kind of get a feel for what life is like down there in the bottom of the earth at the South Pole. You're going to look for things that might have something to do with your design.

JK: As we looked around at different themes we thought about using ideas that were appealing to kids, so we thought about doing units on skateboards, but not everybody in the classroom is going to be a skateboard enthusiast. So you introduce inequity in the classroom. Our approach was to find topics that were equally strange to all children, so Antarctica definitely qualified. Middle schoolers are really at this unique time in their lives where they are still children who can be engaged in role play and fantasy and yet they are adults, thinking about the future and about themselves as adult workers.

Student: MacMerdol(?) Station is actually on the South Pole and the station that you're going to be designing is actually on the coast, but the climate is similar.

CF: The students are working with a computer program called architect. It's a kid-friendly computer aided drafting program, but it does a lot of pretty complex mathematical things.

JK: We give the students a choice of some 40 some variables that they can compare for their design. So in this case the students have chosen to look at the building costs, the heating costs and the outside wall insulation.

CF: Up to this point everyone should have a pretty good design that they are happy with for their research station. Now we are going to do a cost analysis of our design. But we're going to start with a simpler one before we go into your own design, so that everyone has the idea of the proper procedure to go through, because when you go to your designs everyone is going to have different numbers.

JK: When we began designing this program, we were really addressing the very pronounced achievement gap in this country, so we said, let's break that barrier between the outside world and the classroom and create projects so that kids can apply their everyday reasoning to mathematics and then start to excel at mathematics that way.

CF: Today we're going to be looking at the insulation factors. You might have seen this before, where does it go in a building? Student: It goes in between the walls.

JK: That is not an approach that works just for at-risk kids, that's a sensible approach for anyone who is going to be a working adult in the 21st Century.

CF: And what does the insulation do? How does the insulation help? Angela?

Angela: It warms up the room?

CF: Does the insulation actually warm up the room?

Angela: No, it makes it warmer because it's in the walls.

CF: Jan, do you want to add onto that?

Jan: It keeps the cold air from going inside the room. Like it traps the cold air outside. So it won't go inside the room.

CF: So the insulation acts as a protector. So the more the insulation the more protection you are going to have and the less that you're going to have to heat your building. And that's going to be the basis of our investigation today. As a class we do a very simple analysis.

CF: I want you to draw a rectangle that looks like that.

CF: They'll create a simple, 8 meters by 10 meters small building and they'll analyze just one insulation value. So we'll start with just the outside walls and I'll take them step-by-step, going up increments of 5 and recording their information very carefully. So how do you know that your windows are 2 meters long?

Student: You can look on the bottom and it tells you how long your line was.

CF: What's another way that you can tell that your window is exactly 2 meters long?

Student: Count the dots.

CF: How much distance is there between each dot on the computer screen?

Student: 1 meter.

JK: The computer is used approximately 60% of the time during each design project. Teachers have accomplished our curriculum using as few as two computers in a classroom by having kids rotate in and out of stations.

Student: Outside wall insulation (?)

Student: Building costs -

Student: 704.

Student: Heating costs per month?

Student: 15?

JK: They can change these variables and those are the kind of changes that students learn to track and explain.

Student: Calculate?

Student: $72.

JK: Mostly kids would be using 1 or 2 variables at a time, because they want to compare a variable that they are in control of, the independent variable and then the resulting changes in the dependent variable.

CF: And I'd like to take a few minutes to talk about what you notice in the building cost column and what you notice in the heating cost column. If you take a look at the numbers in the building costs, what do you notice? As the insulation value is increasing, what seems to be happening to the building cost?

Student: It goes up.

CF: That's right. If you look at the numbers in the heating cost column, what do you notice about what's happening as the insulation value increases?

Student: They are going down?

CF: What can you tell by looking at the numbers? Let's just look at the first two. We have 184 and we have 120. What's the different right there?

Student: $64?

CF: Perfect, yes, it goes down by $64. What do we have the difference between 10 and 15, between 120 and 96, what's the difference there? How much are we saving?

Student: 24.

CF: It's 24. So look what's happening. We have a difference of 64 between the 5 and the 10, but only 24 between the 10 and the 15, and then what happens as you get to the end of the chart?

Student: They said going down by $1.

CF: Now, you notice our savings gets less and less and less as we get our higher insulation. But if you just look at these numbers, it's not obvious, it's not clear what's happening, what the relationship is between the insulation and the building and the insulation and the heating costs. What could we do to help us out?

Student: You could make it into a graph.

JK: Good.

CF: Well for this specific unit there's so many things you can focus on and I'm really focusing on graphing, being able to create a graph, being able to read a graph, knowing what graphs are good for and how to use them to defend your reasoning for something.

CF: So what is this graph telling us?

Student: As the outside insulation increases, the building cost increases.

CF: That's correct. Every graph gives you a relationship. You should be looking at a graph and you should be able to make a statement like that. You should be able to say as the X axis is doing something then that's affecting the Y axis. Our second graph is the heating costs. Kevin would you mind coming up here and just drawing that little sketch of what the graph looks like on your paper.

CF: Good. What's happening to the heating costs?

Kevin: As the outside insulation gets higher the cost gets lower.

CF: Good. With the heating costs graph, if you notice it's a curve and what that's telling you is that at the beginning you have a lot of savings in your heating cost, but as you get down to the end of the insulation value, the higher the insulation you are not saving as much and that's why it kind of flattens out there on the bottom.

CF: Then in part 2 we actually combine the building costs and the heating costs and we develop the equation or the formula for coming up with the total cost for 10 years.

CF: So we're going to put a plus right here. So in between building costs and heating costs, I want you to put a plus. The next thing we have to consider is the heating costs. We want to heat our building for the entire 10 years for our simple investigation. What are we going to have to do with this heating cost number?

Student: You'll have to multiply the heating costs times 120?

CF: Yes, that's right. Can someone else explain to us exactly why it's 120? Where does the 120 come from?

Student: 12 months per year, then 12 times 10 costs 120.

CF: So on your paper over here, what I'd like you to write is, we're going to take this 184, so take 184, times 120 and then you are going to add that to the building cost which is $34,704. What I'd like you to do next then is to put parenthesis around this heating cost, times 120. Let's see what you have there for that first one. Chris?

Chris: I got 56,784.

CF: Raise your hand if you agree. Do we have an agreement on that first one?

Chris: Okay. Excellent. What I'd like you to do now is to take a few minutes to work with the person sitting next to you to complete the rest of the chart.

JK: In terms of mathematical growth in terms of formalisms, we want to see kids first understanding that formal mathematical representations are something they have control over and then learning to use those adeptly and appropriately.

CF: What's the label going to be on the bottom on the X axis. Dina.

Dina: It goes by 5, 10, 15, 20.

CF: Okay, that's how the numbers are going to be distributed along here, but what are those numbers?

Dina: Outside wall insulation?

CF: That's right. So this is going to be outside insulation. This is going to go from 5 all the way up to what? What's the highest number we're going to have for outside insulation?

Dina: 60.

CF: And then what's the label going to be on our Y axis? What are we - Jessica?

Jessica: Total cost for 10 years.

CF: And then we have to label, or we have to put intervals on our Y axis also. So what we need to do is we need to look for our lowest value that we have. What's the lowest value that you have for total costs?

Student: 45?

CF: 45,000 something right? And if you remember when we distributed our intervals on the other graph, we just went to the lowest one, we went a little bit lower, we started at 45 and if we're going to put 45 and not 45 with the three zeros what else do we have to put on our graph.

Student: On the side where it says total cost of 10 years, we put thousands.

CF: You have to put in thousands, otherwise that's a 45, it's not 45,000. Okay, now what's the highest number that we have on our graph. Christie?

Christie: 56.

CF: 56,000 something, so we're going to go a little bit above that all the way up to 57,000. And remember when you are making your graph, you want to try to spread the information out so it's going to take up as much of the graph as possible.

Student: How come your 25 and your 30 is so low?

Student: It's because I'm using, my boxes are smaller. Student: Oh, and plus my 45 is not on the line where the numbers are.

Student: Because yours is higher and mine's like lower.

Student: Yeah, because my line's down here.

Student: And your 45 is on the blue line and my 45 is (the line that I drew?)

CF: Allowing the kids to feel frustrated or allowing the kids to flounder a little bit, I think is a really important part of the learning process. It takes patients on your part and it takes some faith on their part, you know, that they will get it, but I think it's not something you want to pass up.

CF: We've been leading up to this point until today for the last couple of weeks. Now today is kind of an important day because you are going to be beginning our analysis of your design and you are going to be doing your own cost analysis. For the last few days we've been working on a kind of a basic simple analysis to kind of give you some tools for you to use for this one. Okay, now this is based on this memo, memo 3.

Student: Find the insulation value called the R value that will give the lowest total building and heating costs fir 20 years.

CF: Okay, you are going to have to have a chart that you are going to fill in from your computer. You are going to have to make a graph.

CF: To help them think about how to analyze the graph, what the graph means, I have them complete 2 sentences. As the outside insulation increases, the total cost - blank. So what's happening? What is the relationship? Trying to get them to see the graph as a way to describe a relationship between two things. The other thing I have them do is to look at their graph and to try to come up with what they think would be the best R value.

CF: If I is an R value recommendation, that means we recommend this R value. I also want you to include reasons why. Your reasons should be based on your chart and it should be based on your graph. Now, what I want to do before I send you off to your computers is I want to help you set up the chart. Our chart is going to have a few different columns. So I want you to think about the different columns that you had on the other investigations that we did. So what was the first column that we had?

Student: 5, 10, 15, like the temperature outside.

CF: Okay, this is going to be up to your individual group. You don't have to do 5, 10, 15, 120, all the way up to 60. Okay? You can pick a different interval if you want to. Remember when you are finding the total cost for 20 years, why is this not going to be times 120 this time?

Student: Because it's not for 10 years, it's for 20 years?

CF: Exactly. You have to figure out how many months are there in 20 years. And that's what you're going to multiply this heating cost number by.

CF: The students really, really enjoy working with the computers. When we're working on the Antarctica project every day that we're working on it, when they come in, that's their first question to me, is are we going to be on the computers today?

CF: So you went up in intervals of 15? And which one gave you the lowest?

Student: It was 45, it was 45.

CF: 45 gave you the lowest? Okay, so what I want you to do now is I want you to check, because these are kind of big intervals, I want you to check around the 45, so maybe check 40 and see what that gives you, see if it's lower than that, check 50, see if that's lower than that to see if you can kind of zero in on the one that's going to be exactly the lowest.

Student: Okay.

CF: All right? And then let me know what you get.

Student: We have to do 15 to see (?)

Student: It's 296,000.

Student: 101.

Student: 101. So is that lower?

Student: Let's try (25?).

Student: Why?

Student: Because it might be lower than (?).

Student: Okay.

CF: Each group creates a big graph that they are going to present to the class. And the tricky part about that is when they are doing the scale, because their scale is different from everyone else's.

Student: Because it goes from 35 1 to 35 3. So it does rise a little. So what would be the recommended R value?

Student: We need a low one, because our house is already expensive with all our furniture. So maybe not the cheapest one, but the next to the cheapest one.

Student: This is the 3rd one, this is the 4th one, this is the cheapest one.

Student: That's the lowest price, Shirell.

Student: No it's not. It's between the highest cost and the lowest cost.

Student: (?) to the lowest price?

Student: Here's the highest price, is 14,(?), 419,000 is the lowest, which was $349,000 and there was 2 between them. So we picked one of the two (?) one was higher than the other, we went for the higher quality. We went with the cost of $360,593 because it's kind of in the middle. It's not the highest one, which $349,290. It's not exactly in the middle either, so it's higher than average, but it's not the highest.

Student: That's why we went with that cost for the quality. Do you get it Regelia(?)

CF: All right, you guys did an excellent job of making these graphs, but I just wanted to take a minute just to kind of look at them all together, and talk about the similarities and differences. So if you look at them, can someone name one similarity that they see from the graph?

Student: The outside insulation all goes up to 60?

CF: Okay, good, but if you notice, not all of them are distributed the same way. So this one starts at 15, this one starts at 6, this one starts at 10, that one over there starts at 5. How can you explain that difference?

Student: They all equal 60.

CF: What do you mean by they all equal 60?

Student: If by 15, it's like 15, 30, 45 and 60 and then by 5's it always like, at the end equals 60.

CF: So some people decided to go up by 5, some groups went up by 10s and this group here went up by 15s and then they have an extra point in here to find the low point. What about the shape of the graph that you see that is the same on each graph? Ameri, what do you notice?

Ameri: They all end up like decreasing in the cost?

CF: Okay and what else?

Ameri: It starts to go back up.

CF: And then at some point they start to go up, and that's definitely a similarity that we noticed from all of the graphs. So if you look at this graph it's pretty clear that this, our value of 40 is the lowest point. What does that lowest point actually represent?

Ameri: It's the lowest cost.

CF: This represents the lowest, total cost, because that's what this graph is, this is the total cost for 20 years, the lowest point is saying that this R value of 40 is going to give you the lowest total cost. Okay, what about differences? Shirell, what kind of a difference do you notice from these graphs?

Shirell: On the (side one?) there's different numbers.

CF: And why is that?

Shirell: Like some graphs go all the way up in the 500s, some go in the 400s, some go in the 200.

CF: These are all correct, by the way, but why does this one go up to 450? This one goes up to 900? And this one here goes up to 325? How can those all be different numbers, but all be correct?

Shirell: Because the R value are different. They are different.

CF: Well, the R value might be different, but definitely it's something different, but it has to do with something else that's different. What's different about all these graphs are based on something that's different. Angela, what's different?

Shirell: The houses?

CF: Yes, the houses are different. Each group has a different design and each design costs a different amount. And that's why the numbers on the sides are different. This is a really expensive $900,000 design. So theirs better be a really nice design because that's very expensive. And there's other ones that are a little cheaper, like this one is $325, the range is a little bit lower. But they are all okay. What about one more major difference, at least that I noticed, just looking at the shapes of the graphs. Which one seems kind of different than the rest?

Student: The third one?

CF: Yes, the third one is different. Can someone else explain to me what's different about this one compared to the other graphs?

JK: The graph, like it starts higher than all of the other graphs. Well it doesn't start higher, but it stays high, it doesn't like go down, it just stays down it just stays high.

CF: Yeah, that's right. If you look at this one, it starts kind of high. All of the graphs start really high, but all of them seem to drop down really low whereas this one drops but it doesn't drop very much and it does go up like the rest of them, but it's not as obvious as a drop in going up. Why might that be?

Student: We didn't put enough space. If we put more spaces between the numbers it would have gotten down and it would have been more obvious as the other one. But we only put one space, so that's why it's high.

CF: So you're saying, because you only put one space here -

Student: Yes, between it. Because everyone else put more space and that's why ours is like that.

CF: Okay, well I think that you're on the right track, but what I think, does anyone else have something else that maybe they can see from that also?

Student: They start at like really at their lowest number and then they end at the highest number.

CF: Okay, so really the difference is how this group decided to label their Y axis and that's really important. They way that you label your Y axis has a huge difference for how the graph is going to turn out, because that is a very, very key important concept in graphing. Do we have a volunteer, a group that wants a volunteer to go first, to just give a short presentation on their graph?

Student: It starts to decrease and then also that's the lowest cost and it gives you more installation and (?).

Student: As the outside installation increases, the total cost for 20 years decrease, until 40 and then it starts to increase.

Student: We recommend an R value of 40 because it is the lowest cost and there is more insulation.

Student: Notice you have an extra strip of paper up there on the top? I was wondering if you could explain kind of what happened?

Student: We were supposed to take 7 and times it, but then we took 6 -

Student: We were supposed do divide, because we had too many numbers and we didn't have enough room, so we had to add on a piece.

CF: It was when you were choosing your interval for how many numbers in between on the Y axis, is that way? You chose 6, but you really should have chose 7, is that what you are saying?

Student: Yes.

CF: That's not your fault, the paper's just too small. The paper's small. Okay, thank you.

Student: This is our graph and this is our (?) close to 900,000 and then it slopes down as the insulation gets higher and this is our lowest point right here. But we chose 60 because it's not much higher than here, but it provides a lot more insulation. As the outside insulation increases, the total cost for 10 years, the price of the house decreases, but then it goes back up.

Student: We recommend the R value of 60 for the outside insulation, because it provides lots of insulation for the house, but at a reasonable cost.

CF: There's a point where it slopes down and then all of a sudden it seems like it' at the bottom, which point is that at yours?

Student: That's at the 36.

CF: But I mean, even further to the left, where it seems like it gets to almost the bottom. Or the different really isn't that much.

Student: Oh, you mean right here?

CF: Yeah, at 24.

Student: Yeah.

CF: So if you look at everyone's graph there seems to be a point like that on everyone's graph. You wouldn't want an R value of 5 right, because the heating costs are going to be so expensive it's going to make your total cost be really expensive. And even at 10 and 15, but if you get up to a certain point, it seems like 25 on each of them but at that point the cost is going to be low, right? So everyone chose something that was in the low price range. And really anything past 25 is a pretty good decision as long as you have some reasons for it.

CF(?): After they do the graphs of their outside insulation versus the total cost for 20 years, they do a similar investigation for the roof insulation so that they have two different R values that they are going to recommend. Then they are pulling everything together into a final report. If there's some student that never even thought of being an architect and they remember doing this project and they remember that math is important and they continue to take math because of their experience in 8th grade, working on one of these projects, I mean, I think that makes it all worthwhile.

For further information about the Modeling Middle Mathematics Project, please contact WWW.MMMproject.org.

Major funding for Modeling Middle School Mathematics is provided by the National Science Foundation. END